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03-03-2010, 02:53 AM   #21 (permalink)
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Quote:
 Originally Posted by RobertSmalls The biggest problem I see is with the inputs for the road load equation: M, Ma, Cd*A, Crr. M varies quite a bit. For example, my car sheds 27kg as the fuel tank runs from full to empty. I've removed some parts and added others, so if I wanted to know my car's mass, I'd have to have it weighed.
Very true, although mass loss through fuel burn is something that is directly measurable (since we're measuring fuel consumption after all). Harder is mass change due to passengers, but if the model is good enough we can identify all these things.

I might add that the 27kg mass change is perhaps negligible when considering that the true mass of the vehicle is probably unknown, and other (important) parameters such as air density, wind speed and direction, air temperature, engine temperature, oil temperature, tire pressure, etc... are not included in the model. It's not really that important though, as all these factors tend to average out, leaving you with good results even if you have more uncertainty than you would like.

Some of these above parameters we can easily include in the model once we have the measurements. I'd be especially interested to know if we can get an idea of absolute air pressure changes based on the MAF.

Quote:
 My Cd*A changes with every aeromod I do, and Crr depends on tire temperature, pavement type, and too many other factors. I think the best approach here is to have the user do a high-speed neutral coast to discern Cd*A and a low-speed coast to determine Crr. I think lots of EcoModders would like to have a precise datalogger to help calculate these numbers.
One of the beautiful things about observers is that we can estimate all those parameters. High-speed/low-speed tests are a great way to have a first order approximation, but they miss some essential information that we can reconstruct with a parameter estimating observer.

That being said, an online observer and a parameter estimation observer are slightly different, and I haven't yet written the code to do that with this model. The first order approximation is good enough for my purposes right now.

Quote:
 I would love to study the BSFC of the first-gen Honda Insight. I'd love to quantify the effect of lean burn, find the optimal rate of acceleration, and study exhaust and intake mods.
You can lean burn on the Insight? Cool. I had no idea. How do you do this? What are you using to measure the effects of lean burning? Do you have a CHT or EGT gauges?

Quote:
 How long does it take to fill the 2GB SD card?
Quite a while, although it's speed dependent. Figure roughly 1MB/min, and you've got an idea of how long you can go before emptying the card.

 Today Popular topics Other popular topics in this forum... View the most popular topics in this subforum by views or by posts
 03-03-2010, 08:46 AM #22 (permalink) Left Lane Ecodriver     Join Date: Aug 2008 Location: Buffalo, NY, USA Posts: 2,258 Prius C - '12 Toyota Prius C Thanks: 79 Thanked 283 Times in 196 Posts The car leaves the factory with a stratified charge lean burn scheme, that allows AFR as low as 22:1. It happens automatically when the catalyst is warm and the engine is lightly loaded, and the driver notices, because less power is produced at the same throttle position. Also, the fuel economy gauge shoots upward when you enter lean burn mode. I'm sure Honda used EGT and NOx sensors while developing the engine, but the car only needs a wideband oxygen sensor and lots of emissions equipment as a result of lean burn. Having a fuel economy gauge is very helpful, but a fuel economy gauge will tell you heading down a hill at very high speeds is good, braking is not bad, and heading up a hill at full throttle is bad. We know all these things are untrue. I'd like cars to come with a BSFC gauge, and a road load gauge. The BSFC gauge would tell you how efficiently you're producing power, and the road load gauge would tell you how much power you're using. Although your project doesn't (yet?) give the output in real-time, it would still improve my driving. I would like to install one of your boards in my car. Perhaps I can help find where your road load model deviates from reality.
03-03-2010, 09:24 AM   #23 (permalink)
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Quote:
 Originally Posted by RobertSmalls Having a fuel economy gauge is very helpful, but a fuel economy gauge will tell you heading down a hill at very high speeds is good, braking is not bad, and heading up a hill at full throttle is bad. We know all these things are untrue.
The goal of our research, of which this efficiency estimation is only a small part, is to be able to tell you what you should do going up a hill, in real-time.

As you say, a simple fuel economy gauge is difficult to understand in an optimal way. I'd like to see an energy gauge, one that calculates how much energy is being expended instead of just saying average economy. This would take into account potential energy and kinetic energy, for instance.

