If that makes perfect sense to you, something is seriously wrong not only with you ability to discern the difference between fuel efficiency (percentage) and fuel economy (mileage), but your common sense is seriously lacking as well. Do you have any idea how much power would be required to run your car at 234mph? You do understand that if the fuel efficiency is fixed, and you require twice as much horsepower to travel at some speed above 78mph, the amount of fuel consumed per second doubles? If your speed does not also AT LEAST double along with that doubling in fuel consumption, you're mileage will DECREASE even though your fuel efficiency has remained the same.
As Chuck and others pointed out, the fuel efficiency is also tied to the load (the amount of power-torque you're demanding from the engine), and will increase as you open the throttle provided you're gearing the car to remain at the same engine speed.
Let's take a practical example here to analyze this further. Hopefully you will begin to see the difference between fuel efficiency (percentage) and fuel economy (mileage) and get a glimpse into how they are really related.
First of all, let's define "fuel conversion efficiency," which is exactly the same thing as "engine efficiency," which is what is being talked about at the web site you referred to on the Otto Cycle.
(of about 900, no kidding, this is covered at the beginning of chapter 2):
"The fuel energy supplied which can be released by combustion is given by the mbutt of fuel supplied to the engine per cycle times the heating value of the fuel. The heating value of a fuel, Qhv, defines its energy content. Is is determined in a standardized test procedure in which a known mbutt of fuel is fully burned with air, and the thermal energy released by the combustion process is absorbed by a calorimeter as the combustion products cool down to their original temperature.
This measure of an engine's "efficiency," which will be called the fuel conversion efficiency nf, is given by:
nf = Wc-mf Qhv = (Pnr-N)-(mdto nr-N)Qhv = P mdot Qhv))
Where mf is the mbutt of fuel inducted per cycle. Subsbreastution for P-mdotf from the equation gives:
nf = 1 (sfc*Qhv)"
----
Ok, that's the end of the quote. It goes on to show that a typical heating value for gasoline is 44MJ-kg or 19,000 Btu-lbm. In fact, later on in the book he actually calculates the heating value of several fuels from the chemical equations for them, but we don't need to go there.
Here's what all this means, Dave: If you burn 1kg of gasoline, you will release 44MJ of energy. The faster you burn that fuel, the more power you can get. I.e., if you burn two kg of gasoline every second you can get twice as much power as if you only burned one. And of course, like most other things in science, there is an equation that calculates this:
Horsepower = MJ-sec * 1341.02209
This equation is derived from the very definitions of power and energy, so it is as true as 2 + 2 = 4.
Using the equation, how much potential power could be harnessed if you burned 1kg of gasoline per second?
Horsepower = MJ-sec * 1341.02209 Horsepower = 44 * 1341.02209 Horsepower = 59,004
Wow, that's a lot of horsepower, isn't it? Gasoline is some pretty powerful stuff. 1Kg of gasoline is a lot of gas though. 1 gallon of gas is 2.64kg, so to get 59,000 horsepower in a 100% efficient engine you'd need to burn a gallon of gasoline about every three seconds .
These numbers are pretty huge so let's look at a bit more practical example that applies more to typical cars under freeway cruising conditions. Another equation from Heywood's book is:
HP = ((Cr*Wv + 0.0025*Cd*Av*MPH^2) * MPH) 375
Where: Cr = Coefficient of rolling resistance (between 0.012 and 0.015 typically) Wv = Weight of the vehicle Cd = Coefficient of drag Av = Frontal area of the vehicle MPH = miles per hour.
HP = the amount of power required to maintain whatever MPH you pop into that equation given the other stuff above.
