Rankine Cycle and its derivative cycles belong to pure thermodynamic cycle. The earth gravity has no influence on those thermodynamic cycles. On the contrary, if the earth gravity and heat plays a vital role in the conversion process of energy, the conversion cycle is called Thermodynamic and Gravitational Cycle (TGC).

TGC phenomena are common in natural world. For example, water evaporates on the earth surface and rises to upper air due to heated by sun; water steam condenses and forms a cloud while rising; then cloud rains and drop to the earth surface when meeting cold air. The rainwater fallen in highland or mountain is the source to produce hydroelectric power. We can also find TGC phenomenon in some artificiality, such as gravity assisted heat pipe, and MLC (Mist Lift Cycle) experiment devices [3].

This paper will discuss two kinds of TGC, i.e. Thermodynamic and Gravitational Open-Cycle (TGOC) and Thermodynamic and Gravitational Closed- Cycle (TGCC) to develop the TGC theory.

In particular, MLC is an open-cycle OTEC concept originally developed by Dr. Stuart Ridgway in 1970s [3]. MLC uses a finely perforated titanium plate (mist generator) at the bottom of vacuum chamber. Warm seawater is introduced into the vacuum chamber through 0.1 mm diameter holes in the titanium plate. The low pressure in the vacuum chamber is maintained such that some of the water flash evaporates-creating a fine mist. Farther up the column, cold seawater is injected into the chamber. The cold seawater lowers the pressure in the top of the chamber. This creates a pressure gradient that drives the mist from the bottom of the chamber to the top. The vapor component of the mist is completely condensed into the cold water by the time the lift is complete. The lift height is such that water is lifted above sea level, and then the water is drained into the ocean.

Obviously, MLC is very different from Rankine Cycle. MLC introduced gravitation to thermodynamic open-cycle. That can be explained by the concept of Thermodynamic and Gravitational Open-Cycle (TGOC) whose process theory is shown in Figure 1 comparing with Rankine Cycle.

As we know, Rankine is a close cycle with 4 typical stages shown in Figure 1. The stage 1 - 2: The working fluid is pumped from low to high pressure. As the fluid is a liquid at this stage, the pump requires little input energy. The stage 2 - 3: The high pressure liquid enters a boiler where it is heated at constant pressure by an external heat source to become a dry saturated vapor. The stage 3 - 4: The dry saturated vapor expands through a turbine, generating power. This decreases the temperature and pressure of the vapor, and some condensation may occur. The stage 4 - 1: The wet vapor then enters a condenser where it is condensed at a constant pressure to become a saturated liquid.

Fundamentally different from Rankine cycle, the concept of TOGC mist lift has 3 primary stages shown in Figure 1. The stage 1 - 2: warm seawater from sea surface get a high pressure at bottom of vacuum chamber, seawater is injected into low pressure vacuum chamber through the mist generator. Here is a very large pressure drop. The stage 2 - 3: the mist contains a mass of droplets and some vapor generated by flash evaporation; vapor expands through the vacuum chamber, and lift droplets to the top. The stage 3 - 4: The vapor component of the mist is completely condensed into cold water by cooler spraying water at the top.

So, MLC uses little vapor expansion and condensation to lift mass of droplets, while Rankine Cycle uses dry vapor expansion to drive steam turbine. Wet vapor with small dryness can lift to a high altitude, which is the key to develop the theory of TGC. Although the thermodynamic feasibility of mist lift has been established based on much experiments and researches [5]-[11], its characteristics are poorly understood and there is no sound theoretical support. The most critical issue of OTEC MLC is ignored till now, i.e. without heat exchanger, the temperature difference (17 K) is not enough to lift all warm seawater introduced into vacuum chamber. As we know, the latent heat of water evaporation is 2500 kJ/kg, while the specific isobar heat of water is 4.2 kJ/kg K. One unit mass of water evaporation need the heat of 35.3 unit mass of seawater with 17 K drop of temperature. Obviously, the dryness of mist (X < 0.028) is extremely small. Although some researches show that it is realizable to lift 10.0 unit mass of mist to high altitude using 1 unit mass of vapor if provide a much bigger inlet pressure [7]. It is indeed impossible to lift the 33.5 unit mass of mist using 1 unit of steam. That is the reason MLC haven’t been realized in OTEC till now from the opinion of this paper. MLC should be realized with higher temperature difference.

