Yesterday I saw someone joking online about how if you apply dimensional analysis to fuel efficiency, you end up with an area. Why? Because fuel efficiency is expressed (in Canada and Europe, anyway) as liters per 100 kilometers. The liter is a unit of volume, or length^{3}. The kilometer is a unit of length. If you divide length^{3} by length, you end up with length^{2}, or area. (Similar reasoning applies to American-style fuel efficiency expressed as miles per gallon.)

I tried this with a concrete example. Suppose your car consumes 10 liters per 100 kilometers:

10 L * (1000 cm^{3}/L) = 10,000 cm^{3}

100 km * (1000 m/km) * (100 cm/m) = 10,000,000 cm

10 L per 100 km is therefore equivalent to 10,000 cm^{3}/10,000,000 cm, or 0.001 cm^{2}

Weirdly, this means that the fuel efficiency of your car is 0.001 cm^{2}, or 0.1 mm^{2}.

At first this struck me as bizarre at best, and an abuse of dimensional analysis at worst, but the more I thought about it, the more sense it made. It’s actually quite intuitive if you look at it a certain way.

Imagine a long tube of fuel stretched out along the surface of the road. Now suppose that your car doesn’t have a fuel tank, but is instead equipped with some kind of scoop that gathers fuel from this tube as your car travels down the road. How large should this tube be? In order for your car to continue traveling down the road, the tube must have a minimum cross-sectional area of — you guessed it — 0.1 mm^{2}. Cool, huh?

To confirm this, simply calculate the quantity of fuel contained in the tube, assuming the tube is 100 km long. The volume of the tube is equal to the cross-sectional area times the length. We already know the value of both of these quantities expressed in terms of centimeters. The volume is thus equal to 0.001 cm^{2} * 10,000,000 cm, or 10,000 cm^{3}, which is equal to 10 L. QED.

I’d appreciate a page break.

Yes, that would be good. I can’t edit it, now that it’s been posted. Could you insert a page break after the first paragraph?

For completeness, here’s how you would do it for American miles per gallon:

Suppose your car gets 20 miles per gallon.

20 miles * (5,280 ft/mile) * (12 in/ft) = 1,267,200 in

1 gallon = 231 in^3 (approximately)

20 mpg is therefore equal to 1,267,200 in/231 in^3

Taking the reciprocal to get gallons per mile, we have

231 in^3/1,267,200 in, for an answer of about 0.00018 in^2

So the fuel efficiency is 0.00018 square inches.

As a sanity check, that converts to around 0.12 mm^2, which is in the same ballpark as the fuel efficiency of the car in the OP.

keiths,I’ll try

Thanks.

Nitpick (well maybe not). What keiths’s OP is referring to is fuel consumption rather than fuel efficiency. Internal combustion engines only manage to extract 20 to 40 (tops) percent of the stored energy in oil-based fuels.

And what about carbon footprints? How long does it take to offset the manufacturing element in a new electric car as opposed to carefully maintaining and continuing to use an old IC vehicle?

Alan:

It is indeed a nitpick, and it’s also incorrect. Here’s the very first result of googling “fuel efficiency”:

Heehee. You are priceless.

Alan,

You are

terribleat bluffing. Care to tell us what was ‘priceless’ about that?Look, perhaps this will help. I’ll make some mistakes, you can jump on them, and then maybe you’ll feel better. Let’s get this out of your system.

Goe fore itt.

Not going to indulge you, there. A discussion on the inefficient use of the word “efficiency” would be an inefficient use of my time.

On the other hand, if anyone is also having to consider options regarding vehicle use and climate change, I’d like to hear their views, ideas, and experience.

keiths:

Alan:

Bluff failed.

Ever heard of things called graphs? Not particularly weird.

Nowadays it’s called data visualisation where you can express anything as any shape you deem convenient.

Having trouble solving an unpredictable issue with engine cutting out which my local garage couldn’t solve, I armed myself with an ODB (onboard diagnostics reader) and a suitable app on my phone. As well as fault diagnosis, it outputs various parameters in real time that display selected sensor outputs as line graphs or histograms etc. parameters such as intake pressure, throttle angle, etc.

