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.
Saturday, May 26, 2007
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.
Saturday, October 07, 2006
Mass
It's been almost two months since my last post due to some surgery that made writing and typing difficult. I hope I haven't lost my devoted readers. Right. In any case, onward and upward.
The trend in my mileage has been a significant increase in standard deviation, together with a slight decreasing trend in mileage. This, despite the installation of the K&N high flow air filter noted in my last post. Starting June 19, there was a major downtrend in my mileage from which I've never really recovered, followed by a trendless few tank fulls with large variation.
A comment was left in my post about Dr. Steven Dutch and his article about the 200 mile per gallon car. At the end of that post, I stated that it was my belief that "in order to achieve major reductions in oil consumption without going to vehicles such as the scooter I discussed a couple of posts back, large-scale changes must be made in the technology of internal combustion engines or other propulsion methods must be employed."
Bill Anderson, host of the blog entitled "mental radiation," commented that large gains can be made in automotive gas mileage by reducing the weight of vehicles. He stated that two thirds of the energy used at the wheels is used to overcome weight, and concluded that by reducing weight the amount of energy required to get from point A to point B can be reduced. Dr. Dutch implied a similar conclusion.
What about this? Well, obviously, since F=ma, that is, Force equals mass times acceleration, it takes more force to get a heavier (more massive) vehicle up to a given speed. But at speed, on level road, acceleration is zero and hence, the sum of forces acting on the vehicle must be zero. These forces are dissipative (drag, rolling friction, driveline friction, engine friction) and force applied to the road by the engine. With the likely exception of rolling friction, seemingly none of these are a function of mass, though engine friction must increase with engine size, which in turn typically increases with vehicle weight. Though this is probably not necessary by the laws of physics, a certain capacity for acceleration must be provided by its manufacturer to make the vehicle saleable.
And since I concluded in a series of earlier posts that engine friction is a very significant component of energy usage, heavier vehicles must use more fuel even in unaccelerated travel, though it isn't a direct correlation. Added to this, it takes more fuel energy to lift a heavier vehicle up a hill, energy which is not fully recovered in the descent due to disspative forces. Further, it seems very likely that heavier vehicles produce higher tire rolling resistance. In fact, this is almost certainly the largest contributor to increasing fuel consumption with increasing weight. Finally, unaccelerated travel on level roads for long periods is not the norm.
Thus, I agree that weight reduction is an effective means of increasing fuel economy, but at freeway speeds a large percentage of the force the engine must overcome is produced by drag. Reduction in "flat plate area" can be achieved by making cars smaller as well, but there is a limit - we still want a driver's seat and a passenger seat. I doubt we'll see tandem seating anytime soon.
The trend in my mileage has been a significant increase in standard deviation, together with a slight decreasing trend in mileage. This, despite the installation of the K&N high flow air filter noted in my last post. Starting June 19, there was a major downtrend in my mileage from which I've never really recovered, followed by a trendless few tank fulls with large variation.
A comment was left in my post about Dr. Steven Dutch and his article about the 200 mile per gallon car. At the end of that post, I stated that it was my belief that "in order to achieve major reductions in oil consumption without going to vehicles such as the scooter I discussed a couple of posts back, large-scale changes must be made in the technology of internal combustion engines or other propulsion methods must be employed."
Bill Anderson, host of the blog entitled "mental radiation," commented that large gains can be made in automotive gas mileage by reducing the weight of vehicles. He stated that two thirds of the energy used at the wheels is used to overcome weight, and concluded that by reducing weight the amount of energy required to get from point A to point B can be reduced. Dr. Dutch implied a similar conclusion.
What about this? Well, obviously, since F=ma, that is, Force equals mass times acceleration, it takes more force to get a heavier (more massive) vehicle up to a given speed. But at speed, on level road, acceleration is zero and hence, the sum of forces acting on the vehicle must be zero. These forces are dissipative (drag, rolling friction, driveline friction, engine friction) and force applied to the road by the engine. With the likely exception of rolling friction, seemingly none of these are a function of mass, though engine friction must increase with engine size, which in turn typically increases with vehicle weight. Though this is probably not necessary by the laws of physics, a certain capacity for acceleration must be provided by its manufacturer to make the vehicle saleable.
