Background

Although the invention of internal combustion engines (ICE) dates back to before 19th century, their widespread applications were hindered until mid 1850s when commercial drilling and production of petroleum began [1]. Since then, fossil fuel has been the main source of energy in transportation and travel. However, as the world became more aware of the limited supplies of fossil fuel and resource depletion as well as economical and ecological impacts of its usage, concerns began to grow. As a result, for many decades, at least as far back as the time of Great Depression in the USA, the nation has been concerned with fuel efficiency and conserving energy resources. However, the issue is not quite as simple and straightforward as it seems and there are challenges. For instance, improving fuel efficiency triggers an economic reaction causing additional travel that partially offsets the original energy saving. In other words, as the energy becomes cheaper, it provides an incentive to increase its use, a phenomenon known as the ‘rebound effect’ [2].

In general energy usage has direct impacts on the economy [3]. Furthermore, almost in every country in the world, transportation takes up a large portion of the fossil fuel consumption. For example, in the USA today, 70% of the total oil consumed is directed to fuels used in transportation - gasoline, diesel, and jet fuel [4]. In Nigeria, 96% of the total gasoline consumed and 40% of the diesel fuel consumed go to road transportation [5]. In addition to economical consequences, fossil fuel consumption results in the emission of toxic gases (CO, CO2, NO x , SO2, HC, and PM) [6] that affects local and regional air quality, specially in urban areas, which can pose serious health hazards and cause other ecological issues. Today the emission of CO2, which is directly related to the consumption of carbon-based fuel, is regarded as one of the most serious threats to the environment through greenhouse effect [7], and transportation alone is responsible for about 21% of the global CO2 emission [8].

Shortages of gasoline and other petroleum fuels from time to time, as well as ecological concerns, have renewed interest in identifying factors that affect fuel efficiency of vehicles. To this end, various proposals for reducing fuel consumption have been made. These include simple modification of vehicle components such as replacement of bias-ply tires with radial tires and development of alternative fuel sources and new engine designs to optimize combustion. In addition, a noticeable trend toward smaller and more efficient vehicles has appeared in the last few decades. This is done through reducing vehicle weight by using new materials and by improving vehicles’ exterior design to achieve lower aerodynamic drag [9]. Thus, fuel efficiency has become a selling point in the automotive industry. Although most of the proposals for conserving fuel resources have been focused around vehicle design changes, some external factors have been considered as well. The national reduction of highway speed to 55 mph (88.5 km/h), mandated in the USA in the 1970s, is one example. Other examples include maintaining a constant highway speed or maintaining a constant throttle setting [10].

Oil dependency and global warming have stimulated research and development activities toward utilization of secondary biofuels and renewable energy sources including solid waste and vegetable oil [11–13]. In addition, alternative fuel sources and engines have become popular in recent years, and the endeavor toward production of the so called ‘greener’ vehicles continues. These include electric, hybrid, and fuel cell vehicles. Today, hybrid passenger cars are widely in use and quite popular. Electric cars have zero emission but suffer from the disadvantage of having to be recharged frequently with a long recharging time and having a relatively short range. The fuel cell vehicles, on the other hand, combine the advantages of an electric vehicle, i.e., almost zero emission, high efficiency, and silent operation, with the advantages of conventional internal combustion engine vehicles, i.e., long range and short refueling time. In addition, they have about twice the efficiency of an internal combustion engine during a typical driving cycle [14]. Although hydrogen is the most common fuel, other fuels such as natural gas and methanol have occasionally been used in fuel cells [15]. Hydrogen fuel cell vehicles, however, are very expensive due to the cost of the platinum used in them as catalyst, which has hindered their large-scale production. While fuel cell vehicles may someday replace those with internal combustion engines, various disadvantages, specially the cost of the fuel cells, mean that the market is unlikely to see their large-scale production for decades [16].

