# Submerged Archimedes Screw Heat Engine Air Compressor

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The Submerged Archimedes Screw Heat Engine Air Compressor (SASHEAC) is an elegant and possibly efficient means of harnessing a low $\Delta T$ temperature gradient to produce compressed air. Instead of raising water through the air, this screw pulls air downward through the water. The deeper the air goes, the more the water compresses it. The compressor comprises a pair of screws which are rigidly connected together and rotate about the same axis (i.e., they are coaxial). They are of opposite sense, so that as the submerged rotor turns, one screw pulls air downward while the other lets it rise. Now, cold air is easier to compress than hot air. A given amount of hot air in one screw exerts enough force to make the rotor turn as the air rises. The turning rotor carries a larger amount of cold air downward in the other screw. At the bottom, part of the cold air is harvested while the rest is heated and fed into the opposite screw to continue the process.

The purpose of this project on the Renewable Energy Design Wikia is to prepare a detailed design of this machine and to evaluate its feasibility.

An Archimedes' screw is a well-known device for pumping water. It dates to ancient Egypt, where Archimedes probably invented it. If it was invented earlier in ancient Babylon, the design was probably not handed down. Here I describe a new use for the Archimedes screw, as a heat engine and air compressor. Two very long intertwined screws (rigidly connected) convert compressed air to work and back to compressed air with high efficiency. With an adequate method of keeping the hot air hot and the cold air cold (e.g., heat pipes), this yields a large scale heat engine which yields high amounts of power from a heat source and a heat sink which are not far apart in temperature. The power can be realized as rotation of the screw, or as more compressed air produced than consumed.

 Construction of the water screw described by Vitruvius

The screw described in Roman times was constructed from a cylindrical wooden shaft, flexible flat wooden sheets, and a wooden outer casing bound with iron bands. The flat sheets were glued rigidly in the shape of a helix between the shaft and the outer casing. Modern screws omit the fixed outer casing, using a half-cylinder trough instead, where the blade moves with respect to the trough and water can leak slowly through the narrow gap. The ancient design did not leak, but it would be difficult to clean debris out if any was entrained in the water. The modern design is easily cleaned.

To convert to an air compressor (not yet a heat engine), the Roman design (with an attached outer shell) is used, but the central shaft is omitted. The whole screw but a small portion is submerged. As the screw turns, air is trapped at the top and is carried downward. The air is compressed by the weight of the water above it. The water in the screw is not forced to flow downward at the same rate as the air. If the central shaft were left in place, then the water would be forced to flow with the air, and inefficiency would develop after the volume of the air changed sufficiently.

It is a simple fact that cold air is easier to compress than hot air. If compressed cold air is heated, its pressure rises, and if it can do more work in expanding than it could while cold. If a submerged water screw containing hot compressed air is allowed to turn, the air rises through the water, and expands because the water exerts less pressure on it.

I envision a very large screw made of rigid neutrally buoyant segments and assembled at sea. The device would turn very slowly but being very large would entrap a large amount of air with each turn. The segments have to be sealed together, but they will not be subjected to high pressure. The hydrostatic pressure inside and outside the screw is always equal. Indeed, the ends are open.

## The Ericsson thermodynamic cycle Edit

The second Ericsson Cycle consists of two isothermal (constant temperature) stages and two isochoric (constant pressure) stages. The isothermal stages are realized by a pair of coaxial screws, and the isochoric stages are realized by a countercurrent heat exchanger. There are no valves in the main gas circuit. The rate of gas flow is controlled by the rate of rotation of the coaxial screws.A single screw is well suited to realize isothermal compression or expansion. The screw contains a more water than air, and the water flows freely because the “threads” are on the outside wall and the center is open. Heat pipes run through the inside of the screw to keep the air and water at constant temperature. The screw is tilted at perhaps 45 degrees.

Gas starts at low temperature and pressure and intermediate volume (point A on the P-V chart). Moving from A to B, it undergoes isothermal compression, i.e., it is kept cold during compression until it reaches the maximum pressure. On the P-V chart, the gas moves upward and to the left along the lower hyperbola from point A to point B. Physically, it is moving downward in the compressor screw as work is supplied (from the coaxial expansion screw), and it is heated by the compression but loses the heat to the heat pipes which carry the heat to the cold heat sink.