And since I concluded in a series of earlier posts that engine friction is a very significant component of energy usage, heavier vehicles must use more fuel even in unaccelerated travel, though it isn't a direct correlation. Added to this, it takes more fuel energy to lift a heavier vehicle up a hill, energy which is not fully recovered in the descent due to disspative forces. Further, it seems very likely that heavier vehicles produce higher tire rolling resistance. In fact, this is almost certainly the largest contributor to increasing fuel consumption with increasing weight. Finally, unaccelerated travel on level roads for long periods is not the norm.
Thus, I agree that weight reduction is an effective means of increasing fuel economy, but at freeway speeds a large percentage of the force the engine must overcome is produced by drag. Reduction in "flat plate area" can be achieved by making cars smaller as well, but there is a limit - we still want a driver's seat and a passenger seat. I doubt we'll see tandem seating anytime soon.
Saturday, August 19, 2006
Mixed messages
After my fill-up on August 10 (calculated miles per gallon of 23.90 so climbing back up where I had grown to expect) I installed a K&N high performance air filter. I wanted to see if such a product could increase my fuel efficiency. I was amazed to see that the instant mileage indicator, which I consult frequently when driving, seemed to show a distinct and significant increase. For example, on normal, relatively level stretches of freeway at 55 miles per hour I typically see the indicator jump between 31 and 32 miles per gallon. After installing the K&N filter, the same stretches yielded indications of 33 and 34 miles per gallon. Amazing and quite significant. This should be my best tank full yet, huh?
Think again. Though the driving I did for this tank full was quite typical of my average driving regime, the fill-up from yesterday (August 18) yielded 21.16 miles per gallon. This surprised me as the instant mileage indications led me to expect something like 24 miles per gallon early in the tank. As the fuel quantity indicator went down though, it was clear that I wasn't going to see a good number.
I surmise that one of the major contributors to the variance I see in my results (current standard deviation, the square root of variance, in calculated miles per gallon evaluated at fill-up is 1.71) is inablility to fill the tank to a precise level. That is, if my previous fill-up was short of normal and the current one is above, I'd calculate a lower mileage. The opposite is also true. Such effects clearly average out over a period of time and thus I follow my five tank and ten tank moving average to account for this.
But it's hard for me to see how it could be the entire explanation of this particular fill-up. I put 19.104 gallons in the tank after driving 404.2 miles, thus getting 21.16 m.p.g. For me to have gotten 23 m.p.g. I would have to have only needed 17.565 gallons. That's a difference of 1.539 gallons. It could have happened if the earlier fill-up was, say, about 0.77 below the "average" spot and the most recent was 0.77 above that spot. But it seems like that's an awful lot of variation.
Because I'm aware that the fill-up point is so important to the calculated mileage, I fill it absolutely as full as possible because that seems to be the most accurately repeatable amount. Though I know one is not supposed to "top off," I invariably do. I top off until it's only possible to add a couple of cents worth of fuel (say 0.006 gallons) before the automatic shut-off turns it off. I know that if I didn't top off, the variations would average out to an accurate number over time but I'm too impatient to wait for the averaging out process.
But I'm having a hard time reconciling the low mileage calculated for this tank full with the noticeably higher freeway instant miles per gallon readings. I guess I'll give it another tank full or two before I delve into it more deeply, though I don't know what that delving would invlove.
Think again. Though the driving I did for this tank full was quite typical of my average driving regime, the fill-up from yesterday (August 18) yielded 21.16 miles per gallon. This surprised me as the instant mileage indications led me to expect something like 24 miles per gallon early in the tank. As the fuel quantity indicator went down though, it was clear that I wasn't going to see a good number.
I surmise that one of the major contributors to the variance I see in my results (current standard deviation, the square root of variance, in calculated miles per gallon evaluated at fill-up is 1.71) is inablility to fill the tank to a precise level. That is, if my previous fill-up was short of normal and the current one is above, I'd calculate a lower mileage. The opposite is also true. Such effects clearly average out over a period of time and thus I follow my five tank and ten tank moving average to account for this.
