Hydrostatic Stirling cycle air compressor

The Hydrostatic Stirling cycle air compressor is a nearly reversible heat engine with no large solid moving parts which produces high-pressure compressed air. It uses heat from a warm heat source to keep expanding compressed air warm while raising water, and then uses a cold heat sink to keep newly captured air cold while descending water compresses it. By holding the temperature of the air constant while changing the pressure, the compressor captures more molecules of cold air in the high-pressure tank than it released while raising the water.

Overview
To accomplish its task of temporarily raising water only to lower it and compress more air, the compressor permits warm compressed air to expand, but only while doing work, which cools the air. Heat energy from a warm heat source flows into the cooled air and warms it back up. The air does more work, establishing a flow of heat from the heat source through air and into work against gravity which raises water. Separately, the compressor lowers water, forcing the water to compress cold air as it descends. The water does work on air, dispersing the gravitational energy as heat, warming the air and compressing it. The heat, which came from the potential energy of the water in the upper reservoir, is rejected into the cold heat sink. The air is squeezed into a smaller volume at constant temperature. A fraction of the compressed air, at most $$\left(1 - \frac{T_\mathrm{cold}}{T_\mathrm{hot}}\right)$$, is diverted to the output. The remainder will be heated and used to raise more water.

The cycle is an open one because fresh ambient pressure air is constantly captured and compressed by the compression process, and because some of the high-pressure compressed air is diverted from the cycle and saved for output. Considerable expense for huge air tanks is avoided by releasing low-pressure expanding air instead of keeping it confined in a closed cycle. The compressor has no solid moving parts except for valves.

The machine is hydrostatic for two reasons: because potential energy is stored in the position of water pumped up into a high reservoir, and because a small additional force is used to compress air at many different pressures simultaneously by applying the force at the top of a tall column of water with tanks attached at different levels. The added force causes a little water to flow into each tank, compressing the air in each tank.

Stirling cycle applies because the working gas follows the Stirling cycle, in which cold low pressure air is compressed isothermally, warmed isochorically, allowed to expand isothermally, and cooled isochorically.

The high-pressure compressed air is produced at the temperature of the cold heat sink. Paradoxically, this output air contains less internal kinetic energy than the ambient temperature air which was captured to start the process. However, when confined in a suitable apparatus, it is capable of gathering heat energy from its surroundings and concentrating it into a force moving in a single direction.

See High-pressure compressed air for a discussion of its paradoxical properties and uses.

See Work produced by adiabatic and isothermal expansion of an ideal gas for a discussion and calculations regarding that topic.

Names of parts
The Hydrostatic Stirling cycle air compressor is a complicated machine with many parts and an intricate control program. It comprises many pipes, valves, high-pressure tanks, low-pressure tanks, water reservoirs, and other parts. During design, names has been developed to refer unambiguously to these parts.


 * Hydrostatic : Tanks and manifolds which are directly connected to a tall column of water without valves are called hydrostatic.


 * Hydrostatic manifold : One of a pair of manifolds connected to either the even- or odd-numbered hydrostatic tanks. The manifold is full of water, and extends to the level of the upper reservoirs.  The position of the upper surface of the manifold is adjustable depending on which upper vacuum tank is connected.


 * Vacuum tank : One of a group of water reservoirs connecting to the top of the water column of either hydrostatic manifold. These reservoirs are initially full of water but a vacuum forms at the top as water flows downward during compression.  Water flows back up into the tanks during expansion of previously compressed gas.


 * Hydrostatic tank : One of a series of evenly spaced tanks leading from the bottom reservoir to the top reservoir. Each hydrostatic tank is connected by the bottom port to a hydrostatic manifold and by the top port to valves leading up and down to the top ports of its two neighboring tanks.  Water flows into and out of the bottom port but never through the upper valve.  The valves open only when air pressure is nearly equal on each side.  Of course, the uppermost hydrostatic tank has no upper neighbor.  The corresponding valve is open to atmospheric pressure and is used to admit atmospheric pressure air to the system when compression is occurring and to exhaust atmospheric pressure air when expansion is occurring.

Design goals: Return on investment
Our actual goals are not simply reversiblilty or efficiency however defined. Our goal is to produce the maximum amount of compressed air, and we want to build a machine which does this at minimal cost. So we must figure out what properties decrease the amount of air compressed, or increase the cost.

