Tuesday, December 28, 2010

Cost per Unit Distance

Back in May, I posted an article about some of the maths you can use to estimate the cost of owning an Electric Vehicle.  Although there was a lot there to absorb, the calculation of fuel economy in terms of miles per gallonequivalent means that you're using 2 estimated commodity prices to equate a value that does not include those in its calculation.  Instead, it was suggested to me at the time that a better calculation is to measure things in terms of miles per unit energy or unit energy per mile.  Of course, the energy of gasoline, released as heat, is not totally converted to car motion in the Carnot cycle within the Internal Combustion Engine (ICE) cylinder, but a theoretical maximum energy possible would be captured by the Higher Heating Value (HHV) of total reaction cool-down.  That said, typically the energy of gasoline is measured by the Lower Heating Value (LHV), which terminates the calculation of energy at 150℃.  However, in fairness to the ICE's best-possible score, I will use the value for Gasoline's HHV to equate engine fuel economy per gallon to energy.  Specifically, the Oak Ridge National Laboratory quotes 132 MJ per gallon of gasoline, HHV, where MJ stands for Megajoule, or 1,000,000 Joules of energy.  The corresponding LHV value for Gasoline is 121 MJ, 7.6% less efficient than the HHV.

Energy Efficiency

As I established in the earlier post, the inherent energy in electrical potential at a given current can also be quantified using the standard value of kilowatts of power exerted in one hour's time, kWh.  The equivalency is given by 3.6 MJ per kWh.  Again, this is an ideal, with the practical loss being an artifact of engine and transmission inefficiencies; the real value would be a small percentage less.  Thus, if we replace gallons with 132 MJ and kWh with 3.6 MJ we can put the 2 energy sources on near-even footing.  By converting everything to maximum available energy and comparing to how far that will allow the vehicle to travel, we can compare both ICE cars and Electric Vehicle (EV) fuel-based efficiency in the common terms of the energy required to go a certain distance for each engine design.

Energy Efficiency of Internal Combustion Engine Calculator

U.S. Units Metric Units
Fuel Economy: mpg l100 km
Energy per unit distance: Unknown, Unknown.
Distance per unit energy: Unknown, Unknown.

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Energy Efficiency of Electric Vehicle Calculator

U.S. Units Metric Units
Total Battery Capacity: kW⋅h MJ
Driving Range: mi km
Energy per unit distance: Unknown, Unknown.
Distance per unit energy: Unknown, Unknown.

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Energy Cost

The nice thing about evaluating all fuel sources purely by their inherent energy content is that it allows us to freely switch between each energy storage unit.  Indeed, if one simply considers the cost of a given energy storage medium, one can use these equivalencies to determine how much the cost of an object of equivalent energy storage would be.  We can then compare the cost of both gasoline and of the stored charge of a battery.

Convert Between Different Units of Energy

$ per gallon:
¢ per litre:
¢ per Kilowatt⋅Hours:
¢ per Megajoules (106 Joules):
Megajoules per $:

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Travel Cost

Finally, we can compare the cost to move an EV and an ICE car by having each one take one of the 2 volatile commodity prices, gasoline or electricity, into account.  The result is the most practical question of all: how much does it cost for this car to go a certain distance.  Alternatively, we can answer the question of how far a unit of currency will get you in that car.  For instance, if you know how much it costs per mile of travel in your vehicle, and you know how far your commute to and from work is, you can simply multiply the 2 numbers and get your daily commute cost.  Do the same calculation in a high mileage ICE car and an efficient EV and see which will be cheaper to run.  Of course that won't include oil changes or battery checks, but as a day-to-day measure, it's a good way to budget your needs in these difficult economic times.

Travel Cost of Internal Combustion Engine Calculator

U.S. Units Metric Units
Fuel Economy: mpg l100 km
Fuel Cost: $gal ¢l
Cost per unit distance: Unknown, Unknown.
Distance per monitary unit: Unknown, Unknown.

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Travel Cost of Electric Vehicle Calculator

U.S. Units Metric Units
Total Battery Capacity: kW⋅h MJ
Driving Range: mi km
Cost of Residential Electricity: ¢kW⋅h ¢MJ
Cost per unit distance: Unknown, Unknown.
Distance per monitary unit: Unknown, Unknown.

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Conclusion

It's interesting to note that depending on your area, the price of energy in the form of gasoline may be very close to, if not cheaper than the cost of electricity.  For example, a rate of 7.033¢ per kWh represents gasoline selling for $2.57 910, a mere penny below the price I quoted back in May.  Seven months later, the price of gasoline is pushing $3.00 per gallon, but my electric rate has also fallen to 4.187¢ because demand in the winter months is much lower.  Needless to say, based purely on energy cost, gasoline is competitive with electricity, even if the later is still a bit cheaper.  The real advantage of an EV derives from the fact that an electric motor is much more efficient at converting energy to torque than an ICE.

Based purely on their efficiency of energy usage, a typical modern EV can get about twice the distance for a given unit of equivalent energy fuel in an ICE car with high efficiency.  This translates to a 50% savings in fuel cost for an EV.  However, it should be noted that although gasoline is not as efficient an energy source as modern consumer electricity in the U.S., it is still much lighter for a given unit of energy, as well as a lot more dense than the currently available EV batteries.  No vehicle is perfect in every respect and that's one reason why it's not a good idea to buy an EV if you're often traveling long distances.  While the extra weight of the ICE that isn't needed in an EV does allow some room for battery weight to grow, eventually that weight cannot equal the weight saving in storing your energy as gasoline.  Added weight in a vehicle means linear increases of Rolling Resistance as well as in the force required to climb a hill.  What's more, as the mass of a car increases, the energy needed to accelerate it increases quadratically.

But potentially the worst issue is increases in volume required to store fuel.  The volume can be engineered into a streamlined body so as not to introduce more drag into the vehicle.  However a lot of volume would require a lot of extra design and as such may lead to compromises in the aerodynamic efficiency of the car frame.  It's likely this would be heavily engineered away, but there is still the issue of cars being restricted by width and height on standard automotive routes and roads.  Any air resistance from the increased volume that wasn't engineered away would have the most detrimental effect of cubically increasing the power required to overcome drag.

But in the current crop of EVs, a balance has been struck that puts them out ahead of the ICE design meaning that, when that affordable electric car comes to your area, get it and get it NOW for instant savings during your daily commute!