But it's hard for me to see how it could be the entire explanation of this particular fill-up. I put 19.104 gallons in the tank after driving 404.2 miles, thus getting 21.16 m.p.g. For me to have gotten 23 m.p.g. I would have to have only needed 17.565 gallons. That's a difference of 1.539 gallons. It could have happened if the earlier fill-up was, say, about 0.77 below the "average" spot and the most recent was 0.77 above that spot. But it seems like that's an awful lot of variation.
Because I'm aware that the fill-up point is so important to the calculated mileage, I fill it absolutely as full as possible because that seems to be the most accurately repeatable amount. Though I know one is not supposed to "top off," I invariably do. I top off until it's only possible to add a couple of cents worth of fuel (say 0.006 gallons) before the automatic shut-off turns it off. I know that if I didn't top off, the variations would average out to an accurate number over time but I'm too impatient to wait for the averaging out process.
But I'm having a hard time reconciling the low mileage calculated for this tank full with the noticeably higher freeway instant miles per gallon readings. I guess I'll give it another tank full or two before I delve into it more deeply, though I don't know what that delving would invlove.
Monday, August 07, 2006
Cruise horsepower
This will be a short one. I cited a web site a couple of posts back that has some very interesting articles on automobile physics. The author uses coefficient of drag, estimates of tire rolling friction, etc. to determine that his Corvette needs about 26 horsepower to cruise the highway at 60 miles per hour.
I can get at this problem from a different point of view. I get about 31.5 miles per gallon at 55 miles per hour. This is using fuel at the rate of 55/31.5 or 1.75 gallons per hour. Those 1.75 gallons contain about 1.75*125,000,000=218,000,000 joules of heat energy in the chemical bonds I release by burning them in my cylinders. About 25% or 54,600,000 joules goes to maintaining my 55 miles per hour, the rest is wasted as heat expelled to the environment (an inevitable consequence of the second law of thermodynamics, i.e., you can't break even).
So I'm using 54,600,000 joules per hour to maintain motion, or 54,600,000/3600=15,200 joules per second, otherwise known as watts. Now a horsepower is 746 watts, so I'm using 15,200/746=20.4 horsepower. Pretty darn close to the web site author's number, especially considering I'm looking at a speed 5 miles per hour lower. I really love it when different approaches to a problem confirm each others' results.
To carry the analysis a little further (at the risk of causing any remaining reader to throw up his or her hands in despair), force X speed = power, thus power/speed=force. So: 55 miles per hour is 24.6 meters per second; 15,200 watts/24.6 meters/second = 618 Newtons (the metric unit of force). 618 Newtons is 139 pounds of force. That's all it takes, applied continuously, to move me down the highway at 55 miles per hour in my Jeep Grand Cherokee Limited. I'm going to take an informal survey. I bet most people think it's a whole lot more.
I can get at this problem from a different point of view. I get about 31.5 miles per gallon at 55 miles per hour. This is using fuel at the rate of 55/31.5 or 1.75 gallons per hour. Those 1.75 gallons contain about 1.75*125,000,000=218,000,000 joules of heat energy in the chemical bonds I release by burning them in my cylinders. About 25% or 54,600,000 joules goes to maintaining my 55 miles per hour, the rest is wasted as heat expelled to the environment (an inevitable consequence of the second law of thermodynamics, i.e., you can't break even).
So I'm using 54,600,000 joules per hour to maintain motion, or 54,600,000/3600=15,200 joules per second, otherwise known as watts. Now a horsepower is 746 watts, so I'm using 15,200/746=20.4 horsepower. Pretty darn close to the web site author's number, especially considering I'm looking at a speed 5 miles per hour lower. I really love it when different approaches to a problem confirm each others' results.
To carry the analysis a little further (at the risk of causing any remaining reader to throw up his or her hands in despair), force X speed = power, thus power/speed=force. So: 55 miles per hour is 24.6 meters per second; 15,200 watts/24.6 meters/second = 618 Newtons (the metric unit of force). 618 Newtons is 139 pounds of force. That's all it takes, applied continuously, to move me down the highway at 55 miles per hour in my Jeep Grand Cherokee Limited. I'm going to take an informal survey. I bet most people think it's a whole lot more.
