Speed Costs Power:

In evaluating transportation choices, energetic economy is an important and well-characterized consideration commonly expressed in miles per gallon of gasoline. Average speed is also important, since people are paid by the hour and “time is money.”  Others have considered this interplay between vehicle speed and energetic economy.   

In a seminal technical article entitled, "What Price Speed?" Giuseppe Gabrielli and Theodore von Karman [1] defined the specific resistance of a vehicle, e, as maximum motor output power P, divided by total vehicle weight W multiplied by maximum speed V_M. 

Specific resistance was determined for various vehicles operating at a range of speeds.  An empirical limit for the minimum specific resistance as a function of maximum speed was found for any isolated vehicle.  This was described by Eq. 2, where A is 0.000175 hours per mile.  This relationship is depicted by the diagonal line in Fig. 1, which has become known as the Gabrielli-von Karman limit line of vehicular performance. 
 

(2)        (e)_min = AV_M

Gabrielli and von Karman [1] used gross vehicle rather than payload weight because, “exact information regarding the useful load of vehicles was not available to the authors.” Their analysis was reconsidered, with regard to payload weight and fuel consumption, by Stamper [2].  This latter treatment was more relevant to the economical application of energy resources.  Here, “useful transport work” was defined as the product of payload weight and distance traveled.  “Transport efficiency” was defined as the ratio of useful transport work to thermal energy expended. 

In a subsequent analysis by Teitler and Proodan [3], a quantity was defined as the specific fuel expenditure e_F, where z is the energy per unit volume of fuel, h is the distance traveled per unit volume of fuel, and W_P is the weight of the vehicle payload. 
 

(3)         e_F  =  z  / hW_P

The reciprocal of e_F  was defined as the fuel transport effectiveness, and related to vehicle cruising speed V_C by Eq. 4.  Following Stamper’s definition, the reciprocal of e_F was referred to as, “the dimensionless ratio of useful work output to energy input.” C_F was referred to as a “factor of proportionality.” As shown in Fig. 2, C_F was applied as a limit to what is technologically possible, rather than as a performance parameter to be applied generally to individual vehicles.  The dashed diagonal line was referred to as, “the next level of fuel transport effectiveness to be used as a future standard.”

(4)         (e_F )^-1  =  C_F^-1V_C^-1
 

Other writers referencing Gabrielli and von Karman [1] have also applied A or C_F as a factor describing an experiential performance limit, while treating e_F ^-1 or e_F as a general performance parameter.  These include Greenwalt [4], Carson [5], Minetti et al. [6], Young et al. [7].

There is another vehicle performance parameter that is expressed in units of time or velocity.  As described by Sutton [8], specific impulse, I_S, is a universally accepted parameter used to describe rocket motor performance.  It is defined as shown in Eq. 5, where F is the motor thrust force, m' is the propellant mass flow rate, and w' is the propellant weight flow under constant thrust conditions. 
 

(5)        I_S = F/(m'g_o) = F/w'

Force divided by propellant mass flow is also used to describe rocket motor performance.  This is known as the effective exhaust velocity, and has units of speed.  Ascribing mass to propellant gives a physical insight into this performance parameter; it is proportional to rocket motor exhaust velocity (Sutton [8]).

Analysis

In combining speed and energy expenditure to yield an energetic performance parameter, e_F is more economically informative than e.  The specific fuel expenditure e_F  takes payload weight and motor efficiency into account under cruising conditions, and is more representative of actual use and indicative of resultant benefit. C_F ^-1 is convenient because an increase corresponds to a performance improvement.   

Defining useful transport work as the product of travel distance and payload weight is misleading, since mechanical work is conventionally defined by the vector dot product of force and spatial displacement.  For horizontal displacement near the earth’s surface, payload weight is a force perpendicular to the direction of motion, so that the work done in this fashion is zero.  The dot product of the payload weight and distance traveled is vertical displacement multiplied by the force exerted by the payload mass.  This dot product divided by thermal energy expenditure is a true measure of lifting efficiency, and is always less than unity.   

Treating the payload as a mass (denoted M_P) rather than a weight yields a performance parameter Q with units of time.  For cruising conditions, Q_C is defined by Eq. 6, where g_o is the acceleration due to gravity.

