In my last post I tried to calculate the top speed of my Land Rover LR3 HSE. In doing so, I used the horsepower found in the vehicle's specifications, i.e., 300 horsepower. Earlier in my musings about the LR3, I was marginally successful in some calculations relating to the car's (truck's?) fuel consumption. Now I'd like to calculate the vehicle's horsepower using known facts. These facts are the engine displacement and red line r.p.m. (6200)and the r.p.m. at which it was rated as specified (5500).
I'll use the method that was employed before, i.e., I'll determine how much fluid volume (air is a fluid) is going through the engine at those r.p.m.'s and how much fuel would be in this fluid. From there, I'll use the energetic content of the fuel and some thermodynamic calculations of efficiency to see what's available at the flywheel. Here we go. One advantage I'll have is that I can estimate the efficiency by figuring out how much heat energy per second comes from putting air/fuel mixture through the engine at 5500 r.p.m. and utilizing the rated brake horsepower of 300 at that r.p.m. This should give me an approximation of how many joules per second go to the flywheel versus how many are discarded to the environment.
A straight ratio of (6200/5500)*300 tells me that I should be able to produce 338 horsepower at 6200 r.p.m. As mentioned in the earlier post linked above, Car and Driver stated
that the LR3 is governor limited to a top speed of 121 m.p.h. Let's see what kind of overall efficiency is indicated if 5500 r.p.m. produces 300 horsepower. To do so, I need an estimate of manifold pressure - for starters I'll assume wide open throttle and maybe something like 0.2 p.s.i. losses for an absolute pressure of about 14.3 p.s.i.
So at 5500 r.p.m., the engine (as per my policy, I'll spare readers many of the actual calculations) will pump about 0.0147 kilograms of fuel air mixture through the engine each second. This takes into account manifold pressure of 14.3 p.s.i. as mentioned above. Oxidation of this mass of gasoline will release about 705,000 joules of energy each second. Since joules/second are watts, a measure of power, we can convert to horsepower. Doing so, if the heat of oxidation of that amount of gasoline could be converted to mechanical energy with 100% efficiency, we would develop 946 horsepower. The rated horsepower at 5500 r.p.m. is 300, implying an efficiency of about 32%. This seems very reasonable.
OK, so what's available at 6200 r.p.m.? Well, since all these calculations are based on mass flow of fuel air mixture through the engine, the simple calculation above gets close, at 338 horsepower. However, I assume that there's a slight improvement in volumetric efficiency, in other words, a slightly higher manifold pressure than at 5500 r.p.m. Making this assumption and calculating as above, it seems possible that, if full throttle at redline can be achieved, an absolute maximum of 358 horsepower could be measured at the flywheel.
Tuesday, June 26, 2007
Sunday, June 24, 2007
Top speed
Those who may have stumbled upon my little blog have probably noticed that I enjoy putting numbers to things. The Land Rover LR3 HSE is certainly an example, though the figures that I've been able to derive have borne only a passing resemblance to measured values. So I think I'll derive one that I won't be able to disprove myself - the maximum speed of the vehicle.
As we've previously noted, in unaccelerated travel the sum of the forces acting on the vehicle must be zero. The rated power of the 4.4 liter V8 engine in the LR3 is 300 horsepower. We'll convert that to 223710 watts using the Google Calculator. Now I'm going to estimate, based on various sites I've visited, that the efficiency of transmitting the engine's power at the flywheel to the road is 78%. This may seem a little low to some, typical figures are often in the 80% to 85% range. But this is a full time four wheel drive vehicle, so losses will be higher.
That leaves 174794 watts to the road. Now, power is force times speed, so if I add the external forces and multiply them by the speed, I'll have the power being utilized to overcome forces. The aerodynamic drag reduces to 0.7491*s^2 and the road load is estimated to be 14.65*s, where s is the speed of the vehicle. If anyone leaves a comment that they would like to know where these figures came from, I'll be happy to oblige.
In any case, since those are forces and speed times force equals power, we'll multiply by speed (s) and equate it to 174794 Thus: 174794=0.7491*s^3+14.65*s^2. I have various computer algebra systems, but the simplest is called Derive 6. Much to my regret, Texas Instruments will stop development and shipment of the program this week. I've had most versions, and where truly exotic math isn't required, I prefer it to Mathcad, Maple, and Mathematica, as capable as those programs are.
In any event, Derive easily solves this cubic equation and determines that s=55.65 meters per second, or 124 miles per hour. For purists, this is the solution in the real domain. I'm not sure how realistic this is - I haven't come anywhere close to topping out the speed in the LR3, nor do I intend to do so. This is the basis of my contention at the outset of this post that I won't be able to disprove the number. However, Car and Driver's site gives the top speed as 121 m.p.h. but refers to it as "governor limited." According to my calculations, you'd have to be going down a hill to exceed this by more than 2.5% so I'm not sure why a governor is needed. I still find this published number satisfying.
As we've previously noted, in unaccelerated travel the sum of the forces acting on the vehicle must be zero. The rated power of the 4.4 liter V8 engine in the LR3 is 300 horsepower. We'll convert that to 223710 watts using the Google Calculator. Now I'm going to estimate, based on various sites I've visited, that the efficiency of transmitting the engine's power at the flywheel to the road is 78%. This may seem a little low to some, typical figures are often in the 80% to 85% range. But this is a full time four wheel drive vehicle, so losses will be higher.
That leaves 174794 watts to the road. Now, power is force times speed, so if I add the external forces and multiply them by the speed, I'll have the power being utilized to overcome forces. The aerodynamic drag reduces to 0.7491*s^2 and the road load is estimated to be 14.65*s, where s is the speed of the vehicle. If anyone leaves a comment that they would like to know where these figures came from, I'll be happy to oblige.
In any case, since those are forces and speed times force equals power, we'll multiply by speed (s) and equate it to 174794 Thus: 174794=0.7491*s^3+14.65*s^2. I have various computer algebra systems, but the simplest is called Derive 6. Much to my regret, Texas Instruments will stop development and shipment of the program this week. I've had most versions, and where truly exotic math isn't required, I prefer it to Mathcad, Maple, and Mathematica, as capable as those programs are.
In any event, Derive easily solves this cubic equation and determines that s=55.65 meters per second, or 124 miles per hour. For purists, this is the solution in the real domain. I'm not sure how realistic this is - I haven't come anywhere close to topping out the speed in the LR3, nor do I intend to do so. This is the basis of my contention at the outset of this post that I won't be able to disprove the number. However, Car and Driver's site gives the top speed as 121 m.p.h. but refers to it as "governor limited." According to my calculations, you'd have to be going down a hill to exceed this by more than 2.5% so I'm not sure why a governor is needed. I still find this published number satisfying.
Saturday, June 23, 2007
The Almanac
One of my favorite books is the annual World Almanac and Book of Facts. Some of the book covers topics which don't interest me in the slightest, e.g., entertainment facts and much (but not all I hasten to admit) sports information. But it is chock full of nuggets to please a fact and number junkie such as myself.
I thought I'd look into per capita energy consumption in the United States and see how it's increased in the range of time covered by the Almanac for such information. In 1960 the U.S. consumed 45.09 "quads." A quad is a quadrillion, or 10^15, btu's (british thermal units). This is total consumption of every kind - industrial, agricultural, commercial, transportation, etc., and from all sources. In 2006 the consumption was 100.41 quads.
To get these figures to a unit with which we are familiar, energy per time (quads/year) is power, so the figure for 1960 converts to 1.51*10^12 watts. In 2006 the figure is 3.36*10^12 watts. The population in 1960 was 179,323,175 for a per capita power usage (that is, rate of energy consumption per unit of time) of 8421 watts per person. In 2006 with a population of 298,444,215 we consumed energy at the rate of 11,248 watts per person, a 33.6% increase in consumption rate. Frankly, this is a smaller increase than I would have guessed.
Just to put this into perspective, this is as if, in 2006, each of us had 112 100 watt light bulbs lit 24 hours per day, seven days per week. Or, since 11,248 watts is 15.1 horsepower, each of us had about 15 lawn mowers following us around 24/7.
Let's take a look at a comparison of the U.S. with China. In 2006, the U.S. consumed energy at a rate of 11,248 watts per capita. China, with a total consumption of 59.57*10^15 quads and a population of 1,313,973,713 consumed energy at a rate of 1,514 watts per capita, or about 13.4% of the U.S. rate of consumption. I'd be curious to know how much of China's figure relates to production of consumer goods for shipment to the United States.
Worldwide, in 2005 humanity consumed 446*10^15 quads, or used energy at the rate of 1.49*10^13 watts. With a population of about 6.4 billion, we consumed energy at the rate of about 2330 watts per capita, about 21% of the U.S. rate. This is very scary stuff. Standard of living is very strongly correlated with rate of energy consumption, so to bring the world to "our" standard of living, we'd have to approximately quintuple our rate of energy consumption on a worldwide basis. This seems ludicrous. Never mind climate change, there is no chance of converting (all human energy use is conversion, never creation) energy at such prodigious rates.
How about looking at it from the other direction? How much energy consumption could we, as a society, forgo? I think I personally could struggle by on 1/2 the personal energy consumption. Remember, though, that this would include reducing the energy content of my consumer purchases, my food, etc., since the numbers above are all-inclusive. As I look around, listen to the television on downstairs, listen to the waterfall in my pool as water is pumped through the filtration system, listen to my wife in the shower as the heater provides hot water, etc., I know there's a long way to go.
I thought I'd look into per capita energy consumption in the United States and see how it's increased in the range of time covered by the Almanac for such information. In 1960 the U.S. consumed 45.09 "quads." A quad is a quadrillion, or 10^15, btu's (british thermal units). This is total consumption of every kind - industrial, agricultural, commercial, transportation, etc., and from all sources. In 2006 the consumption was 100.41 quads.
To get these figures to a unit with which we are familiar, energy per time (quads/year) is power, so the figure for 1960 converts to 1.51*10^12 watts. In 2006 the figure is 3.36*10^12 watts. The population in 1960 was 179,323,175 for a per capita power usage (that is, rate of energy consumption per unit of time) of 8421 watts per person. In 2006 with a population of 298,444,215 we consumed energy at the rate of 11,248 watts per person, a 33.6% increase in consumption rate. Frankly, this is a smaller increase than I would have guessed.
Just to put this into perspective, this is as if, in 2006, each of us had 112 100 watt light bulbs lit 24 hours per day, seven days per week. Or, since 11,248 watts is 15.1 horsepower, each of us had about 15 lawn mowers following us around 24/7.
Let's take a look at a comparison of the U.S. with China. In 2006, the U.S. consumed energy at a rate of 11,248 watts per capita. China, with a total consumption of 59.57*10^15 quads and a population of 1,313,973,713 consumed energy at a rate of 1,514 watts per capita, or about 13.4% of the U.S. rate of consumption. I'd be curious to know how much of China's figure relates to production of consumer goods for shipment to the United States.
Worldwide, in 2005 humanity consumed 446*10^15 quads, or used energy at the rate of 1.49*10^13 watts. With a population of about 6.4 billion, we consumed energy at the rate of about 2330 watts per capita, about 21% of the U.S. rate. This is very scary stuff. Standard of living is very strongly correlated with rate of energy consumption, so to bring the world to "our" standard of living, we'd have to approximately quintuple our rate of energy consumption on a worldwide basis. This seems ludicrous. Never mind climate change, there is no chance of converting (all human energy use is conversion, never creation) energy at such prodigious rates.
How about looking at it from the other direction? How much energy consumption could we, as a society, forgo? I think I personally could struggle by on 1/2 the personal energy consumption. Remember, though, that this would include reducing the energy content of my consumer purchases, my food, etc., since the numbers above are all-inclusive. As I look around, listen to the television on downstairs, listen to the waterfall in my pool as water is pumped through the filtration system, listen to my wife in the shower as the heater provides hot water, etc., I know there's a long way to go.
Sunday, June 17, 2007
Turning off the engine
I have linked a blog called "Daily Fuel Economy Tip in the right column of this blog. I enjoy reading it, and have left quite a few comments. Brian Carr's (the host of the blog) most recent post is a follow up to a previous post regarding turning off the engine at stop lights. I also do this, and I have done a fair amount of googling on the topic.