Next, moving rightward from B to C, the gas undergoes isochoric heating, i.e., it stays at constant pressure while being heated. Of course, the gas expands, and work is extracted from it as it expands, but the pressure does not increase (much). On the P-V chart, the gas moves rightward along the upper horizontal line from B to C. Physically, it is moving through the countercurrent heat exchanger, and the expansion is pushing fully heated gas forward into the next stage.

Moving from C to D, the gas undergoes isothermal expansion, i.e., it is kept hot during expansion until it reaches the original low pressure. On the P-V chart, the gas moves downward and to the right along the upper hyperbola from point C to point D. Physically, it moves upward in the expansion screw, supplying work to the screw and absorbing heat from the heat pipes which bring heat from the hot heat source.

Finally, moving from D back to A, the gas undergoes isochoric cooling, i.e., it stays at constant pressure while being cooled. Of course, the gas contracts, and work is done on it as it contracts, but the pressure stays constant. On the P-V chart, the gas moves leftward along the lower horzontal line from D to A. This completes the cycle.

## Two coaxial screws: realizing the Ericsson cycleEdit

 Schematic illustration of a segment of a rotor with two coaxial screws of opposite sense. The whole rotor would be much longer. This rotor rotates clockwise as viewed from the top. The rotor segment is powered by the expansion of three pockets of hot air (not shown) trapped in the inner screw while it compresses three pockets of cold air (not shown) trapped in the outer screw. The air pockets form at a local maximum height of the screw and its enclosing cylindrical covering. A portion of each cylinder is cut away to show the interior.

To realize the Ericsson Cycle (isothermal compression, isochoric heating, isothermal expansion, isochoric cooling), two screws of opposite sense of rotation (so one pumps air downward while the other permits it to rise) are rigidly connected, built one inside the other. They need to be rigidly connected for maximal efficiency in transferring work from the expansion screw to the compression screw. They need to be at the same depths in the water because they deal with the same air and water pressures. This dictates that one is inside the other (they are coaxial) and they rotate together as a unit. This unit, known as the rotor, is a submerged neutrally buoyant cylindrical structure which rotates about its axis while moving air upward and downward through the water. The structure is open to external pressure.

The rotor realizes the two isothermal stages. The water, air and heat pipes in the outer screw are cold while the water, air and heat pipes in the inner screw are hot. A non-rotating coaxial pair of pipes fits like a cap over the upper end of the screw, and supplies cold air to the outer screw and captures hot air escaping from the inner screw. There is no need to seal the connection between the cap and the screw because the air and water pressures inside are equal to the water pressure outside the vessel.

There is a need to prevent heat flow so the clearance between the cap and the screw is close, and there is a mechanical connection with bearings from the axis of the screw to the axis of the cap, so that the cap does not jam.

A similar cap made of coaxial pipes interfaces to the bottom of the rotor, collecting high pressure cold air from the outer screw and supplying high pressure hot air to the inner screw. The high pressure air at the bottom and the low pressure air at the top run through a constant-pressure countercurrent heat exchanger, which realizes the two isochoric stages of the Ericsson cycle.

The countercurrent heat exchanger consists of a series of large low-pressure tanks (because the low pressure air occupies a large volume) through which the high-pressure air moves in coils of small diameter thick-wall copper pipes. (Copper is likely to be the best and most cost-effective material because of its thermal properties). The whole countercurrent heat exchanger is located physically in shallow water so that the water pressure outside the low-pressure tanks is equal to the air pressure inside. The cold high pressure air moves upward from the bottom of the compression rotor in deep water (where the water pressure is exactly equal to the high air pressure) through small pipes that can contain the pressure when it is not balanced by water outside, and the hot high pressure air moves back downward through similar pipes to the bottom where it enters the hot expansion rotor.

The expanding high pressure gas flows forward into the expansion rotor, doing work on the rotor. The cooling low pressure gas (which would otherwise lose pressure) is compressed by hot low pressure gas coming from the expansion rotor. This extracts some work from the rotor, but calculations show that the amount of work supplied to the rotor by the expansion of heated gas in the high pressure side of the countercurrent heat exchanger exactly matches the work extracted from the rotor to compress the cooling gas in the low pressure side. Also, the energy available from the hot low pressure gas (which was heated as it expanded to keep it a constant temperature) is exactly the amount needed to heat the cold high pressure gas (which was cooled as it was compressed so that the amount of work needed to compress it would be minimal). To the extent the heat energy does not balance, the gas delivered to the rotor will not be at the proper temperature, but the heat pipes in the rotor will quickly correct that.

