Energy Differences in Hilly Driving for Electric Vehicles:

Transcription

1 Energ Differences in Hill Driving for Electric Vehicles: Introduction Consider the idealized hill (an isosceles triangle) shown in the bo, with the left and right angle being. Let d sides of the triangle and l the base of the triangle. Then h l 2 tan dsin the perpendicular distance of the peak to the base. Consider the case of an electric vehicle traveling up and down the hill a distance 2d at a constant speed v and compare it to the case of an electric vehicle traveling on the flat a distance 2d. At constant speed, v, the kinetic energ, 2 mv2, is constant, also. The electric vehicle s energ, which comes from a batter, moves the vehicle up the hill. It will be shown that no batter energ is used to provide energ to move the vehicle down the hill at constant speed if the hill angle is greater than some value. Some fraction f of the available energ when the vehicle goes down the hill can be converted b a generator into electrical energ (regenerated) to charge the batter. The gravitational potential energ gained going from the bottom to the top of the hill is E g mgh mgd sin and the same amount of gravitational potential energ is available going from the top to the bottom of the hill to convert into heat energ or charge the batter. Assume that the onl eternal frictional forces of consequence are the air drag due to the velocit v of the vehicle and the rolling resistance of the tires and internal friction. For air drag: where F d 2 c da v 2 mcv 2 c d drag coefficient (unitless), A projected area perpendicular to the velocit air densit. 2 kg (Depends on the temperature, air pressure and the meter 3 amount of water vapor in the air.), m mass of the vehicle (The mass is onl needed in the equation directl below; it will cancel out in the final equations of interest.). C 2 c da /m. Eample: Chevrolet Bolt EV: Area 2.34 m 2, mass 6 8 (driver) kg,

2 c d. 32 C m 2.2 kg m m 2 7 kg For rolling resistance of the tires): F r Rmg. The assumption is that the rolling resistance is proportional to the weight of the vehicle. Eample: F r. 7mg. Then, for Bolt: F r. 7gm. 7 7kg 9. 8 m s kg m s J Thus, the total dissipative force is F F d F r. Traveling Over the Hill Since the speed remains constant going up the hill, the vehicle s batter must provide the energ lost to heat b the air drag and rolling resistance going up the hill at constant speed v, which is E ud F d F r d m Rg Cv 2 d. Similarl, gravit provides the energ lost to heat b air drag and rolling resistance going down the hill at constant speed v, which is E dd F c F r d m Rg Cv 2 d. The total energ that must be supplied b the vehicle s batter going up the hill is the sum of the air-drag/rolling-resistance energ loss and the increase in gravitational potential energ: E u E ud E g m Rg Cv 2 d mgd sin, since the vehicle s kinetic energ remains constant going up the hill. The total energ change going down the hill is E d E g E dd mgd sin m Rg Cv 2 d, since the vehicle s kinetic energ remains constant going down the hill. That is, gravit provides energ and energ is lost due to air drag and rolling resistance. Note that if mgd sin m Rg Cv 2 d, then the vehicle energ change will be positive going down the hill; simplified, this condition is gsin Rg Cv 2 (Note that the equation ields v in meter s ; to get it in mph multipl b ) If the hill angle arcsin Rg Cv 2 /g, then the vehicle cannot travel down the hill at constant speed v; it will lose energ to heating the environment and, thus slow down as it goes down the hill. Then our condition of constant speed would not hold. If the hill angle arcsin Cv 2 /g, then the vehicle cannot travel down the hill at constant speed v; it will slow down as it goes down the hill. Here is a plot of the minimum angle (degrees) versus vehicle speed (mph) for the Nissan LEAF such that it can go down the hill at constant speed: 2

3 So, some energ is available to charge the sstem s batter going down the hill for hills of large enough angle depending on the speed. This etra energ beond what is lost to air drag and rolling resistance could then be used to charge the batter to help move the vehicle over the net part of the trip. Assume that some of the energ supplied b the batter going up the hill is recovered b charging the batter when going down the hill. Let f the fraction of gravitational energ (regeneration) provided b the batter going up the hill that is stored in the batter when going down the hill f. The energ stored in the batter going down the hill is E b fe u fmgdsin. Then the energ epended b the batter going over the hill is E h E u E b dm gsin Rg Cv 2 fmgd sin or E h dm Rg Cv 2 f g sin. Note that some energ is lost due to heating the environment. Traveling on the Flat The energ epended b the batter when traveling the same distance at constant speed v on flat ground (, air drag and rolling resistance for the entire distance 2d and no regeneration) is E f 2dm Rg Cv 2. Comparison of Hill Travel to Flat Travel The energ epended traveling over the hill divided b the energ epended traveling on the flat is: r E h E f Rg Cv2 f g sin 2 Rg Cv 2. This ratio will be greater than when: Rg Cv 2 f gsin 2 Rg Cv 2 or f g sin 2 Rg Cv 2 Rg Cv 2 Rg Cv 2 or sin Rg Cv2 f g or v g f sin R C 3

4 For the Nissan LEAF, f.8, g 9. 8 m s 2, C m and R. 2. Minimum Hill Angle vs Vehicle Speed and Regeneration Here is a plot of the minimum hill angle (degrees) versus vehicle speed (mph) in order that hill driving to be more efficient than flat driving: 2 7 The following graph shows the result if rolling resistance is neglected: 2 7 Here is a plot of the minimum hill angle (degrees) versus regeneration, f, going down the hill for 6 mph speed in order that hill driving to be more efficient than flat driving: 4

5 Maimum Vehicle Speed vs Hill Angle and Regeneration The following graph shows the maimum vehicle speed (mph) versus hill angle (degrees) in order that hill driving is more efficient than flat driving: The following graph shows the result if rolling resistance is neglected:

6 7 2 3 Here is a plot of the maimum vehicle speed (mph) versus fraction of regeneration, f, going down the hill for a hill angle of 2 degrees in order that hill driving to be more efficient than flat driving: The ratio of energ for hill driving to flat driving for the Nissan LEAF is r f,,v sin v v 2. Ratio versus hill angle for vehicle speeds mph and 6 mph (lower curve): 6

7 Above the horizontal line for r hill driving is more efficient than flat driving. Ratio versus vehicle speed (mph) for hill angles 2 degrees and degrees (lower curve): Above the horizontal line for r hill driving is more efficient than flat driving. Conclusion So, for the assumptions given in this article, an electric vehicle can use less batter energ going over a hill than it would use traveling on the flat for the same distance of travel if the hill angle is greater than a certain value for a given vehicle speed or the vehicle speed is less than a certain value for a given hill angle. References 7

8 &t L. David Roper, c:science/phsics/vehicles 8