buttuming we have a 3500lb car with a 24 sq ft frontal area, a coefficient of drag of 0.38, and a rolling resistance coefficient of 0.15, here's the amount of power required to travel at speeds from 20 to 100 mph. I'm buttuming the drivetrain is 85% efficient, so if you do the calculations yourself, divide the above HP number by 0.85. This way, we're seeing the amount of power at the flywheel instead of the driving wheels so later on our fuel calculations are more accurate. Also, I threw in values for 200mph and 300mph just for fun:
20 mph 3.9 hp 40 mph 11.2 hp 60 mph 25.3 hp 80 mph 49.8 hp 100 mph 88.0 hp 200 mph 605.2 hp 300 mph 1980.7 hp
We'll refer to these values later. Just for the sake of interest, here is the amount of power required for the rolling resistance due to tire deformation (the 0.015 coefficient of rolling resistance) as well as the amount required to overcome pure aerodynamic drag ("wind" below):
20 mph 3.3 hp rolling 0.6 hp wind 40 mph 6.6 hp rolling 4.6 hp wind 60 mph 9.9 hp rolling 15.5 hp wind 80 mph 13.2 hp rolling 36.6 hp wind 100 mph 16.5 hp rolling 71.5 hp wind 200 mph 32.9 hp rolling 572.2 hp wind 300 mph 49.4 hp rolling 1931.3 hp wind
Ever wonder what percentage of drag is due to tires versus rolling resistance? Here are the values for this car:
20 mph 85.2% rolling 14.8% wind 40 mph 59.0% rolling 41.0% wind 60 mph 39.0% rolling 61.0% wind 80 mph 26.5% rolling 73.5% wind 100 mph 18.7% rolling 81.3% wind 200 mph 5.4% rolling 94.6% wind 300 mph 2.5% rolling 97.5% wind
Just for interests sake, note that at 60mph 39% of the drag is due to the tires. That's pretty significant. If you changed tire pressure or tires to lower the coefficient of rolling resistance to 0.012, here's what happens at 60-100mph:
60 mph 23.4 hp 80 mph 47.2 hp 100 mph 84.7 hp
60 mph 33.8% rolling 66.2% wind 80 mph 22.3% rolling 77.7% wind 100 mph 15.6% rolling 84.4% wind
How much of an improvement is this?
60 mph 8.1% 80 mph 5.5% 100 mph 3.9%
Those percentages very roughly indicate the improvement in fuel economy (mileage, not "fuel efficiency", which will actually decrease!) with the lower drag tires. But not quite... As Chuck pointed out, these values would be at different throttle settings and influence the fuel efficiency. Here, you'd be closing the throttles slightly to maintain the slightly lower power output, and most likely in doing so you would would LOWER "fuel efficiency" and simulataneously RAISE FUEL ECONOMY. Did you catch that? You're getting less horsepower per gallon of gas (less fuel efficiency), but at the same time you require less power to maintain these speeds so are most likely getting an increase in MPG. Fuel efficiency went down, and fuel economy went up.
Anyway, because of the throttling effect Chuck talked about, the real percentage in fuel economy (miles per gallon, not "fuel efficiency") gained is going to be quite a lot lower than the numbers above, but at least it gives a ballpark figure. And again, I can't stress enough that fuel efficiency went one direction while fuel economy went the other. You and I while driving our cars around don't really care about fuel efficiency, which is a percentage, at all. We care how much it costs to get from point A to point B. That's fuel economy, which is miles per gallon. As I and several others said before, they are not the same thing. They can move in opposite directions.
Ok, moving right along to the subject of the current debate. Let's see if we can get some rough estimations of fuel mileage. To really do this accurately we would need BSFC numbers at different loads (which accounts for the change in throttle position) and different engine speeds (not vehicle speeds), as Chuck and others have correctly stated several times already. Just for the heck of it, we'll go ahead and try some estimations and see if anything interesting pops up.
Ok, Dave, what we're going to do is buttume that we're following your rules here. We will gear the car so that the engine is always operating at peak "fuel conversion efficiency" (same as the "engine efficiency" you refer to, per the Heywood definition). Then we will attempt to analyze what effect this might roughly have on fuel economy, to see if indeed fuel economy is maximized when the engine is running at the speed at which maximum fuel conversion efficiency exists at part throttle (the 40% of redline bit).
In other words, are fuel efficiency and fuel economy the same thing in the end? Let's see if we can find out.
Let's stick to the 60mph, 80mph, and 100mph speeds because that's a hot spot with the the speed limit debate. Personally, I like high speed limits and like to drive fast on the freeway, but at the same time I think a valid argument for raising speed limits can not be found in an analysis on fuel economy.