## Flexing force generated by unbalanced buoyancy Edit

It is important to avoid unbalanced forces with long lever arms, which result in large bending forces. Since the machine is to be rigid, these forces must be kept below the threshold where the structure begins to flex. Otherwise the efficiency drops and the material suffers fatigue. The buoyance of the machine is supposed to remain neutral all along its length. To achieve this, at each point along the length, the volume of water displaced by the moving air pockets must remain essentially constant.

A "multi-blade" screw has more pockets of air moving, and a shorter distance between them. If the screw has N blades, then after 1/N turn, the screw is in the same condition as it was initially.

One possibility for buoyancy control is to keep the air pockets at maximum fullness, permitting air to spill upward if for some reason the volume is larger than expected. There are complications: if the air was too hot but is cooling, then the volume will decrease below the maximum, reducing buoyancy.

Calculations needed: what is the effect of varying the hot temperature? We can assume the cold temperature will always be 4 °C because water at that temperature has maximal density, but the hot temperature can change. Given a fixed depth for the rotor, does the volume of the air pockets change with hot temperature?

## Torque generated by a pair of air-pockets Edit

It is not yet clear exactly what materials should be used, nor how much flexing can be tolerated. It is clear that the volume of the air in the two helices needs to be kept nearly equal so that there is no transfer of torque from one part of the long rotor to another.

## What makes the rotor turn? Edit

Assume that the rotor is moored so that its axis is held in fixed bearings. The rotor has a slight positive buoyancy and the bearings are attached by taut ropes to a fixed structure below.

The rotor turns in order to minimize gravitational potential energy. Air inside the rotor is displacing water. If the rotor turns freely in a way which permits the air to rise, water moves lower. The water gains kinetic energy at expense of potential. This is irreversible.

If the rotor is constrained, able to turn only by expending energy, then a (nearly) reversible process can occur. In that case, when the rotor turns, some air rises and an equal volume moves downward, and the amount of water which rises equals the amount which dropped.

This can also be analyzed in terms of unbalanced forces and kinetic theory of gas.

## Runaway rotor Edit

If there is no air at the top of the rotor to feed the descending helix, then the helix can turn without displacing additional water. This permits air in the rising helix to rise, displacing less water, and the balance is lost. The forces increase and the rotor starts to spin rapidly.

## Compressed air has positive buoyancy at depths over 8 km Edit

The rotor depends on air at the ambient pressure to provide neutral buoyancy. The air is trapped under the pocket formed by the helix and the cylinder wall, and is in contact with seawater at the ambient pressure. Here I calculate at what depth air has the same density as seawater, so that it cannot provide any buoyancy at all.The force it uses is a pulling force & sometimes it is a push.

As air is compressed, more molecules fit into the same volume, and the density increases. The mole fraction of water vapor does not increase but depends only on temperature. (Excess water condenses during compression).

The following facts apply:

• Mass of 1 m3 of air at depth meters of seawater and 280 kelvins
```   mass = 1025 kg = mw PV/RT
= (0.0289644 kg/mol) (1 atm + depth 1.025 m H2O) (1 m^3) / (R 280 K)
= (0.0289644 kg/mol) 1 atm (1 m^3) / (R 280 K) +
depth * (0.0289644 kg/mol) 1.025 m H2O (1 m^3) / (R 280 K)
= 1.2606323 kg + depth * 0.12505941 kg
```

Solving for depth, we find that air is buoyant at depths over 8 km:

```   depth = ((1025 kg - 1.2606323 kg) / (0.12505941 kg)) m
= ((1023.7393677 kg) / (0.12505941 kg)) m
= 8186.0243 m = 8.2 km
```

Pressure at 8 km of seawater:

```pressure = 1 atm + 8000 m 1.025 H2O
= 101.325 + 80414.53 kPa
= 80516 kPa = 794.63 atm = 11678 psi
```