Some of these considerations lead to design tradeoffs, i.e. to finding the optimum size where it is better to build another whole machine than to make this one bigger, etc.

Other considerations lead to rejection of a possible design altogether, i.e. a compressor which opens a large valve at the bottom of a tall and wide column of water into a tank filled with low-pressure air. The water would continue to accelerate until enough pressure builds up to oppose the force of gravity, but by then a sudden stop is required. We must avoid sudden changes in the forces.

On the other hand, if the air were in a U tube so that the water would start flowing upward and be slowed by gravity before the pressure matches, then the system would oscillate without a shock. If the friction is minimal, nearly all of the energy could be recovered.

Rules of thumb

 * Avoid friction and turbulence : they change work back to heat.
 * Avoid shock and large changes in forces: they reduce the useful life of the machine and change work to heat.
 * Carefully match pressures before opening valves : avoid excess forces and runaway acceleration. This eliminates wear and tear on valves and essentially eliminates energy expenditure moving valves.
 * Match rate of heat flow with rate of doing work : this should happen automatically.
 * Avoid irreversible processes, : yet make the machine run at a finite speed.
 * Change hydrostatic pressure with little work: move water horizontally and replace it with air, or use valves under low load.
 * Another reason for compressing rather than pumping: the pumped storage application requires a lower reservoir because there simply isn't enough energy stored in a small height change.  Therefore multiple levels of pumping tanks are needed:  some water raised at the top is lowered while lower water is raised.  This is more complicated and not a direct connection.  A gas must be used to transmit the force to avoid the weight of water in a column.  The gas temperature will increase near the water which rises, and decrease near the water which is being pushed down.

Processes that reduce the output

 * Expansion without doing work : (i.e., of gas into a vacuum) is irreversible and inefficient. Less compressed gas is produced with no decrease in cost.  The inverse would be compression without doing work which doesn't happen.
 * Compression without removal of heat : can be reversible but is inefficient. It increases the work invested in compression going forward but leaves the amount of heat transferred the same (assuming the final temperature is the same).  Less gas is compressed.
 * Compression with excess force : leads to acceleration of the fluid piston doing the compression. The process will overshoot the equilibrium position and then expand.  If gravity is involved things are slightly different.

Processes that are not infinitely slow
No real process is actually reversible. It follows or seems to follow from the definition that a reversible process is infinitely slow: at any step is must be possible to cause the process to move in the opposite direction by an infinitesimal change in conditions.

However, two finite rate processes taken together might meet this requirement, if each is undoing the results of the other. Then an infinitesimal change in one can reverse the progress, even though both processes are proceeding at a finite rate.

Then it becomes hard to see what is special about infinitesimal rates. The two processes are undoing all effects we care about, for example, together they are moving heat from one part of the cold reservoirs to another, but the net transfer into the hot and cold reservoirs is zero.

It is true that taken separately, neither process is reversible, but taken together they become reversible. All reversible processes are equally efficient in the sense of amount of useful work vs. heat transfer. Perhaps some irreversible processes could be this efficient too? And if so, what is the important property? Is it zero acceleration? Or do the separate processes with separate heat transfers lead to a different efficiency definition?

Thermodynamic theory forbids certain things, but perhaps the premises do not always apply. If we want to convert all of the chemical energy in fuel to work by way of heat, but do not require that the process used be cyclic, is it possible?

Why many separate compression tanks?

 * Avoid repeated pressurization / depressurizaiton cycles, which reduce useful lifetime of the machine.
 * Need to match forces to avoid inefficiency. Do this by using separate hydrostatic pressures.
 * Need to avoid cost of very large low pressure tanks by permitting small amounts of low pressure air to be compressed a little at a rapid pace, moving it on to a smaller tank and starting a new batch immediately.
 * Need to keep the required forces constant so that a constant flow of work and heat is achieved. After all, the equipment can handle certain maximum flows.  To the extent that those flows are not used, the equipment is underused.  If more heat flow is needed than possible, the temperature across the heat flow increases and the effective difference between hot source and cold sink decreases, reducing the available work.
 * More tanks means more heat flow.
 * A huge tank which can withstand high pressure would be required if the air is compressed from low to high pressure in the same tank. Better to split the tank up so that the tanks will end up full of gas, not full of water with a tiny bubble of gas.

