“Be kind, for everyone you meet is fighting a hard battle” - Often attributed to Plato but likely from Ian McLaren (pseudonym of Reverend John Watson)

Saturday, August 30, 2008

Moving beyond hypermiling

Several times I've cited the Ecomodder web site. It was started by the owner of a Geo Metro (actually a Suzuki badged as a Pontiac) who'd created a site to discuss modifications both to his car and his driving style to maximize fuel economy. The questions and email he received at that site convinced him that a more general mileage dedicated site with forums, a mileage log, etc., would be popular. He was right. I'm a fairly active participant at the site and recommend it highly. I've acquired a large amount of very informative and sometimes useful information there.

Many of the denizens of that site extensively modify their vehicles. Such modifications range from minor things such as replacing factory original side view mirrors with smaller ones to complete transformations that make the vehicle nearly unrecognizable. Possibly the most extreme is the Honda Civic owned and modified by an Ecomodder using the screen name "Basjoos." He achieves 95 miles per gallon and is frequently stopped by police, queried by bystanders, and even occasionally interviewed by the media.

My vehicle is owned by my company and is used, on occasion, to visit and transport clients and associates and hence is not a suitable candidate for such an extensive makeover. But what could I do to, for example, achieve an overall ("highway" and "city" combined) fuel economy of 25 m.p.g. (I'm currently at 21.6 m.pg.) without making my vehicle a spectacle? I'd have to work to reduce the aerodynamic drag coefficient, or Cd. As I've mentioned in a variety of previous posts, the current Cd of the LR3 is 0.41. If I make some assumptions, I should be able to find out how large a reduction in drag coefficient would be required to achieve a given fuel economy. The assumptions are necessary because drag reduction is most effective at highway speeds. I'll assume that my drag reduction ONLY affects my highway mileage, and that I'm doing highway driving 60% of the time. That should be enough, together with various other estimates, to determine what it would take to get to 25 m.p.g. If I were to be able to accomplish this, it would save me about 147 gallons of fuel annually compared to the current 21.6 m.p.g. I'm getting. Currently, that's worth about $588.

Using my previous calculation of highway mileage and calculating from there, I estimate that my "non-highway" mileage is 19.10 m.p.g. Surprisingly, I've never determined this number before, and it's much higher than I would have thought. Anyway, I now have to determine the Cd that would enable me to achieve a highway mileage of 28.67 m.p.g. Frankly, this seems out of the question, but let's see.

I have to make a few assumptions (as usual). I'll assume that, at 55 m.p.h., 25% of the energy in my fuel turns my wheels and that there are 125*10^6 (125 million) joules of energy in a gallon of gasoline. Thus, a gallon of gasoline delivers (125*10^6)/4 or 31.25*10^6 joules to the wheels. I'll assume that rolling resistance is a function of the tire coefficient and vehicle weight only. I'll assume that the tire coefficient of rolling resistance is 0.12. This may be low. In any case, if I invert the fraction (28.67 miles/31.25*10^6 joules) I'll have energy divided by distance. This is force and, when appropriately converted, will be in newtons. From there, plugging in the known (or estimated) numbers for air density, area, speed, mass, gravitational acceleration, and coefficient of rolling resistance, I can solve for the necessary coefficient of drag.

I know the suspense is killing my readers, the required Cd is slightly under 0.32. Is it possible to reduce the coefficient of drag of my LR3 from 0.41 to 0.32 without making obvious alterations? The only areas I can work with are the grill, under the hood, and the under body. I strongly suspect that grill blocks and belly pans will not result in a 22% reduction in Cd. Still, they will presumably result in a reduced Cd and are cheap and easy. Further, there's a wiki called Instructables that has an article on measuring the drag coefficient of your car. That will enable me to track my progress, which I'll then correlate with my (hopefully) increasing gas mileage.

Friday, August 29, 2008

Google Analytics teaches me something about sociology

This will certainly be outside of my typical topic space. I've plugged in the code to use Google Analytics to see who, if anyone, has visited my little corner of cyberspace and how they might have come here. In looking at the report for the last month, the site has wandered around as usual (I'm too embarrassed to reveal the actual numbers). But yesterday (August 28, 2008) the number of visitors was down by 90%.

This is almost two sigma below the mean. It took but a minute to realize that this was likely to be because people work in the daytime and surf at night. What happened last night? Barack Obama gave his acceptance speech. I was amazed to find that the intersection between fuel economizing web surfers and members of the Obama cult of personality was so large.

I wonder how many of the set of people who would otherwise be perusing my site, or such sites as Ecomodder have actually drunk the Kool-Aid and how many were trying to get to know more about Obama so as to make up their mind?

My gut feeling is that it's more of the former and less of the latter, but the scientist in me is unhappy with leaving it at that. So instead, I've used this opportunity to try a blogger feature I haven't previously used: a poll. Take a moment and let me know, if you surfed less than usual last night, if it was because of Obama's speech and, if so, whether you went there to cheer on your man, or to learn more about a potential recipient of your vote. Clearly this is nonscientific, but it's certainly of interest.

Monday, August 18, 2008

Forces on my LR3 at 56 m.p.h.

