It has been known since Sadi Carnot's work in the 1830's that heat engines produce work from the transfer of heat from a hot reservoir to a cold reservoir, without any requirement for fuel. Small toy Stirling cycle engines exist which will sit on top of a TV set and turn using heat energy from the TV, but the torque is low and not much energy is involved. Recently a 55kW Stirling engine / generator has been developed, but it still needs a large temperature difference between the hot and cold reservoirs.
The points which everyone should understand are these
confined gas can be used as an energy sponge, and
there is a vast amount of energy available at moderate temperatures. The total power from the sun is 174 billion megawatts.
Energy from the sun is captured by Earth's atmosphere in the daytime, exists as heat for a while, and radiates back out into space at night. This amount of energy is so large that a method which can capture only a tiny fraction of it would produce far more energy than we need. So do not scoff at designs which extract a microscopic amount of work from a huge heat flow. The necessary heat is available in the atmosphere.
A few facts are important
Gasses are made up of molecules. These may be regarded as tiny billiard balls.
The temperature of a gas is directly proportional to the average kinetic energy of these tiny billiard balls.
The gas is to be confined inside a solid cylinder between two liquid pistons.
The cylinder and the liquid are also made up of molecules. The temperature of the cylinder and the liqud also reflects how fast the molecules are vibrating.
When a gas molecule strikes the cylinder wall, it is most likely to end up moving with the kinetic energy specified by the temperature of the wall.
Since the cylinder wall cannot move, the repeated collisions of gas molecules do not result in any coherent motion of the cylinder wall. They only result in heat exchange until the temperatures are equal, with no work exchange.
Since the pistons can move, repeated collisions of gas molecules do result in coherent motion.
If the pressure of the gas is lower than the pressure applied to the other side of the piston, then the piston will move to confine the gas in a smaller volume.
The gas temperature will increase to reflect work done by the piston on the gas.
The energy of the work then flows as heat into the cylinder walls and away into the heat sink.
If the pressure of the gas is higher than the pressure applied to the other side of the piston, then the piston will move to allow the gas to expand.
The gas temperature will decrease to reflect work done by the gas on the piston.
Energy required to replace the energy used for work flows as heat from the cylinder walls into the gas.
Heat flows from the hot heat source into the cylinder walls to replace the heat which flowed into the gas.
Thus, we conclude
If the applied pressure on a confined gas is slightly lower than the gas pressure, then
the gas will expand doing work, and
its temperature will drop.
Heat will flow from the container walls into the gas until the temperatures are equal.
Heat will also flow from the hot heat source into the container walls.
The heat which flows into the gas this way is being converted to useful energy.
If the applied pressure on a confined gas is slightly higher than the gas pressure, then
the gas will be compressed, absorbing work, and
its temperature will increase.
Heat will flow from the gas into the container walls until the temperatures are equal.
Heat will also flow from the container walls into the cold heat sink.
The heat which flows out of the gas came from useful energy doing work on the gas.
Now if the compression and the expansion take place at the same temperature, there is no net gain of useful energy, since all of the energy produced by expansion would be used up compressing the gas again. But if the cold heat sink is colder than the hot heat source, it takes less work to compress the cold gas than will be extracted when the hot gas is allowed to expand. So there is a net output of useful energy.
In more detail
Given a clever enough arrangement of pipes and valves and liquid and gas, energy can be extracted from a warm heat source and an amount (depending on the temperature difference) of it converted to work, while the rest is rejected into a cold heat sink. The fraction extracted as work is probably less than one percent of the heat flow, but that heat flow exists whether we capture any or not.
What is this clever arrangement?
Mercury serves as liquid piston. (The earth's mercury has to be somewhere. Why not collect it into this machine?)
Many small diameter glass tubes serve as cylinder for the liquid piston. Together, they are called a manifold.
The tubes are kept in a pool of mercury, so that the hydrostatic pressure outside the tubes is equal to the pressure inside. Therefore the tubes can have thin walls.
The tubes are bent into the shape of a hyperbola. They are nearly vertical at the bottom and nearly horizontal at the top.
The shape of the tubes is dictated by the equation of state of the gas.
For a non-ideal gas, the tubes would deviate very slightly from hyperbolic shape.
Gas bubbles are confined in the glass tubes between moving slugs of mercury.
Glass is used because mercury is repelled from the glass surface so that when gas bubbles separate slugs of mercury in the tube, the slugs remain separate.
The gas and liquid move through the tubes as a unit, at exactly the same rate of speed.
Two manifolds are used in the machine.
A hot expansion manifold has hot mercury and bubbles of hot gas moving upward through hot glass tubes.
The hot manifold is in a pool of hot mercury which carries heat from the hot heat source to the manifold.
A cold compression manifold has cold mercury and bubbles of cold gas moving downward through cold glass tubes.
The cold manifold is in a pool of cold mercury which carries heat from the manifold to the cold heat sink.
How do the bubbles get created?
One glass capillary tube is positioned in the mercury flow inside each glass manifold tube.