As anyone reading this will likely be aware, the web is awash in sites discussing fuel saving methods. A lot of these discuss avoidance of idling, and there are large differences in the "break even" times for turning off the engine and avoiding burning fuel while idling versus extra fuel used to start the engine. Brian estimates an increase in m.p.g. of about 5.8% from this technique alone. On his site, I left a comment contemplating whether that was a plausible number, I'll copy the comment here:
"I also do this, however, I wonder if a significant portion of your increased mileage in the second period comes from this. Let’s see if it’s plausible:
If you had driven the 1559.9 miles in the second period with the miles per gallon (31.25) from the first, you would have used 49.9 gallons. Instead you used 47.2 gallons, a savings of 2.7 gallons. Now, I’ve had a Jeep Grand Cherokee Limited with a 4.7 liter engine and now have a Land Rover LR3 with a 4.4 liter engine. The Grand Cherokee used 0.38 gallons per hour at idle, the LR3 about 0.5 gallons per hour. Your car probably uses less than either of these at idle but let’s assume 0.4 gallons per hour. It would have taken you 6.75 hours of idling at stoplights to burn 2.7 gallons. In a 30 day month, that would be about 13.5 minutes of sitting at stoplights each day, every day. Assuming that a stop is an average of 40 seconds (don’t have data, just a guess), that would mean that, every single day, you were stopped at a little over 20 stoplights.
Now, if you have a much smaller engine (I forgot what you drive), that would mean you’d have to be spending even more time at stoplights to have turning the vehicle off at stoplights be primarily responsible for your savings. Do you think that this is the case?
Also note that these calculations assume absolutely NO extra fuel use on startup. I’ve searched the web and can find no figures for extra fuel used on startup, but the assumptions above are clearly the most favorable for fuel savings by engine shutoff."
It would seem that his savings must be due to some other factors as well, yet the numbers above aren't completely out of the question. As mentioned there, I know well how much fuel I'm burning as I sit at a light (or coast to it) but it's been very difficult finding any information on extra fuel used to start a spark ignition internal combustion engine. Various sites claim six seconds, 10 seconds, 30 seconds, and one minute as the break even point. No one cites any data to back up their number, and they vary by a factor of 10. This is not especially helpful.
However, after replying to Brian Carr's post, I tried changing my google search string to "idling versus shutting down engine". This led me to this site. It's the closest I've found to giving me the answer I need to determine whether shutting off the engine at stop lights is a fuel saver. It says the following: "Our research showed that a V6 restart takes about the same fuel as 5 seconds of idling. We expect a V8 to save more and a 4-cylinder less." It doesn't state the nature of the research, whether it was actual measurement by the authors, by associates of the authors, or was located in a literature search. It ain't much but it's all I've found.
As I've mentioned before, I haven't done experiments by varying a single parameter and keeping all others as constant as possible to determine the effect of that parameter. Brian Carr says he has done so and the article from the ASME Florida Section provides some verification. So I suspect that, while Brian's entire 5.8% increase in fuel economy isn't due to this procedure, a significant portion of it is.
I am going to run an experiment to determine this once and for all. I don't have a way to directly measure the fuel consumption on startup, but I'll find a vacant stretch of road forming a closed course a couple of miles long. I'll run it for an hour or so stopping and idling for a measured period every mile and find the total fuel consumed. Then I'll drive the same distance, turning the car off and back on every mile - it won't matter how long the shut off period is. I'll be concentrating very hard on duplicating the acceleration, cruising, and deceleration regime of the first series. I'll start both with a full fuel tank.
At the end of this process, I should have enough data to approximate the break even time and from there, the excess fuel used on startup. No doubt this number won't reflect what's happening before the car warms up, but it should provide reasonably definitive data on the question of turning the car off at lights. I can't take credit for formulating this methodology - I read it on someone's web site. I'd credit the site, but I can't find it now.
As anyone reading this will likely be aware, the web is awash in sites discussing fuel saving methods. A lot of these discuss avoidance of idling, and there are large differences in the "break even" times for turning off the engine and avoiding burning fuel while idling versus extra fuel used to start the engine. Brian estimates an increase in m.p.g. of about 5.8% from this technique alone. On his site, I left a comment contemplating whether that was a plausible number, I'll copy the comment here:
"I also do this, however, I wonder if a significant portion of your increased mileage in the second period comes from this. Let’s see if it’s plausible:
If you had driven the 1559.9 miles in the second period with the miles per gallon (31.25) from the first, you would have used 49.9 gallons. Instead you used 47.2 gallons, a savings of 2.7 gallons. Now, I’ve had a Jeep Grand Cherokee Limited with a 4.7 liter engine and now have a Land Rover LR3 with a 4.4 liter engine. The Grand Cherokee used 0.38 gallons per hour at idle, the LR3 about 0.5 gallons per hour. Your car probably uses less than either of these at idle but let’s assume 0.4 gallons per hour. It would have taken you 6.75 hours of idling at stoplights to burn 2.7 gallons. In a 30 day month, that would be about 13.5 minutes of sitting at stoplights each day, every day. Assuming that a stop is an average of 40 seconds (don’t have data, just a guess), that would mean that, every single day, you were stopped at a little over 20 stoplights.
Now, if you have a much smaller engine (I forgot what you drive), that would mean you’d have to be spending even more time at stoplights to have turning the vehicle off at stoplights be primarily responsible for your savings. Do you think that this is the case?
Also note that these calculations assume absolutely NO extra fuel use on startup. I’ve searched the web and can find no figures for extra fuel used on startup, but the assumptions above are clearly the most favorable for fuel savings by engine shutoff."
It would seem that his savings must be due to some other factors as well, yet the numbers above aren't completely out of the question. As mentioned there, I know well how much fuel I'm burning as I sit at a light (or coast to it) but it's been very difficult finding any information on extra fuel used to start a spark ignition internal combustion engine. Various sites claim six seconds, 10 seconds, 30 seconds, and one minute as the break even point. No one cites any data to back up their number, and they vary by a factor of 10. This is not especially helpful.
However, after replying to Brian Carr's post, I tried changing my google search string to "idling versus shutting down engine". This led me to this site. It's the closest I've found to giving me the answer I need to determine whether shutting off the engine at stop lights is a fuel saver. It says the following: "Our research showed that a V6 restart takes about the same fuel as 5 seconds of idling. We expect a V8 to save more and a 4-cylinder less." It doesn't state the nature of the research, whether it was actual measurement by the authors, by associates of the authors, or was located in a literature search. It ain't much but it's all I've found.
As I've mentioned before, I haven't done experiments by varying a single parameter and keeping all others as constant as possible to determine the effect of that parameter. Brian Carr says he has done so and the article from the ASME Florida Section provides some verification. So I suspect that, while Brian's entire 5.8% increase in fuel economy isn't due to this procedure, a significant portion of it is.
I am going to run an experiment to determine this once and for all. I don't have a way to directly measure the fuel consumption on startup, but I'll find a vacant stretch of road forming a closed course a couple of miles long. I'll run it for an hour or so stopping and idling for a measured period every mile and find the total fuel consumed. Then I'll drive the same distance, turning the car off and back on every mile - it won't matter how long the shut off period is. I'll be concentrating very hard on duplicating the acceleration, cruising, and deceleration regime of the first series. I'll start both with a full fuel tank.
At the end of this process, I should have enough data to approximate the break even time and from there, the excess fuel used on startup. No doubt this number won't reflect what's happening before the car warms up, but it should provide reasonably definitive data on the question of turning the car off at lights. I can't take credit for formulating this methodology - I read it on someone's web site. I'd credit the site, but I can't find it now.
Saturday, June 16, 2007
Drafting
In thinking about what I've done to attempt to minimize fuel consumption there are three things that stand out in my mind as potentially dangerous. I say "potentially dangerous" because for all three of them it is "common knowledge," stated all over the web, that they are. For the sites and blogs that I frequent (devoted to fuel saving measures and techniques) this is typically in the context of "while this may save fuel, it is very dangerous and no amount of fuel savings is worth a serious injury or your life."
So what are the three? The first is filling my tires (slightly) beyond the recommended maximum. The second is turning off the engine on long downhills. The last and most controversial is drafting trucks. It is this that is the subject of this post.
There are two obvious question. The first is "does it work?" The second is "is it dangerous?" As to its value in fuel savings, while there are some naysayers, most references agree that it is quite effective. Coincidentally, the Discovery Channel hit show Mythbusters covered this topic recently. Their results are summarized here. As is their common practice in such "myths," Kari, Tori, and Grant started with a scale model in a wind tunnel and achieved very encouraging results. Their full scale testing, while it didn't achieve quite the drag reduction of the wind tunnel tests, indicated significant fuel savings even at a following distance as large as 100 feet.
I have played with this idea off and on, both in the Grand Cherokee and in the LR3. However, because I have been so very unsuccessful with the LR3 in achieving the dramatic enhancements to gas mileage I was able to accomplish in the Jeep, I recently decided to pursue this avenue more aggressively. I know that this is controversial, many will call it selfish - if I crash, it will cause a huge backup for those behind me. I agree that this technique is extremely selfish if there is any significant chance that I will crash into a truck and tie up a freeway. I'll get back to that momentarily.
But does it work? According to the information from my Scan Gauge II, it does. I attempted to judge how far I was behind trucks by using my stopwatch to see what fraction of a second elapsed between the back of a truck crossing a highway mark and the front of my car crossing the same mark. Needless to say, tailgating a truck while looking at a Scan Gauge II reading and timing intervals with my stopwatch is sort of living on the edge, but I concluded that I was about 35 feet behind the truck. According to Mythbusters' results, I can look for an increase in miles per gallon of somewhere between 20% and 27%. As best I could tell, I saw an increase from about 21.5 m.p.g. to about 25 m.p.g., an increase of about 16%. These are pretty fuzzy figures - the readouts are constantly changing with changes in slope, etc., and the trucks typically don't maintain a particular speed quite as efficiently as the LR3 (due to the huge mass of the trucks no doubt). Nevertheless, every time I try it I get significant indications of increased fuel efficiency.
So apparently it works, how dangerous is it? Obviously, the worst case scenario is for the truck to apply maximum braking suddenly with no advance indication to me. I don't know if trucks have anti-lock braking systems, for the analysis to follow I won't assume they do. Judging from the condition of a lot of the trucks I see, this is a valid assumption. It should go without saying that, when following a truck at 35 feet, my eyes are focused on the truck's brake lights and my foot is ready to instantly hit the brake pedal. So the question is, when I see his (male pronouns are to be understood as gender-free) brake lights when he implements maximum braking, can I avoid hitting him?
I have to calculate the circumstances under which the distance between my front and his rear decreases to zero. The data necessary for this calculation is his maximum deceleration rate, my maximum deceleration rate, and my time to go from his brake lights on to my application of maximum braking. I will obviously limit my calculations to dry pavement in good condition - I'm not suicidal.
Truck braking, like everything else when looked at by academics, is ridiculously complex, see here if you are skeptical. But the nugget for my purposes in that paper is that the best that an empty semi-tractor trailer can do in deceleration is right at 20 ft/sec^2, or 6.10 m/s^2. For my LR3, it's (maybe, subject to correction) about 8.0 m/s^2. My reaction time, when paying close attention (as I do when tailgating a large truck) was determined using this online reaction timer to be 0.303 seconds in an average of 10 trials. With a lot of practice and knowing what to expect, I was able to bring the average way down, to around 0.20 seconds, but I'll leave it at 0.303 to be conservative.
All right then, we have what we need. The initial conditions are that the truck and my LR3 are each going 60 m.p.h. or 26.8 m/s. The truck hits his brakes maximally and decelerates at 6.10 m/s^2. 0.303 seconds later, I hit my maximum brakes and decelerate at 8.0 m/s^2. How far back must I be to avoid hitting the truck? I suspect that detailed expositions of mathematics bore those who read here (though I'd certainly appreciate any feedback on this). So I'll just state that the answer is that I must be a minimum of 9.67 m or 31.7 feet behind the truck. Now, this assumes I can meet my reaction time and instantly apply maximum braking. I should add, say, a 50% safety factor for a total of 47.6 feet. Call it 50 feet. For those that would say that that's not enough of a safety factor, remember that I assumed that the truck applied perfect braking as well.
While I won't proselytize that drafting trucks is safe and should be done by one and all, I do think that with extreme caution and maximum alertness it can be safely done. I wouldn't want to try to get closer than 50 feet, and I will adjust my procedures accordingly, but I'm keeping this weapon in my arsenal.
So what are the three? The first is filling my tires (slightly) beyond the recommended maximum. The second is turning off the engine on long downhills. The last and most controversial is drafting trucks. It is this that is the subject of this post.
There are two obvious question. The first is "does it work?" The second is "is it dangerous?" As to its value in fuel savings, while there are some naysayers, most references agree that it is quite effective. Coincidentally, the Discovery Channel hit show Mythbusters covered this topic recently. Their results are summarized here. As is their common practice in such "myths," Kari, Tori, and Grant started with a scale model in a wind tunnel and achieved very encouraging results. Their full scale testing, while it didn't achieve quite the drag reduction of the wind tunnel tests, indicated significant fuel savings even at a following distance as large as 100 feet.