Sunday, August 06, 2006
More on fuel used to run the engine
I've spent a couple of posts attempting to determine where the heat energy in the cylinders from burning fuel is used. I talked about the air conditioner and about the fuel used to keep the engine turning. I was rather surprised at the amount used for the latter and decided to look into it a bit more.
Several posts back I wrote about the web site of Dr. Steven Dutch, a Professor at the University of Wisconsin at Green Bay. He has an article debunking the fantasy 200 mile per gallon car. To be clear, he doesn't claim that no such vehicle exists or is possible, only that it's not possible to use simple bolt-on parts or additives to achieve this kind of mileage with "off the shelf" cars.
In any case, Dr. Dutch uses several strategies to infer the force required to turn the engine against the friction of the moving parts (probably primarily the pistons in the cylinders I would guess). The one I'm considering is his analysis involving the cranking power of an automotive battery. He concluded that turning the engine over requires 3600 joules per second or 3600 watts.
If that's true, it means that the energy needed to turn the engine for an hour is 3600*3600 or 12,960,000 joules. However, the engine only uses about 25% of the heat energy in gasoline to do useful work, the rest is wasted as heat expelled to the environment. In fact, some of the "useful work" is turning the fan and the water pump to dissipate the heat. In any case, this means I need the energy of 4*12,960,000 or 51,840,000 joules of heat. This is the amount available in 51,840,000/125,000,000 or about 0.41 gallons of gasoline.
Let me repeat that. Dr. Dutch's calculations imply that I consume about 0.41 gallons per hour to turn the engine. My observations on the road lead me to conclude it's about 0.38 gallons per hour. Absolutely amazing that the agreement is so close. And it reinforces my conclusion that a large amount of the fuel burned in a car is used to operate the engine.
Several posts back I wrote about the web site of Dr. Steven Dutch, a Professor at the University of Wisconsin at Green Bay. He has an article debunking the fantasy 200 mile per gallon car. To be clear, he doesn't claim that no such vehicle exists or is possible, only that it's not possible to use simple bolt-on parts or additives to achieve this kind of mileage with "off the shelf" cars.
In any case, Dr. Dutch uses several strategies to infer the force required to turn the engine against the friction of the moving parts (probably primarily the pistons in the cylinders I would guess). The one I'm considering is his analysis involving the cranking power of an automotive battery. He concluded that turning the engine over requires 3600 joules per second or 3600 watts.
If that's true, it means that the energy needed to turn the engine for an hour is 3600*3600 or 12,960,000 joules. However, the engine only uses about 25% of the heat energy in gasoline to do useful work, the rest is wasted as heat expelled to the environment. In fact, some of the "useful work" is turning the fan and the water pump to dissipate the heat. In any case, this means I need the energy of 4*12,960,000 or 51,840,000 joules of heat. This is the amount available in 51,840,000/125,000,000 or about 0.41 gallons of gasoline.
Let me repeat that. Dr. Dutch's calculations imply that I consume about 0.41 gallons per hour to turn the engine. My observations on the road lead me to conclude it's about 0.38 gallons per hour. Absolutely amazing that the agreement is so close. And it reinforces my conclusion that a large amount of the fuel burned in a car is used to operate the engine.
Saturday, August 05, 2006
Dissipative forces
There's a principle in physics called "conservation of energy" which is of awesome utility. There are also what are termed "conservative forces" (gravity is one). Please note carefully that I am talking about PHYSICS and NOT POLITICS. Anyway, gravity being a conservative force, theoretically, I can get all of the energy I use to climb a hill back when I descend. As we all know, this isn't the nature of our real world. The fly in the ointment is the phenomenon of dissipative forces. Such forces in our example include friction of all types and fluid mechanical drag.
I was reading a site that discusses the physics of automobiles (a very interesting site by the way) wherein the author stated that "Mechanical Drag is due to all the moving mechanisms in the vehicle that have frictional losses, most specifically the wheel bearings, but is actually nearly entirely due to the action of the tires on the road surface. In nearly all actual situations, all the other causes of mechanical drag factors can be ignored, and just the Tire Resistance considered, regarding the Mechanical Drag."