Quote:
 I'd like cars to come with a BSFC gauge, and a road load gauge. The BSFC gauge would tell you how efficiently you're producing power, and the road load gauge would tell you how much power you're using. Although your project doesn't (yet?) give the output in real-time, it would still improve my driving. I would like to install one of your boards in my car. Perhaps I can help find where your road load model deviates from reality.
I admit it sounds interesting, but you're driving an Insight, and I have no way of measuring power flow in the electric circuit. To make matters more interesting, you have a CVT, which again makes life a little harder. If there's a way to log all that data while driving, then I can work with it. Otherwise, I'm afraid that the only way this would work on the hybrid would be if you could completely disable the electric motors for your tests.

 03-03-2010, 09:38 AM #24 (permalink) Left Lane Ecodriver     Join Date: Aug 2008 Location: Buffalo, NY, USA Posts: 2,258 Prius C - '12 Toyota Prius C Thanks: 79 Thanked 283 Times in 196 Posts Actually, I have a five-speed manual transmission. Also, I drive around with the electric motor disabled all the time, because I save gas by avoiding the constant motor->battery-> motor energy conversion losses. The battery pack has a current sensor that outputs a voltage proportionate to current. The efficiency of the inverter and motor, once warmed up, vary with RPM and load in a knowable manner. So it would be possible to account for the torque from the electric motor, but you'd probably need a lot of dyno time to come up with the efficiency map of the electric motor for each model of hybrid you want to support. I've come across maps for the Prius, first-gen Insight, and Honda Accord V6 hybrid.
 03-03-2010, 01:40 PM #25 (permalink) ...beats walking...   Join Date: Jul 2009 Location: . Posts: 6,191 Thanks: 179 Thanked 1,521 Times in 1,122 Posts ...when the "integrated" version becomes available, I'll get one; but, I'll just be a curious spectator during its' "component" development stage. ...talking on a cell-phone is already dangerous; don't need something *interesting* to distract me even more (ha,ha).
 03-03-2010, 06:34 PM #26 (permalink) EcoModding Lurker   Join Date: Jan 2010 Location: Los Angeles, CA Posts: 54 Thanks: 0 Thanked 20 Times in 8 Posts I commend you on the effort. It is an interesting area of study and I hope you continue. However, I do have some comments/concerns about the paper. First, and minor, reference [3]. It is very doubtful that you are achieving +/- .1% flow measurements from the electrical pulse width controlling the injectors. When commercial automotive fuel systems are tested gravimetrically, the variation is already larger. Also, it is a mistake to assume that this method is readily applicable to modern fuel efficient designs. Consider, some systems intentionally alter injectory opening and closing times conditionally, others will leave the injectors on some cyls closed while still cycling others. Second, I have some real concerns about your basic mathmatical model. My prediction would be that it would generate a fairly innacurate instrument of engine efficiency, and this seems to actually match your own data. Consider figure 7, efficiency calculated from dyno readings. It does rougly match the expected V/P curve for a conventional combustion engine. And also, as expected, you have an 'efficiency island' near peak torque. However, your efficiency island is elongated from typical engine efficiency models. Mathmatically, this deviation in the curve should be exaggerated by under sampling and an EKF techique, and that is what we see in your data. Consider figure 6. You point out that peak matches, but that is the lowest possible hanging fruit, like finding lambda 1.0 on a UEGO sensor. The question is, what is the typical error as we move away from the easily identifiable set point? Based on your included data, pretty high. Notice how much larger and elongated the efficiency island is on figure 6. The expected V/P slope is not even really visible on the chart. If we did a scatter plot of point to point deviation between the two it would appear that, aside from peak, the instrument would almost always over estimate efficiency, and in a non-trival operational envelope, extremely so. My third concern is that the paper never really establishes a link between the acheived measurement and the stated goal. Look at part of your first paragraph: "optimal control to make great reductions in emissions and fuel consumption" Now, we could debate rather or not your measurement of engine efficiency is accurate enough to be of value. But the more fundemental question is, is the data even really connected to either fuel consumption or emissions? Virtually everyone in the field knows that every conventional combustion engine has a fuel economy sweet spot - wide open throttle (so there is no induced vaccuum reducing VE) at peak torque. But this is not an available decision for routine operation of a vehicle. In fact, most people here (myself included) go out of our way to avoid this point of operation, because it is terrible from both the perspective of emissions and *operational* fuel economy. At peak torque wide open throttle, the mixture is generally quite rich, to keep CHTs and pressure down so that destructive abnormal combustion does not occur. But this also means that the vehicle is operating way out side of the cat efficiency envelope, making emissions soar. I'm not saying this to discourage you. Just the opposite. You are attempting to inexpensively measure a very narrow definition of effiency. What I am suggesting is that you try an experiment to demonstrate rather or not your target metric is of any practical use in either the area of fuel economy or emissions. Good Luck! -jjf
 03-03-2010, 07:34 PM #27 (permalink) ...beats walking...   Join Date: Jul 2009 Location: . Posts: 6,191 Thanks: 179 Thanked 1,521 Times in 1,122 Posts ...my thoughts were: "...too many 'user-supplied' inputs for the average-Joe Blow-driver to have reasonable knowledge of..." ...and, unfortunately, I don't believe the automotive companies would help by making ANY of their "proprietary Intellectural Properties" available...to anyone. Last edited by gone-ot; 03-03-2010 at 07:48 PM..
03-03-2010, 07:46 PM   #28 (permalink)
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 Originally Posted by kubark42 It's not really that important though, as all these factors tend to average out, leaving you with good results even if you have more uncertainty than you would like.
Sorry, I missed this earlier. I would contend that that is almost wholly false. The factors tend to work additively, and many act exponentially, which is why you over estimate effiency over so much of the operational curve.