I have played with this idea off and on, both in the Grand Cherokee and in the LR3. However, because I have been so very unsuccessful with the LR3 in achieving the dramatic enhancements to gas mileage I was able to accomplish in the Jeep, I recently decided to pursue this avenue more aggressively. I know that this is controversial, many will call it selfish - if I crash, it will cause a huge backup for those behind me. I agree that this technique is extremely selfish if there is any significant chance that I will crash into a truck and tie up a freeway. I'll get back to that momentarily.
But does it work? According to the information from my Scan Gauge II, it does. I attempted to judge how far I was behind trucks by using my stopwatch to see what fraction of a second elapsed between the back of a truck crossing a highway mark and the front of my car crossing the same mark. Needless to say, tailgating a truck while looking at a Scan Gauge II reading and timing intervals with my stopwatch is sort of living on the edge, but I concluded that I was about 35 feet behind the truck. According to Mythbusters' results, I can look for an increase in miles per gallon of somewhere between 20% and 27%. As best I could tell, I saw an increase from about 21.5 m.p.g. to about 25 m.p.g., an increase of about 16%. These are pretty fuzzy figures - the readouts are constantly changing with changes in slope, etc., and the trucks typically don't maintain a particular speed quite as efficiently as the LR3 (due to the huge mass of the trucks no doubt). Nevertheless, every time I try it I get significant indications of increased fuel efficiency.
So apparently it works, how dangerous is it? Obviously, the worst case scenario is for the truck to apply maximum braking suddenly with no advance indication to me. I don't know if trucks have anti-lock braking systems, for the analysis to follow I won't assume they do. Judging from the condition of a lot of the trucks I see, this is a valid assumption. It should go without saying that, when following a truck at 35 feet, my eyes are focused on the truck's brake lights and my foot is ready to instantly hit the brake pedal. So the question is, when I see his (male pronouns are to be understood as gender-free) brake lights when he implements maximum braking, can I avoid hitting him?
I have to calculate the circumstances under which the distance between my front and his rear decreases to zero. The data necessary for this calculation is his maximum deceleration rate, my maximum deceleration rate, and my time to go from his brake lights on to my application of maximum braking. I will obviously limit my calculations to dry pavement in good condition - I'm not suicidal.
Truck braking, like everything else when looked at by academics, is ridiculously complex, see here if you are skeptical. But the nugget for my purposes in that paper is that the best that an empty semi-tractor trailer can do in deceleration is right at 20 ft/sec^2, or 6.10 m/s^2. For my LR3, it's (maybe, subject to correction) about 8.0 m/s^2. My reaction time, when paying close attention (as I do when tailgating a large truck) was determined using this online reaction timer to be 0.303 seconds in an average of 10 trials. With a lot of practice and knowing what to expect, I was able to bring the average way down, to around 0.20 seconds, but I'll leave it at 0.303 to be conservative.
All right then, we have what we need. The initial conditions are that the truck and my LR3 are each going 60 m.p.h. or 26.8 m/s. The truck hits his brakes maximally and decelerates at 6.10 m/s^2. 0.303 seconds later, I hit my maximum brakes and decelerate at 8.0 m/s^2. How far back must I be to avoid hitting the truck? I suspect that detailed expositions of mathematics bore those who read here (though I'd certainly appreciate any feedback on this). So I'll just state that the answer is that I must be a minimum of 9.67 m or 31.7 feet behind the truck. Now, this assumes I can meet my reaction time and instantly apply maximum braking. I should add, say, a 50% safety factor for a total of 47.6 feet. Call it 50 feet. For those that would say that that's not enough of a safety factor, remember that I assumed that the truck applied perfect braking as well.
While I won't proselytize that drafting trucks is safe and should be done by one and all, I do think that with extreme caution and maximum alertness it can be safely done. I wouldn't want to try to get closer than 50 feet, and I will adjust my procedures accordingly, but I'm keeping this weapon in my arsenal.
Tuesday, June 05, 2007
Another comparison
I'm still trying to understand the differences between the Jeep Grand Cherokee Limited I had until late November of 2006 and that was the topic of many of my posts, and the Land Rover LR3 HSE I have now. Right now, I'm thinking about the 31 m.p.g. the Jeep exhibits at 55 m.p.h. versus the 21.5 shown by the Land Rover. My last post dealt with the LR3 from the point of view of a big fuel burning air pump. I decided to compare what the two vehicles "should" require.
I gave some figures that should help out in this effort a few posts back. Let's run a few numbers. The LR3 has a frontal area of 3.15 m^2 and a coefficient of drag of 0.41. We have a dynamic pressure at 24.6 m/sec (approx. 55 mph) and air density of 1.16 kg.m^3 of 351 newtons/m^2. Thus, for this condition and drag coefficient, my aerodynamic drag is approximately 453 newtons (dynamic pressure times frontal area times drag coefficient).
Tire rolling resistance (as best I've been able to find) is about .015 times vehicle weight, or about 392 newtons. Total external forces to be overcome by the engine at 55 mph are therefore about 743 newtons or 167 pounds force. Now, force times speed is power, so the engine must provide 743 newtons at 24.6 m/s or about 18,280 watts. This equates to about 24.5 horsepower.
Obviously, much of the energy in the gasoline is lost to heat, engine and driveline friction, and pumping various fluids (refrigerant if the a.c. is on, water in the cooling system, air through the fan, oil, etc.). So enough gas must be burned per second to overcome all of these "dissipative" forces and still provide 24.5 horsepower.
For the Grand Cherokee, frontal area is 2.48 m^2, coefficient of drag is 0.44. Running through the same calculations, I get the external forces to be overcome by the Jeep at 55 mph are 383 newtons of aerodynamic drag and 289 newtons of rolling resistance for a total of 672 newtons or 151 pounds force. Enough energy must come from the fuel per second to overcome the dissipative forces and provide about 22.1 horsepower to maintain 55 mph against the external forces. So, if all else were equal, the LR3 should burn about 10.9% more fuel per mile at 55 mph. In fact, it burns about 44% more fuel. Or so the gauges say. So obviously, all else isn't equal.
These are the types of things I'm trying to understand.
I gave some figures that should help out in this effort a few posts back. Let's run a few numbers. The LR3 has a frontal area of 3.15 m^2 and a coefficient of drag of 0.41. We have a dynamic pressure at 24.6 m/sec (approx. 55 mph) and air density of 1.16 kg.m^3 of 351 newtons/m^2. Thus, for this condition and drag coefficient, my aerodynamic drag is approximately 453 newtons (dynamic pressure times frontal area times drag coefficient).
Tire rolling resistance (as best I've been able to find) is about .015 times vehicle weight, or about 392 newtons. Total external forces to be overcome by the engine at 55 mph are therefore about 743 newtons or 167 pounds force. Now, force times speed is power, so the engine must provide 743 newtons at 24.6 m/s or about 18,280 watts. This equates to about 24.5 horsepower.
Obviously, much of the energy in the gasoline is lost to heat, engine and driveline friction, and pumping various fluids (refrigerant if the a.c. is on, water in the cooling system, air through the fan, oil, etc.). So enough gas must be burned per second to overcome all of these "dissipative" forces and still provide 24.5 horsepower.
For the Grand Cherokee, frontal area is 2.48 m^2, coefficient of drag is 0.44. Running through the same calculations, I get the external forces to be overcome by the Jeep at 55 mph are 383 newtons of aerodynamic drag and 289 newtons of rolling resistance for a total of 672 newtons or 151 pounds force. Enough energy must come from the fuel per second to overcome the dissipative forces and provide about 22.1 horsepower to maintain 55 mph against the external forces. So, if all else were equal, the LR3 should burn about 10.9% more fuel per mile at 55 mph. In fact, it burns about 44% more fuel. Or so the gauges say. So obviously, all else isn't equal.
These are the types of things I'm trying to understand.
Saturday, June 02, 2007
First principles
This post is a continuation of the previous one in which I tried to reason the cause of my inability to coax better mileage out of the LR3. I was able to show through an analysis of manifold pressure, engine frequency (r.p.m.) and the chemical mixture requirements for combustion that the vehicle was burning fuel at a rate that equated to 13.3 m.p.g. at 55 m.p.h. on level ground. While this is in the ballpark, it's certainly not close to the base, so I tried reasoning from first principles, as the mathematicians say.
I started with the assumption that the fuel/air mixture in the manifold (and the cylinder during the intake stroke) is an ideal gas. I carried through an analysis on that basis (from the details of which I'll spare my patient readers) utilizing the absolute pressure and the manifold air temperature as reported by the Scan Gauge II I have attached to my engine.
I ran my analysis using the approximation that gasoline is normal heptane and that air is 22% O2 and 78% N2. From there, I proceeded to calculate from the ideal gas law. As Scotty used to say on Star Trek, "you canna change the laws of physics." I didn't expect to determine a fuel consumption number that matched the indicated m.p.g. from the Scan Gauge II to the nearest 0.1 m.p.g., but I did hope for something within, say, 10%.
Nope. The calculated result was worse than my previous calculation, coming in at 11.06 m.p.g. So what gives?? I have begun reading a treatise entitled "The Internal-Combustion Engine, Theory and Practice" by Taylor, since it's obvious that my level of understanding of the physics of internal combustion engines is inadequate to the problem at hand. It's a two volume tome, and exhaustive in scope and detail. When I learn enough to see where I've gone astray with my analyses, hopefully I'll also understand the mysteries of the LR3 vs. Grand Cherokee Limited engine comparison.
I started with the assumption that the fuel/air mixture in the manifold (and the cylinder during the intake stroke) is an ideal gas. I carried through an analysis on that basis (from the details of which I'll spare my patient readers) utilizing the absolute pressure and the manifold air temperature as reported by the Scan Gauge II I have attached to my engine.
I ran my analysis using the approximation that gasoline is normal heptane and that air is 22% O2 and 78% N2. From there, I proceeded to calculate from the ideal gas law. As Scotty used to say on Star Trek, "you canna change the laws of physics." I didn't expect to determine a fuel consumption number that matched the indicated m.p.g. from the Scan Gauge II to the nearest 0.1 m.p.g., but I did hope for something within, say, 10%.
Nope. The calculated result was worse than my previous calculation, coming in at 11.06 m.p.g. So what gives?? I have begun reading a treatise entitled "The Internal-Combustion Engine, Theory and Practice" by Taylor, since it's obvious that my level of understanding of the physics of internal combustion engines is inadequate to the problem at hand. It's a two volume tome, and exhaustive in scope and detail. When I learn enough to see where I've gone astray with my analyses, hopefully I'll also understand the mysteries of the LR3 vs. Grand Cherokee Limited engine comparison.
Saturday, May 26, 2007
Understanding a new engine
I mentioned in my last post that I'm trying to understand the factors that make it impossible for me to achieve the fuel savings in my Land Rover LR3 that I did in my Jeep Grand Cherokee Limited. I'm beginning to think my understanding of the internal combustion engine is sadly lacking.
For example, in the post linked above, I tabulated some of the relevant numbers for each vehicle. The LR3 uses a smaller engine to produce a higher rated horsepower than the Jeep. In the highest gear (at freeway speeds) the engine rpm is lower as well so one would think that the smaller engine turning more slowly would burn less fuel and hence derive heat to perform work at a slower rate. I know that the LR3 has a higher compression ratio. Could this be the explanation?
It's well known that the maximum theoretical efficiency, E, of an engine using an idealized Otto cycle is E=1-r^(1-y) where r is the compression ration and y (should be be the Greek letter gamma) is the ratio of the constant pressure to constant volume heat capacities. For the LR3 with its 10.5:1 compression ratio, this works out to 0.61. For the Grand Cherokee Limited at 9.3:1 it is 0.59. So what does this mean?
It means that for a given amount of heat from burning fossil fuels, the LR3 engine would be able to do ((0.61-0.59)/0.59)*100%=3.4% more work per joule of heat from burning gasoline than the Jeep if they were both working at the maximum theoretical efficiency. In order to see what how this affects our consumption, let's see how much heat per second from burning fuel is available to each engine.
In one second, at 1750 rpm in the Jeep, the engine moves 68.5 liters ((1750/2)/60) * 4.7 liters of fuel/air mixture through the engine. (The division of 1750 rpm by two is necessary because the rpm readout is crankshaft rpm, the crankshaft in a four stroke engine revolves twice for each engine cycle). The LR3 at 1660 rpm will move 60.9 liters per second. In these mixtures will be fuel, and I will assume that the richness of the mixture is the same for each engine, since the ECS (engine control system) will try to maintain the so-called "stoichiometric" ratio (14.7:1 by air mass to fuel mass). This mystifies me because the LR3 should burn less fuel since it's moving a smaller volume of fuel/air mixture through the engine at a presumed identical mixture. Yet the Grand Cherokee indicates instant mileage of approximately 31 m.p.g, the LR3 shows about 21.5.