Why compression tanks separate from expansion tanks?

 * The compression process needs heat removal, while expansion needs heat supply. Gravity feed heat pipes provide one or the other.
 * With separate tanks, both processes can occur at once if both hot and cold are available at once.

Detailed description of the process
This machine is a large-scale distributed Stirling cycle heat engine which raises and lowers water continuously, using separate tanks, pipes, and valves for compression and expansion and for raising and lowering water. Any given pressure tank handles only a narrow range of pressures dictated by the vertical distance from the tank to the upper reservoir surface. Gravity feed heat pipes are used to carry heat upward from warm heat sources to expansion tanks, or from compression tanks to heat sinks. The energy to raise water comes not from compressed air, but from the heat entering the compressed air as it expands. The energy extracted by lowering water appears temporarily as an increase in the temperature of the air being compressed, but as the air is cooled, the heat pipes carry this energy out of the tank and up to the heat sink, from whence it returns to the atmosphere.

The compressor uses an open cycle in which atmospheric pressure air is constantly captured, cooled to cold heat sink temperature, compressed by descending water while being kept cool, warmed, allowed to expand while raising water and being kept warm, and finally, released back to the atmosphere. Unlike a heat engine which does work without later undoing it, the compressor (which always lowers the water it raised) eventually returns all of the heat energy absorbed from the warm heat source to the cold heat sink. The act of compressing gas at constant temperature is always a conversion of work to heat, and this compressor delivers all of that heat to the heat sink. Also, the act of expanding gas at constant temperature is always a conversion of heat to work. This compressor always undoes the work produced then.

Because the heat source is merely warm and not hot, it is vitally important that the work from expansion of warm compressed gas be converted to potential energy and back to work with nearly perfect efficiency in a nearly reversible manner. The absence of solid moving parts makes highly efficient energy conversion feasible.

In addition to efficient conversion of potential energy to work, the conversion of work to compressed gas and vice-versa must be nearly reversible. This is necessary because irreversible expansion allows compressed air to escape without capturing enough heat as potential energy to continue the cycle, and irreversible compression allows potential energy to escape as heat without compressing enough gas to replace the initial investment.

In general, irreversibility is associated with turbulent accelerations of the water. Rapidly moving water under automatic control is not necessarily irreversible. The important point is that the forces must nearly balance. Nearly balanced forces always result in nearly zero acceleration, but not necessarily nearly zero velocity.

Compression phase
An intricate collection of tanks, pipes, valves, and manifolds is used to maintain nearly balanced forces throughout the system, resulting in smooth flow of descending water, smooth intake of atmospheric pressure air, and the efficient compression of air at many different pressures at once.

Intake of atmospheric pressure air into the uppermost hydrostatic tank
After every other compression step, air is moved from the even numbered tanks to the odd numbered tanks. Tank number zero is the atmosphere, and air is moved from the atmosphere into tank one at the same time as air moves from tank two into tank three, etc. Air flows into tank one because the hydrostatic pressure in the tank is less than atmospheric pressure, so that water flows downward out of the tank as fresh air flows in.

Compressing air at many different pressures at once
Each compression step begins with all valves connecting hydrostatic tanks closed. Half of the tanks (say the even numbered tanks) are effectively full of water, with only negligible amounts of air left. The other half (the odd tanks) are full of air at a pressure dictated by the height of water in the vacuum tank connected to the odd hydrostatic manifold.

In order to compress air, a different vacuum tank with a higher water level must be connected to the odd manifold. Then, water will flow out of that vacuum tank and up into all of the odd numbered hydrostatic tanks. The flow stops or nearly stops, and the next higher level tank is connected until no vacuum tank has a higher level than the one connected to the odd manifold.

