# Fuel consumption as an area

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Hi folks. I've not posted in a while but thought of this site when a topic cropped up elsewhere. Apologies if it has already cropped up here but I didn't find anything with a quick search.

It is standard in Europe to quote fuel consumption for a liquid-hydrocarbon-powered vehicle in volume of fuel burnt per distance covered. A small car may achieve 5 litres per 100 kilometres, for example.

But if we convert both volume and distance to the same unit, millimetres, we get 5,000,000 cubic millilitres over 100,000,000 millimetres. This gives us an area of 0.05 square millimetres.

With me so far?

Edited by Arthur Smith
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46 minutes ago, Arthur Smith said:

Hi folks. I've not posted in a while but thought of this site when a topic cropped up elsewhere. Apologies if it has already cropped up here but I didn't find anything with a quick search.

It is standard in Europe to quote fuel consumption for a liquid-hydrocarbon-powered vehicle in volume of fuel burnt per distance covered. A small car may achieve 5 litres per 100 kilometres, for example.

But if we convert both volume and distance to the same unit, millimetres, we get 5,000,000 cubic millilitres over 100,000,000 millimetres. This gives us an area of 0.05 square millimetres.

With me so far?

Yes I think so. What insight does this division provide?

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58 minutes ago, Arthur Smith said:

With me so far?

So far, it's obvious.

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What is the cross section of a pipe filled with fuel, that you consume by moving along it instead of moving the fuel through the pipe?

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As I feared, the question is trivial. If you wanted to establish instantaneous consumption, you could simply reduce the distance of travel from one millimetre (which consumes 0.05 cubic millimetres of fuel) to zero. Which is not possible.

Though I guess you could model instantaneous fuel consumption by keeping flow to the engine constant and reporting the required cross section of supply pipe. Foot off, downhill and cross-section is zero.

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Why use such an arbitrarily large unit as millimeters? You can use nanometers, picometers, or even better, calculate how far a device will travel per molecule of fuel..

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39 minutes ago, Sensei said:

Why use such an arbitrarily large unit as millimeters? You can use nanometers, picometers, or even better, calculate how far a device will travel per molecule of fuel..

Which sort of molecule?

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1 hour ago, Arthur Smith said:

As I feared, the question is trivial. If you wanted to establish instantaneous consumption, you could simply reduce the distance of travel from one millimetre (which consumes 0.05 cubic millimetres of fuel) to zero. Which is not possible.

Though I guess you could model instantaneous fuel consumption by keeping flow to the engine constant and reporting the required cross section of supply pipe. Foot off, downhill and cross-section is zero.

Instantaneous consumption is given by the rate of consumption, isn't it?  The units of which are volume per unit distance, sure, but it does not seem to me to add anything to express this as an area, even though dimensionally that is what it is.

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Well, I guess a histogram would do it. Start with a graph of horizontal scale 100 km and rectangles of one millimetre. Height of each will be proportional to the fuel used over the distance interval. Play with scales and you have a representation of fuel consumption over distance as the area under a line following the height of the rectangles. Why do I feel I'm differentiating?

2 hours ago, md65536 said:

What is the cross section of a pipe filled with fuel, that you consume by moving along it instead of moving the fuel through the pipe?

Sorry, didn't spot these replies as I just noticed a reply flagged on activity and it took me straight to exchemist's comment.

I'm guessing "what" is a typo for "that", and I agree.

1 hour ago, Sensei said:

Why use such an arbitrarily large unit as millimeters? You can use nanometers, picometers, or even better, calculate how far a device will travel per molecule of fuel..

Is there a limit? Are we in Zeno's paradox?

1 hour ago, John Cuthber said:

Which sort of molecule?

The european molecule 😉

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2 hours ago, John Cuthber said:

Which sort of molecule?

That's yet another question. The easiest way to answer it is when someone uses a biofuel: bioethanol.

When someone uses the word "fuel" it is too ambiguous. 5L/100km, liters of what?

1 hour ago, Arthur Smith said:

Is there a limit? Are we in Zeno's paradox?

You can't burn less than one molecule. But it can be only partially oxidized (i.e., releasing only part of its potential energy)..

3 hours ago, Sensei said:

or even better, calculate how far a device will travel per molecule of fuel..

As an exercise, try doing it with ethanol ("biofuel").

Edited by Sensei
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4 hours ago, Arthur Smith said:

Hi folks. I've not posted in a while but thought of this site when a topic cropped up elsewhere. Apologies if it has already cropped up here but I didn't find anything with a quick search.

It is standard in Europe to quote fuel consumption for a liquid-hydrocarbon-powered vehicle in volume of fuel burnt per distance covered. A small car may achieve 5 litres per 100 kilometres, for example.

But if we convert both volume and distance to the same unit, millimetres, we get 5,000,000 cubic millilitres over 100,000,000 millimetres. This gives us an area of 0.05 square millimetres.

With me so far?

Yes, but why would you do this? What insight does it provide?

You don’t always simplify units. Torque and energy are both N-m in metric units, and N-m is a joule, but it’s not considered proper to express torque in joules, because it’s not energy.

