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

Saturday, May 26, 2007

Understanding a new engine

I mentioned in my last post that I'm trying to understand the factors that make it impossible for me to achieve the fuel savings in my Land Rover LR3 that I did in my Jeep Grand Cherokee Limited. I'm beginning to think my understanding of the internal combustion engine is sadly lacking.

For example, in the post linked above, I tabulated some of the relevant numbers for each vehicle. The LR3 uses a smaller engine to produce a higher rated horsepower than the Jeep. In the highest gear (at freeway speeds) the engine rpm is lower as well so one would think that the smaller engine turning more slowly would burn less fuel and hence derive heat to perform work at a slower rate. I know that the LR3 has a higher compression ratio. Could this be the explanation?

It's well known that the maximum theoretical efficiency, E, of an engine using an idealized Otto cycle is E=1-r^(1-y) where r is the compression ration and y (should be be the Greek letter gamma) is the ratio of the constant pressure to constant volume heat capacities. For the LR3 with its 10.5:1 compression ratio, this works out to 0.61. For the Grand Cherokee Limited at 9.3:1 it is 0.59. So what does this mean?

It means that for a given amount of heat from burning fossil fuels, the LR3 engine would be able to do ((0.61-0.59)/0.59)*100%=3.4% more work per joule of heat from burning gasoline than the Jeep if they were both working at the maximum theoretical efficiency. In order to see what how this affects our consumption, let's see how much heat per second from burning fuel is available to each engine.

In one second, at 1750 rpm in the Jeep, the engine moves 68.5 liters ((1750/2)/60) * 4.7 liters of fuel/air mixture through the engine. (The division of 1750 rpm by two is necessary because the rpm readout is crankshaft rpm, the crankshaft in a four stroke engine revolves twice for each engine cycle). The LR3 at 1660 rpm will move 60.9 liters per second. In these mixtures will be fuel, and I will assume that the richness of the mixture is the same for each engine, since the ECS (engine control system) will try to maintain the so-called "stoichiometric" ratio (14.7:1 by air mass to fuel mass). This mystifies me because the LR3 should burn less fuel since it's moving a smaller volume of fuel/air mixture through the engine at a presumed identical mixture. Yet the Grand Cherokee indicates instant mileage of approximately 31 m.p.g, the LR3 shows about 21.5.

The density of air at typical temperatures, pressures, and relative humidities is about 1.16 kilograms/meter^3 or 0.00116 kilograms/liter. This density is reduced in the intake manifold due to throttling effects, in fact, that's how the throttle works. I have equipped my LR3 with a Scan Gauge II so that I can measure absolute manifold pressure. At a steady 55 m.p.h. on level ground, manifold pressure is 68% of ambient. Since density is proportional to pressure, the ambient density of 0.00116 kilograms/liter is reduced in the cylinders to 7.84*10^(-4) kilograms/liter. We'll assume that the mixture is air as an approximation, since it's about 94% air in reality. Therefore, in one second, the LR3 moves 0.0477 kilograms of mixture through the engine and in that fluid, there should be (1/15.7)*0.0477=0.00304 kg. of gasoline. Burning this gasoline will release about 143,000 joules of heat energy. Hats off to me, this is great information. There's only one problem.

Since a gallon of gasoline weighs about 2.65 kilograms, this implies that I'm burning (0.00304*3600)/2.65=4.13 gallons/hour. At 55 m.p.h., this is about 13.3 m.p.g. The LR3 is no economy car but it isn't as bad as that. As I stated earlier, I expect about 21.5 m.p.g. at 55 m.p.h. on the freeway. Clearly, something is wrong in my assumptions. I'm not sure what it is, the mass flow calculation seems pretty straightforward.

Saturday, May 19, 2007


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.

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


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.