Moving air to the other set of hydrostatic tanks
After there is no higher water level vacuum tank available to connect, no further compression is possible until after the air is moved to the lower level tanks. This is accomplished with very little loss of potential energy and very little change in air pressure, by closely matching the pressures before opening valves. The next lower level vacuum tank is connected to the other hydrostatic manifold. When the higher tanks are full of air and the lower tanks full of water up to the valve, the water levels are about equal in each pair of tanks. Therefore, if the water levels in the vacuum tanks attached to the two manifolds is about equal, the pressures on each side of the valves connecting the paired tanks will be about equal, and the valves will open easily. In order to transfer the air to the lower tanks, a vacuum tank with a slightly lower level is connected to the lower manifold. Water will flow up into the air-filled tanks, air will flow down into the water-filled tanks, and water will flow up into the lower manifold and into the vacuum tank, until the water levels are equal. Then, the next lower level tank is connected, and the process continues until the upper tanks are full of water and the lower tanks are full of air.

Transferring air into the storage tanks
Compressed air is stored in a number of hydrostatic tanks at the lowest level and all filled with water to the same height. Assuming an odd number of non-storage tanks, the storage level is assigned an even number and the bottom ports of all of the tanks on the storage level are attached to the even hydrostatic manifold. The storage tanks are arranged so that the height difference from top to bottom is an absolute minimum, and the bottom of the storage tanks is level with the bottom of the lowest odd tank.

When air is being transferred from odd tanks to even tanks, all regular even tanks are full of water. There is air in the storage tanks, but the water level in the storage tanks is still above the water level of the lowest odd tank, because the storage tanks are higher than the next regular tank would have been. Air will move from the lowest odd tank into the storage level, and an equal volume of water will flow out the bottom of the storage level into the even manifold. When the storage level is nearly empty, the air moving from the lowest odd tank into the storage tanks will expand slightly because it ends up at a higher level under less hydrostatic pressure.

Expansion phase
Some of the high-pressure compressed air is diverted to the output. The rest is allowed to expand against a carefully balanced hydrostatic head of slightly lower pressure than the gas pressure. This raises water and cools the expanding air, which causes heat flow which warms the air to the temperature achieved by the heat pipes inside the expansion tanks, while causing condensation and downward flow of refrigerant in the heat pipes and additional heat flow from the heat source to evaporate the refrigerant.

Transfer of air from the storage tanks into the lowest expansion tank
The expansion process is carried out in a separate set of hydrostatic tanks with a separate pair of hydrostatic manifolds also called even and odd. The vacuum tanks are shared. Water pumped upward during expansion is lowered during compression...


 * What is the net water flow?

Independent action of compression and expansion
The cold compression and hot expansion processes operate independently, coupled by a hydrostatic reservoir comprising many tanks at different hydrostatic pressures, and a large series of low-pressure pumping tanks which are used to raise water to the top of the hydrostatic reservoir. The total volume of the reservoir represents how far ahead of the cold compression process the hot expansion process can get.

The cold compression process and the hot expansion process are located at physically separate places and connected by a well-insulated high pressure gas line which transmits force but not volume. It is like the rope in a well matched tug-of-war contest which moves back and forth, but never very far.

Links
A predecessor to this design was discussed on halfbakery.com.

Related design
This machine can produce either compressed gas or electricity, depending on what is done with the extra energy.
 * If only enough liquid is raised by hot gas to recompress the same volume of cold gas, : then excess compressed gas is left over. It could be heated and used to drive a turbine and generator, leaving low pressure working gas suitable for a closed cycle.  Otherwise, if atmospheric air is captured and used as working gas, the excess compressed air could be shipped away through a pipeline, forming an open cycle.
 * If all available gas is heated and used to raise liquid, : then some liquid can be let down via a turbine and generator while still recompressing the same amount of working gas. In that case, the working gas can be kept sealed and pure, and more choices of materials are available.

Mercury and gas heat engine
Design points:
 * Closed cycle with pure working gas plus mercury vapor. The working gas could be chosen for optimum performance.  Hydrogen or helium might have advantages.
 * Operating pressure is not dictated by needs of consumer of compressed gas, since the output is electricity. Instead, the achievable heat flow into a given volume dictates the minimum volume and hence maximum pressure.
 * Mercury serves as liquid piston and gravitational potential energy store.
 * Mercury is raised by expanding gas pushing downward into U tubes. Rate of expansion, controlled by adjusting the hydrostatic head above the manifold, is controlled to keep gas temperatures (in multiple tanks with different pressures) at set point.
 * High temperature provided by desert solar collectors.
 * Low temperature provided by heat pipeline up the mountain.
 * Turbine driven by descending liquid mercury generates electricity.

