Blogs Car Info Our Show Deals Mechanics Files Vehicle Donation

"square/cube rule" and engine efficiency

In the thread about EV efficiency…http://act…, a “prime mover” engine was mentioned. 10 cyl, 8’ stroke, 50% efficiency.

This made me think of something called the “square/cube rule.” From flying, it references what happens when a flying design is scaled up (let’s say doubled in all dimensions): weight goes up per volume (3D, so 2X2X2=8), where lift is dependent on wing area (2D, so 2X2=4).

From this, it follows that: (1) small objects are easier to “fly” than big ones, and (2) for a given level of engine and structural tech, there exists a maximum size for a flyable object–any bigger and it can’t get off the ground!

Which leads up to a very large engine having very high efficiency. I wonder if there isn’t a “square/cube rule” in effect here: power is dependent on cylinder volume (3D), whereas parasitic losses are dependent on (among other things) the interaction of piston to cylinder wall (2D).


1. Is engine efficiency generally increasing per increasing cyl displacement?

2. Is such gain due to a “cube/square law”?

3. Is there a theoretical engine “too small to idle,” thus ruling out my dreams of a gasoline-powered wristwatch?

Yes, yes, and sort of.

The job of an engine is to convert heat into energy. Every BTU of heat that escapes to the combustion chamber walls only to be removed by the cooling system is a BTU of heat that the engine will never be able to convert into mechanical energy.
Doubling all the dimensions of a cylinder increases its volume by eightfold yet only quadruples its surface area so there is less surface area for the heat to escape to while the engine expands the hot gases in the cylinder.
In addition to large cylinders, the surface area of the piston crown and combustion chamber is further minimized by making the engine highly undersquare.
For this reason, great big industrial engines are inherently more efficient than tiny little engines.

As far as too small to idle, all I can say is that it’s pretty hard to get a throttled Cox .049 cubic inch model airplane engine to idle reliably below 5000 rpm even when burning fuel that’s 30% nitromethane to keep the glow plug lit.

Scale effect is also why a model airplane engine can easily rev 20,000 rpm while a car engine can’t. Why hummingbirds can flap their wings 70 times a second and a bald eagle can’t. Why a St. Bernard pants in 50 degree weather and a Chihuahua shivers if the temperature drops below 75.

I think technolgy can attenuate these type of rules, in paticular the rule of bigger displacement giving better efficiency. With the cars I used to work with BMW went to a 5.0l engine for 500hp with better mileage than the high cubic inch/high horsepower engines of the 70’s. I try to keep an automotive slant on these type of questions.

Bigger is better for efficiency, but there’s more - that huge ship engine has other systems to increase efficiency, like exhaust heat extraction and electricity generation. I think they figure all those in the 50% efficiency. As for ‘too small’, they now make miniature four-cycle engines for small planes (this one’s 0.40 cu. in.).

Increasing the size of an engine cylinder to decrease the internal surface area to volume ratio certainly will reduce heat loss. On the other hand, very large, very efficient diesel engines run very slowly which will permit more time for heat loss to the cylinder walls to take place. This is contrary to efficiency but possibly not contrary enough to defeat the area-volume ratio advantage or is there more going on that has not been mentioned to increase efficiency by increasing size?

You can decrease idle RPM by providing more flywheel mass. My old two cylinder BMW motorcycle had a huge flywheel which made it so I could reduce the idle speed to where I could count the firing pulses.

“The job of an engine is to convert heat into energy.” Wrong. In an internal combustion engine, heat is wasted energy.

In an internal combustion engine, heat is wasted energy.

Not exactly. I think I understand what you are trying to say, and you are correct, but it does not conflict with B.L.E.'s outstanding response.

Not stated, but implied in most of the responses is the balance of engine loading.  I believe most responses have assumed a the larger engine is assumed to be outputting more power.  Clearly if an application called for the power produced by a model airplane engine, they are not going to compare that to a locomotive engine producing the same output as the model airplane engine. :-)

The only reason the pressure in the cylinder is higher after combustion than before is because the air in the cylinder is hotter. The number of molecules of stuff in the cylinder is actually lower after combustion so once the gases cool back to room temperature, they actually take up less space than they did before.

Let’s burn a molecule of methane as an example. One molecule of methane CH4 needs three molecules of oxygen O2 to burn into one molecule of CO2 and two molecules of H2O. We had four molecules of gas before the burn and after the burn we only have three molecules of gas.

Years ago, (1974) I took an internal engines course (as part of the auto mechanics school I was at). At the end of that course, we compared an “049” model plane engine to an ocean liner’s engine (where the pistons were large enough to park a VW on and piston travel was 20-30 feet).

While I do forget many of the details, I do remember the two engines were strikingly similar in their Brake Specific Fuel Consumption, Brake Horsepower, and efficiency. The point our instructor was trying to make was you can take a given type of engine (gas, diesel) and its characteristics remained similar, regardless of size.

Somewhere buried up in our attic I have my notes and equations that we used to derive those results that day.

I think piston travel on those marine diesels is in the under 10 ft range, just a detail,nothing more.

You are correct. It was the size of the piston and connecting rod that were several stories high, not the stroke. Thanks for that clarification.

“The job of an engine is to convert heat into energy.”

The job of an engine is to convert heat energy into mechanical (kinetic) energy.

How’s that?

True for methane, not true for gasoline, with its longer-chained molecues. Take octane as as example, many more molecules after than before.

I appreciate your reference to the square/cube rule, but like all theories, taking them to extremes without regard to other factors as everyone has been alluding to makes little sense. The shear interaction with a common fuel when motors change size would in my opinion too as illustrated, be extremely important. That includes o2 as well.
I’ll limit my expertise to “sails” as motors and I can tell you there are many important factors other than size where efficiency is concerned.

Feh! you call that small? I got a O.S. .26 cubic inch four stroke in a little model of a Piper J3 Cub. And, even that’s gigantic compared to this.

Yes, burn a pound of gasoline and you get about 20,000 BTU of heat. A BTU of heat would be 778 ft-lb of energy if the engine could convert all of the heat into energy. Every BTU of heat that escapes to the cooling system is a BTU of heat that the engine failed to convert into 778 ft-lb of energy. Every BTU of heat that comes out of the exhaust system is a BTU of heat that the engine failed to convert into 778 ft-lb of energy.
The gasoline engine is actually a member of a larger group of machines called heat engines.
Other heat engines include the Stirling cycle, Brayton cycle, Rankine cycle, and the theoretically most efficient but not really practical Carnot cycle.

Do yourself a favor and look up ‘heat engines’ on Google.

I bow to your four stroke minitudiness!

The job of an engine is to convert heat energy into mechanical (kinetic) energy.

How’s that?

That would be wrong.

That would be wrong.