Returning to the internal combustion engine vehicles, other factors that influence fuel consumption and exhaust emission in transportation are those related to driving style and independent driving pattern [6, 7, 17, 18]. It has been shown that nine driving pattern factors have important effect on fuel consumption and emission. These factors include acceleration and deceleration, gear changing, as well as the speed of the vehicle [18].

As a result of the ever-increasing cost of fossil fuel as well as the air pollution concerns, vehicle fuel consumption and emission have become two critical aspects in the transportation planning process of highway facilities. To this end, several mathematical models have been developed that predict vehicle fuel consumption and emissions using instantaneous speed and acceleration [19–21]. It is a well-known fact that fuel consumption (and emission) increases substantially as the speed of a vehicle increases, an issue that has been extensively discussed in the literature [22, 23]. Some references have referred to this increase as being ‘exponential’ [24] although technically, this statement is not correct.

Although the effect of speed on fuel consumption of a vehicle has been extensively investigated, the effect of its air speed has not been studied. This is because all of the investigations in the past have focused on vehicles moving in still air, in which case the ground speed becomes the same as the airspeed. However, if the vehicle moves upwind or downwind, the results would be quite different. In fact, any experienced truck driver would testify that fuel consumption of the vehicle varies substantially between driving upwind or downwind all day and, to some extent, driving through crosswind. In this article we show that what really determines the fuel consumption of a vehicle under normal driving conditions is the airspeed of the vehicle rather than its ground speed. However, as mentioned above, if the vehicle travels in still air, its airspeed and ground speed are obviously the same.

The objective of this paper is to develop a theoretical model for the rate of fuel consumption of a vehicle (especially of a highway vehicle) as a function of its airspeed, taking into account the fuel consumption due to the moving parts of the vehicle as well. To the best of our knowledge, the effect of these factors on fuel consumption of a vehicle has not been investigated or addressed in the literature, not at least from a quantitative point of view. We then compare this model to the actual experimental data that we have collected and show that they are in good correlation. We also show that this model is consistent with those previously developed as well as the existing experimental data for vehicles moving in still air. Finally, based on our results, we suggest a new design for highway construction which could substantially reduce the rate of fuel consumption of vehicles on busy highways.

Methods

Theoretical model

The fuel consumed by a vehicle is responsible for four different types of work: (a) vehicle acceleration, which increases the kinetic energy of the vehicle, (b) vehicle climbing a hill, which increases the gravitational potential energy of the vehicle, (c) work done against all internal frictional forces as well as the rolling friction of the wheels, and (d) work done against air resistance. Here we are not interested in parts (a) and (b). We only consider cases where the vehicle travels on a straight level road with a constant speed.

Consider a vehicle traveling on a straight level road, which we take to be the x direction, at a constant speed. Assuming no air resistance and no internal friction between the moving parts of the vehicle, according to the Newton’s first law of motion, the net force on the vehicle would be 0 and the vehicle would consume no fuel. But in reality, there is internal friction between the moving parts as well as the rolling friction of the tires. However, these frictional forces are all proportional to the normal forces acting between the moving parts, and the proportionality constants are all, to a good approximation, independent of the speed of the moving parts [25–27]. Therefore, for a vehicle traveling on a straight level road, to a good approximation, the contribution from these frictional forces is a constant k0, independent of the speed of the vehicle. The value of k0 depends on the details of the design of the vehicle.

We now turn our attention to air resistance. For a solid object moving in a fluid, the functional form of the force of air resistance is determined by the dimensionless Reynolds numbers [28, 29],

Re= Lρv η
(1)

which is the ratio of inertia to viscous forces on the object. In this equation, L is a characteristic length of the object, ρ is the fluid density, and η is its dynamic viscosity. In general, when Re < < 1, the viscous forces are much greater than the inertia forces and the latter can be neglected. In this case the magnitude of the drag force is proportional to the first power of the speed of the object relative to the fluid,

F d = k 1 v
(2)

which is known as the Stokes regime. On the other hand, when Re > > 1, the inertia forces dominate and the viscous forces can be neglected. In this case the drag force is in the Newton regime and its magnitude is given by a quadratic equation,

F d = k 2 v 2 .
(3)

For intermediate values of the Reynolds number, both the linear and the quadratic terms are present.

