Thermodynamic Process for a Low Temperature, Highly Efficient Heat Engine

Ericsson Cycle Adaptation                The Galt Cycle™

 

 

 

Background

 

Gas Compression and Expansion

Isentropic Compression or Expansion is the process of compressing or expanding a compressible fluid, without the addition or removal of heat.  Isentropic Expansion and Compression of an Ideal Gas is defined by the Pressure and Volume relationship PVk = Constant.  In this case, the variable (k) is the ratio of Cp/Cv for a specific gas.  The absolute work: required or delivered by this process is the area under the curve defined by said PV relationship, between any two volumes.

 

Fig. 1) Expansion Comparison Isentropic, Isothermal and Polytropic

 

Polytropic Compression or Expansion is the process of compressing or expanding a compressible fluid with the addition or removal of heat, at a rate which plots a family of curves defined by the Pressure and Volume relationship PVn=Constant.  In this relationship, n is an exponent value that is set by the addition or removal of heat, to manipulate the slope of the curve, where 1<n<k.

 

If the expansion or compression is contained within an adiabatic container, then n = k, and the process is Isentropic.  If n is manipulated to (1), then the process is an Isothermal process.  Note that as n decreases, the area under the curve, between any two volumes is larger, which implies more work can be done by a Polytropic Expansion process, than by the Isentropic process.  The added work, comes at a price and is equal to the added or subtracted heat to the process during expansion or compression.

 

The ideal situation is the case of a poly-tropic expansion which follows an Isothermal profile because the expanding working substance is held at the ceiling temperature during expansion.  This indicates that the maximum amount of heat from the system was converted to work output.

 

Conventional Heat Engine

A heat engine is a mechanical system which employs a compressible fluid (working substance) to convert heat to useful work, such as rotational shaft work.

 

A heat engine is based on a thermodynamic process which operates across a heat potential and typically involves four phases know as Compression, Heat Addition, Expansion and Heat Rejection, making a complete cycle.  Useful work is produced during the Expansion Phase.  Two key components are predominantly relevant to the useful and efficient operation of a heat engine: Ceiling Temperature and Ceiling Pressure.  Some predominant Heat Engine Cycles are: Carnot, Otto, Diesel, Brayton, and Rankine.

 

The ceiling temperature is the maximum temperature which the working substance will reach in the process and is generally limited by the available temperature in a passive heat source, or by the mechanical properties of the heat engine itself.  The efficiency of this type of heat engine increases with an increase of ceiling temperature.

 

The ceiling pressure is the highest pressure which the working substance will reach in the process and is generally limited by either the ceiling temperature or the mechanical properties of the engine itself.  A higher ceiling pressure allows for greater power production per unit volume (power density).

 

In the compression phase of a conventional heat engine, the working substance undergoes isentropic compression, wherein no heat is added or rejected.  The temperature of a gas which undergoes isentropic compression increases as a function of the compression ration.  The greater the compression ration, the higher the temperature of the working substance prior to the second phase, Heat Addition and the higher the final ceiling temperature and pressure.

 

An important point is to note that engines which work from a passive heat source, must have a heat source temperature which is significantly higher than the temperature of the working substance following compression to effect efficient transfer of heat from the source to the working substance.  Thus the compression ratio of a heat engine is limited by the available temperature of the heat source.

 

 

The Ericsson Cycle

 

The Ericsson Cycle also has four phases, illustrated in Fig. 2 and like the previously mentioned cycles, incorporates Compression, Heat Addition, Expansion and Heat Rejection.  It is the compression and expansion phase’s which make the Ericsson Cycle a very unique process.  The unique design concepts depicted in this invention result in the ability to achieve a very high ceiling pressure and very high thermal efficiency for a given Ceiling Temperature.  Additionally and uniquely this design lends itself to the concepts of heat sharing and thermal regeneration for further thermal efficiency gain.

 

 

Compression Phase

As described above, and illustrated in Fig. 1 from V2 to V1, polytropic compression gives the ability to compress a gas with an endpoint temperature that is lower than if the gas were compressed isentropically. 

 

With polytropic compression, the exponent can ideally be brought down to a value of “1” which yields isothermal compression.  Isothermal compression is a compression process where the temperature of the gas remains constant (also depicted in Fig. 1 from V2 to V1).  No referring to Fig. 2, with isothermal compression as the first process phase, the endpoint temperature of the compressed gas, equals the initial cold-sink starting temperature or (Ambient Temperature).

 

To achieve true Isothermal Compression with a polytropic process, heat must be removed during the compression process and the compression process must proceed at a rate which matches the heat removal soas to maintain a constant temperature.  If the temperature of the gas see saws up and down as the gas is compressed, inefficiencies are introduced.

