Theoretical horsepower

[NateDN10]

I set about to calculate how much power an engine could theoretically produce, assuming the combustion chambers were completely filled with air/fuel mixture, and the air was at atmospheric pressure. I was also interested in what the volumetric air:fuel ratio was (we know the mass ratio is 14.7:1). Here I go...

(engine displacement) = (volume of air) + (volume of fuel)
(volume of air) = (mass of air)/(density of air)
(volume of fuel) = (mass of fuel)/(density of fuel)

substituting these in:
(engine displacement) = (mass of air)/(density of air) + (mass of fuel)/(density of fuel)

substitute in the stoichiometric ratio: (mass of air) = 14.7 * (mass of fuel)
(engine displacement) = (14.7 * mass of fuel)/(density of air) + (mass of fuel)/(density of fuel)

approximate density of gasoline: 0.75kg/L
approximate density of air @ 0C: 0.001294kg/L
let's assume a 2.0L engine

2.0L = (14.7 * mass of fuel)/0.001294kg/L + (mass of fuel)/0.75kg/L

Since dividing by x*kg/L is the same thing as multiplying by 1/x * L/kg,
2.0L = 11360L/kg * (mass of fuel) + 1.333L/kg * (mass of fuel)

Here we can see the volumetric AFR is 11360:1.333, or 8520:1.
Solving for (mass of fuel):
(mass of fuel) = 1.7603*10^-4 kg

Gasoline has 47.2MJ/kg, so this mass of fuel contains 8.31KJ.
Since this amount of fuel would be injected once per revolution, if we assume the engine is spinning at 6000 RPM (100 revolutions per second), it's liberating 831KJ/s, which is equivalent to 1,114 horsepower.

If we assume, the engine is 25% efficient at converting chemical energy to motion, that's still 279hp at the crank.

This is where my question comes in. That's a lot of horsepower! Even if you multiply by 75% to account for volumetric efficiency, that's 209hp. Did I do something wrong, or am I making an incorrect assumption?

Thanks for reading this!

 

[abycat]

I think your air fuel mixture will be alot thicker under full power and your engine might be a tad too efficient but id say thats fairly close. too bad most of any engines efficiancy is blown out the tailpipe or dissipated through the cooling system. If you turbocharged that engine and ran it on methonal it would be pretty crazy.

 

[sasha]

that's close considering an average 2.0l N/A produces 160=210hp.

 

[Scott_Tucker]

There is a formula used to determine that. Look up Brake Mean Effective Pressure.

 

[mahansm]

You have forgotten one important fact about your engine. It's a 4 cycle engine and each cylinder only gets fed fuel on every other revolution. That puts an extra factor of .5 into the power number.

Good cam/valve/intake/exhaust design can approach or even exceed 100% volumetric efficiency in certain speed ranges.

30-40% is a pretty good (above average) range for thermal efficiency of a piston engine.

You have to consider all the losses in the engine system; thermal into the piston/head/cylinder, blowdown (release of pressure on exhaust valve open), all friction and viscous losses (bearings, cam, seals, etc..) and parasitic (water pump, alternator, fuel pump (if mechanical)), etc..

A more traditional look at engine power starts with PLAN where

P Brake mean effective Pressure
L Length of stroke
A Piston Area
N crankshaft speed

This allows one to compare engines of various displacements and RPM ranges on a more even basis.

Pressure for a normally aspirated gasoline fueled internal combustion engine will generally be in the 160 PSI range. Higher than this leads to a very narrow torque band (intake/cam/exhaust tuning) and may be acceptable for racing but is not pleasant for general use.

This is also a good way to figure out how to make the horsepower number you desire. You can increase displacement (bore/stroke/cylinder count), RPM, or BMEP (different fuel/supercharge).

So far the best anyone has done with gasoline fueled normally aspirated piston engine (4 stroke) is 320 BHP/liter. 23,000 RPM, though (no, it's not a typo. 23,000 RPM at peak power, valve float (redline/serious damage) at 24,000).

 

[mechtech2]

Valve opening/closing determines the actual useable displacement of the engine.
Compression ratio is a huge actor for efficiency.
Ram tuning in the intake and exhaust can often happen,throwing out the limits that are on paper.
Full throttle fuel use is about 0.5 lb gas per HP per Hour.
On street cars [they are getting better!] figure 75% to 95% volumetric efficiency.

 

[morris]

i dont know enough say you are wrong, and i think you are right. but james watt set down what a horsepower is. reff. http://www.howstuffworks.com/horsepower.htm.