There happens to be a hill of, as near as I can tell, constant slope on my commute to work on which I can put my car in neutral and coast down at about an unaccelerated 56 m.p.h. Obviously, the calculations herein will be approximate, these are hardly tightly controlled conditions. But using Google Earth, I can find that in a run of 563 feet, I descend from an elevation of 161.5 feet to 144.5 feet. Assuming that my loaded Land Rover LR3 weighs 5900 pounds force ("lbf"), I can use trigonometry to determine the component of the gravitational force acting to accelerate the truck down this hill. That will give me another estimation of the sum of the external forces acting on my truck, that is, its rolling resistance plus aerodynamic drag.

The calculation is sin(arctan((161.5-144.5)/563))*5900 lbf = Fr where Fr is the total is the total resisting force on the car. Of course, at such a small angle, the sine, the tangent, and the angle itself (in radians) are approximately equal, so what we have is (17/563)*5900 lbf. Thus, the downward component of gravity acting on my car and the total resisting force are each about 178 lbf or 792 Nt (Newtons). Startlingly, my calculations using .5*Cd*rho^2*A*v^2+Crr*m*g (see here) found 743 Nt. Now, this was at 55 m.p.h. rather than 56 m.p.h. and used what I have since determined is likely to be a very slightly low number for air density. Plugging in the appropriate numbers, I get 783 Nt, within 1.1% of the number calculated by determining the component of gravitational force acting parallel to the roadway above. As I've mentioned before, I just love it when different approaches to the same problem yield similar (or almost identical) answers.

So what does it mean? Well, it certainly means I'm on the right track in making calculations based on the resisting forces. I like this because I've made many deductions on that basis. The calculations are fairly limited to the case of analyzing the vehicle as the system and "outside the vehicle" as the environment, that is, the truck is a "black box." This is the case because there's no calculation of the forces involved in the many rotating masses in the vehicle, etc., or of the thermodynamic efficiency of the engine. But it clearly shows that the calculation of the external forces has been accurately performed, and thus the previous two posts are reasonable estimations of what it would take to create a very high mileage vehicle. Oh, and the slope? 1.7 degrees.

Sunday, August 10, 2008

Specifics of a high mileage car

In my previous post, I discussed what, outside of the engine and driveline, could be modified to increase fuel mileage. What are the specifics of such a car? Since the laws of physics are unchanging as far as is known and reasonably well known at the macro scale at which cars travel down roads, certain conclusions can be drawn. Let's start with the obvious: fuel is burned to overcome forces acting on the car to take it down the road. So there are two fundamental approaches to high gas mileage, i.e.: put more of the energy in a given amount of fuel to work; and reduce the forces acting on the vehicle.

I'll save maximizing the utilization of energy available in the fuel for another post. Here, I'd like to see what it would take to make a car that gets, say, 75 m.p.g. with currently achievable engine and drive line efficiency by reducing the forces acting on the car. I'll look at achieving this fuel mileage at 55 m.p.h. As I've previously mentioned, force times speed is power, and power is the rate of doing work or, equivalently, using energy.

So, we should be able to say that force times speed equals energy (fuel) divided by time, if the appropriate adjustments are made for units. Or, rearranging, force equals energy divided by speed multiplied by time. And, as would be expected, this simplifies to energy divided by distance. So if I assume 125 million joules/gallon, 25% drive line efficiency, and that I use that gallon in 75 miles, I can determine that the maximum combined force of aerodynamic drag and rolling resistance that I can overcome is about 260 Nt (Newtons). For the SI challenged reader, this is 59.6 pounds.

Referring to my previous post, at a fixed speed the only variables available to control are mass, rolling resistance, frontal area, and drag coefficient. Let's assume that tandem seating isn't a saleable option at this point. What can we do? Well, let's start with vehicle weight. In this article, it's estimated that about 40% of the weight of an average car could be eliminated through replacing steel with carbon fiber. Let's use a conservative estimate of 25%. Then, in this article it's stated that the lowest coefficient of rolling resistance on tires currently available is 0.0062, the highest checked was 0.0152. Let's assume that we can utilize tires with a coefficient of 0.008.

Let's get started. We'll take a small four seat sedan, something like a Toyota Yaris. This vehicle has a curb weight of 2293 pounds, a drag coefficient of 0.29 and a frontal area of (as best I could find) 2.282 meters squared. Let's predict the highway m.p.g. at a steady 55 m.p.h. using, from the previous post, the equation for joules/meter (which is another measure for the inverse of miles per gallon, using the appropriate unit conversions and efficiencies). We'll assume two 170 pound adults to make total weight 2633 pounds. Finally, I'll assume a coefficient of rolling resistance of 0.0115. Running through the calculations, we find that about 355 Newtons are required. To apply this force over a mile, assuming 25% efficiency in the engine, we'd use 0.01828 gallons, or a fuel efficiency of 51.8 m.p.g. Not bad, we're a good part of the way there.