In order to avoid the above weaknesses of Rankine Cycle, MLC (TGOC) and other conventional thermodynamic cycles, this paper introduce a better TGC theory, i.e. thermodynamic and Gravitational Closed-Cycle (TGCC), and then combine it with mist lift.

TGCC is a completely closed thermodynamics and gravitational cycle with completely recycling working medium.

A proposed TGCC schematic diagram for OTEC is shown in Figure 2, working

medium is heated and evaporated into vapor in evaporator under high pressure and temperature condition, then vapor expand and lift up against gravity to the top condenser. Vapor is completely condensed to cold liquid by lower temperature condenser and that continuously produce lower pressure. Cold liquid is continuously collected into a reservoir and keep the reservoir full, while the cold liquid down to the hydro turbine due to the gravitational force. Liquid gravitational potential energy is conversed to mechanical energy, and then conversed to electric energy.

The process of TGCC cycle contains 5 stages shown in Figure 3.

・ The stage 1 - 2: Cold liquid is heated at constant pressure by an external heat source to become a saturated vapor at bottom. The vapor temperature is, the vapour specific enthalpy is, the quantity of heat absorbed from external heat resource is.

・ The stage 2 - 3: The saturated vapor expands and lifts through the lifting pipe to the top against gravity. This decreases the temperature and pressure of the vapor, and some condensation may occur. Here the vapor temperature is, the vapor pressure is, and the specific enthalpy of vapor is.

・ The stage 3 - 4: The wet vapor then enters a condenser where it is condensed at the constant pressure to become a saturated liquid with the same temperature. Here the specific enthalpy of liquid is, the quantity of heat released to external cold resource is.

・ The stage 4 - 5: The liquid flow down in the declining pipe to the bottom due to gravitational force, while liquid pressure becomes higher and gets the maximum at the point of hydro turbine. Liquid temperature and specific

enthalpy is almost invariant.

・ The stage 5 - 1: The liquid in declining pipe is drained into evaporator through hydro turbine. Liquid temperature and specific enthalpy is almost invariant. But the liquid pressure is plunged from to. The output work of liquid almost equals to the produced electric energy.

In the stage 2 - 3, the vapor pressure vary with the lift height, its equation is

Herein, is the original pressure at bottom (h = 0), is the pressure at the height of, e is the natural constant, is the acceleration of gravity, is molar mass of gasmolecular, is the degree kelvin (K), is the Avogadro constant, and is the Boltzmann constant.

In the stage 2 - 3, the vapor temperature also vary with the lift height, its temperature drop equation without condensation is:

Its temperature drop equation with condensation is:

Herein, is the degree of temperature drop, is acceleration of gravity, is specific isobaric heat capacity, is the lift height, is latent heat, is the mass of condensation. Normally, the temperature drop values comply with the sequence.

According to the cycle process above, we can get its theoretic efficiency equation:

Herein, is specific enthalpy of saturated vapour, is specific enthalpy of wet vapour lifted to the top, is specific enthalpy of condensed liquid.

Accordingly, its ideal efficiency depends on the lift height:

Herein, the volume of liquid, is density of liquid, is the mass of liquid (or vapour), is acceleration of gravity, is the lift height, is the vapour pressure at bottom, is the vapour pressure at top.

According to the Equations (1) (2) and (3), the lift height is limited by the temperature difference. For the example, given, then the maximum condensation height of carbon dioxide gas is 1360 meters. Otherwise, higher than 1360 meters, the carbon dioxide will not condensate at the same temperature since pressure will become lower with higher lift height. Table 1 show detail attributes and parameters of carbon dioxide sample.

According to the Equation (5) and Table 1, we can get the maximum efficiency % that seems to be a little better comparing with conversional OTEC efficiency (around 3%). Considering building a TGCC pipe system reach up to the height of 1360 meters is extremely hard and expensive. If the real lift height is reduced to 600 meters, the efficiency is less than 3%. Thus, the efficiency still need be improved further in order realize more economic and practical.

Reducing the height will decrease efficiency, while vapour condensation and mist lift will be easier at the same cooling temperature. That gives a potential way to improve efficiency at a feasible height. So, this paper combine mist lift concept to TGCC theory to significantly improve its efficiency.