To caculate per megawatt as a function of hot and cold temperatures:
 * How much mercury is needed?
 * How much working gas is needed?
 * How much refrigerant in the heat pipeline is needed?
 * How large are the tanks and pipes and valves?

"Good Questions"
These questions are copied from my remarks on halfbakery. Some aren't answered yet.

Rate of heat flow
So one good question to ask is how long will it take for the required amount of heat to flow across the available temperature difference. It is true that heat flows more slowly the smaller the difference. I admit I haven't calculated this yet and it remains a potential gotcha. You haven't calculated it either.

Effect of water vapor
Another good question is how much effect the water vapor has. The amount will change when the temperature changes. I haven't worked that out for this machine but I don't expect it to be a big problem. It is another potential gotcha.

Effect of barometric pressure
A good question to ask about this design is "what is the effect of changes in barometric pressure?"

How is the water heated if necessary
Either the water in tanks stays in about the same place, or a countercurrent heat exchanger is used. For instance, if cold water is being pumped up in contact with hot gas, the water is warmed first by a countercurrent heat exchanger with hot water which flows down a U tube from the top to get rid of heat, then it is cold as it flows back up to reservoir, all in balance.

Why don't the pressure tanks lose pressure as they raise water?
Basically, because they are converting heat to work, not using their own stored energy to raise water.

The pumping mechanism uses multiple reservoirs above the hydrostatic columns to match hydrostatic pressures. To permit air to expand, the top of a column is connected to the vacuum tank with the next lower surface height. Air expands and cools in the hydrostatic tank as water rises into the vacuum tank. Heat flows in to re-warm the air.

After air has expanded so much that the hydrostatic tanks are full of air, an upper reservoir is selected and connected to the other column, so that the hydrostatic pressures on both sides of the valve to the next upper tank are equal, for each pair of tanks. That valve is opened and the hydrostatic pressure under the air is increased slightly. The air flows from the lower tanks into the upper tanks, displacing water from those tanks into the vacuum reservoir at the top of the column attached to the upper tanks. This process is nearly reversible and uses very little potential energy as no air is being compressed. The process stops by itself and another pair of tanks must be selected to continue it until the lower tanks are filled with water. Precisely as much water is raised as lowered during this process, and the final height of the water is barely less than the initial height.

If the process is allowed to proceed faster, the water levels will overshoot and continue to oscillate if the valves are left open, so long as water does not rise so far in the lower hydrostatic tank to reach the valve. Closing the valve when the levels are equal and the flow maximum will cause a shock and dissipate the kinetic energy. Closing the valve when the flow has stopped will result in tanks with unequal levels, always useful, and very little loss of energy.

Air from the topmost tank is allowed to escape as water flows in with just enough pressure to displace the air. The deepest tank with high pressure air in it receives a very small amount of water compared to the amount raised.

Why isn't there a lot of friction and energy loss activating valves?
The valves between pressure tanks are only actuated when the pressure is balanced across them.

Text to copy from
The machine I describe is intended to collect energy, not to produce work. It is naturally very large and occupies otherwise useless space like a mountainside. The amount of energy it handles scales as the volume of the machine but the materials required to build it scales as the surface area.

In the text posted in this idea, I didn't describe the heat collection and rejection apparatus (which I expect is needed only on the hydrostatic expansion / compression tanks, not on the pumping tanks or the bias tank). It includes gravity-feed heat pipes between the interior of the tanks and the heat sink (above the tank) and heat source (below the tank), and provision for air flow: the heat sink heats air and a chimney carries the heat away because hot air is less dense than cold air. And the heat source cools air, which then falls into a downward chimney and carries the cooled air away.

BTW do you know the T-S diagram for the Stirling cycle? (Recalling Stirling engines don't quite use that cycle). I'm looking at the Carnot_heat_engine article on wikipedia, where the T-S diagram is a square, but a Stirling cycle exchanges heat in all phases so its T-S diagram has no vertical lines. I haven't needed that diagram because I don't care much about how much heat goes into and comes out of the machine, since both are supplied by the surroundings. The amount of heat which flows is determined by the heat capacity of the working gas (wet air) and the number of moles stored, and by the requirement that the compression and expansion be isothermal. The number of moles of working gas has to be adjusted in order to be able to pump the desired amount of water.