-jjf

03-04-2010, 06:22 AM   #29 (permalink)
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Very interesting comments. Here are my responses:

Quote:
 Originally Posted by jfitzpat I commend you on the effort. It is an interesting area of study and I hope you continue. However, I do have some comments/concerns about the paper. First, and minor, reference [3]. It is very doubtful that you are achieving +/- .1% flow measurements from the electrical pulse width controlling the injectors. When commercial automotive fuel systems are tested gravimetrically, the variation is already larger.
For instance in http://ecomodder.com/forum/showthrea...tml#post152293, it seems that that is the accuracy that people on this forum are getting.

Furthermore, Luxembourg is home to Delphi’s fuel injection research center (although they might have others). I spoke with one of my colleagues who researches direct gasoline injections, and he felt that this level of repeatability sounds reasonable.

That being said, I don’t believe everything I read on the internet, and my colleague couldn’t give me another citation, so if you have some additional information on this, could you send me the reference?

Quote:
 Also, it is a mistake to assume that this method is readily applicable to modern fuel efficient designs. Consider, some systems intentionally alter injector opening and closing times conditionally, others will leave the injectors on some cyls closed while still cycling others.
Agreed. For instance, some cars can switch between sequential and grouped firing patterns depending on efficiency vs. power requirements. However, this is a trivial problem, as by simply adding a voltage sensor to each injector, instead of only one, the problem is solved.

A bigger problem is what to do with diesels. Most cars sold over here are diesels, and this technique does NOT work for them.

Quote:
 Second, I have some real concerns about your basic mathematical model. My prediction would be that it would generate a fairly inaccurate instrument of engine efficiency, and this seems to actually match your own data. Consider figure 7, efficiency calculated from dyno readings. It does rougly match the expected V/P curve for a conventional combustion engine. And also, as expected, you have an 'efficiency island' near peak torque. However, your efficiency island is elongated from typical engine efficiency models.
Note, Fig. 7 is a 4th order curve fitting, not calculated efficiency. The curve fitting will have a tendency to reinforce this island effect. It’s somewhat worse on the 2nd order model, which is what we’re actually using now because it seems to provide acceptable results for the optimal control problem.

Quote:
 Mathmatically, this deviation in the curve should be exaggerated by under sampling and an EKF techique, and that is what we see in your data. Consider figure 6. You point out that peak matches, but that is the lowest possible hanging fruit, like finding lambda 1.0 on a UEGO sensor. The question is, what is the typical error as we move away from the easily identifiable set point?
Hmm… my initial thought is that the fact that a 17-dimension non-linear observer with asynchronous outputs (data measurements) converges to a correct value is not low-hanging fruit. In future work, it can be refined, augmented, etc… in order to get fruit higher up the tree, but, of course, on principle you are right, what good is this model if it’s telling us what we already know? I’ll try to answer that at the end.