I don't know about that. As I discussed in my last post, my Grand Cherokee burns about 0.38 gallons per hour when idling or coasting in neutral. It idles at 650 r.p.m. I think it's reasonable to assume that: a) all of the fuel burned while idling goes to overcome engine friction and pumping losses; b) these losses are directly proportional to engine r.p.m.
At 55 miles per hour, the engine turns at 1750 r.p.m., so assumption b) above would indicate that I'm burning (1750/650)*0.38 or 1.02 gallons per hour to overcome engine friction and pump fluids. As I also showed in the previous post, I burn 1.75 gallons per hour at 55 miles per hour. This would indicate that 58% of my fuel consumption goes to keeping the engine running at 1750 r.p.m. and 42% goes to overcoming aerodynamic drag, driveline friction and tire rolling friction.
This seems very surprising to me, and I'm sure it would be extremely surprising to the author of the article cited above. Could it be true? Let's suppose that the engine friction and pumping losses are proportional to the square root of r.p.m. If so, I'd use 0.62 gallons per hour at 1750 r.p.m. and about 35% of the fuel burn is to overcome engine friction and pumping losses. If fuel used to overcome these losses doesn't increase at all with r.p.m., an extremely unlikely scenario, it would still mean that almost 22% of the fuel burn at 55 miles per hour is used in keeping the engine going. Very surprising indeed.
It's possible that the fuel is used for something other than overcoming these dissipative forces when idling, but I'm not sure what it would be. Unless there is a governor that absorbs the mechanical energy and turns it into heat, if there is an excessive amount of fuel for the state of energy use in the engine, the r.p.m.'s would increase. In other words, since the r.p.m.'s stay at 650, the system is in equilibrium. So, surprising as it may be, I think the conclusion is accurate, i.e., that a whole lot of fuel is burned to keep the fuel burner burning fuel.
I was reading a site that discusses the physics of automobiles (a very interesting site by the way) wherein the author stated that "Mechanical Drag is due to all the moving mechanisms in the vehicle that have frictional losses, most specifically the wheel bearings, but is actually nearly entirely due to the action of the tires on the road surface. In nearly all actual situations, all the other causes of mechanical drag factors can be ignored, and just the Tire Resistance considered, regarding the Mechanical Drag."
I don't know about that. As I discussed in my last post, my Grand Cherokee burns about 0.38 gallons per hour when idling or coasting in neutral. It idles at 650 r.p.m. I think it's reasonable to assume that: a) all of the fuel burned while idling goes to overcome engine friction and pumping losses; b) these losses are directly proportional to engine r.p.m.
At 55 miles per hour, the engine turns at 1750 r.p.m., so assumption b) above would indicate that I'm burning (1750/650)*0.38 or 1.02 gallons per hour to overcome engine friction and pump fluids. As I also showed in the previous post, I burn 1.75 gallons per hour at 55 miles per hour. This would indicate that 58% of my fuel consumption goes to keeping the engine running at 1750 r.p.m. and 42% goes to overcoming aerodynamic drag, driveline friction and tire rolling friction.
This seems very surprising to me, and I'm sure it would be extremely surprising to the author of the article cited above. Could it be true? Let's suppose that the engine friction and pumping losses are proportional to the square root of r.p.m. If so, I'd use 0.62 gallons per hour at 1750 r.p.m. and about 35% of the fuel burn is to overcome engine friction and pumping losses. If fuel used to overcome these losses doesn't increase at all with r.p.m., an extremely unlikely scenario, it would still mean that almost 22% of the fuel burn at 55 miles per hour is used in keeping the engine going. Very surprising indeed.
It's possible that the fuel is used for something other than overcoming these dissipative forces when idling, but I'm not sure what it would be. Unless there is a governor that absorbs the mechanical energy and turns it into heat, if there is an excessive amount of fuel for the state of energy use in the engine, the r.p.m.'s would increase. In other words, since the r.p.m.'s stay at 650, the system is in equilibrium. So, surprising as it may be, I think the conclusion is accurate, i.e., that a whole lot of fuel is burned to keep the fuel burner burning fuel.
Friday, August 04, 2006
Air conditioning?