For a typical vehicle, the characteristic length is of the order a few meters. The density of air at sea level and normal temperatures is about 1.3 kg/m3, and its dynamic viscosity is about 1.8 × 10-5 Pa s. Therefore, for a typical vehicle even at speeds as low as a few kilometers per hour, say 5 km/h which is about 1.4 m/s, the Reynolds number is of the order of 105 to 106, which is much greater than unity, and the force of air resistance on the vehicle is in the Newton regime.

Therefore, for a vehicle traveling on a straight level road (the x direction) with velocity v, the total retarding force is given by

F=- k 0 - k 2 v 2
(4)

where the first term is the contribution from all internal frictions in the vehicle as well as the rolling friction of the wheels, and the second term is the contribution from air resistance. Once again we stress that in this equation v is the airspeed of the vehicle, not its ground speed.

The work done by a force F → on an object when the object undergoes a differential displacement d r → is given by [30]

dW= F → ·d r →
(5)

Because in the case of our vehicle,

F → =-( k 0 + k 2 v 2 )îandd r → =dxî
(6)

where î is a unit vector in the direction of motion (the x direction), Equation 5 reduces to

dW dx =-( k 0 + k 2 v 2 ).
(7)

Since d W/d x is the energy consumed by the vehicle per unit distance, the quantity -(d x/d W) is proportional to the fuel economy Φ of the vehicle, defined as the ratio of the distance traveled to the volume of gasoline burned [31, 32]. It is worth pointing out here that this quantity has occasionally been incorrectly referred to as ‘fuel consumption rate’ in the literature [33–35]. In terms of Φ, therefore, Equation 7 reduces to

Φ= 1 c 0 + c 2 v 2 .
(8)

Note that in this equation we have used new constants c0 and c2, instead of k0 and k2, because Φ is only proportional to -(d x/d W), not equal to it. This proportionality depends on a number of factors, including the efficiency of the engine and the limitations set by the second law of thermodynamics. This is because in an internal combustion engine, the chemical energy of the fuel is first converted into heat before it is converted into mechanical energy.

Equation 8 relates the fuel economy of a vehicle traveling on a straight level road to its airspeed. The constants c0 and c2 are specific to the details of the construction of the vehicle and vary from vehicle to vehicle. Before further investigation of this equation, for the sake of simplicity, we rewrite it in a slightly different form,

Φ= a b + v 2 ,
(9)

where the new constants a and b are related to c0 and c2 by a = 1/c2 and b = c0/c2. The graph of this equation has the correct functional form for the experimental data that have been collected for various vehicles [24]. Equation 9 also conforms to the functional form of the regression model of Rakha et al. for fuel consumption rate as a function of speed [36] as well as the Oak Ridge National Laboratory experimental data [37], both for vehicles moving in still air.

Results and discussion

During several trips from Wisconsin to Arizona and back in the 1990s, one of us (CHH) noticed a marked difference in fuel consumption between the westward trips out and the eastward return trips. Based on this observation, a 2003 Chevy Suburban was equipped with a Pitot tube [38] and an airspeed indicator (Wag-Aero 0-120 mph). The Pitot tube was mounted in the front of the vehicle, flush with the front bumper and center of the radiator. The device was calibrated by driving the vehicle in still air and comparing the speedometer readings with the airspeed indicator. The agreement was remarkably good, within 2 mph (3.2 km/h). The vehicle was also equipped with an electronic device which was plugged into the OBD2 engine analyzer to read out the instantaneous fuel economy.

After several pre-runs and confirming that downwind driving results in a better fuel economy than upwind driving, from 2010 to 2013, the vehicle was driven from Wisconsin to Arizona and back. The driving mode was a steady ground speed of 70 mph (speedometer reading) or about 113 km/h, looking for flat sections of road to eliminate error due to the change of altitude. A total of 383 measurements were taken on the fuel economy of the vehicle at various airspeeds. The results are shown in Figure 1.