The density of air at typical temperatures, pressures, and relative humidities is about 1.16 kilograms/meter^3 or 0.00116 kilograms/liter. This density is reduced in the intake manifold due to throttling effects, in fact, that's how the throttle works. I have equipped my LR3 with a Scan Gauge II so that I can measure absolute manifold pressure. At a steady 55 m.p.h. on level ground, manifold pressure is 68% of ambient. Since density is proportional to pressure, the ambient density of 0.00116 kilograms/liter is reduced in the cylinders to 7.84*10^(-4) kilograms/liter. We'll assume that the mixture is air as an approximation, since it's about 94% air in reality. Therefore, in one second, the LR3 moves 0.0477 kilograms of mixture through the engine and in that fluid, there should be (1/15.7)*0.0477=0.00304 kg. of gasoline. Burning this gasoline will release about 143,000 joules of heat energy. Hats off to me, this is great information. There's only one problem.
Since a gallon of gasoline weighs about 2.65 kilograms, this implies that I'm burning (0.00304*3600)/2.65=4.13 gallons/hour. At 55 m.p.h., this is about 13.3 m.p.g. The LR3 is no economy car but it isn't as bad as that. As I stated earlier, I expect about 21.5 m.p.g. at 55 m.p.h. on the freeway. Clearly, something is wrong in my assumptions. I'm not sure what it is, the mass flow calculation seems pretty straightforward.
For example, in the post linked above, I tabulated some of the relevant numbers for each vehicle. The LR3 uses a smaller engine to produce a higher rated horsepower than the Jeep. In the highest gear (at freeway speeds) the engine rpm is lower as well so one would think that the smaller engine turning more slowly would burn less fuel and hence derive heat to perform work at a slower rate. I know that the LR3 has a higher compression ratio. Could this be the explanation?
It's well known that the maximum theoretical efficiency, E, of an engine using an idealized Otto cycle is E=1-r^(1-y) where r is the compression ration and y (should be be the Greek letter gamma) is the ratio of the constant pressure to constant volume heat capacities. For the LR3 with its 10.5:1 compression ratio, this works out to 0.61. For the Grand Cherokee Limited at 9.3:1 it is 0.59. So what does this mean?
It means that for a given amount of heat from burning fossil fuels, the LR3 engine would be able to do ((0.61-0.59)/0.59)*100%=3.4% more work per joule of heat from burning gasoline than the Jeep if they were both working at the maximum theoretical efficiency. In order to see what how this affects our consumption, let's see how much heat per second from burning fuel is available to each engine.
In one second, at 1750 rpm in the Jeep, the engine moves 68.5 liters ((1750/2)/60) * 4.7 liters of fuel/air mixture through the engine. (The division of 1750 rpm by two is necessary because the rpm readout is crankshaft rpm, the crankshaft in a four stroke engine revolves twice for each engine cycle). The LR3 at 1660 rpm will move 60.9 liters per second. In these mixtures will be fuel, and I will assume that the richness of the mixture is the same for each engine, since the ECS (engine control system) will try to maintain the so-called "stoichiometric" ratio (14.7:1 by air mass to fuel mass). This mystifies me because the LR3 should burn less fuel since it's moving a smaller volume of fuel/air mixture through the engine at a presumed identical mixture. Yet the Grand Cherokee indicates instant mileage of approximately 31 m.p.g, the LR3 shows about 21.5.
The density of air at typical temperatures, pressures, and relative humidities is about 1.16 kilograms/meter^3 or 0.00116 kilograms/liter. This density is reduced in the intake manifold due to throttling effects, in fact, that's how the throttle works. I have equipped my LR3 with a Scan Gauge II so that I can measure absolute manifold pressure. At a steady 55 m.p.h. on level ground, manifold pressure is 68% of ambient. Since density is proportional to pressure, the ambient density of 0.00116 kilograms/liter is reduced in the cylinders to 7.84*10^(-4) kilograms/liter. We'll assume that the mixture is air as an approximation, since it's about 94% air in reality. Therefore, in one second, the LR3 moves 0.0477 kilograms of mixture through the engine and in that fluid, there should be (1/15.7)*0.0477=0.00304 kg. of gasoline. Burning this gasoline will release about 143,000 joules of heat energy. Hats off to me, this is great information. There's only one problem.
Since a gallon of gasoline weighs about 2.65 kilograms, this implies that I'm burning (0.00304*3600)/2.65=4.13 gallons/hour. At 55 m.p.h., this is about 13.3 m.p.g. The LR3 is no economy car but it isn't as bad as that. As I stated earlier, I expect about 21.5 m.p.g. at 55 m.p.h. on the freeway. Clearly, something is wrong in my assumptions. I'm not sure what it is, the mass flow calculation seems pretty straightforward.
Saturday, May 19, 2007
Comparison
As mentioned in my previous post, I am now driving a 2006 Land Rover LR3 HSE. I have been trying to achieve fuel economies that exceed the EPA rating for the vehicle as I was easily able to do in my 2001 Jeep Grand Cherokee Limited. I have failed utterly.
I'm now trying to analyze the reason for my failure, as well as the reasons for the much lower fuel economy of the LR3 in contrast with the Grand Cherokee. Further, my driving methods seem to make much less difference in the LR3 than they did in the Jeep. I'd sure like to find the reason for that, as it could influence many of my earlier conclusions about the extent to which fuel consumption in the U.S. could be reduced by the large scale adoption of fuel conserving driving techniques.
For reference, the following represents some comparative information on the two vehicles, as best I have been able to determine it. Should anyone have more accurate data or an authoritative source, I'd like to know of it.
So the key suspects seem to be the weight and the frontal area. I am going to hypothesize that the engine friction is directly proportional to r.p.m. and hence, in a given gear, to speed. I will speculate that the force required to pump fluids is proportional to the square of r.p.m., and thus, in a given gear, to speed. I will assume that tire rolling resistance is a constant for a given vehicle weight. Finally, I will declare that aerodynamic drag is proportional to the square of velocity. Thus, the force to be overcome as a function of speed and thus the force to be supplied by the engine to maintain a fixed speed is of the form f(v) = a + b * v + c * v^2. If I know the force required to maintain a given speed, I can calculate power required, since force times speed is power. Then I can compare the power required by the Grand Cherokee versus power required by the LR3 at various speeds.
I am also suspicious of the rated power of the engine - the LR3 is rated at 300hp at 4.4L displacement versus the Grand Cherokee's 235hp at 4.7L displacement. The compression ratios are 10.5:1 in the LR3 versus 9.3:1 in the Jeep. But really, a higher rated power is the same as saying that an engine can burn more fuel per second. After all, power is the rate of doing work and that work is done by the energy released by the burning fuel. Nevertheless, I am not enough of an expert on the physics of internal combustion engines to know how much additional power can be had from an engine by increasing the compression ratio.
This analysis will be continued over the next couple of posts.
I'm now trying to analyze the reason for my failure, as well as the reasons for the much lower fuel economy of the LR3 in contrast with the Grand Cherokee. Further, my driving methods seem to make much less difference in the LR3 than they did in the Jeep. I'd sure like to find the reason for that, as it could influence many of my earlier conclusions about the extent to which fuel consumption in the U.S. could be reduced by the large scale adoption of fuel conserving driving techniques.
For reference, the following represents some comparative information on the two vehicles, as best I have been able to determine it. Should anyone have more accurate data or an authoritative source, I'd like to know of it.
| Jeep | Land Rover | |
| Average Weight (pounds)* | 4338 | 5893 |
| Coefficient of Drag | 0.44 | 0.41 |
| Frontal Area (square feet) | 26.69 | 33.9 |
| Engine Size (L) | 4.7 | 4.4 |
| Rated Power (HP) | 235 | 300 |
| *Normal cargo, single occupant, half full fuel tank | ||
So the key suspects seem to be the weight and the frontal area. I am going to hypothesize that the engine friction is directly proportional to r.p.m. and hence, in a given gear, to speed. I will speculate that the force required to pump fluids is proportional to the square of r.p.m., and thus, in a given gear, to speed. I will assume that tire rolling resistance is a constant for a given vehicle weight. Finally, I will declare that aerodynamic drag is proportional to the square of velocity. Thus, the force to be overcome as a function of speed and thus the force to be supplied by the engine to maintain a fixed speed is of the form f(v) = a + b * v + c * v^2. If I know the force required to maintain a given speed, I can calculate power required, since force times speed is power. Then I can compare the power required by the Grand Cherokee versus power required by the LR3 at various speeds.
I am also suspicious of the rated power of the engine - the LR3 is rated at 300hp at 4.4L displacement versus the Grand Cherokee's 235hp at 4.7L displacement. The compression ratios are 10.5:1 in the LR3 versus 9.3:1 in the Jeep. But really, a higher rated power is the same as saying that an engine can burn more fuel per second. After all, power is the rate of doing work and that work is done by the energy released by the burning fuel. Nevertheless, I am not enough of an expert on the physics of internal combustion engines to know how much additional power can be had from an engine by increasing the compression ratio.
This analysis will be continued over the next couple of posts.
Saturday, May 12, 2007
Confession
Well, it's time to 'fess up. Several posts ago, I mentioned that I was contemplating the replacement of my 2000 Jeep Grand Cherokee Limited that has been the subject of the majority of my posts in this blog.
The fact is that I did so in November of 2006. Did I buy a Prius? No. Did I buy a Civic Hybrid? No. A Diesel Rabbit? No. An Insight? No. Well, did I at least buy a Lexus 400h Hybrid? Yes. Umm... I mean no.
In the end, after driving several vehicles and looking at many more, I wound up in a Land Rover LR3 HSE. This 6000 pound vehicle has a 4.4 liter engine and gets an EPA estimated 14 m.p.g. city and 18 m.p.g. highway. What a hypocrite, huh? Well maybe, maybe not.
A careful reading of my blog (should anyone wish to engage in such extensive self-abuse) will reveal that I never preached that people should buy vehicles with high mileage ratings, rather, I have suggested strategies for consumption reduction in whatever vehicle was driven. I haven't even, as best I recall, recommended reducing driven mileage though this is clearly the most obvious way to burn less fuel.
Now that that's out of the way, let's talk about why I did purchase the Land Rover. First, it's capable of having seven comfortable seats and converting to five with a very large and functional cargo area. Second, it is an unbelievably capable off road crawler and I have a deep and abiding love for the Mojave, Sonoran, and Great Basin deserts, particularly those areas to which no one (except me) ever goes. I wanted the capabilities of the Land Rover for this pursuit, though I have an old (1989) Jeep Comanche pickup that I have extensively modified for extended desert trips (water tank, lift kit, custom over sized fuel tank, spare battery system, cargo carrier, etc.). But I wanted something in which I could take more than one passenger to the desert, given that I have a family of four. I did not have that family when I bought the Comanche.
Now that I have the reasoning (some will say rationalization) out of the way, what has been my experience so far? I started out driving the LR3 with the same methods I had used in the Grand Cherokee. I was able to achieve a combined mileage of about 17.5 m.p.g. Then, in order to see what sort of diminution of mileage a less strict fuel saving methodolgy of driving might produce, I drove in relatively "normal" fashion for a few tank fulls. For these, I saw an average of about 16.1 m.p.g.
In other words, going from normal driving to extreme fuel saving only produced an 8.7% increase in gas mileage. Remember that going from extreme fuel consumption methods to extreme fuel saving methods in the Grand Cherokee produced about a 58% increase in gas mileage. What gives? It's an interesting question, I never drove the Grand Cherokee in a "normal" fashion, only the two extremes. Is it true that I could have gotten almost all of the benefits I achieved by only going from extreme fuel consumption mode (speeding as much as possible, full throttle takeoffs, etc.) to "normal" mode? I don't have the Grand Cherokee, but I passed it down to an employee. I am going to assume he drives "normally" and see what the average mileage indicator shows.Of course, I'll log it here.
There are several questions I'd like to address in subsequent posts. I'd like to know why the 4.4 liter engine in the LR3 burns more fuel per mile than the 4.7 liter engine in the Grand Cherokee. I'd like to know why I can achieve overall fuel economy dramatically higher than even the EPA highway rating in the Grand Cherokee, but not in the LR3. Is this because of changes between 2001 and 2006 in how the EPA performs its evaluations? Is it the aerodynamics of the two vehicles? Differences in the engines? I'll try to find out.
The fact is that I did so in November of 2006. Did I buy a Prius? No. Did I buy a Civic Hybrid? No. A Diesel Rabbit? No. An Insight? No. Well, did I at least buy a Lexus 400h Hybrid? Yes. Umm... I mean no.