Quote:
 Based on your included data, pretty high. Notice how much larger and elongated the efficiency island is on figure 6. The expected V/P slope is not even really visible on the chart. If we did a scatter plot of point to point deviation between the two it would appear that, aside from peak, the instrument would almost always over estimate efficiency, and in a non-trival operational envelope, extremely so.
I’m going to show my ignorance here. The people I work with do observer and optimal control research, and I am a controls engineer (See e. busvelle - Google Scholar , jp gauthier - Google Scholar , u boscain - Google Scholar), so we’re not as strong as we'd like to be in car industry experience. What is the expected V/P slope, and keeping in mind that we’re looking at total system efficiency, where should we expect to see it?

Quote:
 My third concern is that the paper never really establishes a link between the acheived measurement and the stated goal. Look at part of your first paragraph: "optimal control to make great reductions in emissions and fuel consumption" Now, we could debate rather or not your measurement of engine efficiency is accurate enough to be of value. But the more fundamental question is, is the data even really connected to either fuel consumption or emissions?
Someone once told me, “When you can say it better than anyone else, write it. When you can’t, cite it.” I think citations [6,11,14] answer your first question better than I could.

As to the second, I’m not sure I follow. Are you asking if fuel consumption data is “connected to either fuel consumption or emissions?” I think I might be missing the scope of your question.

Quote:
 Virtually everyone in the field knows that every conventional combustion engine has a fuel economy sweet spot - wide open throttle (so there is no induced vacuum reducing VE) at peak torque. But this is not an available decision for routine operation of a vehicle. In fact, most people here (myself included) go out of our way to avoid this point of operation, because it is terrible from both the perspective of emissions and *operational* fuel economy. At peak torque wide open throttle, the mixture is generally quite rich, to keep CHTs and pressure down so that destructive abnormal combustion does not occur. But this also means that the vehicle is operating way out side of the cat efficiency envelope, making emissions soar.
Interesting point about the catalytic converter. Can you give me some references, and perhaps a simplified model? This is exactly the kind of thing that we can/should include in the cost function. (More on cost functions below.)

Quote:
Quote:
 Originally Posted by Originally Posted by kubark42 It's not really that important though, as all these factors tend to average out, leaving you with good results even if you have more uncertainty than you would like.
Sorry, I missed this earlier. I would contend that that is almost wholly false. The factors tend to work additively, and many act exponentially, which is why you over estimate effiency over so much of the operational curve.
I think we’re talking about different things here. You’re interested in absolute efficiency, whereas I’m interested in the shape of the efficiency curve. I going to say we’re both right, as it pertains to our particular problems.

One of the most nagging questions for me during this project is why the efficiency is overestimated. Looking at my equations, an overestimated efficiency implies either an underestimated fuel flow (going back to the +-0.1%, I do not know if this is the case for my car or not, due to a forgetful fiancée and the fact that we refill the car every two months) or an overestimated force. Judging from the dynamometer results, we are overestimating force. Still, it’s not important if the hypothesis that the overall shape is correct and that we have a bias error holds.

Quote:
 I'm not saying this to discourage you. Just the opposite. You are attempting to inexpensively measure a very narrow definition of effiency. What I am suggesting is that you try an experiment to demonstrate rather or not your target metric is of any practical use in either the area of fuel economy or emissions.
I’m not taking it at all as discouraging. Quite the contrary! I appreciate the fresh voice and the experience.

We have real-world trials in the pipeline, hopefully in the coming weeks, but it will be months before we can present the data. Taking your comments into account, I think a new visit to the dynamometer test bench is in order, this time to look at emissions as a function of optimal control.

Quote:
 Good Luck! -jjf
A sincere thanks for your comments, and I hope you will find the time to continue the conversation. In summation, I would like to reiterate the goal of our project, and how this fits in.

The problem comes down to this: people want to drive optimally, but do not know how. Optimal control studies are performed, but they are always constrained to working hand-in-hand with manufacturers, making publishing data difficult at best. Furthermore, the data they publish is for an ideal engine in ideal test conditions. The real-world is far different, most of all because we’re not looking at the engine as a single unit, but as a part of the whole. The transmission, differential, etc… must be included, too. And that’s even before we start talking about differences in manufacturing and wear. Our observer is a method of overcoming these obstacles on a case-by-case basis without investing in hundreds of thousands or millions of dollars of equipment.

So how does it all fit together?