(Yes, I found the fault)

Erik:

Are you seriously equating “can be graphed” with “has units of area”? Ask yourself: Would it be possible to plot (for example) diurnal temperature variation on a graph, even though degrees aren’t units of area? I hope you can see that the answer is yes.

What’s weird about fuel efficiency isn’t that it can be plotted on a graph. What’s weird is that it can be expressed in units of area.

(Sorry, folks, I can’t help myself)

What units are used to express fuel efficiency, I wonder?

ETA.

Fuel economy

Alan,

Your point?

keiths,Well, apart from suggesting that efficiency, economy and consumption are not interchangeable when looking at ICE vehicles, I’m wondering what a fuel gauge indicates, other than the level of fuel remaining in a fuel tank.

Alan:

I’m still not seeing your point.

That fuel consumption can be inferred from a fuel gauge? That fuel efficiency and fuel economy cannot?

Alan,

Haha. You’re still fixated on

that?I already responded.I know. You’re still using words sloppily. It’s no big deal. The daft thing is you want to argue to the death over it.

Anyway, I’m interested in what I should be doing to economize on my motoring costs as well as minimise my carbon use. But it seems no-one else is or there are better places to seek and share helpful info.

Alan:

Says the guy who keeps bringing it up. I laid it to rest yesterday.

Rather than asking rhetorical questions about fuel gauges, why not jump on the spelling mistakes I offered you yesterday?

Good grief.

Anyway, that article I linked to seems to suggests your thoughts on “instantaneous” fuel consumption being represented as an area are not original.

And it should be trivially obvious. Fuel is flowing into the injection system at a certain rate at any one time interval. Reduce that time interval and the fuel consumed becomes a shorter and shorter cylinder of cross-section of the supply pipe. The limit is a two-dimensional disc at zero time interval.

Alan:

Jesus, Alan. We reached the same conclusion, therefore I must have copied from him? How many intuitive interpretations of fuel efficiency as area do you think there are? I can’t think of any, other than the one I presented.

Also, where did “instantaneous fuel consumption” come from? I was talking about

averagefuelefficiency, not instantaneous fuel consumption. When the sticker says “25 mpg”, they are talking about average fuel efficiency.Lol. Yes, there’s no doubt that you took one look at the OP title and thought “that isn’t weird; it’s obvious!” You just decided not to mention that until now.

No, it isn’t. Over a given interval, the flow rate

varies. Even if you assume an ideal situation with a perfectly flat road and no wind, the flow rate will still vary over time simply because you won’t be holding a constant speed.Go out to your car. Pop the hood. Measure the cross-sectional area of the line feeding the fuel injection system. Then do the reverse of the calculation I performed in the OP. Do you really think the result will be equal to your car’s gas mileage? The diameter of the line is

fixed. It does not vary, even though fuel efficiencydoesvary over time. Your scenario completely misses the mark.Alan, your obsession with catching me in errors is

nothealthy. I think you should get it looked at.In the meantime, go after those spelling mistakes. It will make you feel better.

The flow rate varies depending on demand from the injectors. This can be modelled by imagining the pipe expanding and contracting while fuel rate of flow is constant. Consumption, btw, not “efficiency”.

Alan:

First, your statement contradicts itself. In the first sentence you say the flow rate varies, but in the second one you say that it’s constant. To salvage your statement, you would need to change your second sentence by replacing “fuel rate of flow” with something like “linear speed of fuel flow”.

That aside, let’s stipulate that your imaginary fuel line expands and contracts like a boa constrictor. That

stilldoesn’t make your scenario work.To see why, imagine taking two identical copies of your fuel injection system and installing them in two cars. Car A gets better gas mileage than car B. Decide on a target flow rate, and then drive each car so that the flow rate matches the target value. Since the flow rates are equal, the boa constrictor diameters will also be equal.

Is the fuel efficiency the same? No. Over a given interval, car A will travel farther than car B. Yet the same amount of fuel is consumed, and the cross-sectional area of the boa constrictors is the same.