So one good question to ask is how long will it take for the required amount of heat to flow across the available temperature difference. It is true that heat flows more slowly the smaller the difference. I admit I haven't calculated this yet and it remains a potential gotcha. You haven't calculated it either.

Another good question is how much effect the water vapor has. The amount will change when the temperature changes. I haven't worked that out for this machine but I don't expect it to be a big problem. It is another potential gotcha.

I'm calculating the answer for required energy conversion efficiency so that this machine can pump more water up than it lets down. I want to use a single fraction, x, for the conversion efficiency from gravitational potential energy to gas pressure potential energy. This is easy enough to explain (but it took me a while to figure out how). Suppose the ideal cycle puts out K times the work it takes in. The inefficient machine converts gravitational potential W to work x*W. Then the ideal cycle operates using that work plus whatever heat it absorbs and rejects, and puts out work K*x*W. Finally, the inefficient machine converts the work back to gravitational potential energy K*x^2*W. So, an ideal cycle which multiplies the input work by K will yield more gravitational potential than it takes in only when K*x^2 > 1. So if K is 1.1, then x must be > sqrt(1/1.1) = 95.4%.

Let me consider this in more detail, since usually ideal cycles are discussed in terms of the heat in and out rather than the work in and out. The ideal Stirling cycle starts with its working gas at temperature H and volume V. It cools the gas to temperature C volume V putting out the necessary heat to do this. Next it takes energy W in as work, compressing the gas to volume Y while still at temperature C, and putting heat out at temperature C. Then it takes some heat in at temperature H (warming the working gas and keeping it warm) and puts some work out by letting the gas expand to volume V and temperature H, the original state.

How much work does it put out? Can we figure this without knowing the amounts of heat? Using an ideal gas for the working gas, we know that the work is for volume change from V to Y is integral Pdv, with P = nRT/v. This works out to nRT integral dv/v = nRT*log(Y/V). So the work W done compressing gas from V to Y at temperature C is W = C*nR*log(Y/V). The work extracted when the gas expands at temperature H is H*nR*log(Y/V) = W*H/C. The work out is the work in times H/C. Thus, K = H/C. The ideal Stirling cycle machine does multiply work in by a constant. That's why a Stirling engine can't start up by itself.

But you are right, I did not calculate how much heat is involved. That's because I just want to know whether or not the machine operates at a profit or at a loss. For H/C = 1.1, the efficiency has to be over 96%. Below that, and the machine must operate at a loss, and you can't "make it up on volume" as the retail sales joke goes. It seems quite plausible to me that the liquid piston no-moving-parts (except for valves) mechanism I describe could operate at efficiency over 96%.

Next I go over the cycle in terms of the new diagram of pipes, valves, tanks, and reservoirs. The machine converts gravitational potential of water in the pairs of pumping tanks to low-pressure compressed air in the bias tank. That pushes water downward through the hydrostatic manifold and up into half of the hydrostatic tanks (the others are full of water), with a net increase in the air pressure in the hydrostatic tanks relative to atmospheric pressure. Each hydrostatic tank is at a different pressure, but the same bias pressure raises the water level in each tank a little. That increases the pressure in the tank because the volume decreases. It also increases the temperature a little, but a heat pipe carries the extra heat away. Next, the valve to the next lower reservoir is opened. Water flows out of the bias tank down the manifold and into all of the hydrostatic tanks, displacing the air into the next lower hydrostatic tank. This flow stops when pressures balance. Meanwhile, air flows into the bias tank from the large number of pumping tanks, and water flows downward in each pair of pumping tanks. Water from the lower hydrostatic tank of each pair flows into the other manifold and up into the upper reservoir. This completes a half-step, and all valves are closed. Water from the upper reservoir is let into the bias tank to replace the water which flowed into the hydrostatic tanks to displace the air downward into the next lower hydrostatic tank. Also, water is let into the uppermost pumping tank to replace the water which flowed downward into the second pumping tank. This flow has to be metered to prevent flooding of the pumping tanks -- it does not stop automatically. The refill valves are closed, and the other set of pairing valves is opened to create the opposite pairing (odd-even instead of even-odd) of hydrostatic and pumping tanks, and the manifolds are connected in the opposite sense, so that the odd tanks are connected to whatever the even tanks were, and vice-versa. More air flows out of the lower tanks of each pair of pumping tanks, and into the bias tank. I believe this is where we came in.