In the end, after driving several vehicles and looking at many more, I wound up in a Land Rover LR3 HSE. This 6000 pound vehicle has a 4.4 liter engine and gets an EPA estimated 14 m.p.g. city and 18 m.p.g. highway. What a hypocrite, huh? Well maybe, maybe not.
A careful reading of my blog (should anyone wish to engage in such extensive self-abuse) will reveal that I never preached that people should buy vehicles with high mileage ratings, rather, I have suggested strategies for consumption reduction in whatever vehicle was driven. I haven't even, as best I recall, recommended reducing driven mileage though this is clearly the most obvious way to burn less fuel.
Now that that's out of the way, let's talk about why I did purchase the Land Rover. First, it's capable of having seven comfortable seats and converting to five with a very large and functional cargo area. Second, it is an unbelievably capable off road crawler and I have a deep and abiding love for the Mojave, Sonoran, and Great Basin deserts, particularly those areas to which no one (except me) ever goes. I wanted the capabilities of the Land Rover for this pursuit, though I have an old (1989) Jeep Comanche pickup that I have extensively modified for extended desert trips (water tank, lift kit, custom over sized fuel tank, spare battery system, cargo carrier, etc.). But I wanted something in which I could take more than one passenger to the desert, given that I have a family of four. I did not have that family when I bought the Comanche.
Now that I have the reasoning (some will say rationalization) out of the way, what has been my experience so far? I started out driving the LR3 with the same methods I had used in the Grand Cherokee. I was able to achieve a combined mileage of about 17.5 m.p.g. Then, in order to see what sort of diminution of mileage a less strict fuel saving methodolgy of driving might produce, I drove in relatively "normal" fashion for a few tank fulls. For these, I saw an average of about 16.1 m.p.g.
In other words, going from normal driving to extreme fuel saving only produced an 8.7% increase in gas mileage. Remember that going from extreme fuel consumption methods to extreme fuel saving methods in the Grand Cherokee produced about a 58% increase in gas mileage. What gives? It's an interesting question, I never drove the Grand Cherokee in a "normal" fashion, only the two extremes. Is it true that I could have gotten almost all of the benefits I achieved by only going from extreme fuel consumption mode (speeding as much as possible, full throttle takeoffs, etc.) to "normal" mode? I don't have the Grand Cherokee, but I passed it down to an employee. I am going to assume he drives "normally" and see what the average mileage indicator shows.Of course, I'll log it here.
There are several questions I'd like to address in subsequent posts. I'd like to know why the 4.4 liter engine in the LR3 burns more fuel per mile than the 4.7 liter engine in the Grand Cherokee. I'd like to know why I can achieve overall fuel economy dramatically higher than even the EPA highway rating in the Grand Cherokee, but not in the LR3. Is this because of changes between 2001 and 2006 in how the EPA performs its evaluations? Is it the aerodynamics of the two vehicles? Differences in the engines? I'll try to find out.
Sunday, April 29, 2007
New car for "free"?
As mentioned in a previous post, the Jeep Grand Cherokee Limited that is the subject of a large part of this blog was purchased in July of 2000. It has well over 150,000 miles on the odometer, and has been everything I could reasonably want in a vehicle. But to quote George Harrison, "All Things Must Pass."
So, given that I'm able to drive the Grand Cherokee for 23.5 m.p.g., can I purchase a vehicle that, through savings at the pump, will pay for itself? I'm speaking purely of the cash cost to me and not of the impact on overall fuel consumption for "the universe" (fuel to produce and deliver the car, etc.). I expect to write about that aspect in another post.
I'll also leave out insurance and other extraneous considerations. Any vehicle that could conceivably pay for itself in fuel savings would be much smaller and possibly less costly than the Cherokee was. But it would be new, and so insurance might be a wash. Anyway, that's not what this blog is about.
So, in any case, the Grand Cherokee gets about 23.5 m.p.g. at $3.29 per gallon currently. That amounts to $0.14 per mile, and I average about 56.13 miles per day, 365.25 per year. Therefore, the monthly gas expenditure is (365.25 * 56.13 * 0.14)/12 or $239.18. Now, suppose I could get a vehicle in which I could achieve an average of, say, 45 m.p.g. My monthly fuel expenditure would be reduced to $124.91 for a savings of $114.27.
Hmmm... that won't buy much of a car I guess. What if gasoline went to $4.00/gallon as I hear predicted, and by extreme techniques, I could get 50 m.p.g.? Well, the Cherokee would then cost $0.17 per mile and total $290.80 per month and the new car would cost $136.67 per month and save $154.13. Not there yet, though it is starting to eat significantly into the monthly payment for an economy car with very low fuel consumption.
As I've mentioned, my company actually owns the Grand Cherokee, so "business finance" and considerations of internal rate of return, net present value, and payback period are appropriate. I found a 1995 Geo Metro on line for $2,200. The payback period, exclusive of necessary repairs, is on the order of 14 months. I don't know if I can quite deal with that type of accommodation, but it's the only way I can see to getting a car for free. After the 14 month payback period, I'd be essentially adding to the retained earnings every time I drove.
So, given that I'm able to drive the Grand Cherokee for 23.5 m.p.g., can I purchase a vehicle that, through savings at the pump, will pay for itself? I'm speaking purely of the cash cost to me and not of the impact on overall fuel consumption for "the universe" (fuel to produce and deliver the car, etc.). I expect to write about that aspect in another post.
I'll also leave out insurance and other extraneous considerations. Any vehicle that could conceivably pay for itself in fuel savings would be much smaller and possibly less costly than the Cherokee was. But it would be new, and so insurance might be a wash. Anyway, that's not what this blog is about.
So, in any case, the Grand Cherokee gets about 23.5 m.p.g. at $3.29 per gallon currently. That amounts to $0.14 per mile, and I average about 56.13 miles per day, 365.25 per year. Therefore, the monthly gas expenditure is (365.25 * 56.13 * 0.14)/12 or $239.18. Now, suppose I could get a vehicle in which I could achieve an average of, say, 45 m.p.g. My monthly fuel expenditure would be reduced to $124.91 for a savings of $114.27.
Hmmm... that won't buy much of a car I guess. What if gasoline went to $4.00/gallon as I hear predicted, and by extreme techniques, I could get 50 m.p.g.? Well, the Cherokee would then cost $0.17 per mile and total $290.80 per month and the new car would cost $136.67 per month and save $154.13. Not there yet, though it is starting to eat significantly into the monthly payment for an economy car with very low fuel consumption.
As I've mentioned, my company actually owns the Grand Cherokee, so "business finance" and considerations of internal rate of return, net present value, and payback period are appropriate. I found a 1995 Geo Metro on line for $2,200. The payback period, exclusive of necessary repairs, is on the order of 14 months. I don't know if I can quite deal with that type of accommodation, but it's the only way I can see to getting a car for free. After the 14 month payback period, I'd be essentially adding to the retained earnings every time I drove.
Saturday, April 28, 2007
Searching the journals
As is my wont, I have spent a bit of time on the web googling (I've come to accept this term as a verb) on "mathematical model automobile fuel consumption." This search produces 612,000 hits today. Many of them are irrelevant to my obsession ("Pricing for Environmental Compliance in the Auto Industry" for example). Some of them are to abstracts of journal articles and not relevant enough to my interest to cause me to spend the money to buy them. But there is a Ph.D. thesis from 1992 entitled " Automobile fuel economy and traffic congestion" by Feng An. Its abstract is reproduced here :
"An analytical model for automobile fuel consumption based on vehicle parameters and traffic characteristics is developed in this thesis.^This model is based on two approximations: (1) an engine map approximation, and (2) a tractive energy approximation.^This model is the first comprehensive attempt to predict fuel economy without having to go through a second-by-second measurements, simulation or a regression procedure.^A computer spreadsheet program based on this model has been created.^It can be used to calculate the fuel economy of any motor vehicle in any driving pattern, based on public-available vehicle parameters, with absolute error typically less than +/-5%.^Several applications of this model are presented: (1) calculating the fuel economy of motor vehicles in 7 different driving cycles, (2) determining the relationship between fuel economy and vehicle average velocity, (3) determining the vehicle optimal fuel efficiency speed, (4) discussing the effect of traffic smoothness on fuel economy, (5) discussing how driving behaviors affect fuel economy, (6) discussing the effect of highway speed limit on fuel economy, (7) discussing the maximum possible fuel economy for ordinary cars, and finally, (8) discussing the impact of vehicle parameters on fuel economy."
Now that's what I'm talking about. I ordered it from the source referenced on the page and am looking forward to reading it and seeing what it can teach me.
Now, there is another hit from the journal "The Engineering Economist." The title of the article is "The Economic Impact of Obesity on Automobile Fuel Consumption." Fascinating. In the abstract, it is stated that "results indicate that, since 1988, no less than 272 million additional gallons of fuel are consumed annually due to average passenger weight increases." Talk about hitting the jackpot on constructive ways to save fuel! An extremely quick and dirty calculation indicates that, if the above is true, 0.18% of our annual oil consumption can be attributed to weight gain since 1988.
So, in addition to adopting the driving methods I've described over the past months, we can all resolve to achieve our ideal weight. This will not only reduce our oil imports, trade deficit, and carbon footprint (are you getting this Al?), but it will increase our health, reduce our health care expenditures, and make us more attractive to the opposite sex (or, for those so disposed, to the same sex).
Further, the fuel savings don't stop with the reduction in automobile gas consumption. Food that needn't be produced needn't be fertilized, shipped, cooled, etc. I haven't yet set up estimates for the potential savings here, but they have to be huge.
So my recommendation is that, if you are overweight, get on a weight reduction program. It's the patriotic thing to do!
"An analytical model for automobile fuel consumption based on vehicle parameters and traffic characteristics is developed in this thesis.^This model is based on two approximations: (1) an engine map approximation, and (2) a tractive energy approximation.^This model is the first comprehensive attempt to predict fuel economy without having to go through a second-by-second measurements, simulation or a regression procedure.^A computer spreadsheet program based on this model has been created.^It can be used to calculate the fuel economy of any motor vehicle in any driving pattern, based on public-available vehicle parameters, with absolute error typically less than +/-5%.^Several applications of this model are presented: (1) calculating the fuel economy of motor vehicles in 7 different driving cycles, (2) determining the relationship between fuel economy and vehicle average velocity, (3) determining the vehicle optimal fuel efficiency speed, (4) discussing the effect of traffic smoothness on fuel economy, (5) discussing how driving behaviors affect fuel economy, (6) discussing the effect of highway speed limit on fuel economy, (7) discussing the maximum possible fuel economy for ordinary cars, and finally, (8) discussing the impact of vehicle parameters on fuel economy."
Now that's what I'm talking about. I ordered it from the source referenced on the page and am looking forward to reading it and seeing what it can teach me.
Now, there is another hit from the journal "The Engineering Economist." The title of the article is "The Economic Impact of Obesity on Automobile Fuel Consumption." Fascinating. In the abstract, it is stated that "results indicate that, since 1988, no less than 272 million additional gallons of fuel are consumed annually due to average passenger weight increases." Talk about hitting the jackpot on constructive ways to save fuel! An extremely quick and dirty calculation indicates that, if the above is true, 0.18% of our annual oil consumption can be attributed to weight gain since 1988.
So, in addition to adopting the driving methods I've described over the past months, we can all resolve to achieve our ideal weight. This will not only reduce our oil imports, trade deficit, and carbon footprint (are you getting this Al?), but it will increase our health, reduce our health care expenditures, and make us more attractive to the opposite sex (or, for those so disposed, to the same sex).
Further, the fuel savings don't stop with the reduction in automobile gas consumption. Food that needn't be produced needn't be fertilized, shipped, cooled, etc. I haven't yet set up estimates for the potential savings here, but they have to be huge.
So my recommendation is that, if you are overweight, get on a weight reduction program. It's the patriotic thing to do!
Tuesday, April 24, 2007
At what point will fuel prices justify low speed?
The last couple of posts have dealt with the trade off between the value of the fuel saved at low speeds versus the value of the time "wasted." I determined that, for my trip, at my salary, in my vehicle, at $3.40/gallon for gasoline, the speed at which my total cost (my salary + fuel to make the trip) was minimized is 110 m.p.h. Clearly not practical,especially if everyone ran a similar calculation and decided to drive at what I will call their "minimum total expenditure speed (mtes)."
But the process and the mathematics of determining those numbers led me to wonder what gasoline price would make the fuel savings balance the time costs at reasonable speeds. It turns out that the number is extremely high. For example, for gas at $5.00/gallon, the mtes is 97 m.p.h. At $8.00/gallon, the mtes is 87 m.p.h. The mtes never gets to the most efficient speed because the m.p.g. as a function of speed curve is fairly flat in the range near the maximum, which I determined analytically to be 50.8 m.p.h. for the Grand Cherokee Limited.