The cross-sectional area doesn’t correlate with fuel efficiency, so your scenario fails.

Context:

in a real situation, a petrol engine with electronic management system, the fuel is maintained in the common rail (at 6 bar from memory in the car I’m familiar with) by the fuel pump with excess returning to the tank. The injectors (and the software controlling them) transfer a variable amount of fuel to the cylinders. So consumption of fuel and flow through the injectors varies with demand.

But modelling the consumption by imagining constant flow through a pipe, valve or orifice of varying cross section (as happened with carburettor needle valves) achieves the same variable fuel delivery.

Of course, how rate of flow, pressure and resistance are connected to volume delivered in reality is another rich isue.

I’m not talking about efficiency. I’m talking about consumption.

Alan:

Haha. OK, let’s bypass your weird terminological fixation and simply talk about miles per gallon rather than ‘fuel efficiency’ or ‘fuel consumption’.

keiths:

Alan:

Context doesn’t rescue your statement. If the flow rate varies, it isn’t constant. If it’s constant, it doesn’t vary. You contradicted yourself.

Take my advice. In the second sentence, replace “fuel rate of flow” with something like “linear speed of fuel flow”. Your argument will still fail, but at least you won’t be contradicting yourself within the space of two sentences.

I still don’t think you grasp

whyyour argument fails, but do you at least understand that itdoesfail?Here’s how you can tell. If in my scenario you specify the minimum cross-sectional area of the tube of fuel, I can give you the miles per gallon. If I specify the cross-sectional area of your boa constrictor fuel line, you

can’tgive me the miles per gallon. Not even if I hand you a graph of boa constrictor area vs time.Your scenario simply doesn’t work.

Alan,

Here’s a simple explanation of why your scenario fails. Take a specific car with a specific mpg number. Drive it at a specific speed, so that the fuel flow rate is constant.

Let R be the constant flow rate

Let A be the cross-sectional area of the line feeding your fuel injection system*

Let S be the linear speed of the fuel flow

The flow rate is equal to the cross-sectional area times the linear speed:

R = AS

R is fixed, but A is not. For any particular (nonzero) value of A, the equation can be satisfied by the choice of an appropriate value for S. If the fuel line is narrow, the linear speed has to be high in order to achieve the specified flow rate. If the fuel line is wide, the linear speed can be lower and still achieve the specified flow rate.

The miles per gallon is the same regardless of the value of A. Therefore you cannot determine mpg from A, or A from mpg, so your scenario fails.

* It doesn’t matter whether or not it’s a boa constrictor line, because we have specified that the flow rate is a constant.

Alan,

For completeness, here’s why my scenario succeeds where yours fails:

Let the car travel at a constant speed, just as we specified for your scenario. The fuel flow rate is therefore also constant.

Let R be the constant flow rate

Let A be the cross-sectional area of the tube of fuel

Let S be the speed of the car

The equation

R = AS

still applies, just as it did in your scenario. However — and this is the key point — S is fixed. That’s because in my scenario, S is the speed of the car, and we’ve already specified a constant speed so that the flow rate will be constant. In your scenario, S is

notfixed. It can vary inversely with A.R and S are fixed in my scenario, which means that there is only one value of A that will satisfy the equation.

Thatis what allows me to determine mpg from area, and area from mpg.Think of it this way: In both scenarios, there is effectively a tube of fuel entering the fuel injection system. In my scenario, that tube of fuel enters the system with a speed that is equal to the speed of the car. Why? Because the tube of fuel is fixed in place. It isn’t moving with respect to the ground. In your scenario, the tube of fuel can enter the system with a variety of speeds. The boa constrictor just has to expand or contract appropriately in order to keep the flow rate constant.

In your scenario, the fuel’s speed can vary, meaning that the cross-sectional area can vary. The mpg figure remains constant, however. Thus you have no way to determine mpg from area, or area from mpg.

In my scenario, mpg determines area, and area determines mpg. They are two ways of expressing the same thing.