Add a mist generator above the evaporator, we can get a new process of TGCC

cycle contains 8 stages shown in Figure 3.

・ The stage 1 - 2: Cold liquid of mass is heated at constant pressure by an external heat source to become a saturated vapor at bottom. The vapor temperature is, the vapour specific enthalpy is, the quantity of heat absorbed from external heat resource is. Saturated vapor dryness is near to 1.0.

・ The stage 1-1’-2’: Cold liquid of mass is heated at constant pressure, then is pumped into mist generator with a higher pressure (point 1’), and then is sprayed at a lower pressure (point 2’). Mist is produced at point 2’, mist dryness is near to 0.

・ The stage 2 - 6: The saturated vapor expands and lifts to the height where mist generator installed. Vapor flow already gets a very fast velocity.

・ The stage (2’, 6)-6’: Vapor meet with mist, and then combine into liquid-vapor mixture at point 6’. Its mass is.

・ The stage 6’-3’: the vapor expands through the lifting pipe and lift droplets to the top condenser. This decreases the temperature and pressure of the vapor, and some condensation may occur. The dryness of the mixture will be smaller.

・ The stage 3’-4: The mixture enters a condenser where vapor is condensed at the constant pressure. The mixture completely become a saturated liquid at temperature. Here the specific enthalpy of liquid is, the quantity of heat released to external cold resource is.

・ The stage 4 - 5: The liquid of mass flows in the declining pipe down to the bottom due to gravitational force, while liquid pressure becomes higher and gets the maximum at the point of hydro turbine. Liquid temperature and specific enthalpy is almost invariant.

・ The stage 5 - 1: The liquid in declining pipe is drained into evaporator through hydro turbine. Liquid temperature and specific enthalpy is almost invariant. But the liquid pressure is plunged from to. The output work of liquid almost equals to the produced electric energy.

Since a mass of mist is added into vapor of, the ideal efficiency is different from Equation (5), its new equation is as below:

Herein, the liquid volume, is the density of liquid, is the mass of all working medium, is acceleration of gravity, is the lift height, is the vapour pressure at bottom, is the vapour pressure at top, is specific enthalpy of saturated vapour, is specific enthalpy of condensed liquid. is the specific isobar heat capacity of liquid, is the temperature rise. is the heat absorbed by the liquid for producing mist.

If, the dryness of mixture is. According to Equation (6) and the parameters in Table 1, the ideal maximum efficiency is 8.55%.

If, the dryness of mixture is. The maximum efficiency will be 11.06%.

And if, The dryness of mixture is, the maximum efficiency will be 18.17%, which is an exciting value comparing with conventional OTEC efficiency (3% - 4.5%). Even we reduce the lift height to the half (680 m), is still larger than 9%.

In the condition supposed this paper, efficiency increases with more and more mist combined, at the same time dryness decreases accordingly. We can get the maximum efficiency line as shown in Figure 4. Obviously, to left the cycle efficiency trends to a closed MLC without vapor, and to right the cycle efficiency trends to a closed Rankine cycle without mist.

One the one hand, combining more mist with vapour is the primary method to improve efficiency significantly, which will move the stage of 6’-3’ to a new position (6”-3”) trending to zero dryness as shown in Figure 3.

On the other hand, Mist Lift is significantly influenced by the diameter of mist droplets. Many institutes such as the University of Virginia, Dartmouth College and Solar Energy Research Institute have finished much experiments and researches [7]-[12] on mist lift flow rate, velocity, droplet diameter, and inlet pressure, which give full evidences and summarize laws of mist lift. Experiments show that inlet pressure influences mist lift performance [7]. Bigger inlet pressure can get a bigger maximum flow rate before rainout appears. If inlet pressure is 0.2 bar, then the maximum flow rate is 181 kg/s without rainout, dryness is 0.22. If inlet pressure is 5.0 bar, then the maximum flow rate is 945 kg/s without rainout, the dryness is about 0.09. Figure 5 shows the droplet velocity and temperature changing rules in mist lift stage [7]. Droplets velocity can be accelerated by the vapor expanding at bottom and condensing at top. Figure 5 shows droplets

get the maximum velocity at top end of the short lift pipe. We can lengthen the lift pipe to its upper limit so that provide a space for droplet’s moderating process, in order to converse all kinetic energy to gravitational potential energy.