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pkv

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I am trying to understand which thermodynamics process or processes could model a vacuum engine (mostly known by the name "flamelickers") and similar engines..

In a vacuum engine you have a cylinder and piston system that draws hot air from a heat source, cools it and mechanical work is produced from the pressure difference between the cold gas and the atmosphere.

https://en.wikipedia.org/wiki/Vacuum_engine

These are the numbers I calculated for two simple models.

Data:

Working fluid: 1 mol of N2, 1 bar (101 kpa) at 600K. Density 0.564 kg/m3. Volume 0.0493 m3

Steps:

1.- 1 mol of N2 is heated at ambient pressure to working heat.

Vf = 0.0246 ; From PV = nRT at 300K

Q = Delta Ei + W

For constant pressure:

Q = CV * n * Delta T + P * Delta V

Q = 20.8 * 300 + 101000 * 0.0246 = 8724.6 J heat added

2.- N2 is admitted into the cylinder. Piston stops after drawing 49,3 l (1 mol at 600k).

3.- Valve closes, piston stops, and water spraying cools the working fluid to 300K.

This is a constant volume process.

I model this step as an isochoric process, using equation:

Q = (Cv/R) * V * (Pf - Pi)

Result is -6240 J of heat removed from the N2.

I can obtain the same quantity just using temperature delta and CV of N2:

-300 * 20.8 = 6240 J

Pressure now is: PV = nRT -> P = (8.3144598 * 300) / 0.0493 = 50500 pascals

4.-Atmospheric pressure of 1 bar performs compression on the working fluid as piston moves to the bottom. Final pressure of the working fluid is 1 bar.

I am modelling this process for 2 different possibilities: adiabatic and isothermal.

A) Model using simple adiabatic compression.

We know the cylinder will move down until both ends have the same pressure, 1 bar.

Using the adiabatic pressure constant, we can get the final volume.

Pi * (Vi^y) = K const -> 50500 * (0.0493^1.4) = 746.925

Vf^y = K / Pf => 746.925 / 101000 = 0.00739

Vf = 0.00739 ^ (1/1.4) = 0.03 m3

Final temperature will be:

PV = nRT => T = (Pf * Vf) / R => (101000 * 0.03) / 8.3144598 = 365.8K

Work performed on the piston:

W = (K * (Vf^(1-y) - Vi^(1-y))) / (1-y)

W = (746.925 * ((0.03^(1-1.4)) - (0.0493^(1-1.4)))) / (1-1.4)

W = -1368 J

B) Model using isothermal compression.

The water spray continues as the piston compresses the working fluid.

Isothermal work W = n * R * T * log(vf / vi)

Vi = 0.0493

Vf = 0.0246 ; From PV = nRT at 300K

W = 8.3144598 * 300 * log(0.0246 / 0.0493) = -1728.9 J

Heat removed during the process should be equal to work = -1728.9 J

QUESTIONS.

Are my equations well chosen for these processes?

Are the negative works of compression equal to the mechanical working capabilities of the machine? (ignoring looses).

If everything is correct I get thermal efficiencies of:

For adiabatic: 1368 / 8724.6 = 0.156 = 15%

For isothermal: 1728.9 / 8724.6 = 0.198 = 20%

I assume 8724.6 J would be the only heat added during the process, as the isothermal compression is rejecting heat.