Though I doubt anyone reads this blog so carefully (hidden assumption: anyone reads this blog at all), it's possible that someone could have wondered, reading earlier posts, how I estimated the idling fuel consumption of my Grand Cherokee. When coasting in neutral down a mild grade, the car will sometimes reach an equilibrium speed. When this happens for long enough for the instant gas mileage reading on the display to stabilize, the speed in miles per hour divided by the mileage in miles per gallon gives the consumption in gallons per hour. I think this is reasonably accurate since the car is only burning fuel to maintain idling r.p.m. (650).
The number is typically pretty close to 0.38 gallons per hour. However, coasting down the typical grades where I check that, I've noticed lately that I'm burning closer to 0.48 gallons per hour. Now, although my most recent fill-up indicated an improved mileage of 22.1, my previous post was pretty much of a rant about my declining gas mileage of late. Is the increase in idling fuel consumption indicative of a systemic problem that could cause my overall decrease in efficiency? Perhaps it's a symptom of dirty injectors or a clogged air filter.
But another possibility is air conditioning. I try to minimize its use, but it's been quite hot in Southern California and I've had it on quite a bit of late. Could it be that this is the difference I'm seeing? If so, it would indicate that the air conditioner uses about 0.1 gallons per hour when it's on. It certainly seems reasonable that air conditioning fuel use would be a constant function of time and independent of vehicle speed, therefore, the faster I'm going, the less deleterious the effect would be on my fuel consumption per hour.
Let's see if this makes any sense at all. A gallon of gasoline contains heat energy of about 1.25*10^8 joules. Thus, 0.1 gallons per hour would be 1.25*10^7 joules per hour, or about 3500 watts. But the internal combustion engine wastes most of that heat energy - something like 75%. So that means only about 25%, or 875 watts (a tad over 1 horsepower) would be available as mechanical energy to cool the air. This squares niceley with my "gut feeling" about air conditioning.
If it is the air conditioner, I can calculate the effect of a.c. on my gas mileage at , say, 55 miles per hour. Let's see what we find: when driving at 55 m.p.h. on stretches of freeway that appear to be level (i.e., 0% grade) I typically see the instant mileage display oscillate between 31 and 32 miles per gallon. Let's say it's 31.5 m.p.g. This indicates I'm burning 1.75 gallons per hour. But if the air conditioner then burns an additional 0.1 gallons per hour for a total of 1.85 gallons per hour, it should reduce my gas mileage to about 29.7 m.p.g.
This number should be quite noticeable and I don't notice it. It could mean that the 0.1 gallons per hour useage noted above comes from some other cause (e.g., the clogged injectors or air filter) or it could mean that the fuel consumption of the air conditioner is not merely a constant function of time. Or finally, it could be an indication of the error in my determination of the relevant numbers while also performing driving tasks (tuning the radio, answering the cell phone, checking my schedule on the pda, etc.).
I guess I need to get a tune up, see about getting the injectors cleaned, and change the air filter. Then it would be reasonable to check the numbers, followed by a day trip out to the desert somewhere to do some experimentation under more controlled conditions. Stay tuned...
The number is typically pretty close to 0.38 gallons per hour. However, coasting down the typical grades where I check that, I've noticed lately that I'm burning closer to 0.48 gallons per hour. Now, although my most recent fill-up indicated an improved mileage of 22.1, my previous post was pretty much of a rant about my declining gas mileage of late. Is the increase in idling fuel consumption indicative of a systemic problem that could cause my overall decrease in efficiency? Perhaps it's a symptom of dirty injectors or a clogged air filter.
But another possibility is air conditioning. I try to minimize its use, but it's been quite hot in Southern California and I've had it on quite a bit of late. Could it be that this is the difference I'm seeing? If so, it would indicate that the air conditioner uses about 0.1 gallons per hour when it's on. It certainly seems reasonable that air conditioning fuel use would be a constant function of time and independent of vehicle speed, therefore, the faster I'm going, the less deleterious the effect would be on my fuel consumption per hour.
Let's see if this makes any sense at all. A gallon of gasoline contains heat energy of about 1.25*10^8 joules. Thus, 0.1 gallons per hour would be 1.25*10^7 joules per hour, or about 3500 watts. But the internal combustion engine wastes most of that heat energy - something like 75%. So that means only about 25%, or 875 watts (a tad over 1 horsepower) would be available as mechanical energy to cool the air. This squares niceley with my "gut feeling" about air conditioning.