Figure 1
figure 1

Fuel economy as a function of airspeed for a 2003 Chevy Suburban. Fuel economy as a function of airspeed for a 2003 Chevy Suburban driven at a constant ground speed of 113 km/h (70 mph). The markers are the experimental data and the solid curve is the nonlinear least-squares fit of Eq. (9) to the experimental data. The vertical line shows the ground speed of 113 km/h.

Full size image

As can be seen from Figure 1, there is a considerable amount of scatter in the experimental data, which will be explained later. Nevertheless, the trend is quite obvious; the fuel economy decreases as the airspeed increases. A nonlinear least-squares fit of Equation 9 to the experimental data gives the following values for the constants a and b:

a = ( 1 . 804 ± 0 . 044 ) × 1 0 5 km l km h 2 b = ( 1 . 421 ± 0 . 058 ) × 1 0 4 km h 2 .
(10)

The graph of this function is also plotted in Figure 1 as a solid line. To see the goodness of the fit more clearly, we grouped the experimental data in airspeed intervals of 5 km/h. The result together with the theoretical curve with parameters given by Equation 10 is shown in Figure 2, which clearly shows that the theoretical function fits the experimental data quite well. The error bars in this figure correspond to the standard deviations of the means in averaging the airspeeds over 5-km/h intervals [39].

Figure 2
figure 2

Fuel economy as a function of airspeed with data collected at 5-km/h intervals. Same as in Figure 1, except that the experimental data are collected in airspeed intervals of 5 km/h. The error bars indicate the standard deviations of the means as a result of averaging over 5-km/h intervals. The solid curve is the same as that in Figure 1.

Full size image

To see the correlation between ground speed and fuel consumption, we plotted the ground speed (speedometer reading) versus the fuel economy of our vehicle in Figure 3. Since all our data were collected at a ground speed of 70 mph (113 km/h), Figure 3 shows that there can be a large variation of fuel economy for the same ground speed,simply because the airspeed can be very different. In our case, with a ground speed of 113 km/h, the fuel economy varied from about 4.6 to 11.0 km/l. Therefore, if one collects only one data point at each ground speed, there would be essentially no correlation between the fuel economy and the ground speed. However, if we take several readings of the fuel economy at each ground speed and consider their average, we can see some correlation between the two, although this correlation is not as strong and well defined as that in Figure 1. This is due to the fact that through the averaging process, ground speeds simply become the same as airspeeds.

Figure 3
figure 3

Ground speed as a function of fuel economy for a 2003 Chevy Suburban.

Full size image

Finally, to confirm the above results, we equipped a 2005 Volkswagen Passat for the test and collected over 200 measurements at various airspeeds, but at a ground speed of 60 mph (about 97 km/h). These measurements confirmed all the results obtained for the Chevy Suburban, but of course, with different parameters for Equation 9. The graphs for the Volkswagen Passat are qualitatively nearly identical to those for the Chevy Suburban, therefore we do not duplicate them here.

Conclusions

The results of this investigation show that our theoretical model correctly describes the experimental data on fuel economy of a vehicle as a function of airspeed. The theoretical model as well as the experimental data clearly show that the fuel economy of a vehicle does indeed depend on its airspeed. The ground speed of the vehicle, on the other hand, plays little role in its fuel consumption unless the vehicle travels in still air, in which case the ground speed becomes the same as the airspeed. Furthermore, in addition to the air resistance, there is always a contribution to fuel consumption from the moving parts of the vehicle resulting in frictional forces between them as well as the rolling resistance of the wheels. This contribution, however, is a constant, independent of the speed for a given vehicle. The fuel consumption due to this contribution is equal to a/b, which is obtained by setting v = 0 in Equation 9. For the vehicle tested (Chevy Suburban), we have

a b = ( 1 . 804 ± 0 . 044 ) × 1 0 5 ( 1 . 421 ± 0 . 058 ) × 1 0 4 =12.7±0.8 km l
(11)

which would be the fuel economy of the vehicle if it traveled with an airspeed of 0.