An analogous fuel transport effectiveness is defined by Eq. 7, with E_th representing the thermal energy expended to travel a path length, denoted as l. The parameter e _Q^-1 is also referred to as the thermal energetic economy and it has been used by others to compare various modes of transport (e.g., Hobson, [9]).   

(6)        Q_C =  (g_o C_F)^-1 = V_CM_Ph / z

(7)        e _Q^-1  = Q_C / V_C = [lM_P / E_th]_C

As shown in Eq. 8, the differential form of the travel distance to thermal energy ratio becomes the ratio of cruising speed to thermal power expenditure (P_th).  Eq. 9 thus provides a physical interpretation of Q_C as the time during which a total fuel energy release equals twice the payload kinetic energy (E_k). 
 
 (8)        h / z = [l/E_th]_C = [V/P_th]_C

(9)        Q_C = [M_PV² /P_th]_C = 2[E_k/P_th]_C

Propellant weight is the quantity a rocket designer would like to minimize while obtaining the same result.  Since fuel (energy) consumption is the quantity most other vehicle designers endeavor to minimize, and Q_C is an energy divided by an energy flow, Q_C is analogous to I_S.  Ascribing mass rather than weight to matter provides a physical insight into both vehicle performance parameters.   

Using mass rather than weight to describe energetic performance yields a result which can be consistently applied in any environment, as illustrated by considering an extraterrestrial vehicle.  Due to differences in gravitational acceleration and atmospheric density, a vehicle should travel further on Mars (for example) than on earth, using the same quantity of energy.  Defining performance with C_F ^-1 gives a result that decreases because of the atmospheric density difference, and does not change because of the difference in gravitational acceleration.  On the other hand, Q_C increases due to both influences, and is more indicative of the change in conditions.  Whether moving a payload mass through a gravitational field or head wind, Q consistently reflects changes in conditions.  A pedagogically inferior treatment of this discrepancy is to introduce the concept of “standard weight,” which is a measure of mass expressed in units of weight. 

Transportation modes are often compared in terms of fuel economy, or distance traveled per unit volume of fuel, and Q_C can be defined as a related quantity. Q_C can readily be determined from vehicle speed, and automobile person-miles per gallon of gasoline or freight carrier british thermal units per ton-mile.  Given that a gallon of gasoline contains about 133 megajoules of thermal energy (Hobson, [9]), one can readily determine Q_C for a given number of persons in an automobile from the cruising fuel economy rating.  Cruise conditions are similar to those encountered on a long distance highway trip, and it is tempting to use the automobile “highway” miles per gallon rating to determine l/E_th.   

The EPA fuel economy ratings are intended to represent how vehicles are actually used.  The highway rating is meant to model free flow traffic at highway speeds.  This is measured with a dynamometer system over a simulated distance of only ten miles, with no stops and a maximum speed of 60 miles per hour (27 m/s).  Average speed during test is 48 miles per hour (21 m/s).  City driving conditions are simulated over a distance of 11 miles (18 km), with 23 stops and a maximum speed of 56 miles per hour (25 m/s).  Average speed during city driving simulation is only 20 miles per hour (8.9 m/s). Fuel consumption is measured over these short simulated distances by collecting and analyzing exhaust gas [10]. 

By using the EPA fuel economy ratings [l/E_th]_E, measured at an average speed, V_E, we can calculate an effective energetic performance, Q_E, for transporting  given number of persons through the EPA driving routine.  This is shown in Eq. 10 below.  Since the mass of the human payload is typically much less than the mass of the car, it is assumed that [l/E_th]_E does not change based on how many persons are in the automobile.  