100mph in our example vehicle:
60 mph 25.3 hp 80 mph 49.8 hp 100 mph 88.0 hp
Again, we will buttume the car is geared so the engine is operating at 40-45% of the redline where our imaginary engine is producing its best "fuel efficiency" (not economy).
For now, let's pick a fuel efficiency for our imaginary engine and see what happens if it does not change with throttle position. Later, we will use an actual example from a real engine where efficiency changes with throttle position to examine what happens more closely. Dave, from the web site you provided a link to:
"Argonne Labs measured the efficiency of the Japanese Prius engine to be 34% (good for any engine at its peak) at only 13.5 hp."
That's extremely good compared to anything anyone here is probably driving. I'd prefer to use something that is more indicative of what we're used to, so I'll refer to page 723 of "Internal Combustion Engine Fundamentals" where a graph of mechanical efficiency versus percent load can be found for one particular engine. Mechanical efficiency is the ratio of actual power output to what the power output would be without the losses due to friction, i.e., throttle position. In other words, this shows the pumping loss due to throttling. The fuel efficiency (not economy) should scale up and down right along with the graph as the throttle position changes pretty closely. So, we'll buttume that the full throttle fuel efficiency is 34% (rather than part throttle like the Prius is), and scale that down by the mechanical efficiency in the graph to get our "part throttle fuel efficiency."
And again, we're running at 40% of redline here so we are getting the maximum fuel efficiency at this load that we can possibly get, per your insistance.
Before we get the fuel efficiencies at part throttle, we need to find the percent load (the graph shows mechanical efficiency as a function of percent load). Percent load is the percentage of the total available engine power at whatever RPM we're looking at. Let's buttume your engine produces 200 hp at its peak, up near the redline. At the same time, we gear the car to keep the engine speed at 40% of the redline where we'll have the peak part throttle efficiency. We'll buttume our engine has a pretty flat torque curve at part throttle in this mid range and produces a maximum of only 130 hp at these speeds. This is slightly above 1-2 maximum power output at peak rpm and is a reasonable approximation for the purposes of this discussion. Actually, that might be a bit high, which will result in lower MPG calculations later on, but to heck it with it, let's do it anyway.
To summarize, our engine at full throttle at 40% of redline can produce 130 hp. We now need to know the percent loads throughout our speed range. To find this we can simply divide the following horsepower numbers into 130 hp:
60 mph 25.3 hp 80 mph 49.8 hp 100 mph 88.0 hp
The resulting "percent loads" are:
60mph 19.5% 80mph 38.3% 100mph 67.7%
That means that at 80mph, you are asking the engine to provide 38.3% of the total power it is capable of producing in the mid rpm range (130hp).
Looking at the graph, mechanical efficiency (the ratio of actual power output to what the power output would be without the losses due to friction, i.e., throttling losses primarily) varies from 45% to 80% within the range we're interested in. For starters we'll buttume the mechanical efficiency is constant, right smack in the middle of that at 60%. Our best fuel efficiency is 34%, and thanks to all of our throttling, we've reduced that by 60% to 20.4%.
Our "fuel efficiency" at part load in these conditions is 20.4%.
Now, how can we find our "fuel economy"? We are not getting 20.4 "percent" miles per gallon. That makes as much sense as saying I stepped on a scale today and it told me I weighed 150 seconds or degrees.
The heating value of our fuel is 44MJ-kg. Since there's 2.63kg of gas in a gallon, we can say our heating value is 115.72MJ-gallon. We converted MJ per second into horsepower, so let's write an equation that goes the other way and converts horsepower into MJ per second instead.
Horsepower = MJ-sec * 1341.02209
Therefore:
MJ-sec = Horsepower 1341.02209
Now, taking our horsepower numbers from before and using that last equation, let's calculate how many MJ-sec we need our fuel to produce to keep us running along at these speeds:
60 mph 25.3 hp = 0.01887 MJ-sec 80 mph 49.8 hp = 0.03714 MJ-sec 100 mph 88.0 hp = 0.06562 MJ-sec
How many gallons of fuel must we burn per second to produce this? Easy. Just divide the above MJ-sec numbers into 115.72, which is the heating value of our fuel in MJ-gallon.
