LTD temperature using PV = nRT

[Tom Booth]

Using this online calculator:

https://www.calculatorsoup.com/calculat ... as-law.php

I first calculated the number of moles in 44.35 cm3 at atmospheric pressure (101.325 kPa) using an engine inside working fluid temperature of 190°F (hot cup of water)

The results being 0.014975 moles (rounded)

And 44.35 cm3 at an ambient temperature of 70°F

Result 0.0018367 (rounded)

Assuming the number of moles inside the engine remains static I then used the actual PV readings of an LTD from this diagram.

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After spending most of the day plotting temperatures and connecting the dots, I ended up with these "isotherms" superimposed over the PV diagram.

Resize_20230522_031849_9086.jpg (175.66 KiB) Viewed 2155 times

There is approximately a 10° ∆T between Tmin and T max


Rather surprisingly to me the lowest temperature reached by the working fluid appears to be in the return "compression" stroke while the working fluid is still just very slightly below atmospheric pressure (horizontal line)

The temperature appears to drop just 1/10th of one degree below the presumed source heat input temperature (189.98°F)

After the pressure increases above atmospheric pressure, the temperature increases about 1.5° at TDC (full compression, minimum volume 191.56°F)

From TDC pressure and temperature rise the pressure peaking at about 102.1 kPa the temperature risen to about 195°F

From there the temperature continues to rise, peaking at about 200°F. The pressure simultaneously dropping.

Needless to say, I don't see how these values could be correct as the temperature, according to the ideal gas law has risen about 10° above the hot input temperature from a starting point of 190° up to nearly 200° then begins to drop in temperature prior to BDC.

The Ideal gas law, however does not take into account cooling by expansion and/or work output.

Logically, according to the ideal gas law, the more heat a gas takes in the more it expands. So reciprocally, the more it has expanded the hotter it must be, however, it is not logical that the gas in a Stirling engine could heat up, above the source temperature while expanding

Another puzzle is that from BDC as presumably, atmospheric pressure is driving the piston inward, although the working fluid temperature is falling, it stays well above ambient, and also above the heat source temperature almost all the way back until the pressure approaches atmospheric pressure

This seems to make more sense if we assume a source temperature above 200°F. (Say, Boiling water)

My selected starting point and temperature of 190°F at atmospheric pressure lower left where the horizontal 1atm line is crossed were arbitrary.

If a higher source temperature is assumed, then the temperature changes are more or less what would be expected given a 90° displacer advance.

Then at the upper right (hottest temperature) is where the displacer shifts over to expose the sink so temperature and pressure drop.

At the lower left (coldest temperature) is about where the displacer would move back exposing the heat source.

So then how does atmospheric pressure drive the piston inward after BDC if the temperature of the working fluid is still much higher than ambient?

I would suppose, though atmospheric temperature is lower there are more molecules more densely packed.

The working fluid has been heated and expanded and so much of the air has leaked out past the piston, so that the working fluid is "thin" with fewer molecules/cm3

The "weight" of the atmosphere therefore exerts more force on the piston from the outside than the rather thin feeble hot gas inside the engine.

[Tom Booth]

Going back to what is apparently the original source for the PV diagram

https://people.ok.ubc.ca/jbobowsk/Stirl ... tation.pdf

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The actual temperatures for Tc and Th are provided being:

Tc = 24°C (75°F, 297°K)
Th = 95°C (203°F, 368°K)

So my guestimates are a little off.

So,...

Apparently the external, applied ∆T is 128°F but the internal high and low temperatures according to the ideal gas law, taken from the diagram about 10°F ???

When I have the time to spend doing this again, I may recalculate the temperatures starting with one or the other of these actual temperatures, but I don't think the results will be much different.

I suspect the mere 10° internal temperature working fluid temperature change is due to much less than perfect heat transfer at the heating/cooling plates together with the relatively high RPM there is insufficient time for heat transfer from the plates to the working fluid.

I think there is a great deal of room for improved heat transfer using VincentG's extended dwell time mechanism.

[VincentG]

The phasing issues of an unmodified LTD I think makes it extremely difficult to analyze what is happening in real time. There is most definitely a large force available during the hot stroke(Tmax at Vmin), but the only way to achieve a cold return stroke(and have internal pressure > external pressure) is to achieve Tmin at Vmax. The challenge is making an engine that can achieve both. As is these engines are so out of phase there are really no distinct events to speak of.

Edit: Unmodified, these engines do an excellent job of balancing these power strokes. This just comes at the expense of a poor utilization of available delta temperature.

[VincentG]

Something that jumps out at me reading the pdf, are they calculating work done(using cycle time as 7.5hz) as a 2 cycle engine? Because a stirling engine is most definitely a 1 cycle engine.

They say work is the flow of heat from hot sink to cold sink. But to my eyes they are not accounting for flow of heat from hot sink to gas, and then from gas to cold sink.

Isn't the magic of these engines in the molecular activity of gas in relation to small(relatively) increases in temperature? Its like getting the phase change power of water to steam with a fraction of the energy input.

[Tom Booth]

I haven't actually read the PDF, just hunted it down from another source where this PV diagram appeared there was a footnote reference to the above PDF.

"They say work is the flow of heat from hot sink to cold sink"

LOL... Don't get me started.

I'll just say: Conservation of energy. Heat can't be converted to external "work" while also "flowing" out to the sink.

[VincentG]

Beyond that, the engine is unloaded. That in and of itself is enough to drastically change the results. Not to mention the large increase in cold space volume caused by the valve assembly.

Sorry for the tangent. Nice job plotting the temperature out.

[Tom Booth]

I don't mind tangents,

I'm not sure it is a nice job, don't really know if it's actually even possible. Kind of an experiment but thought I'd post the results anyway.

In some ways, I doubt the results. I don't think PV = nRT was intended for this kind of analysis where mechanical expansion and contraction of a gas is involved, or I may be applying it all wrong in some other way, but I really don't know.