It is reminiscent of the earlier post where I contemplated the point of indifference for a woman in a hummer at the bank drive-through window. There, as here, the gasoline price that would economically cause a behavioral change was ludicrously high.
There are many actions that can be taken to reduce consumption, some of which do not have the down side associated with maximally efficient freeway speeds. And there are forms of income other than monetary, so-called "psychic income." This involves benefits such as fulfilling the desire to protect the environment, feeling as if one is doing the best one can to leave a better world for one's children, and other non-monetary considerations that comport with one's philosophy or desires. In my case, it has been the satisfaction of pursuing the goal of minimization of fuel consumption for its own sake. Some call me obsessive, and they are probably correct.
These things are difficult, if not impossible to quantify in a meaningful way, other than the extent to which someone is willing to forgo things that can be quantified to receive the benefit. In my case, it is clear that the enjoyment of running the experiment and writing about it in this blog compensates me for the opportunity cost of getting to my destinations more slowly.
But the process and the mathematics of determining those numbers led me to wonder what gasoline price would make the fuel savings balance the time costs at reasonable speeds. It turns out that the number is extremely high. For example, for gas at $5.00/gallon, the mtes is 97 m.p.h. At $8.00/gallon, the mtes is 87 m.p.h. The mtes never gets to the most efficient speed because the m.p.g. as a function of speed curve is fairly flat in the range near the maximum, which I determined analytically to be 50.8 m.p.h. for the Grand Cherokee Limited.
It is reminiscent of the earlier post where I contemplated the point of indifference for a woman in a hummer at the bank drive-through window. There, as here, the gasoline price that would economically cause a behavioral change was ludicrously high.
There are many actions that can be taken to reduce consumption, some of which do not have the down side associated with maximally efficient freeway speeds. And there are forms of income other than monetary, so-called "psychic income." This involves benefits such as fulfilling the desire to protect the environment, feeling as if one is doing the best one can to leave a better world for one's children, and other non-monetary considerations that comport with one's philosophy or desires. In my case, it has been the satisfaction of pursuing the goal of minimization of fuel consumption for its own sake. Some call me obsessive, and they are probably correct.
These things are difficult, if not impossible to quantify in a meaningful way, other than the extent to which someone is willing to forgo things that can be quantified to receive the benefit. In my case, it is clear that the enjoyment of running the experiment and writing about it in this blog compensates me for the opportunity cost of getting to my destinations more slowly.
Monday, April 23, 2007
What's REALLY the best speed?
My last post looked at possible savings utilizing costs with 55 m.p.h. as a basis. I decided I had enough information to look at what speed minimizes total cost with salary and fuel consideration only. The elimination of the 55 m.p.h. baseline had a surprising effect.
I calculated using the constants developed in my previous post, which enabled me to determine the total fuel cost as miles of the round trip (freeway part only) divided by miles per gallon at the speed in question, times $3.40 per gallon. My salary was easily calculated at miles travelled (36, as noted previously) divided by miles per hour times salary per hour.
As it turns out, the resulting equation has a global maximum at about 110.5 m.p.h., a speed that I can drive (and have driven) in the Grand Cherokee. I will concede that there are many downsides to the strategy of driving 45 m.p.h. over the speed limit and that time lost at the side of the road talking to the Highway Patrol, time lost in court, money lost paying for traffic school, fines, and insurance increases (assuming that, after a few reckless driving tickets, insurance could be obtained at any price) would have a severe impact on the accuracy of the cost calculations.
For these reasons, I am unlikely to adopt this habit. Nevertheless, I am going to become dramatically more aggressive in reducing the impact of the salary component by finding ways to produce on the road. Failing that, the emphasis may have to fall back to those aspects of fuel consumption reduction that do not result in increased trip times. Unfortunately, as best I can estimate, all of those together have approximately 25% of the impact of low freeway speeds. This is beginning to feel like a manifestation of the second law of thermodynamics ("you can't break even").
I ran the numbers for someone making $60,000 per year (significantly less than my salary) and the optimum speed for that particular drive in that particular vehicle is 80 m.p.h. So presumably, for a minimum wage worker, 55 m.p.h. might be the right choice.
These are all rather fuzzy numbers, since I will get to work and complete my job, and the hourly wage guys will put in their hours. Therefore, it's actually "personal time" that is being used during the commute. However, the best surrogate I have for that is the rate at which people are paid, since that amount of money will cause them to leave home and family to go to work. In any case, it is clear that there is a high price to be paid for minimizing fuel consumption by driving slowly.
I calculated using the constants developed in my previous post, which enabled me to determine the total fuel cost as miles of the round trip (freeway part only) divided by miles per gallon at the speed in question, times $3.40 per gallon. My salary was easily calculated at miles travelled (36, as noted previously) divided by miles per hour times salary per hour.
As it turns out, the resulting equation has a global maximum at about 110.5 m.p.h., a speed that I can drive (and have driven) in the Grand Cherokee. I will concede that there are many downsides to the strategy of driving 45 m.p.h. over the speed limit and that time lost at the side of the road talking to the Highway Patrol, time lost in court, money lost paying for traffic school, fines, and insurance increases (assuming that, after a few reckless driving tickets, insurance could be obtained at any price) would have a severe impact on the accuracy of the cost calculations.
For these reasons, I am unlikely to adopt this habit. Nevertheless, I am going to become dramatically more aggressive in reducing the impact of the salary component by finding ways to produce on the road. Failing that, the emphasis may have to fall back to those aspects of fuel consumption reduction that do not result in increased trip times. Unfortunately, as best I can estimate, all of those together have approximately 25% of the impact of low freeway speeds. This is beginning to feel like a manifestation of the second law of thermodynamics ("you can't break even").
I ran the numbers for someone making $60,000 per year (significantly less than my salary) and the optimum speed for that particular drive in that particular vehicle is 80 m.p.h. So presumably, for a minimum wage worker, 55 m.p.h. might be the right choice.
These are all rather fuzzy numbers, since I will get to work and complete my job, and the hourly wage guys will put in their hours. Therefore, it's actually "personal time" that is being used during the commute. However, the best surrogate I have for that is the rate at which people are paid, since that amount of money will cause them to leave home and family to go to work. In any case, it is clear that there is a high price to be paid for minimizing fuel consumption by driving slowly.
Sunday, November 19, 2006
"Optimum" economic speed
For a math guy like me, this one will be strictly fun. In "Use of time" I discussed various matters related to the time "lost" by traveling at 55 m.p.h. on the freeway. One of the comparisons I developed was that, at the fuel prices in effect at that time (April of 2006), the time I lost in a day was valued by my company at $12.39 whereas I saved fuel valued at $4.57.
This would seem to indicate that, from a purely "dollars and cents" point of view, the faster I go, the better. Let's take an analytical look at that. I'll ignore limits on engine performance, speed limits and law enforcement, the physics of negotiating curves, and all other real-world matters.
The aerodynamic force resisting my vehicle's forward motion is proportional to the square of velocity (I insist) so additional speed increases fuel use dramatically. The time gained increases as my speed increases. Can I go so fast that I burn extra fuel worth more than the dollar value of my time savings?
Since I do in fact have to go to work and must use fuel, I can't merely calculate when fuel burned per minute equals my salary per minute. I have to use a baseline. I'll choose to use 55 m.p.h. So the problem is to determine how fast I must go to burn so much more fuel than I would burn at 55 m.p.h. that it is worth more than the value of my salary for the time that I'm traveling at the high speed.
The reader should note that I typically contemplate the problems about which I write "on the fly," hence I don't know what the answer will be until I have set it up in the blog. This is no exception, but I anticipate that the the speed will be a ludicrously high one.
So let's get started. I can easily calculate that the value of the time saved by driving faster than 55 m.p.h., as determined by my salary (and hence by my company's valuation of my time) to be $62.95-(3462/x) where x is my speed in miles per hour and the result is the savings for a single day's driving to and from work. So, for example, at 75 m.p.h. I save time worth $16.79.
The excess gasoline costs are not nearly so easy. To get a handle on this, I started with the model on the "How Stuff Works" web site. There, Marshall Brain models the power required as p = a*v + b*v^2 + c*v^3. Now, using some physics definitions and the chain rule from elementary calculus, we get to c = d + e*v + f*v^2, where c is "consumption" in appropriate units (say, gallons per mile) and d, e, and f are constants for a given vehicle. To determine the three constants, I need to know the consumption in gallons per mile (the inverse of miles per gallon) at three different speeds.
Well, I have 31 m.p.g. or 1/31 gal./mile at 55 m.p.h. so there's one. I can't use my idling fuel consumption since it is at zero m.p.h. and consequently would lead to an undefined consumption in gallons per mile, since we would be dividing by zero. So I acquired two more points, one at 40 m.p.h. and one at 70 m.p.h. Utilizing those numbers in the same way and solving the three simultaneous linear equations for the constants d, e, and f, and plugging them into the equation enables me to equate the dollars saved on my salary to the dollars spent in extra fuel, assuming gasoline at $3.40/gallon. At that speed, any faster and I would start losing money since losses on the fuel side would exceed gains on the salary side.
I hope I've built the suspense enough. The break even point occurs at 170 m.p.h. Now, if someone were to think that a Jeep Grand Cherokee Limited isn't capable of those speeds, that person would be correct. So what does it mean? It further emphasizes the need to be productive on the road, because gas isn't expensive enough, by far, to make up for the time lost in driving slowly.
Of course, I'm relatively well paid and gas could go up, so I guess the next step would be to develop a table for the break even speed at different salaries and fuel costs. But even then, a change in the miles driven at freeway speeds (such as 170 m.p.h.) would change the table. But it is clear that, from a purely economic point of view, I'm not doing myself any favors.
This would seem to indicate that, from a purely "dollars and cents" point of view, the faster I go, the better. Let's take an analytical look at that. I'll ignore limits on engine performance, speed limits and law enforcement, the physics of negotiating curves, and all other real-world matters.
The aerodynamic force resisting my vehicle's forward motion is proportional to the square of velocity (I insist) so additional speed increases fuel use dramatically. The time gained increases as my speed increases. Can I go so fast that I burn extra fuel worth more than the dollar value of my time savings?
Since I do in fact have to go to work and must use fuel, I can't merely calculate when fuel burned per minute equals my salary per minute. I have to use a baseline. I'll choose to use 55 m.p.h. So the problem is to determine how fast I must go to burn so much more fuel than I would burn at 55 m.p.h. that it is worth more than the value of my salary for the time that I'm traveling at the high speed.
The reader should note that I typically contemplate the problems about which I write "on the fly," hence I don't know what the answer will be until I have set it up in the blog. This is no exception, but I anticipate that the the speed will be a ludicrously high one.
So let's get started. I can easily calculate that the value of the time saved by driving faster than 55 m.p.h., as determined by my salary (and hence by my company's valuation of my time) to be $62.95-(3462/x) where x is my speed in miles per hour and the result is the savings for a single day's driving to and from work. So, for example, at 75 m.p.h. I save time worth $16.79.
The excess gasoline costs are not nearly so easy. To get a handle on this, I started with the model on the "How Stuff Works" web site. There, Marshall Brain models the power required as p = a*v + b*v^2 + c*v^3. Now, using some physics definitions and the chain rule from elementary calculus, we get to c = d + e*v + f*v^2, where c is "consumption" in appropriate units (say, gallons per mile) and d, e, and f are constants for a given vehicle. To determine the three constants, I need to know the consumption in gallons per mile (the inverse of miles per gallon) at three different speeds.
Well, I have 31 m.p.g. or 1/31 gal./mile at 55 m.p.h. so there's one. I can't use my idling fuel consumption since it is at zero m.p.h. and consequently would lead to an undefined consumption in gallons per mile, since we would be dividing by zero. So I acquired two more points, one at 40 m.p.h. and one at 70 m.p.h. Utilizing those numbers in the same way and solving the three simultaneous linear equations for the constants d, e, and f, and plugging them into the equation enables me to equate the dollars saved on my salary to the dollars spent in extra fuel, assuming gasoline at $3.40/gallon. At that speed, any faster and I would start losing money since losses on the fuel side would exceed gains on the salary side.
I hope I've built the suspense enough. The break even point occurs at 170 m.p.h. Now, if someone were to think that a Jeep Grand Cherokee Limited isn't capable of those speeds, that person would be correct. So what does it mean? It further emphasizes the need to be productive on the road, because gas isn't expensive enough, by far, to make up for the time lost in driving slowly.
Of course, I'm relatively well paid and gas could go up, so I guess the next step would be to develop a table for the break even speed at different salaries and fuel costs. But even then, a change in the miles driven at freeway speeds (such as 170 m.p.h.) would change the table. But it is clear that, from a purely economic point of view, I'm not doing myself any favors.
Saturday, November 11, 2006
Run for the light?