 

[Duffman77]

I roughed out some numbers to follow your example, mahansme is correct that you didnt divide by 2 for 4 stroke calculations.

I neglegted the volume of fuel in my calculations for simplity, your number of 831 kJ/s was 827.5 kJ/s for me (so not a bad assumption) if I also forget to divide by 2.

Seems about right to me, I would use 30% thermal efficiency and 80% for more modern average performance engines.

 

[Dominic]

The highest commercially produced road car HP/L that I am aware of is the Honda S2000. 240hp from a 2.0L or 120HP/L

This is about the theoretical maximum on a street engine that's enjoyable to use. Race motors in F1 cars reach much higher limits thanks to high RPM tuning; for example F1 motors of 2.4L are capable of 740hp which is 308HP/L -- but they make it at 19,000RPM.

Anything over 100HP/L is fantastic in the normally aspirated world.

 

[SteveSRT8]

http://www.caranddriver.com/features/2012-10best-highest-specific-output-engines-features


Some newer data about specific output

 

[Shannow]

Originally Posted By: morris
i dont know enough say you are wrong, and i think you are right. but james watt set down what a horsepower is. reff. http://www.howstuffworks.com/horsepower.htm.


So how much air and gas are fed into said conceptual horse ?

 

[Shannow]

Problem with the "bottom line" is assuming 25% efficiency. It's a circular argument, assuming a number which is an approximation on a calculated number in the first place.

Sadi Carnot defined the maximum efficiency of a thermal engine (they all are thermal), when operating between an upper and a lower temperature limit, and the "carnot efficiency" is the efficiency that is less than Carnot's limit.

The very first part of the equation, requires that an engine full of air first has to be compressed to whatever the compression ratio (plus the work due to heat from the cylinder/chamber is), which takes energy.

Consider a 2hp air compressor, and how much energy that you actually get out of the compressed air in the tank.

Then there's the pumping exhaust, and sucking the inlet...two "wasted" strokes.

To the OP, you've really grasped a key fundamental, and something I haven't seen on this topic on BITOG, for very many thousand posts, an actual first principals approach to an engineering fundamental, usually covered in 2nd year thermodynamics...

Grab points are
* Carnot efficiency
* Otto Cycle
* bmep (4 and 2 stoke)

Then mess with
* Pumping loops
* Inertial/cam timing effects.

 

[NateDN10]

Thanks, all! I forgot about dividing by two for a 4-cycle engine. Adding that to the equation and massaging the efficiency factors gets you to some very believable power numbers.

And it also makes you realize that there must be some pretty cool engineering going on in normally aspirated engines making over 100hp/L.

 

[mechtech2]

Note the high HP/ci for motorcycles and race cars.
How?
Beaucoup RPM!

 

[sciphi]

I don't know enough about the theoretical side. Sounds about right, considering there are some 2.0 liter 4 cylinders right around 200 hp naturally aspirated.

 

[Cujet]

Another way to look at HP is through BSFC. Or simply the weight of the fuel consumed, per HP, per hour.

For example, my Lycoming IO-360 air cooled aircraft engine makes a certified 200HP at 2700RPM (standard temp/pressure) at 0.55 BSFC. Or, roughly 1/2 pound of fuel per hour for every HP produced.

What's interesting is that the engine can achieve much better BSFC numbers in cruise flight, with a peak efficiency of 0.38 BSFC. (better than a prius engine)

 

[Shannow]

All goes back to the fundamentals.

An engine can really be modelled from first principals, rather than coming backwards from empirical rules of thumb.

Sir Harry Ricardo was achieving BSFC of 0.45lb/hphr in the 20s, running kerosene, and 6:1 compression ratios...amazing.

 

[Cujet]

A bit off topic, but this is one reason why I don't like CAFE rules. Engines have not become significantly more efficient.

 

[Shannow]

I get continually slammed for saying it, but the 1948 holden seated 5, travelled well above the speed limit, and got over 30MPG, as does the 2012 Commondore.

 

[Duffman77]

Originally Posted By: Cujet
A bit off topic, but this is one reason why I don't like CAFE rules. Engines have not become significantly more efficient.


Actually they have gotten vastly more efficient we just expect so much more out of them. If people would accept a family sedan with only 120 hp and it only weighed 3000 lbs you would get a sizable bump in MPG. Also the emissions levels are orders of magnitudes less for the same power outputs.