The hydrostatic pressure at that point is equal to the distance upward to the mercury surface, plus the gas pressure of the gas above the mercury.
The pressure of the gas entering the capillary tube is slightly higher than the hydrostatic pressure at the end of the capillary.
Hence the gas will flow out of the capillary tube into the mercury flow.
After a bubble of the proper size has formed in each tube of the manifold, the gas pressure is reduced to slightly below the hydrostatic pressure and a slug of mercury is allowed to form.
The process continues indefinitely.
Hot high pressure gas is injected through small diameter tubes into the upward flow of mercury inside glass tubes at the bottom of the expansion manifold.
Cold low pressure gas is injected into the downward flow of mercury inside the nearly horizontal glass tubes at the top of the compression manifold.
How does the confined gas serve as an energy sponge?
Cool gas absorbs heat from warmer tube walls.
Hot gas loses heat to cooler tube walls.
One set of long curved glass tubes serves as the expansion manifold.
The outside of these tubes is in thermal contact with the hot heat source.
The tubes are nearly vertical at the bottom, and nearly horizontal at the top.
Hot mercury flows upward constantly into this manifold.
At the top, mercury flows into a closed reservoir with gas confined above the surface.
Hot high pressure gas bubbles are injected into the mercury stream at the bottom of the manifold.
Slugs of mercury separate the gas bubbles, and remain separate as the stream moves up the tube.
The gas bubbles up a short distance to the mercury surface as the mercury leaves the manifold and enters the reservoir.
The hydrostatic pressure at any point in a tube is equal to the vertical distance up to the mercury surface, plus the pressure of the gas above the mercury.
The volume of the gas bubbles will change whenever the gas pressure is not equal to the hydrostatic pressure.
As the mercury and gas rise through the long tubes, the gas expands and cools.
The level of mercury in the reservoir rises slightly because of the gas expansion.
As the gas particles collide with the warm tube walls, they regains heat.
This process continues for the duration of the trip through the expansion manifold.
If the pressure at the top is half the pressure at the bottom, then the volume of a bubble at the top is twice the volume at the bottom.
The hot low pressure gas flows into a countercurrent heat exchanger, exchanging heat with cold high pressure gas.
The hot mercury flows into another countercurrent heat exchanger, exchanging heat with cold mercury.
Another set of long curved glass tubes serves as the cold compression manifold.
The outside of the tubes is in thermal contact with the cold heat sink.
The tubes are nearly vertical at the bottom, and nearly horizontal at the top.
Cold mercury flows from the countercurrent heat exchanger into the compression manifold.
Cold low pressure gas is injected into the cold mercury stream just after it enters the individual small nearly horizontal glass tubes of compression manifold, creating a downward moving stream of gas bubbles separated by slugs of cold mercury.
At the bottom of the compression manifold, the gas bubbles are collected in a tank which leads to the gas countercurrent heat exchanger.
The cold high-pressure gas is warmed at constant pressure using heat from the hot low-pressure gas which is being cooled.
After being warmed, the resulting hot high-pressure gas flows into the capillary tubes at the bottom of the hot expansion manifold, creating bubbles as described earlier.
The cold mercury flows into the bottom side of the mercury countercurrent heat exchanger, to be warmed up before entering the hot expansion manifold.
How is useful energy actually extracted?
The short answer is that the flow of mercury would speed up indefinitely unless excess energy were extracted. Energy can be extracted by allowing mercury to flow downward without compressing gas but instead turning a turbine, for example.
The machine as described has four tanks of mercury.
The two upper tanks are low-pressure hot and cold.
The two lower tanks are high-pressure cold and hot.
The cold upper tank and the hot lower tank
are fed from the output of the mercury heat exchanger,
contain only mercury, and
feed into the corresponding manifold.
The hot upper tank and the cold lower tank
are fed from the output of the corresponding manifold,
contain both gas and mercury, and
feed from the bottom into the appropriate mercury heat exchanger input, and
feed from the top into the appropriate gas heat exchanger input.
Mercury from the bottom of the hot upper tank flows through the countercurrent heat exchanger to the cold upper tank.
Mercury from the cold upper tank flows down through the compression manifold to the cold lower tank.
Cold low-pressure gas is injected into the nearly horizontal downward flow and is carried along with it.
The descending mercury does work on the cold low-pressure gas, compressing it.
The energy transferred to the cold gas raises its temperature causing heat flow to the tube walls, so all of the energy used compressing the gas is delivered as heat to the cold heat sink.
Mercury and gas separate in the cold lower tank.
Mercury flows through the fluid countercurrent heat exchanger and into the hot lower tank.
Gas flows through the gas countercurrent heat exchanger, through the capillary tubes, and into the expansion manifold.
Mercury flows from the hot lower tank up into the expansion manifold.
Hot high pressure gas from the countercurrent heat exchanger is injected into the expansion manifold at a point where the hydrostatic pressure is lower than the gas pressure.
As the gas expands, it does work on the ascending mercury.
All of the energy used to raise the ascending mercury higher than the descending cold mercury would push it comes from heat absorbed by the gas from the tube walls, which came from the hot heat source.