Let me summarize.

I say divide volume by length and you get area.

Anyone disagree?

Instead of “has”, I’m talking about “can be expressed as”, which is in your title.

I don’t think it is weird. Actually I would expect fuel efficiency to map to space somehow. I expect the same of carbon footprint.

🧐

So my car can achieve a sustained 100 km per hour while burning petrol at a rate of 5 litres per hour and emitting water vapour and carbon dioxide as waste products and converting the chemical energy to a turning force at the point of contact on the road (which is straight autoroute and devoid of traffic). The turning force is equal to the total of friction and air resistance which would otherwise bring the car to a halt.

I could simulate things if I had a rolling road and anchored the car to a fixing point with a strain gauge. I have a brake on the rolling road that I can vary to simulate the friction and air resistance. I use my speedometer and computer readout so my “instantaneous” fuel consumption is 5 litres per 100 km [at a steady speed of 100 kph] I measure the force produced at the wheel by reading the strain gauge.

Do I now have enough information to calculate the thermal efficiency of the car, its economic efficiency and its carbon footprint?

Anyone can answer.

Supplementary information if needed. The car is a 23 year old VW Golf. It is virtually valueless in market value. It has worn valve guide seals resulting in engine oil consumption of 1 litre per 1,000 kilometres. It will be due for its third timing belt change in 10,000 km.

My other car isn’t a Golf.

And whilst I am using numbers (round figures, I might say colloquially), one should not assume they are exact measurements.

Just for context, 😉

keiths:

Erik:

OK, but I still don’t see what that has to do with graphability. You wrote:

Could you elaborate? Surely you’re not saying that if something is graphable, it therefore isn’t weird, right? How do you get from “graphs aren’t weird” to “fuel efficiency expressed as an area isn’t weird”?

keiths:

Erik:

I would too, given that volume and distance are both spatial concepts. I wouldn’t expect to mash those together and get kilograms as a result, or degrees Fahrenheit. But the fact that miles per gallon can be expressed as an area

doessurprise me.Tell me honestly: Suppose that last week, before you had seen this OP, a friend of yours had told you excitedly about her new car: “It gets great gas mileage. 0.05 square millimeters!” Wouldn’t that have struck you as a bit… weird?

I’m willing to bet that it would have.

Sitting here enjoying my mid-morning coffee, so:

Instead of my usual fuel tank, I substitute a 100,000 metres-long clear plastic tube. What cross section does it need to be if I want to drive 100km and I know my rate of fuel consumption will be around 5 litres over that 100km. Converting all to the same units, that’s 0.005 cu metres of fuel in a tube 100,000 mètres long. So the cross section needs to be .005 divided by 100,000. Let’s switch back to millimetres, so multiplying by 1,000,000, we get a cross-section of 0.5 [square] millimetres, unless my mental arithmetic is as poor as I know it to be.

ETA thanks for correction, keiths.

But what about my carbon footprint? Should I look at an electric vehicle? No question I should look at a photovoltaic panel installation. I have room for 9KVA on a south-facing section. The sun shines a lot here and there is a buy-back scheme. With a home-charging point, what’s the downside.

Manufacturing costs of the vehicle and panels in carbon emissions, rare metal extraction and the unconceived alternative?

Alan:

0.5 millimeters is a length, not an area.

Also, there’s a shortcut. You can simply scale the calculation in the OP to get your answer. 10 liters per 100 km in the OP corresponded to 0.1 square millimeters, so 5 liters per 100 km in your example corresponds to 0.05 square millimeters.

Correct. Though the fact I multiplied by 1,000,000 might have suggested it was a typo.

keiths:

Alan:

The answer you gave was 0.5 millimeters. The correct answer is 0.05 square millimeters.

You got the number wrong, and you got the units wrong. Your answer was completely wrong. Accept it and move on.

Oh, I admit my mistake. My mental arithmetic is abysmal.

I should have fetched a pen and paper.

Using pen and paper

But what does represent? If we un-divide and return to over we see represents the volume of fuel consumed in one millimetre of travel.