But the car is rated at 36 m.p.g., what gives? Well certainly the EPA tests are more demanding than a steady 55 m.p.h. on level ground. Beyond that, it could be that the new tires with fresh tread have a higher coefficient of rolling resistance. Or, it could be that the engine is able to deliver significantly less than 25% of the energy available in the fuel. If we assume a rolling resistance coefficient of 0.0130 and 20% efficiency, the figure is 36.7 m.p.g. This seems close, and is typical of the types of iterative calculations that are necessary. I'm going to stay in the middle, since I should calculate better than the EPA mileage, due to the rigors of their test. I've verified this in my own LR3. I'm going to assume that the Yaris has a rolling resistance coefficient of 0.0122 and is able to deliver 22% of the energy in the fuel it burns to the wheels. This yields 43.0 m.p.g. Close enough.

Now, what do we get if we reduce the weight by 25%, use tires with a coefficient of rolling resistance of 0.009, and a coefficient of drag of 0.24? Running the numbers, we get 60.8 m.p.g. This is not good, let's see what the maximum credible reductions of coefficient can give us. Using 0.0062 and 0.16 for the coefficients of rolling resistance and drag respectively, we get 90.3 m.p.g. Thus, we conclude that a small car like the Yaris, with the maximally achievable modifications for efficiency, can exceed the target 75 m.p.g. But remember that we've replaced most of the steel with carbon fiber, taken every conceivable measure to reduce drag, and installed tires that are exceptionally efficient and may not wear well, handle well, or be very comfortable. And the fact of the matter is that I very much doubt if a vehicle can be brought to market with a 0.16 coefficient of drag. Let's see what we get with 0.22 and call it good. After all, tires with rolling resistance coefficient of 0.0062 currently exist according to the above-cited article. The answer is 71.5 m.p.g., slightly below the target.

So we conclude that it can be done but the price, both economic and in terms of comfort, is quite high. Clearly, attention to the engine is warranted, as is consideration of drive train modifications. A hybrid engine, combined with pulse and glide driving techniques, could greatly increase efficiency of fuel utilization but it would increase the weight. There is just no free lunch. Tandem seating anyone?

What does a high fuel economy car look like?

To quote Scotty, "you canna change the laws of physics." I'm going to look at what a high fuel economy car would look like, with no assumptions about engine technology breakthroughs. Therefore, there are four fundamental things that we can control: vehicle weight (affects fuel used for acceleration to speed and amount of rolling resistance); tire coefficient of rolling resistance; vehicle frontal area; vehicle shape, reflected in the drag coefficient.

Let's look at cruising. In this case weight only comes in as a factor in rolling resistance, while frontal area and vehicle shape are the factors affecting drag. I've seen an equation that alleges to combine these components - the equation is: Fr=0.5*rho*Cd*A*v^2+Crr*m*g*v where rho is air density, Cd the coefficient of drag, A the frontal area, v is velocity, Crr is coefficient of rolling resistance, m is mass of vehicle, and finally, g is the acceleration of gravity.

I don't buy it. My analysis shows that, at least for first order effects, rolling resistance is not a function of velocity, so let's use Fr=0.5*rho*Cd*A*v^2+Crr*m*g. This is dimensionally correct with both coefficients dimensionless. It is, therefore, plausible and I'm going with it.

So, what can be changed here? We can't change rho or g, and v is whatever the driver chooses to use. I'll list the variables we can change and what would be done:

1. Reduce Cd. This can be done by the manufacturer, there have been vehicles with Cd as low as 0.16, though not many. There are those who modify their vehicles themselves to reduce Cd. To see this in action, visit the Aerodynamics forum at the ecomodder web site. I'd suggest looking for posts by "basjoos" to see the extremes to which this can be taken. A blog post about his vehicle can be found here. If you choose to do this, be careful because aerodynamics can be non-intuitive.

2. Reduce A, frontal area. This means a smaller vehicle in general. For a two-seater, tandem seating might be an option. There are concept vehicles out there that take this route and they will certainly have a low so-called "drag area," the product of Cd and A. Market acceptance is clearly a question.

3. Reduce Crr. This is the amount of force used up by tire rolling resistance. There are low rolling resistance tires out there, and California is contemplating requiring manufacturers to list Crr for tires sold here. The rolling resistance depends in a complicated way on a number of factors, but tires primarily use energy in so-called "hysteresis losses," i.e., flexing portions of the tire without full energy recovery as the tire rotates. Steel wheels on trains have extremely low Crr's since they barely flex at all. For a look at low rolling resistance tires, check here.

4. Reduce m, mass. Obviously, reducing A helps here since, in general, smaller cars weigh less. Lighter materials, less room for storage, smaller fuel tanks, etc. can also be utilized, as can minimally sized engines for the mission at hand. These reductions are synergistic - lighter vehicles need smaller engines, which can utilize lighter drive line components, which can utilize smaller fuel tanks for less fuel weight, etc.

So, a composite, tandem seating car, optimally shaped with little or no trunk and a small fuel tank would appear to be the best prescription. Of course, as is usually the case, the easiest savings coming from driving less and sharing the ride.

As I stated at the outset, this doesn't address possible gains from engine efficiency. In my opinion, dramatic gains aren't likely here. I'll address engine issues in another post.