 

For a polytropic process which realizes an isothermal compression profile, the compression ratio is only limited by the structural design of the heat engine, yielding a maximum allowable compression ratio.

 

Fig. 2) Galt Adaptation of the Ericsson Cycle PV Diagram

 

In the PV Diagram of Fig. 2 the working substance is initially compressed isothermally from the Ambient Pressure, Volume and Temperature conditions at point (1), to the Ceiling Pressure, Minimum Volume and Ambient Temperature of point (2).  The work required to complete this step, in ft-lbs, is the area under the PV Diagram curve in Fig. 2 from point (1) to (2). 

 

For an Ideal Gas, the equation for this work is:               Work (ft-lb) = R T ln (Rc)

                                                                                                R -           Gas Constant (ft-lb/lb-DegR)

                                                                                                T -           Absolute Temp Deg Rankine

                                                                                                Rc -         Compression Ration (Pceiling/Pambient)

 

The amount of heat which must be rejected, Q(out) is equal to the amount of work performed, and can be expressed in the units (btu) by the relation Q (btu) = W (ft-lb)/778 (ft-lb)/(btu).

 

Heat Addition Phase

Since the temperature of the working substance at point (2) in Fig. 2 is still ambient, heat addition can be accomplished using a much lower temperature heat source.  The Mechanical Design Ceiling Pressure has already been attained by the working substance, so heat addition is done in an Isobaric Process, which is to say, that heat is added at a constant pressure.  When heat is added to a gas under constant pressure, it expands to a larger volume.  This expanded volume is noted in the PV Diagram of Fig. 2 during the heat addition phase from point (2) to (3).  The temperature of the working substance increases during this phase.  It is the intent of this invention to add sufficient heat so as to reach the Mechanical Design Ceiling Temperature for the working substance at point (3).

 

The heat added in this phase Q(2-3) can be calculated for an Ideal Gas with the following equation:

                Q (btu) =                Cp (Tceiling – Tambient)

                                                Cp-          Heat Capacity for Constant Pressure of the Gas (btu/deg R)

                                                Tc-          Absolute Temperature in Deg. Rankine for the Ceiling Temperature

                                                Ta-          Absolute Temperature in Deg. Rankine for the Ambient Temperature

 

Expansion (Work) Phase

At the start of this phase, the working substance is at its Ceiling Pressure and Ceiling Temperature, shown on the PV Diagram in Fig. 2 at point (3). 

 

Isentropic Expansion is the natural expansion for work out of a conventional thermodynamic heat engine.  It was shown in the discussion above regarding Gas Compression and Expansion and illustrated in the PV Diagram of Fig. 1, that Isentropic Expansion has the least amount of work output.

 

More work out can be yielded from expansion of the working substance using a polytropic expansion process where the polytropic exponent is less than the Ideal Gas constant (k).  Clearly shown in the Expansion profiles of Fig 1, the lower the exponent attained in a polytropic expansion, the greater the work out achieved.  If the Polytropic exponent of “1” is attained during expansion, an Isothermal Expansion Phase is accomplished and the final temperature of the working substance after expansion is still the Ceiling Temperature.

 

In the Ericsson Cycle, during the expansion phase, using a polytropic process to achieving Isothermal Expansion is the ideal goal, in that the working substance remains at the Mechanical Design Ceiling Temperature Limit and the Maximum amount of system work is performed.

 

To achieve polytropic expansion of an exponent less than (k), with the ideal goal of attaining Isothermal Expansion, heat must be added during the expansion phase along the path between states (3) and (4) on the PV Diagram in Fig. 2.  A unique nuance of this invention is that Heat Addition takes place, not only in the second phase, but also during the Expansion (Work) phase, and translates directly into more work out.

 

For the Ideal Model of Isothermal Expansion during the Work phase, the amount of work out can be visualized by the area under the curve of the PV Diagram of Fig. 2, from point (3) to (4).  The calculations for this work out, is the same formula as for the compression phase as follows:

 

 For an Ideal Gas, the equation for this work is:              Work (ft-lb) = R T ln (Rc)

                                                                                                R -           Gas Constant (ft-lb/lb-DegR)

                                                                                                T -           Absolute Temp Deg Rankine

                                                                                                Rc -         Compression Ration (Pceiling/Pambient)

 

The amount of heat which must be added, Q(3 - 4) is equal to the amount of work performed, and can be expressed in the units (btu) by the relation Q (btu) = W (ft-lb)/778 (ft-lb)/(btu).