(10)        Q_E = M_PlV_E / E_th = M_PV_E² / P_th

As described in the appendix, the physical meaning of Q changes when average speed is substituted for steady state cruising speed.  The numerator in Eq. 9 becomes twice the effective useful payload kinetic energy, which is evaluated at the average speed.  This differs from the average payload kinetic energy, which is determined from the root mean square (rms) velocity.  Because rms velocity is always greater than or equal to average speed, the payload kinetic energy evaluated at the average speed is the minimum possible average kinetic energy for any velocity profile.  It may be difficult to obtain rms velocities, and these higher values would not represent an improvement in the utility of the trip.  A vehicle traveling at constant velocity on an elevated express lane is assumed to be no more or less useful than a vehicle traveling at the same average speed through a series of stops.  To use rms speed would inflate the value of Q_E in the latter case.  The denominator in Eq. 9 is still the average thermal power expenditure, which is measured directly.  Using the average speed to determine performance thus takes the effectiveness of the velocity profile into account. The parameter Q_E is applicable to intermodal comparisons of transportation energy use. 

For some modes of mass transit, a passenger may spend a considerable amount of time captive within the system, perhaps while not even being present on the vehicle or while the vehicle itself waits for other vehicles.  This describes air travel in particular.  For this situation, V_E is determined by dividing the distance between airports by the average time between passengers entering the departure airport and leaving the arrival airport.   

Table 1 gives energetic economy, greenhouse gas emissive economy, payload mass, speed and energetic performance for various modes of human transportation. Energetic economy is determined by estimating the number of payload kilogram-kilometers obtained per Joule of thermal energy present in the fuel consumed.  Typical human mass is assumed to be 70 kilograms. The human body is assumed to be 25% efficient in converting the caloric content of food into mechanical work (DeLong [11]).  Moped vehicles are assumed to be ridden without pedaling. Since carbon dioxide emission is directly proportional to energy consumption, this column was easily added.  It has been estimated that 10.7 kilograms of CO₂ is emitted per gallon (3.78 liter) of gasoline burned (Google [12]).  Emissions associated with the use of electrical energy were determined for electricity obtained in 2007 Denver, with its assortment of generation facilities, load factors and distribution patterns.  In an analysis published by the City of Denver [13] it was estimated that 0.796 kilograms of CO₂ is emitted per kilowatt-hour of electricity delivered.  Table 1 was sorted first by payload mass, then by energetic performance.  This indicates a trend for personal vehicles traveling at low speeds, where vehicles with highest payload to vehicle mass ratios tend to perform better.  At higher speeds, aerodynamic effects predominate.  Fig. 3 is a logarithmic representation of the data from Table 1.   

The most complex economy determinations were those of electric vehicles.  Since electrical energy is a more organized form than thermal energy, it is important to determine how much thermal energy was expended in creating the electricity used to charge the vehicle batteries.  By measuring charger energy input, the battery charge, storage, and discharge efficiencies are accounted for.  Electrical powerline transmission efficiency was assumed to be 96%.  Net efficiency of the generating facility at the other end of the powerline is typically 33% (El-Wakil [14]). The range of the prototype Neodymics Cyclemotor electric bicycle is 17.7 kilometers at 11.2 meters per second. Fully charging the four DeWalt lithium iron phosphate battery packs (model DC9360) required 360 Watt-hours of electrical energy fed into the battery chargers.  So, one may travel 17.7 kilometers on an electric bicycle using 4.1 megajoules of thermal energy released at a typical electrical powerplant.   

In a similar manner, economy of the Segway^TM I2^TM personal transporter was determined from the manufacturer’s specifications (Segway [15]).  This device uses the same battery chemistry as the Neodymics Cyclemotor, so battery efficiency was assumed to be the same.  It is suspected that much of the energy consumed by the I2^TM is used to keep it upright.  

Fig. 3 also compares economy in the movement of petroleum and people.  Most petroleum is transported by ship or pipeline.  These means are about 1000 times more efficient than a single occupant SUV.

Table 1. Thermal energetic economy and energetic performance of various transportation modes. 

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Discussion

This analysis shows that light personal vehicles perform far better than heavy ones.  Energy used to produce vehicles and transportation infrastructure was not considered here, and such an analysis would make light vehicles appear even more attractive.  For all payload weight classes, the payload to vehicle mass ratio of the best performers is between three and four.  Since most people tend to travel individually when possible, and energy resources are becoming increasingly scarce with respect to demand, it would appear that personal vehicles of the future will be very light by today’s standards.  A challenge in the development of urban transportation infrastructure will be to allow for safe use of these personal vehicles amidst heavier cargo and mass transit vehicles.  There is a related challenge to improving economy ratings for personal vehicles in developed countries.  Here, people tend to view a light vehicle as impractical because of it lacks the crumple zone they have become accustomed to.  A light personal vehicle such as a bicycle is often seen as an exercise toy to be carried on top of a car.  Future changes in environmental and economic considerations may induce a sea change in this attitude.