Of course, some of the modifications to my driving technique save gallons per tank full; in particular, slow acceleration to a maximum of 55 m.p.h. Without getting into statistics, there's no question that large improvements in fuel economy have been made - I used to have to fill up at about 280 to 290 miles, now it's more like 430 to 450. I'm reasonably sure most of it comes from the speed and acceleration reductions.
Some of the things I do may save, literally, only milliliters per tank full. For example, my driveway slopes severely to the street. I can roll down, turn into the street, use the momentum to turn 180 degrees onto the adjacent street, and roll to the stop sign for the main road before turning on the engine, saving about 30 seconds and 180 feet of running the engine. In a tank full period I may do this 6 times, thereby saving something like 2.5 fluid ounces of fuel. In a year, the savings could amount to a gallon. Not enough to save the world.
And I do other things whose savings make that seem huge by comparison, such as turning off the engine and coasting into a parking space when I have the space made. People chuckle, shake their heads in pity and say "tsk tsk" when they see me do this, but I'm strong and can take it!
But I do these things because I'm trying to do everything possible, no matter how trivial. In so doing, I often am faced with the decision of how to treat stoplights. There are several issues to contemplate but the one I have in mind today is how to treat a light that is currently green but that may change to red before I get there. Under what circumstances should I accelerate and run for it?
It's clear that the question is the balance between fuel wasted while stopped at a red light versus that wasted by hitting the throttle to get through the light. Looming in the background is the horrifying risk of hitting the throttle to get through the light and missing it anyway. Worse still is the doomsday scenario of running for the light, having it change, being unable to stop and getting a traffic ticket. We'll ignore this remote possibility.
It's a complicated problem since lights have different durations, my knowledge is typically imperfect (though I know some lights quite well and can therefore make more informed decisions), I may or may not be able to keep the speed I generate in running for a light (depending on traffic conditions, whether or not I am turning, etc.), the continuum between a slight, gentle acceleration and "stomping on it," and many other factors.
But to at least get started, let's suppose I estimate that, if I run for it there's an 80% chance I'll make it. If I don't make the light, I'll spend 35 seconds stopped while it's red. For the purposes of the analysis, let's say that I'm going 25 m.p.h. Let's further assume that I use hard but reasonable acceleration - say, 2.7 meters/second^2, or 0.28g to accelerate to 45 m.p.h. For this scenario, let's assume that I am not turning and can keep my momentum or at least coast to the appropriate speed without braking if I make the light. For the accelerate and make it scenario, we have to make still more assumptions. I'll assume that I'm at 25 m.p.h., I accelerate to 45 m.p.h. at 2.7 m/s^2 and coast back down to 25 m.p.h. at 0.22 m/s^2. Finally, let's assume that, without acceleration there's a 30% chance that I will make the light.
OK, we should be able to get some comparative numbers here. It will be probabilistic and deal with so-called "mathematical expectation" since I have to incorporate the 20% chance of not making the light if I accelerate and the 70% chance of not making it if I don't. I'll spare my patient readers (reader?) the details of most of the calculations, but there are 4 situations: accelerate, make it; accelerate, miss it; don't accelerate, make it; don't accelerate, miss it. These scenarios have probabilities 80%; 20%; 30%; 70%.
I think the easiest way to go about this is to figure how much fuel is used in each case to get, say, one mile past the light with no further stopping given each of the scenarios above. So without further ado:
Don't accelerate, make light uses 0.0458 gallons
Don't accelerate, miss light uses 0.0523 gallons
Accelerate, make light uses 0.0546 gallons
Accelerate, miss light uses 0.0611 gallons
Surprisingly, acclerating and MAKING the light uses more fuel than not accelerating and missing the light. Therefore, it could not possibly pay to try to make the light using this specific scenario. Obviously, other assumptions regarding light durations, speeds, accelerations, etc. could change this. And in case anyone incorporates my driving techniques, the numbers above were derived using fuel consumption numbers for my Grand Cherokee Limited. As they say in chatrooms, ymmv (your mileage may vary).
To close the chapter, the mathematical expectations (under this set of assumptions) are:
Don't accelerate: 0.3*0.0458+0.7*0.0523=0.0503 gallons
Accelerate: 0.8*0.0546+0.2*0.0611=0.0559 gallons.
So there you have it. If I accelerate to make a light, I can expect to use about 11% more fuel at the intersection than if I just maintain my normal speed. Again, the circumstances for this calculation are quite specific but not abnormal. Every now and again though, at lights where I know the duration of the red is long and where I know a short burst will get me through and the lack thereof won't, I'll give it a try.
Some of the things I do may save, literally, only milliliters per tank full. For example, my driveway slopes severely to the street. I can roll down, turn into the street, use the momentum to turn 180 degrees onto the adjacent street, and roll to the stop sign for the main road before turning on the engine, saving about 30 seconds and 180 feet of running the engine. In a tank full period I may do this 6 times, thereby saving something like 2.5 fluid ounces of fuel. In a year, the savings could amount to a gallon. Not enough to save the world.
And I do other things whose savings make that seem huge by comparison, such as turning off the engine and coasting into a parking space when I have the space made. People chuckle, shake their heads in pity and say "tsk tsk" when they see me do this, but I'm strong and can take it!
But I do these things because I'm trying to do everything possible, no matter how trivial. In so doing, I often am faced with the decision of how to treat stoplights. There are several issues to contemplate but the one I have in mind today is how to treat a light that is currently green but that may change to red before I get there. Under what circumstances should I accelerate and run for it?
It's clear that the question is the balance between fuel wasted while stopped at a red light versus that wasted by hitting the throttle to get through the light. Looming in the background is the horrifying risk of hitting the throttle to get through the light and missing it anyway. Worse still is the doomsday scenario of running for the light, having it change, being unable to stop and getting a traffic ticket. We'll ignore this remote possibility.
It's a complicated problem since lights have different durations, my knowledge is typically imperfect (though I know some lights quite well and can therefore make more informed decisions), I may or may not be able to keep the speed I generate in running for a light (depending on traffic conditions, whether or not I am turning, etc.), the continuum between a slight, gentle acceleration and "stomping on it," and many other factors.
But to at least get started, let's suppose I estimate that, if I run for it there's an 80% chance I'll make it. If I don't make the light, I'll spend 35 seconds stopped while it's red. For the purposes of the analysis, let's say that I'm going 25 m.p.h. Let's further assume that I use hard but reasonable acceleration - say, 2.7 meters/second^2, or 0.28g to accelerate to 45 m.p.h. For this scenario, let's assume that I am not turning and can keep my momentum or at least coast to the appropriate speed without braking if I make the light. For the accelerate and make it scenario, we have to make still more assumptions. I'll assume that I'm at 25 m.p.h., I accelerate to 45 m.p.h. at 2.7 m/s^2 and coast back down to 25 m.p.h. at 0.22 m/s^2. Finally, let's assume that, without acceleration there's a 30% chance that I will make the light.
OK, we should be able to get some comparative numbers here. It will be probabilistic and deal with so-called "mathematical expectation" since I have to incorporate the 20% chance of not making the light if I accelerate and the 70% chance of not making it if I don't. I'll spare my patient readers (reader?) the details of most of the calculations, but there are 4 situations: accelerate, make it; accelerate, miss it; don't accelerate, make it; don't accelerate, miss it. These scenarios have probabilities 80%; 20%; 30%; 70%.
I think the easiest way to go about this is to figure how much fuel is used in each case to get, say, one mile past the light with no further stopping given each of the scenarios above. So without further ado:
Don't accelerate, make light uses 0.0458 gallons
Don't accelerate, miss light uses 0.0523 gallons
Accelerate, make light uses 0.0546 gallons
Accelerate, miss light uses 0.0611 gallons
Surprisingly, acclerating and MAKING the light uses more fuel than not accelerating and missing the light. Therefore, it could not possibly pay to try to make the light using this specific scenario. Obviously, other assumptions regarding light durations, speeds, accelerations, etc. could change this. And in case anyone incorporates my driving techniques, the numbers above were derived using fuel consumption numbers for my Grand Cherokee Limited. As they say in chatrooms, ymmv (your mileage may vary).
To close the chapter, the mathematical expectations (under this set of assumptions) are:
Don't accelerate: 0.3*0.0458+0.7*0.0523=0.0503 gallons
Accelerate: 0.8*0.0546+0.2*0.0611=0.0559 gallons.
So there you have it. If I accelerate to make a light, I can expect to use about 11% more fuel at the intersection than if I just maintain my normal speed. Again, the circumstances for this calculation are quite specific but not abnormal. Every now and again though, at lights where I know the duration of the red is long and where I know a short burst will get me through and the lack thereof won't, I'll give it a try.
Sunday, November 05, 2006
Traffic jams
Though I've had trouble finding the original source, it seems that the consensus on the web (see here for example) is that Americans waste 2.3 billion gallons of motor fuel in traffic jams annually. Another figure that seems to be well accepted is that Americans use about 100 billion gallons of fuel. So approximately 2.3% of the motor fuel in the United States is wasted in traffic jams.
This is truly awful, but is it significant? As noted in a previous post a barrel of oil produces 19 gallons of motor fuel, so we waste the gasoline from 2.3 X 10^9 / 19 = 121 million barrels of oil per year in traffic jams. We use about 21.9 million barrels per day or 7.9 billion barrels per year. Thus, fossil fuel wasted in traffic jams represents about 1.5% of our annual fossil fuel usage. Of course, the barrels of oil producing the 19 gallons are actually 42 gallons each. The remainder goes to various other uses and thus this figure of 1.5% overestimates the reduction in oil usage that could be achieved by the elimination of traffic jams. Figure about half of that or so.
In an earlier post I noted that the United States could save almost 29% of our personal transportation motor fuel if everyone implemented the measures I have undertaken to save fuel. Of course, I have also explored the likelihood that all, most, or even a significant portion of the population of U.S. drivers would take these measures. In the current vernacular: I'm sure we'll do it..... NOT!!
Still, if a way could be found to motivate Americans to take up these driving habits, to avoid unnecessary car trips by telecommuting, carpooling, combining trips, etc. I believe we could cut our use of fossil fuel for personal transportation by 50% and our overall fossil fuel usage by upwards of 20%. Increasing the so-called "CAFE" (corporate average fuel economy) requirements could increase this further still.
But in order to accomplish anything like this, a general awareness of the urgency of the situation would have to be generated. These things could be done at one time, the sacrifices of World War II come to mind, but in today's completely fragmented society, I am more than skeptical. The facts of our spiralling trade deficit, increasing population, and diminshing availability of cheap and easy fossil fuels will have to hit us on the head.
It will not be painless.
This is truly awful, but is it significant? As noted in a previous post a barrel of oil produces 19 gallons of motor fuel, so we waste the gasoline from 2.3 X 10^9 / 19 = 121 million barrels of oil per year in traffic jams. We use about 21.9 million barrels per day or 7.9 billion barrels per year. Thus, fossil fuel wasted in traffic jams represents about 1.5% of our annual fossil fuel usage. Of course, the barrels of oil producing the 19 gallons are actually 42 gallons each. The remainder goes to various other uses and thus this figure of 1.5% overestimates the reduction in oil usage that could be achieved by the elimination of traffic jams. Figure about half of that or so.
In an earlier post I noted that the United States could save almost 29% of our personal transportation motor fuel if everyone implemented the measures I have undertaken to save fuel. Of course, I have also explored the likelihood that all, most, or even a significant portion of the population of U.S. drivers would take these measures. In the current vernacular: I'm sure we'll do it..... NOT!!
Still, if a way could be found to motivate Americans to take up these driving habits, to avoid unnecessary car trips by telecommuting, carpooling, combining trips, etc. I believe we could cut our use of fossil fuel for personal transportation by 50% and our overall fossil fuel usage by upwards of 20%. Increasing the so-called "CAFE" (corporate average fuel economy) requirements could increase this further still.
But in order to accomplish anything like this, a general awareness of the urgency of the situation would have to be generated. These things could be done at one time, the sacrifices of World War II come to mind, but in today's completely fragmented society, I am more than skeptical. The facts of our spiralling trade deficit, increasing population, and diminshing availability of cheap and easy fossil fuels will have to hit us on the head.
It will not be painless.
Sunday, October 29, 2006
New car considerations
I've had the 2000 Jeep Grand Cherokee Limited that has been the subject of the experimentation described (ad nauseum) in this blog since August of 2000. It has about 142,000 miles on it and has been a truly wonderful car. I know some have had lots of trouble with their Jeeps, but I've changed the oil in mine, and the brake pads and rotors once. Other than that, it's been maintenance free. But it's getting a little long in the tooth, with no navigation system, the cd changer in the back of the truck rather than in dash, etc. I decided to start looking for a replacement.