60 mph 25.3 hp = 0.000163 gallons per second 80 mph 49.8 hp = 0.000321 gallons per second 100 mph 88.0 hp = 0.000567 gallons per second
Now, Dave, remember how you said a lot of that heat goes straight out the exhaust because even the Otto Cycle engines are horribly inefficient? You're right. If we had 100% efficient engines we would only need to burn the amount of fuel shown above to maintain those speeds in our example car. However, as we determined before, our fuel efficiency is 20.4% in these conditions. Therefore, we need to scale these values up by a factor of 1 0.204 to find the real fuel consumption rates.
Results:
60 mph 25.3 hp = 0.000799 gallons per second 80 mph 49.8 hp = 0.001574 gallons per second 100 mph 88.0 hp = 0.002779 gallons per second
Or (multiplying by 3600 seconds in an hour):
60 mph 25.3 hp = 2.8764 gallons per hour 80 mph 49.8 hp = 5.6664 gallons per hour 100 mph 88.0 hp = 10.0044 gallons per hour
That is how much fuel we must burn every hour in order to maintain those speeds, including the effects of "fuel efficiency," which is at our peak in ALL OF THE ABOVE SITUATIONS. I.e., we have geared the car to keep the engine running at 40% of redline for 60, 80, and 100mph.
fuel every hour than we will at 60mph. However, obviously we are not going 3.47 times further every hour (100mph is not 3.47 times 60mph). Therefore, "fuel economy" (miles per gallon) has decreased substantially, even though "fuel efficiency" remained constant. The only way you are going to get more fuel economy (miles per gallon) at 100 mph than 60 mph is if your fuel efficiency increased, due to a change in throttle position which produces a lower pumping loss (less energy wasted rather than something gained as you mentioned earlier), by a whole bunch. We'll get to that in a minute using the part throttle data from Heywood's book to see if that's what indeed happens as you say.
Now... BURN THIS INTO YOUR SKULL... The "fuel efficiency" is exactly the same at all of the above speeds. It's 20.4%. The "fuel economy" is "gallons per mile," which is different. To get the fuel economy we can just divide our speed by the fuel consumption rate:
60 mph 60-2.8764 = 20.9 MPG (miles per gallon) 80 mph 80-5.6664 = 14.1 MPG (miles per gallon) 100 mph 100-10.0044 = 10.0 MPG (miles per gallon)
See that, Dave? Even though "fuel efficiency" stayed the same, the "fuel economy" changed dramatically as speed was increased.
Now, the astute reader will notice that the mileage in these calculations dropped substantially more with increasing speed than it does in reality. The reason for this was covered already by Chuck. Regardless of whether or not we regear the car to operate at 40% of the redline where the peak fuel efficiency is maximized for each of the above speeds, the fact is you must open the throttle more. Obviously at any given engine rpm you must open the throttle more to produce 88hp versus 49.8hp or 25.3hp. This is where we get back to page 723 of the book.
In reality, as Chuck and the web site you referred to pointed out, the efficiency increases as the throttle is opened. The mechanical efficiency is a function of percent load. We calculated the load percentages already. Here they are again:
60mph 19.5% 80mph 38.3% 100mph 67.7%
Here are the mechanical efficiencies for an engine at all of those load percentages. I'm not making these numbers up, they are measurements from a real engine that was tested throughout the entire throttle range at mid range engine speed; the speeds in question:
60mph 50% 80mph 70% 100mph 85%
Now, our fuel efficiency scales up and down right along with these mechanical efficiences. Our peak, full throttle fuel efficiency is 34% at this "ideal 40% of redline." Therefore, here are our fuel efficiencies with throttling effects included at the different engines speeds, including the effects of changes in gear ratio:
60mph 17.0% 80mph 23.8% 100mph 28.9%
The fuel efficiency increased with speed here because we are opening the throttle more, which is reducing engine friction (primarily throttling work loss). Dave, this equates to the comment you made about how when you open the throttle, you are "wasting less energy" rather than "gaining more." The end result is you're getting more power per unit fuel being burned, so whether you're "wasting less" or "gaining more" is irrelevant. The end result is the same. You get more POWER per gallon. But do you get more MILES per gallon? That's a different question.