The above describes a step of one quarter (isothermal compression) of the Stirling thermodynamic cycle executed by this machine once each day. This happens in the early morning during a period of low temperature. The second quarter (constant volume heating) happens all morning and half of the afternoon while the temperature increases. The air in the lowest hydrostatic tank increases in pressure and the water level decreases, but the volume changes very little because the tank is tall and thin (not shown). The third quarter (isothermal expansion) of the Stirling cycle occurs when the temperature is at a maximum, in the late afternoon. The process is almost identical to that described above, proceeding in reverse. The main difference is the water level in the bias tank. It is above the level in the upper reservoir instead of below, and the bias pressure is slightly below atmospheric. This causes water to be "sucked up" into the lowest pumping tank. Actually, atmospheric pressure pushes down on the surface of the lower reservoir and pushes the water up into the lowest tank where the pressure above the water level is a little below atmospheric. Similarly, atmospheric pressure above the water in each lower tank of a pair of pumping tanks pushes water upward into the upper tank of the pair where the air above the water is at the bias pressure. Some air flows into the bias tank.

more text to copy from
Where does energy come from to move rest of water used for compression. Archimerged, Apr 20 2006

[jhomrighaus]: A very good question. If you understand the answer, you will see how the machine works. As arranged, there is too much hydrostatic pressure for the gas to expand. But if the arrangement is changed in a carefully designed system, then when a valve is opened, some work will be done and some water will flow until the pressure balances again, at a slightly lower pressure _and temperature_. That is the crucial point.

When a gas expands against an exactly balancing force, i.e., very slightly less hydrostatic pressure than gas pressure, and the pipes are carefully arranged so as the gas expands, the hydrostatic pressure increases, _not_ decreases (since otherwise the expansion would run away and cause a water fountain but lots of water would be left behind in the tank), then the temperature goes down. The gas did work and the energy for the work came from the temperature decrease. Agreed?

Now, when the temperature of a gas is lower than the surroundings, energy will flow out of the surroundings and into the gas. Still with me?

Assuming you agree, then you see where the energy to raise all of that water is going to come from. It comes from the surroundings. It is necessary to repeatedly decrease the hydrostatic pressure applied to the gas to match the gas pressure, such that as the gas expands, the hydrostatic pressure will increase so the flow will stop by itself. This can be done without expending significant energy, if the piping is arranged properly, by opening a valve and waiting for the flow to stop. The temperature goes down and heat flows in and a little more water flows but it stops pretty soon. Then a few other valves are adjusted, the crucial valve is opened, and the same thing happens again in a slightly different place with slightly lower pressure.

All of those "extra" tanks and valves are needed in order to arrange for the energy from the surroundings to flow into the gas and raise the water. This process is shown on the PV diagram (P is vertical axis, V is horizontal axis) as the two long hyperbolic curves which plot PV=nR(275K) and PV=nR(305K). The process of lowering all of the water is represented by the lower hyperbola, running from the lower right hand corner point P = 1 atm, V = nR(275K) / 1 atm to P = 300 atm, V = nR(305K) / 300 atm, in the upper left but below the next point. The process which raises the little bit of water shown in your diagram is represented on the PV diagram as the (very short) vertical line running up from the point with P = 300 atm, V = nR(275K) / 300 atm, to the highest point in the upper left corner with P = 300 atm, V = nR(305K) / 300 atm. The process of raising the remainder of the water and the extra water is represented by the upper hyperbola running from upper left to lower right (but above the lowest point). The process of cooling the gas is represented by the short vertical line at the right side running down to the lower right corner, in which P and T decrease and V is constant. The net work done by a complete cycle is the area inside the closed curve formed by the three curves mentioned, and the fourth which runs from P = 1 atm, V = nR(305K) / 1 atm to P = 1 atm, V = nR(275K). The area between the X axis and the lower hyperbola is the work extracted from the water and then redone to the water to pump up the same amount as was dropped. The area between the two hyperbolas is the work done to raise the extra water. n is the number of moles of air in the tank, and R is the gas constant. This area is equal to nR(30K) ln(300 atm/1 atm).