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Intro to Otto vs Prius Atkinson Thermodynamics

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Posted by bwilson4web on May 14, 2011, 9:29 pm
Someone  asked about compression ratios and engine efficiency. At the
time, I reccommended Wiki and Mr. Google not realizing they needed to
understand what to ask for . . . thee basics. So I replied:

I'm sorry, I should have been more specific about the effect of
compression ratio on engine efficiency. So let's start with this URL:


They don't really discuss compression ratio but this is the take away:

". . .
The thermodynamic limits assume that the engine is operating in ideal
conditions: a frictionless world, ideal gases, perfect insulators, and
operation at infinite time. The real world is substantially more
complex and all the complexities reduce the efficiency. In addition,
real engines run best at specific loads and rates as described by
their power band. For example, a car cruising on a highway is usually
operating significantly below its ideal load, because the engine is
designed for the higher loads desired for rapid acceleration. The
applications of engines are used as contributed drag on the total
system reducing overall efficiency, such as wind resistance designs
for vehicles. These and many other losses result in an engine's real-
world fuel economy that is usually measured in the units of miles per
gallon (or fuel consumption in liters per 100 kilometers) for
automobiles. The miles in miles per gallon represents a meaningful
amount of work and the volume of hydrocarbon implies a standard energy

Most steel engines have a thermodynamic limit of 37%. Even when aided
with turbochargers and stock efficiency aids, [B]most engines retain
an average efficiency of about 18%-20%[/B] . . . ."

Practical, commonly sold engines have abysmal thermal dynamic
efficiencies. To be perfectly blunt, they are rubbish compared to what
they could (and should) achieve such as with the Prius Atkinson
cyycle. So lets look at two university web pages that go into details
addressing the effect of compression ratio:


Both papers show the math to derive the same efficiency formula for a
'perfect' Otto cycle engine:

efficiency = 1 - ( 1 / (r ** (k-1) ) )

r - compression ratio
k - specific heat ratio, a measure of energy in the fuel

For simplicity, here is the MIT chart:

Here is the Northwestern chart:
Note they have have different scales with the MIT showing the full
range and the Northwestern a more practical range we find today.

Now everything in these classical, Otto cycle lessons derives from the
P-v diagram:

The top curve coming from point 3 towards point 4 is the power stroke
curve and the longer it is, the more energy extracted. The x-axis is
the "v" expansion ratio. A greater compression ratio, the longer the
expansion stroke and more energy extracted. But higher compression can
lead to detonation and hammer the engine to pieces. The key to
efficiency is the longest possible expansion  ratio to extract
mechanical energy.

The Prius Atkinson cycle changes the compression stroke so the fuel-
air charge is not compressed to ignition. Itt takes the line from 1 to
2 and breaks it into two sections:

1 to 1.5 - this is a flat line as the intake valve is kept open and
part of fuel-air mix goes back into the intake manifold to be sucked
in the next cylinder.

1.5 to 2 - this is the shortened compression stroke which being
smaller, also means less compression losses as well as avoiding
detonation or knock.

Qs - the energy added is less because there is less fuel-air to burn

Qout - is the same

4 - is moved to the right about 50% further (aka., 13 to 1 expansion
versus 9 to 1 typical compression ratio.) So the Prius has a much
longer power stroke to extract more energy.

I do not like this P-v chart but it is 'close enough:'
I do not like it because the segment 3-4 implies a substantial
increase in volume without work being done and that doesn't happen. If
you stretch 4 over to combine it with 3, you'll have an accurate,
Atkinson cycle P-v diagram.

Bob Wilson

Posted by Bruce Richmond on May 14, 2011, 10:55 pm

Funny, I just used some of the same links in a discussion in


I agree the p-v chart is wrong and the normal transition can be seen
in figure 3.8 of your previous link.


The horizontal line from 3 to 4 is usually used in the cycle of a
diesel.  The rational is that in a gasoline engine the fuel burns
almost instantly giving a vertical line for the rise in pressure.
With the diesel the fuel is injected over a period of time which
mantains a constant pressure during the first part of the power
stroke.  Here is a p-v diagram for an Atkinson diesel:


Here is an article that gives some info about how the Atkinson cycle
is applied to the Prius.


I also found out that a true Atkinson cycle engine involves more than
just variable valve timing, it actually changes the lengths of the
strokes, thus avoiding the pumping losses of drawing air in and then
pumping it out before closing the intake valve.


Still the valve timing trick is a lot better than nothing.

Posted by bwilson4web on May 14, 2011, 11:23 pm
 . . . <good stuff>

Thanks! I'll share these links with some of the folks over at

Bob Wilson

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