Energetic performance was determined for widely different modes of transportation.  Streamlined human powered vehicles excel in personal vehicle energetic performance because of the relatively efficient human engine and the designer's careful attention to aerodynamics.  Commercial airliners also perform well because people are willing to crowd themselves into an aerodynamically optimized fuselage for fast, long distance travel.  There is no upper limit on energetic performance, and for a vehicle in a frictionless environment, it approaches infinity as trip length is extended. By terrestrial standards, a one-way trip into the void of interstellar space can be extremely fast and efficient.  Using the chemical energy release of the launch vehicle, and the present displacement from earth, Q_E for the Voyager 1 spacecraft is on the order of 10⁸ seconds.  The gravitational assist in propelling Voyager 1 is acknowledged as free, since it was not paid for with chemical combustion.  Solar energy captured by various means and used for earthly transport without terrestrial combustion is, in a similar sense, free. 

This analysis compared vehicle performance in terms of the thermal energy content of vehicle fuel only.  Primary energy used to obtain vehicle fuel was not considered.  In the case of human powered vehicles, this may be especially misleading.  On average, the thermal energy content of food consumed in the US is only about one-tenth the primary thermal energy used to produce, process, distribute, store and prepare the food.  A more complete comparison of primary energy use in personal transportation can be found in the Knol entitled Energy, Global Warming and Electric Bicycles.    

Acknowledgments

The author gratefully acknowledges receiving encouragement, helpful comments or suggestions from A. Babson and Dr. D. Faber; graphics review from D. Bahr, Dr. W. Connors, L. Dore, H. Noeldner, P. Nonn, Dr. G. Penn and Dr. C. Sprott; mathematical review from T. Schmidt and J. Snyder; and data refinement from R. Finley, G. Hertzler, T. Millett, R. Parker, Dr. R. Steeves, M. Weaver, C. Vetter and R. Zitarosa. 

Appendix:   Integral Definition of Effective Energetic Performance 

The effective (or time averaged) speed V_E is defined by the integral of the path length dl divided by the elapsed time T. 

V_E = ( ∫₀^Tv dt ) / T  = ( ∫₀^T(dl/dt) dt ) / T     (A.1)

V_E = ( ∫₀^Tdl ) / T         (A.2) 

Substituting into Eq 7 and solving for Q, we obtain: 

Q_E = ( M_P (l_T - l₀) ( ∫₀^Tdl ) / T) / ( ∫₀^TdE )     (A.3)

Q_E = M_P l V_E / E_th     (A.4₎ 

Dividing numerator and denominator by elapsed time, we obtain the differential form: 

Q_E = ( M_P (( ∫₀^Tdl ) / T)²) / ( ( ∫₀^TdE ) / T )  (A.5)

Q_E = M_P V_E² / <P_th>      (A.6) 

But the exact time averaged payload kinetic energy is: 

<E_k> = M_P V_rms² / 2   (A.7) 

Where V_rms is the root mean square velocity: 

V_rms² = ( ∫₀^Tv² dt ) / T   (A.8) 

The root mean square velocity may also be defined statistically for any time dependent speed profile by using the standard deviation, σ_V . 

V_rms² = V_E² + σ_V²    (A.9) 

By both definitions, we see that the minimum root mean square velocity for any time dependent velocity profile is the effective speed, and that these two quantities are equal when velocity is constant over time. 

V_rms² ≥ V_E²    (A.10) 

It follows that the minimum average payload kinetic energy for transport at the same effective speed becomes: 

<E_k>_min = M_P V_E² / 2   (A.11) 

Effective energetic performance becomes the following ratio: 

Q_E = 2 <E_k>_min / <P_th>   (A.12) 

References 

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[15]  Segway, Inc., Segway I2 Specifications.  http://www.segway.com/downloads/pdfs/i2-specs.pdf.  (accessed Oct 19, 2007).