Most who are aware of what I've been doing to minimize fuel consumption in the Jeep assumed I'd look at Prius, Insight, Civic Hybrid, or other mileage maximizing vehicles. Today I looked at the Acura RL, the Acura RDX, and the Lexus RX350. Huh? The best of these achieves E.P.A. ratings of 20 city, 26 highway. Isn't this hypocritical?
Well, for starters, they all are rated higher than my Jeep (15/20). But the fact of the matter is, I like big engines, acceleration, and luxury. The fact that I never utilize the horsepower available to me in the Jeep doesn't mean that I couldn't (and haven't). And there are those who say "you've squeezed all the blood out of that turnip" in reference to my experiment. Occasionally, I do think it would be nice just to drive, enjoy the performance of a nice car, and not concern myself with trying to extract the last possible foot out of each milliliter of gasoline.
But the fact is, none of the vehicles I looked at today excited me. The RL is nice, high tech, has real time traffic through XM satellite radio, is comfortable, and handles well. The RDX just didn't thrill me at all, nor did the RX350. In fact, my experience today led me to realize how much I really do like my Jeep. It's exceptionally comfortable, has power in case I should ever decide to go back to using it, and has demonstrated flawless reliability. I'd like to have satellite navigation, but that can be achieved with a Garmin portable unit or similar. Blue tooth hands free calling would be nice, but I can get an earpiece. An iPod port would be great but I can get an adapter. XM satellite radio is cool, but I can get a portable unit. All in all, I'm just still very pleased with my Jeep.
One of my business partners got a BMW X5 last week, though I haven't been in it yet. I'm not a BMW kind of guy. They seem like a "look what I can buy" kind of vehicle, though Brian assures me that he isn't a "look what I can buy" kind of guy. It's been suggested that I look at the Infiniti M class. I don't know much about the car, either in terms of features or gas mileage. But unless it knocks my socks off, I'm thinking very seriously of staying with old faithful.
Most who are aware of what I've been doing to minimize fuel consumption in the Jeep assumed I'd look at Prius, Insight, Civic Hybrid, or other mileage maximizing vehicles. Today I looked at the Acura RL, the Acura RDX, and the Lexus RX350. Huh? The best of these achieves E.P.A. ratings of 20 city, 26 highway. Isn't this hypocritical?
Well, for starters, they all are rated higher than my Jeep (15/20). But the fact of the matter is, I like big engines, acceleration, and luxury. The fact that I never utilize the horsepower available to me in the Jeep doesn't mean that I couldn't (and haven't). And there are those who say "you've squeezed all the blood out of that turnip" in reference to my experiment. Occasionally, I do think it would be nice just to drive, enjoy the performance of a nice car, and not concern myself with trying to extract the last possible foot out of each milliliter of gasoline.
But the fact is, none of the vehicles I looked at today excited me. The RL is nice, high tech, has real time traffic through XM satellite radio, is comfortable, and handles well. The RDX just didn't thrill me at all, nor did the RX350. In fact, my experience today led me to realize how much I really do like my Jeep. It's exceptionally comfortable, has power in case I should ever decide to go back to using it, and has demonstrated flawless reliability. I'd like to have satellite navigation, but that can be achieved with a Garmin portable unit or similar. Blue tooth hands free calling would be nice, but I can get an earpiece. An iPod port would be great but I can get an adapter. XM satellite radio is cool, but I can get a portable unit. All in all, I'm just still very pleased with my Jeep.
One of my business partners got a BMW X5 last week, though I haven't been in it yet. I'm not a BMW kind of guy. They seem like a "look what I can buy" kind of vehicle, though Brian assures me that he isn't a "look what I can buy" kind of guy. It's been suggested that I look at the Infiniti M class. I don't know much about the car, either in terms of features or gas mileage. But unless it knocks my socks off, I'm thinking very seriously of staying with old faithful.
Sunday, October 22, 2006
Effects of weight
I've scoured the internet for the last year (plus) looking for mileage and fuel economy related sites. One of the "rules of thumb" I've seen quoted is that "you will lose 1% to 2% of your gas mileage for every 100 pounds of excess weight you carry" (see here for example). Since I tend to be a pack rat and that tendency extends to the cargo area of my Grand Cherokee, it's one of the areas where further savings may be possible.
First, suppose it is true. In that case if I removed, say, 200 pounds of stuff from the car I could expect to improve from the 23.6 m.p.g. I am currently averaging to about 24.1 m.p.g. In the course of the approximately 20,250 miles I drive per year, I could expect to save about 17.8 gallons. At current Southern California prices, that represents a savings of a little over $43. Maybe dinner at Islands for two, but no movie afterward. Of course, long-term readers of my blog (lol) will realize that this likely exceeds the savings realized by eschewing the drive through window. That means I MUST do it if I can demonstrate that it's a plausible number. Let's see what we can do.
A couple of posts back I discussed mass as it relates to mileage. As related there, I think there are probably three detrimental effects of a more massive vehicle on gas mileage. Only two can be controlled by eliminating weight from a given vehicle: the energy cost of lifting mass up hills and not receiving full repayment on downhills; and tire rolling friction.
I will make an educated guess that increased dissipative losses on hills due to increased weight are a so-called "second order effect" and that the primary effect of increased weight on fuel economy is based on the increased rolling friction. I have cited a web site several times where the author discusses the physics of automobiles, and on that site the author contends that rolling friction is approximately 1.5% of vehicle weight at freeway speed. His discussion is actually more detailed, but that's my estimate based on the information he provided. In another post I've shown that his calculations agree with the ones I've made based on fuel consumption, so I think it's reasonable to use his figures.
Thus, I can estimate that 200 extra pounds would result in 3 extra pounds of rolling friction. Since fuel expended to maintain speed is proportional to the total resistive force, which I calculated using the rate of fuel consumption in a previous post as approximately 139 pounds, and that I will calculate in a subsequent post using a different method as 170 pounds, 3 pounds represents somewhere between 2.2% and 1.8% of total resistive force. The "second order effect" mentioned above will only add to the savings, though likely by a minor amount. But that's pretty close to the 2% to 4% predicted by the rule of thumb, so, out comes the junk.
First, suppose it is true. In that case if I removed, say, 200 pounds of stuff from the car I could expect to improve from the 23.6 m.p.g. I am currently averaging to about 24.1 m.p.g. In the course of the approximately 20,250 miles I drive per year, I could expect to save about 17.8 gallons. At current Southern California prices, that represents a savings of a little over $43. Maybe dinner at Islands for two, but no movie afterward. Of course, long-term readers of my blog (lol) will realize that this likely exceeds the savings realized by eschewing the drive through window. That means I MUST do it if I can demonstrate that it's a plausible number. Let's see what we can do.
A couple of posts back I discussed mass as it relates to mileage. As related there, I think there are probably three detrimental effects of a more massive vehicle on gas mileage. Only two can be controlled by eliminating weight from a given vehicle: the energy cost of lifting mass up hills and not receiving full repayment on downhills; and tire rolling friction.
I will make an educated guess that increased dissipative losses on hills due to increased weight are a so-called "second order effect" and that the primary effect of increased weight on fuel economy is based on the increased rolling friction. I have cited a web site several times where the author discusses the physics of automobiles, and on that site the author contends that rolling friction is approximately 1.5% of vehicle weight at freeway speed. His discussion is actually more detailed, but that's my estimate based on the information he provided. In another post I've shown that his calculations agree with the ones I've made based on fuel consumption, so I think it's reasonable to use his figures.
Thus, I can estimate that 200 extra pounds would result in 3 extra pounds of rolling friction. Since fuel expended to maintain speed is proportional to the total resistive force, which I calculated using the rate of fuel consumption in a previous post as approximately 139 pounds, and that I will calculate in a subsequent post using a different method as 170 pounds, 3 pounds represents somewhere between 2.2% and 1.8% of total resistive force. The "second order effect" mentioned above will only add to the savings, though likely by a minor amount. But that's pretty close to the 2% to 4% predicted by the rule of thumb, so, out comes the junk.
Sunday, October 15, 2006
Efficient speed
It's been well over a year since I began my experiment to increase gasoline mileage in my Jeep Grand Cherokee Limited. Without any doubt, huge increases can be achieved. At the outset of the experiment, in August, 2005, my average mileage indicator on the display was at 14.9 m.p.g. It is currently at 23.6 m.p.g., a whopping 58.4% increase, and an estimated 31.1% above the EPA estimate for the vehicle (18 m.p.h. combined). I should add that the information on the average mileage indicator is confirmed by an extremely detailed, tank full by tank full spreadsheet. I calculate mileage by tank full, five and ten tank full moving averages, standard deviation, and estimated annual savings in gallons and in dollars.
In an earlier post (More on acceleration) I estimated that about 14% of my savings come from reduced rate of acceleration. It might be wondered where the rest comes from. As loyal readers may recall, the other steps I've taken are to utilize cruise control at 55 m.p.h. on highways and freeways; anticipate stops and slowdowns to enable coasting to stops and speed reductions so as not to waste energy by braking; minimize use of "appliances" (air conditioning, headlights, seat heaters, defroster, etc.); coasting downhill out of gear (the savings here are controversial - some maintain that modern computerized cars are more efficient coasting in gear); filling the tires to 2 p.s.i. above recommended maximum; avoidance of drive through windows; and turning the engine off on long downhills and at long stoplights.
Of these, I think it's very clear, based on both theory and the evidence of the instant mileage indicator, that the main contributor to my increased fuel efficiency comes from my reduced highway speeds. My understanding of the physics involved leads me to conclude that the reason for the dramatic decrease in mileage per gallon at speeds above 55 m.p.h. is that aerodynamic drag increases as the square of speed (as noted previously, others say cube, which would make it even more dominant).
I have done a lot of "googling" using search terms involving fuel efficiency, minimizing fuel consumption, etc. and there are many people on forums and blogs who contend that their vehicles are much more efficient at 70 m.p.h., and even 80 m.p.h. than at 55 m.p.h. As I noted in my original article on acceleration (To floor it or not to floor it) there is general agreement that fuel efficiency (m.p.g.) increases as speed increases up to a point where the aerodynamic drag increase overrides the increase in efficiency from utilizing fuel for motion rather than merely running the engine. A further complication is that the gearing and engine parameters for a particular vehicle may make it utilize fuel to develop power more efficiently at some relatively high engine speed.
So is it possible that the above-mentioned posters are correct? It would imply an extremely low coefficient of drag, combined with an engine and drivetrain combination that would lead to terrible low speed performance. Since I don't have such a vehicle, I can't do any experimentation, but I suspect it's wishful thinking on the part of the drivers of those vehicles in an effort to rationalize their behavior. I'm not a psychologist, so I have no comment on why they would have a need to engage in such rationalization.
In an earlier post (More on acceleration) I estimated that about 14% of my savings come from reduced rate of acceleration. It might be wondered where the rest comes from. As loyal readers may recall, the other steps I've taken are to utilize cruise control at 55 m.p.h. on highways and freeways; anticipate stops and slowdowns to enable coasting to stops and speed reductions so as not to waste energy by braking; minimize use of "appliances" (air conditioning, headlights, seat heaters, defroster, etc.); coasting downhill out of gear (the savings here are controversial - some maintain that modern computerized cars are more efficient coasting in gear); filling the tires to 2 p.s.i. above recommended maximum; avoidance of drive through windows; and turning the engine off on long downhills and at long stoplights.
Of these, I think it's very clear, based on both theory and the evidence of the instant mileage indicator, that the main contributor to my increased fuel efficiency comes from my reduced highway speeds. My understanding of the physics involved leads me to conclude that the reason for the dramatic decrease in mileage per gallon at speeds above 55 m.p.h. is that aerodynamic drag increases as the square of speed (as noted previously, others say cube, which would make it even more dominant).
I have done a lot of "googling" using search terms involving fuel efficiency, minimizing fuel consumption, etc. and there are many people on forums and blogs who contend that their vehicles are much more efficient at 70 m.p.h., and even 80 m.p.h. than at 55 m.p.h. As I noted in my original article on acceleration (To floor it or not to floor it) there is general agreement that fuel efficiency (m.p.g.) increases as speed increases up to a point where the aerodynamic drag increase overrides the increase in efficiency from utilizing fuel for motion rather than merely running the engine. A further complication is that the gearing and engine parameters for a particular vehicle may make it utilize fuel to develop power more efficiently at some relatively high engine speed.
So is it possible that the above-mentioned posters are correct? It would imply an extremely low coefficient of drag, combined with an engine and drivetrain combination that would lead to terrible low speed performance. Since I don't have such a vehicle, I can't do any experimentation, but I suspect it's wishful thinking on the part of the drivers of those vehicles in an effort to rationalize their behavior. I'm not a psychologist, so I have no comment on why they would have a need to engage in such rationalization.
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