To find out, let's go back to our 100% efficient engine where we found out how many gallons of fuel we burned per hour (not per mile):
60 mph 25.3 hp = 0.000163 gallons per second 80 mph 49.8 hp = 0.000321 gallons per second 100 mph 88.0 hp = 0.000567 gallons per second
Scaling this up to gallons per hour (multiply by 3600 seconds in an hour):
60 mph 25.3 hp = 0.5868 gallons per hour 80 mph 49.8 hp = 1.1556 gallons per hour 100 mph 88.0 hp = 2.0412 gallons per hour
Ok, now, to compensate for the fact that our fuel efficiency is changing due to changing throttle position, we will use the data we derived from the real data on the part throttle engine. All of those fuel efficiency values (the percentages) can now be divided into the above fuel consumption rates, which are buttuming a 100% efficient engine. Here we're doing the same calculation before where we divided everything by 20.4% by buttuming the fuel efficiency didn't change with throttle position, but instead using the different fuel efficiencies at different throttle positions that now compensate for throttle position.
Fuel efficiences (again, these are percentages and not miles per gallon): 60mph 17.0% 80mph 23.8% 100mph 28.9%
Actual fuel consumption rates including variations in throttle position: 60 mph 3.4518 gallons per hour 80 mph 4.8554 gallons per hour 100 mph 7.0630 gallons per hour
Now, again, "fuel economy" is something entirely different, and we can now calculate what the trend in fuel economy variation with speed will be. As before, all we need to do is divide the speed by the consumption rate. Here are the results:
60 mph 17.38 MPG (miles per gallon) 80 mph 16.47 MPG (miles per gallon) 100 mph 14.16 MPG (miles per gallon)
The mileages here are lower than you're used to seeing, probably because of a mismatch between the engine data graph and the hypothetical engine in this example, as well as the rather large 24 sq foot frontal area, fairly high drag coefficient, powerful engine, and rather inefficient drivetrain (85% at part throttle), but what's important here is that even when you take into account all of these effects, the "fuel economy" is decreasing with increasing speed, while the "fuel efficiency" is increasing very sharply because of the throttling changes. The variation in "fuel efficiency" with ENGINE SPEED is very different from the variation in "fuel economy" with VEHICLE SPEED.
FUEL ECONOMY is DECREASING while FUEL EFFICIENCY is INCREASING as vehicle speed increases and engine speed is held constant, due primarily to the throttling effect both you and Chuck have referred to.
-------------- In summary:
Dave, as you've been told by virtually everyone in this thread, "fuel efficiency" and "fuel economy" are not the same thing. Through the use of gearing, placing the ENGINE rpm at "peak fuel efficiency" (40-50% of redline in the Otto Cycle, or any percentage of redline in any engine or cycle at all) does not also place the VEHICLE and engine in combination at "peak fuel economy." In fact, as can be seen above, the two variables move in opposite directions at part throttle at constant speed as that speed is altered.
If you continue to argue this point, you deserve to be horribly abused as a blatantly ignorant moron like Laura Bush end whatever. I shall be forced to taunt you incessantly as a raging idiot from this day forwards if you do so. ;-)
Furthermore, if you bother to continue to argue without reading and giving an honest attempt to understand my analysis, which I've now spent a significant amount of time on, I will be personally offended to a level I care not to describe. In addition, your failure or success will be a public exhibition of your level of intelligence. Please don't disappoint us. If there is something you don't understand, ask. But it would be idiotic in the extreme for you to continue to tell me I'm wrong at this point because of your damn 40% scientific fact, which does not refer to fuel economy, but rather fuel efficiency, which both move in opposite directions anyway in the case we're debating as just illustrated.
"Peak FUEL EFFICIENCY is at 40% of redline?" Yes, that's probably about right. I don't argue with that.
"Peak FUEL ECONOMY (mileage) is at 40% of redline?" Utter nonsense. If it is in your case, it's a coincidence.
I must note on the side that it is refreshing to see that nobody else required such a lengthy explanation. Their faculties of common sense and unbiased experience were all that were required to come to the same conclusions I have through a reasonably detailed analysis. Hats off to you guys. Good to see some smart folks around this group.
Todd Wbutton Performance Simulations