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Another example, Hubble parameter in cosmology. Its units are speed per distance, which is 1/time, which is units of frequency. It does not make the Hubble parameter a frequency of anything. Even if one expresses it in Hz.

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40 minutes ago, swansont said:

Yes, but why would you do this? What insight does it provide?

You don’t always simplify units. Torque and energy are both N-m in metric units, and N-m is a joule, but it’s not considered proper to express torque in joules, because it’s not energy.

Having come across the "problem" elsewhere, I asked to confirm my take that it is a pointless exercise. Thanks for confirming

I chose "brainteasers" presuming it to be not for serious issues.

1 hour ago, Sensei said:

That's yet another question. The easiest way to answer it is when someone uses a biofuel: bioethanol.

When someone uses the word "fuel" it is too ambiguous. 5L/100km, liters of what?

The question is the same for any fuel. I have a car that I run on petrol/gasoline with 10% "bio" ethanol added. I get, on average, about 5 litres burnt for 100 km of travel. My larger-engined turbo diesel also returns around 5L per 100km. Of course the calorific content of fuel and the thermal efficiency of internal combustion engines is a whole subject (as is carbon emissions)in itself. The more interesting question is overall fuel economy, thermal efficiency, and carbon footprint. I can bypass all of that in buying an electric vehicle. But factor in the construction of a new vehicle when comparing carbon footprints and the comparison is not so easy.

But I'm derailing my own thread...

48 minutes ago, Arthur Smith said:

Torque and energy are both N-m in metric units, and N-m is a joule, but it’s not considered proper to express torque in joules, because it’s not energy.

If I wanted to measure the actual thermal efficiency of an IC vehicle moving at a reasonable constant speed (say 100 kph) and I had access to a rolling road, and able to apply a variable braking load (that I could measure as a drag force) that I could regulate so that the "instantaneous" fuel consumption settles at 5l/100km, do I then have enough information to calculate the torque at the driving wheel, the net power output of the engine and, hence the thermal efficiency of the vehicle and fuel?

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58 minutes ago, Arthur Smith said:

If I wanted to measure the actual thermal efficiency of an IC vehicle moving at a reasonable constant speed (say 100 kph) and I had access to a rolling road, and able to apply a variable braking load (that I could measure as a drag force) that I could regulate so that the "instantaneous" fuel consumption settles at 5l/100km, do I then have enough information to calculate the torque at the driving wheel, the net power output of the engine and, hence the thermal efficiency of the vehicle and fuel?

You could measure the effect of varying braking but simply subtracting or extrapolating that effect wouldn’t give you the answer, since there is air resistance as well, so you haven’t accounted for all of the dissipative losses.

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32 minutes ago, swansont said:

You could measure the effect of varying braking but simply subtracting or extrapolating that effect wouldn’t give you the answer, since there is air resistance as well, so you haven’t accounted for all of the dissipative losses.

Ah but am I not simulating total drag force by using the output from my "instantaneous" fuel consumption gauge on the dash to set the power equivalence to that necessary to drive the car at 100 kph (and speedo to 100 kph) as I already know the car consumes an average of 5L/100  kph? If the rolling road is set to simulate that, surely it simulates total drag, including air resistance.

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2 hours ago, Arthur Smith said:

Having come across the "problem" elsewhere, I asked to confirm my take that it is a pointless exercise. Thanks for confirming

I chose "brainteasers" presuming it to be not for serious issues.

The question is the same for any fuel. I have a car that I run on petrol/gasoline with 10% "bio" ethanol added. I get, on average, about 5 litres burnt for 100 km of travel. My larger-engined turbo diesel also returns around 5L per 100km. Of course the calorific content of fuel and the thermal efficiency of internal combustion engines is a whole subject (as is carbon emissions)in itself. The more interesting question is overall fuel economy, thermal efficiency, and carbon footprint. I can bypass all of that in buying an electric vehicle. But factor in the construction of a new vehicle when comparing carbon footprints and the comparison is not so easy.

But I'm derailing my own thread...

If I wanted to measure the actual thermal efficiency of an IC vehicle moving at a reasonable constant speed (say 100 kph) and I had access to a rolling road, and able to apply a variable braking load (that I could measure as a drag force) that I could regulate so that the "instantaneous" fuel consumption settles at 5l/100km, do I then have enough information to calculate the torque at the driving wheel, the net power output of the engine and, hence the thermal efficiency of the vehicle and fuel?

Yes. If the rolling road measured the power (torque x revs) output of the engine at the driving wheels, then if you measure the rate of fuel consumption and you know the calorific value of the fuel, you can work out what proportion of the calories burnt end up as mechanical power. It will be 25-30%, I expect.

There will be some  errors due to frictional losses in the transmission, so you will slightly underestimate the output of the engine itself. In a mechanical transmission those losses are small  - <2% if I recall correctly - , but with a torque convertor they may be more significant.  There may also be some errors due to the fuel not being 100% burnt to CO2.

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3 hours ago, swansont said:

since there is air resistance as well, so you haven’t accounted for all of the dissipative losses.

Wind can also accelerate a thing, depending on the direction. Airplane pilots often use this knowledge to reduce fuel consumption during flight.

