How to use the road load equation [wiki talk page]
 

[This is a proposed Wiki article providing an introduction to how to calculate road load. By understanding road load, you can better understand the physical phenomena governing this aspect of fuel economy. Hopefully it's written at an introductory level and helps explain certain concepts and why they matter to fuel economy. I'm wide open to suggestions as to how to improve the article. Are there any parts that are not clear enough?

Saand, if you would like to add it to the wiki, please do so, and link and format it as appropriate. I do ask that you remove the page titled Theoretical Weight Energy, as this article corrects errors on that page. Feel free to discuss that below.]



The road load equation describes all the forces applied to your car: aerodynamic drag, rolling resistance, and braking. EcoModder's calculator can crunch these numbers for you for steady state driving on flat ground, if you know a few parameters such as your car's mass, CdA, and CRR. Looking through the equations behind the calculator will give you insight into what aspects of your car and driving affect your fuel economy.



Aerodynamic drag is given by F = ½ * Cd * A * ρ * V².

Cd*A is drag coefficient times frontal area. You can look up these values for your car in the wiki [link]. Cd describes the smoothness of the vehicle's shape, but frontal area is just as important. These variables never appear seperately from each other in the physics. Much of our work on EcoModder is an effort to improve our Cd, to move through the air while disturbing as little of it as possible.

ρ is the density of air, which is around 1.3kg/m³, but varies with temperature and barometric pressure. Your car will cut through the air better when the air is thinner, e.g. when it's hotter, or at higher elevations. Don't ignore this term.

V² is your vehicle's airspeed, SQUARED. This means that driving twice as fast means four times as much aerodynamic drag. A headwind or even a crosswind will give you an airspeed higher than the value on your speedometer. A crosswind will also increase the CdA of a car that's optimized for driving forward, such as a bus, tractor trailer, or a Prius.

Cars and bicycles generally spend a great majority of their energy overcoming aerodynamic drag. You can improve your fuel economy by reducing any of the factors in the above equation: slower speeds, a more slippery shape, a narrower or shorter car, thinner air. Note that aerodynamic drag is not affected by mass (assuming that mass doesn't deflect your suspension).



Rolling resistance = CRR * weight = CRR * mass * gravitational acceleration

CRR is your coefficient of rolling resistance, a property dependent on your tire and the road surface. Low rolling resistance (LRR) tires are an excellent way to reduce your CRR.

USCS people will probably use lbs for weight. The metric system makes it clear that mass is an amount of material (measurable in kg), while weight is a force due to gravity.

Note that the force of rolling resistance is the same regardless of vehicle speed, while aerodynamic drag varies with V². The amount of rolling resistance per mile depends only on your vehicle's weight and CRR.



So, the road load equation for steady state (constant speed) driving on flat land with no wind is:

F = ½*CdA*ρ*V² + CRR*m*g.

This makes for simple math, but it really only applies to highway driving, and even then not all the time. Still, studying your steady state road load gives you insight into what parameters of your car and driving cause you to spend the most gas.




[The material below probably belongs in a second article]

Gravitational and inertial loads

Energy spent to overcome rolling and aerodynamic drag is irreversibly lost. Not so for gravitational and inertial loads. Climbing a hill takes energy, but once you start to descent, 100% of that energy is recovered. Likewise, bringing a car up to speed takes energy, and you get 100% of it back when you coast to a stop.

Gravitational potential energy is calculated as: E = m*g*ΔH, where ΔH is the change in elevation.

For example, consider a car with a curb weight of 1835lbs, with a 160lb driver and a 33lb toolbox sitting in the trunk. Also, the car has 8 gal less than a full tank, thus saving 48lb off the curb weight. This adds up to 1980lbs, or 900kg. When you ascend a 1000 foot ridge, you will find there is about 2.69MJ of gravitational potential energy stored in your car. This amount of energy will need to be put to use or burned off with the brakes on the way down the hill.

Take the same car and accelerate from 0-60mph. Kinetic energy = ½*m*V², so you start with zero kinetic energy and end with about 0.323MJ of kinetic energy. Again, when you want to slow down, you will need to use this stored energy to coast to a stop, or burn energy off with the brakes.

There is a V² term in kinetic energy, so doubling your speed quadruples the amount of kinetic energy you're carrying around. I view kinetic energy as a liability on any road with red lights or traffic, as you may need to waste that energy in order to slow down as quickly as you require to remain safe, legal, and curteous.



Comparing force to energy

Force times distance equals energy. Suppose you took the road load equation "F = ½*CdA*ρ*V² + CRR*V*m*g" and determined there was a force of 250N at 60mph and 367N at 75mph. The amount of energy required to travel a mile at 60mph is 250N*1mi = 0.40MJ, or 367N*1mi = 0.59MJ at 75mph.



[make this equation accessible or delete it]
E = integral(½*CdA*ρ*V² + CRR*m*g)*dx + braking energy + accessory loads

[conclusion goes here?]

This is very interesting and a great read for people who are interested in physics.

Good info RobertSmalls :)

I would add it to the wiki myself but I am packing for a trip :)

If Saands has not done it I will do it next week when I return. You can always add it yourself, we will edit or remove if needed.

Excellent writeup Matt :thumbup:

Make sure that you Cd is CD total, and therefore includes the form drag rather than pure frontal. :thumbup:

You show the form of the road load equation to be

F = ½*CdA*ρ*V² + CRR*V*m*g.

This indicates that the resistive force due to rolling resistance is directly proportional to the velocity of the vehicle. I am surprised at this and indeed the online calculator -
Aerodynamic & rolling resistance, power & MPG calculator - EcoModder.com indicates that rolling resistance is constant with respect to velocity.

Which is correct?

@cr45: Nice catch. I was thinking about power to overcome RR, which is linear with velocity.

Unfortunately, for passenger car tires, that is only true at slower speeds. RR climbs exponentially at higher speeds. 50 mph seems to be the breakoff point - which is also unfortunately within normal driving speeds.

I suggest you add a note that the RR part is only true below 50 mph and explain what happens to RR at higher speeds.

What causes increased RR at higher speeds? Is it due to more heat buildup from tire deflection at such high rpms?

And I noticed you excluded other tires such as truck tires. The main difference coming to mind is the thicker sidewalls and tread of a truck tire vs a passenger vehicle.

Since I am an amature at mathematics can some one answer the following questions? Is the term g ,in the rolling resistance equation essentially one, on the on the surface of earth. With minor variations due to elevation that are small enough to be ignored for fuel economy purposes? Is wind resistance an exponental equation due to the v squared term and the rolling resistance equation linear with constant slope due to v or no other them being squared? Feel free to be wordy with your answer I welcome details. A suggested conclusion to the wiki being like the introduction that understanding the forces acting on your car you can be a more efficent driver and modifications to your car more effective.