One of the great weakness of optimal control is defining the cost. At this moment, our cost is only expressed in terms of gasoline used. Of course we can use far less gasoline if we run in extremely lean conditions, but then there’s too much NOx production. So it would be interesting to know what else could be included in the cost. I welcome any and all suggestions, keeping in mind that for the purposes of making a complete synthesis we need to be able to idealize.

Optimal control results are heavily dependent on the solution method. One of the most common, dynamic programming, is one of the worst when it comes to accuracy. Dynamic programming is easy, you hardly need to think about the math at all. However, it takes tremendous computing costs and is completely impractical for a small microcontroller implanted in a car. NLP (non-linear programming) comes out a little better, but still needs a fair amount of processing power.

For optimal control to work in cars, we need a way to get an answer without throwing a supercomputer at the problem. By fitting the observed efficiency map to a polynomial, we are capable of finding analytic solutions, instead of being forced to find only numeric solutions.

Our goal is not to provide a bullet-proof efficiency solution, that will have to be left to people who have better access to the sensors, and better models. We hope that by publishing this observer, researchers will be able to focus on the more interesting parts of optimal control by saying, “the solution to the problem of mapping efficiency has already been demonstrated, so now let’s get on to the fun stuff.”

So in the end, once we find the analytic solution for a given polynomial, finding the best-fit constants can become more important. For the moment, it suffices to show that our approach finds a polynomial that is sufficiently good. After that, if it is really worth it, we, or others, can spend time refining the model. After all, that’s what happened with the MPGuino: it launched a method that was lacking in certain refinement, and my project was to take it a step further. I’m certain there’s far more to be done, and I thank you for the encouragement.

03-04-2010, 02:27 PM   #30 (permalink)
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Sorry, I'm going to be terse and may miss some issues, I'm trying to help a plant get to rich burning LP engines the size of houses within the upcoming emissions compliance today because we're anticipating a lot of rain.

Quote:
 Originally Posted by kubark42 it seems that that is the accuracy that people on this forum are getting.
Think about that. 1. They calibrate measured to actual with a fudge factor. This covers a number of variables, including driver behavior. 2. They are measuring their results with fuel fillups. Are you certain that they are filling their stock tank systems at existing gas stations to the level of accuracy you are asserting?

Quote:
 Originally Posted by kubark42 Furthermore, Luxembourg is home to Delphi’s fuel injection research center (although they might have others). I spoke with one of my colleagues who researches direct gasoline injections, and he felt that this level of repeatability sounds reasonable.
Then he probably does not work on the fluid dynamics side. I'd strongly recommend that you read some of the SAE research papers in this area. Or, just try an external, and more precisely controllable, tank feeding the conventional one and measure for yourself.

Remember, if you can't measure it, it isn't science!

Quote:
 Originally Posted by kubark42 However, this is a trivial problem, as by simply adding a voltage sensor to each injector, instead of only one, the problem is solved.
No. Look at the trend in efficiency over the last 3 decades. A big key is higher compression. This means that peak fuel delivery has to high, which makes operating injectors fast enough at idle speeds difficult. There are several technologies in use, but look at peak-hold injectors, which are expressly meant to address this problem.

Now look at modern control methods of peak-hold injectors, which include techniques like PWM. In PWM control, open and close times are conditionally varied - that is, they cannot be simply averaged out, they need to be measured and modeled continuously. This makes for a considerably more demanding measurement system and computations. More sophisticated device, multipled by every injector... It seems contrary to your basic assertion. Add the current trend for muliple injectors and even duel fuels, and it sees dead end.

Quote:
 Originally Posted by kubark42 A bigger problem is what to do with diesels. Most cars sold over here are diesels, and this technique does NOT work for them.
But there are relatively low cost methods that do. For example, wideband UEGO measurement is becoming mroe common in modern vehicles. If you combine good lambda measurement with, say, MAF, you have volume of air and combustion ratio, so you know the volume of fuel burned.

Quote:
 Originally Posted by kubark42 Hmm… my initial thought is that the fact that a 17-dimension non-linear observer with asynchronous outputs (data measurements) converges to a correct value is not low-hanging fruit.
Then you may be thinking about the wrong end of the problem. Look at your simplified model and consider the influence of torque. Now look at your instrumentation, it is all, generally speaking, at best accuracy at peak torque.

Now look at the torque curve and the weighting of your data predictor. You have the least error in your sample prediction and your calculation at peak torque.