If it is the air conditioner, I can calculate the effect of a.c. on my gas mileage at , say, 55 miles per hour. Let's see what we find: when driving at 55 m.p.h. on stretches of freeway that appear to be level (i.e., 0% grade) I typically see the instant mileage display oscillate between 31 and 32 miles per gallon. Let's say it's 31.5 m.p.g. This indicates I'm burning 1.75 gallons per hour. But if the air conditioner then burns an additional 0.1 gallons per hour for a total of 1.85 gallons per hour, it should reduce my gas mileage to about 29.7 m.p.g.
This number should be quite noticeable and I don't notice it. It could mean that the 0.1 gallons per hour useage noted above comes from some other cause (e.g., the clogged injectors or air filter) or it could mean that the fuel consumption of the air conditioner is not merely a constant function of time. Or finally, it could be an indication of the error in my determination of the relevant numbers while also performing driving tasks (tuning the radio, answering the cell phone, checking my schedule on the pda, etc.).
I guess I need to get a tune up, see about getting the injectors cleaned, and change the air filter. Then it would be reasonable to check the numbers, followed by a day trip out to the desert somewhere to do some experimentation under more controlled conditions. Stay tuned...
Friday, July 28, 2006
Mystified
It's been a very frustrating period. Over the last few weeks, my five tank moving average mileage has decreased by more than two miles per gallon. The troubles began with the brake job reported in an earlier post. The pads definitely dragged on the rotors for a few hundred miles after that - I could feel it in the behavior on coasting to a stop and, in the beginning, I could smell the burning pads. But these are no longer the case. The car rolls just like it always has, the highway instant miles per gallon seems normal, and yet my mileage has deteriorated. Why?
I got involved in an argument with a guy who has a blog called "mental radiation." (Note that that link is to the argument, I believe he actually maintains his blog at a wordpress site here.) The nature of the argument was over the extent to which the use of ethanol as a motor fuel eliminates the impact of driving on fossil fuel usage. As is common in internet arguments, neither of us was convinced. But in the process of our discussion, he looked at my blog and made several comments.
One of them was in response to my posting about my learning curve, in which I opined that the increasing trend in my mileage, as shown by the five and ten tank moving averages, indicated that I was learning to better maximize fuel efficiency. Bill Anderson, publisher of "mental radiation," was skeptical. His belief is that there are likely other reasons, e.g., different brands and different seasonal blends. I'm beginning to wonder.
I've bought Chevron, mostly from the same station, for the duration of the experiment. There have been perhaps three exceptions when I patronized Arco. The Arco purchases are not distinguishable on the graphs or in the raw data, but the downward trend from the last few weeks is obvious. I don't know if there has been a change in gasoline blends in June and July, certainly I haven't read of one. Maybe something's wrong with the vehicle. But even at that, why does my highway instant mileage still indicate what it did when my trend was upward?
I got involved in an argument with a guy who has a blog called "mental radiation." (Note that that link is to the argument, I believe he actually maintains his blog at a wordpress site here.) The nature of the argument was over the extent to which the use of ethanol as a motor fuel eliminates the impact of driving on fossil fuel usage. As is common in internet arguments, neither of us was convinced. But in the process of our discussion, he looked at my blog and made several comments.
One of them was in response to my posting about my learning curve, in which I opined that the increasing trend in my mileage, as shown by the five and ten tank moving averages, indicated that I was learning to better maximize fuel efficiency. Bill Anderson, publisher of "mental radiation," was skeptical. His belief is that there are likely other reasons, e.g., different brands and different seasonal blends. I'm beginning to wonder.
I've bought Chevron, mostly from the same station, for the duration of the experiment. There have been perhaps three exceptions when I patronized Arco. The Arco purchases are not distinguishable on the graphs or in the raw data, but the downward trend from the last few weeks is obvious. I don't know if there has been a change in gasoline blends in June and July, certainly I haven't read of one. Maybe something's wrong with the vehicle. But even at that, why does my highway instant mileage still indicate what it did when my trend was upward?
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