(Taking off and landing against wind direction is another story)

Edited by Sensei
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14 hours ago, Arthur Smith said:

Well, I guess a histogram would do it. Start with a graph of horizontal scale 100 km and rectangles of one millimetre. Height of each will be proportional to the fuel used over the distance interval. Play with scales and you have a representation of fuel consumption over distance as the area under a line following the height of the rectangles. Why do I feel I'm differentiating?

Because you are. Drop down one dimension and the problem is similar to finding the area under a curve. Your questions about the "instantaneous fuel consumption" might be equivalent to asking what is the area of an infinitesimally wide line under the curve. I believe it was thinking along these lines that lead Newton to develop calculus, and that if you look into Newton's reasoning, it might help make sense of the fuel consumption idea without apparent paradoxes.

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Integration, not differentiation, for the area under a curve. Sorry to have misled you.

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23 hours ago, swansont said:

Yes, but why would you do this? What insight does it provide?

I do this all the time with both my own petrol car and my brother's all electric car.

Simple question How many litres of petrol or Kwh of electricity do I need to tget to Bristol / Plymouth and back ?

Really I want to know whether to top up before I go or not.

So yes please @Arthur Smith  What are you actually trying to do  -   The answer is unclear to me.

On 10/4/2023 at 8:37 AM, Arthur Smith said:

But if we convert both volume and distance to the same unit, millimetres, we get 5,000,000 cubic millilitres over 100,000,000 millimetres. This gives us an area of 0.05 square millimetres.

With me so far?

Not so because this is incorrect maths .

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2 minutes ago, studiot said:

I do this all the time with both my own petrol car and my brother's all electric car.

Simple question How many litres of petrol or Kwh of electricity do I need to tget to Bristol / Plymouth and back ?

Really I want to know whether to top up before I go or not.

How does expressing fuel efficiency as an area do this?

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It works the other way around as well:

Area as a fuel consumption.

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46 minutes ago, swansont said:

How does expressing fuel efficiency as an area do this?

I didn't say it did.

But I also indicated that clearing up the OP misunderstandings about units in general would help bring clarity to the question.

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Z

2 hours ago, studiot said:

I do this all the time with both my own petrol car and my brother's all electric car.

Simple question How many litres of petrol or Kwh of electricity do I need to tget to Bristol / Plymouth and back ?

Really I want to know whether to top up before I go or not.

So yes please @Arthur Smith  What are you actually trying to do  -   The answer is unclear to me.

Not so because this is incorrect maths .

There is indeed a typo in my comment. I meant "cubic millimetres" where I wrote "cubic millilitres". I believe the arithmetic is correct, though.

And to clarify, I think representing fuel consumption as an area is trivial and unhelpful.

I'd be much more interested in how to compare holding on to a reasonably economic, well maintained IC car or swapping for an all-electric, taking into account all the relevant factors, including the overall carbon footprint.

Eta including the carbon cost of manufacturing an electric car and its batteries.

Edited by Arthur Smith
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45 minutes ago, Arthur Smith said:

Z

There is indeed a typo in my comment. I meant "cubic millimetres" where I wrote "cubic millilitres". I believe the arithmetic is correct, though.

And to clarify, I think representing fuel consumption as an area is trivial and unhelpful.

I'd be much more interested in how to compare holding on to a reasonably economic, well maintained IC car or swapping for an all-electric, taking into account all the relevant factors, including the overall carbon footprint.

Eta including the carbon cost of manufacturing an electric car and its batteries.

So your question has nothing to do with golliwogs per square root  of carrot or any other strange measurement unit.

Unfortunately real life is more complicated than your wish, especially as no one yet knows the 'carbon cost' of electric vehicles, which must vary according to your source of electricity.

Here are some real world figures from England.

Petrol

My car does approximately 9.3 miles per litre, pretty reliably since I do very few journeys of less than 10 miles.

Petrol in Bristol and Plymouth are usually about £0.1  per litre more expensive than where I live, so I prefer to fill up near home.

My tank holds about 35 litres which give me a range of 325 miles.

Electricity

My brother's car does about 4.3 miles per kWH but that can decrease by 50 % in worst driving conditions (winter, very wet, night time)
It should be remembered that all the auxiliary machinery in an electric car (heater, wipers, lights etc) are run from the battery which is the only energy source.

Now here comes the awkward bit.

When my brother visits he pay me £0.3 per kWH to plug into my english mains and recharge.

His car has a capacity of 64kWH  but this can only be achieved with heavy duty 3 phase industrial supplies.

Lesser supplies can only charge up to 80% that is 51 kWH.

The commercial cost of electricity varies from £about £0.5  to £0.8  per kWH. but only the most expensive will achieve a recharge time of 30 mimnutes (as compared to 3 minutes for filling my car).

Further my brothers car currently attract zero road fund licence (tax) compared to my several hundred pounds.
Also petrol is heavily taxed in England, far more so than elctricity.
Heaven only knows what it will cost to run an electric car taxwise when the chancellor has forced everyone into them.

My mains supply takes over 24 hours to fully recharge his car from flat.

So there you have it.

Some facts and fiures from the last 3 years driving experience.

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