On the flip side, forget the physics, math, and sensing and look at the data! Ultimately, you have to explain why your chart has the lowest error, essentially zero, at peak torque, but huge amounts of error elsewhere!

If the cause of error is not understood and established, then there is no reason to presume that incrememtal improvement can occur. In other words, we can't assume it is a matter of better math. It could easily be the precision and limits of your underlying measurements, or even a flaw in your foundational models and premises. In science, everything is on the table until the data is explained and the results replicated.

Quote:
 Originally Posted by kubark42 What is the expected V/P slope, and keeping in mind that we’re looking at total system efficiency, where should we expect to see it?
Actually, you are not looking at total system efficiency, you are looking at a fairly narrow area of system efficiency, post drive train (but while ignoring many real world events)

As far as the V/P curve and slope, that is too big a question to properly answer here. I'd recommend starting with basic texts in engine design. But, in super brief, think of a single cyl in an engine. Mechancially, we have a constantly changing volume (pistone up, piston down). In combustion, we are creating additional pressure which also follows a curve (picture the flame front radiating from the ignition source while generating an envelope of gases).

If we could somehow create a perfect engine, plotting these two on a two axis graph, we would basically have a repeating rectangle. But chemistry and simle mechanics do not allow this, so we get a deflated and slighly twisted football (American). We do different things to try to draw portions closer to an ideal rectangle, like turbulence in the mix giving faster burns at higher RPMs, but, ultimately, the optimum point of peak pressure to occur is fixed, literally built into the cyl. So all engines have a fairly small peak efficiency island.

Quote:
 Originally Posted by kubark42 As to the second, I’m not sure I follow. Are you asking if fuel consumption data is “connected to either fuel consumption or emissions?” I think I might be missing the scope of your question.
No, I'm saying that your model appears to be too simple to be of much use in actually predicting operational economy. Consider a seemingly simple question, why do people here hate driving in winter?

Simple, fuel economy is worse. But why?

There are actually multiple reasons, even the fuel composition is different, but let's look at just two, emissions and aircharge.

The effiency of a cat is very narrow, both in terms of gas composition and in terms of required exhaust temp. So a modern automobile runs, as much as possible, at lambda 1.0 (actually, the vehicles are closed loop to equivelency ratio, but we often talk about the reciprocal, lambda). This gives both peak EGT, and a cat friendly gas composition.

Now, it is winter, and the air charge entering the engine is denser, so it takes more fuel to reach the same stoichiometric ratio. Simply by virtue of air being dense, you have to burn more fuel at even the lightest loads to keep emissions systems operating.

If you are a small plane pilot, you love cooler denser air, because you are taking off and generally climbing at wide open throttle. That denser air means you climb faster (for multiple reasons, some more important than others, but a big one is the plane is probably normally aspirated and the density altitude is lower).

If you are piloting an big car, it is another story. Like the plane, your peak performance is improved, but you aren't using peak performance, you are driving the same speed as always. How do you do this? You throttle back.

What does a throttle create behind it? Vacuum. So the VE of the engine is lower. That is, to obey the rules of the road, the engine is operated farther from peak efficiency...

And this is just the tip of the iceberg. The point of all this is that I doubt that a typical user here would be able to, say, save any fuel using your efficiency measurement, because your model simplifications mean that you inherently are least accurate in the parts of the opeartional envelope where they have no choice but to operate.

I also doubt that it is much use on the vehicle development side. If you make a criteria that a car only has to drive on surface streets, you can make it more fuel effient because you can put a smaller engine operating at higher effiency at relatively low speeds. Once the vehicle goes on a highway, you have to carry more weight to protect the occupants, and you need a bigger engine to propel the increased mass at higher speeds. Inherently, when that vehicle is operated at lower speeds, you will, by nec. be farther from the efficiency island of the engine.

The directions we have been going are smaller, turbo charged, higher compression engines, and you are not really compatible with those fuel delivery technologies - and making certain forms of very ineffient operation more efficient, and you are, again, least accurate in these parts of the operational envelope with your measurements and calcuations.

I'd be very happy to proved wrong, but you will need to do some experimentation and collect some data to do so!

Good Luck
-jjf

P.S. I mean it, I short changed a lot of important subjects above, but I didn't want to let it sit for days, since your paper is still relatively fresh in my mind today.

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