Unlike in an aircraft, in an automobile, the Pitot tube is parallel to the airstream only when driving exactly upwind or downwind. Consequently, the automobile Pitot tube responds only to the parallel component of the airstream. Thus, in a 90° crosswind, the airspeed indicator in an automobile would read the same as the speedometer, regardless of the wind speed. This phenomenon contributes to data error, because a strong crosswind requires front wheels ‘crabbing’ into the wind to hold the road lane, contributing to more fuel consumption, while the airspeed indicator shows the same as the speedometer. This is one reason for the scatter in our data as shown in Figure 1. Another is the difficulty of finding a level stretch of road long enough to get stable airspeed and readings of the fuel economy.

In view of the above results, the question is how can the fuel economy of vehicles be improved? Of course, the first factor is to streamline the vehicle. This has been done by all manufacturers to the point that all new automobile profiles resemble each other. Streamlining the vehicle simply reduces the drag coefficient. Thus, the value of k2 in Equations 3 and 4 are lowered as much as possible, which maximizes a and minimizes b in Equation 9 accordingly. The second factor is to control the driving environment in order to expose vehicles to lower airspeeds. This is where we introduce the concept of parallel corridors. To this end, an enormous opportunity exists on heavily traveled roads to reduce fuel consumption by significant amounts without changing the vehicle or driving habits. In addition, parallel corridors could eliminate crosswind effects as well.

Recently, on a trip west, one of us (CHH) drove on the middle lane of seven lanes of traffic leading east out of Phoenix, Arizona, on Highway 60. While driving in the center lane between closely spaced vehicles, all traveling at 70 mph (113 km/h), the airspeed indicator registered 30 to 40 mph (48 to 64 km/h), illustrating the fact that vehicles carry air along with them.

Several years of driving experience with an airspeed indicator has revealed that natural conditions sometimes resemble the parallel corridor concept. In the mountainous states, for example, lanes are sometimes separated and flanked by cuts or berms, where a substantial reduction in airspeed is noted if traffic is heavy. This unique situation was first observed on Interstate 17, between Phoenix and Flagstaff.

The first phase might be to construct a 10-km test strip on a busy lane to confirm the benefits. Design guidelines could be established for exits, snow accumulation, and barrier heights, etc. A cost-benefit ratio could be obtained to determine payback time. Various barrier types might also be evaluated. In some areas, a single center barrier might suffice.

Finally, for anyone wishing to explore the parallel corridor concept personally, we would suggest mounting the indicator in such place that both airspeed and speedometer could be read at the same time. It is customary in mounting the Pitot tube (any tube will do) that the first bend behind the opening to the air be inclined upward to prevent water (from rain) to accumulate and block pressure. In our case, a small diameter silicone rubber tube connected the Pitot tube to the airspeed indicator.

The results of this investigation apply to any vehicle regardless of its size, shape, engine design, or type of fuel consumed as well as electric vehicles.

Authors’ information

CHH received his degree in Industrial Engineering in 1947, from Iowa State College. As a child during the Great Depression of 1929, he became motivated and impressed with the need for resource conservation as he grew up. He was actively involved in the Southeastern Wisconsin industries in various areas, including making fuel pumps for oil burners and making silicon rubber parts for the medical industry. He retired in 2007. As an amateur pilot, CHH became an expert in aviation and flight dynamics. He has always been interested in and an advocate of energy conservation.

PM is a professor of physics at the University of Wisconsin-Parkside. He received his PhD in Materials Science and Engineering from the University of California, Berkeley, in 1975. He has published extensively in a wide variety of areas, ranging from bicycle stability to cancer research. His current research interest is thermodynamics and statistical mechanics of small systems.