 

Heat Rejection

The final phase of the Ericsson Cycle is Heat Rejection, where in the working substance has been completely expanded to Ambient Pressure and is at its maximum volume and some elevated temperature which is ideally the ceiling Temperature.  In reality, the working substance is exhausted to the ambient environment at this time to Isobaric cooling back to the initial state of Ambient Pressure, Temperature and Volume.

 

The amount of heat rejected back to the ambient environment show as Q(4 - 1) in the Fig. 2, for an Ideal Gas can be calculated by the same formula used during  Heat Addition at Constant Pressure and is as follows:

 

                Q (btu) =                Cp (Tceiling – Tambient)

                                                Cp-          Heat Capacity for Constant Pressure of the Gas (btu/deg R)

                                                Tc-          Absolute Temperature in Deg. Rankine for the Ceiling Temperature

                                                Ta-          Absolute Temperature in Deg. Rankine for the Ambient Temperature

One point worth noting is that the amount of heat rejected Q(4 - 1) to the environment is the same amount of heat which is added Q(2 - 3) during the Heat Addition phase and the Temperature endpoints are the same, but opposite.  This creates a significant opportunity for Heat Regeneration to improve the Thermal Efficiency of the applied system.  For various design embodiments using the Ericsson Process, it is very feasible to achieve a Regeneration Coefficient (Rg) of greater than 1/2, implying that much of the exhaust heat, can be re-applied to the system during heat addition.

 

 

Ericsson’s Flaw

While the Ericsson Cycle, in limited application, can achieve the highest engine efficiency, it has not been possible to produce a practical engine of this type.

 

The key problem with the Ericsson Cycle is easy to understand.  As explained earlier, a Heat Engine has four phases, one being Compression.  It is common knowledge that when air is compressed, it heats.  For the compression phase of the Ericsson Cycle, this heat must be dissipated, which takes time and an available heat sink.  For operation, the engine must have a very slow operational speed to allow time for heat dissipation, a low compression ratio, and have ready access to a large heat sink.  These are extreme limiting factors, which lead to a dead end for the Ericsson Cycle.

 

 

Galt Cycle

It is first important to accept that the Ericsson Cycle, or any heat cycle discussed herein, is based on a continuous cycle where a given volume of air is compressed, heated, expanded to produce work, then cooled back to its original state, creating a repeatable cycle.

 

Further, it is possible to approximate any of these cycles by compressing (n) volumes of air, in advance, into a storage container, then at a later time, extract the stored, compressed air, in small packets to complete the final three phases of  Heat Addition, Working Expansion and Cooling.  This is referred to herein as Disassociated Compression.

 

An important aspect of the system design is that in this scenario, heat addition is also required to the storage container during operation.  With the removal of each packet, in effect, the container is expanding from its original volume to a volume which also includes the packet size.  This expansion not only executes the first amount of work out, in to the turbine, but will also act to cool the contents to the container slightly due to the Isentropic Expansion.  Since the goal of this process is to maintain Isothermal Expansion, even the container must be heated as packets are drawn, to keep its temperature constant.

 

For the initial conditions of a container, pressure is at maximum (Pm) and temperature is at ambient (To), and heat is added to reach the ceiling temperature of (Tc).

 

Variation I

A packet of fixed mass is drawn from the container, heated to Tc then expanded Isothermally doing work through the entire process.  The exhausted, expanded air is still at Tc.  This hot exhaust is then reapplied to preheat the next packet drawn, as well as to supply heat to the container.

 

To truly achieve full Isothermal Expansion, heat must be added to the container at a rate which not only counters the cooling effect of expansion, but also to steadily increase the temperature of the container above To, so as to maintain the pressure in the container at Pm.  As packets are drawn, the temperature inside the container is increased to maintain peak pressure.

 

This process continues until the container reaches the ceiling temperature Tc.  From this point on, when a packet is drawn, it no longer requires heat addition since it is already at the ceiling temperature.  At this point, heat is added for the Isothermal Expansion and the heat of the exhaust is regenerated to maintain the canister at Tc.

 

This scenario results in an equivalent process of treating the container as a single volume and executing the Ericsson Cycle. 

 

A significant advantage to this variation is for increased thermal efficiency through regeneration.  In the original Ericsson Cycle, regeneration plays an important role in its high efficiency, since the hot exhaust can preheat the compressed phase, referring to Fig. 2, from point (2) to (2’).  This reduces the amount of external heat required for the heat addition phase of (2 – 3) in Fig. 2 and in effect reduces it to heat addition (2’ – 3).

 

However in this variation, regeneration is not only to assist in preheating, it also contributes to the heat maintenance of the container, which is a ready sink.

 

Variation II

 

It may not be structurally feasible to add heat to a container of compressed air, to increase its internal temperature much above ambient.  This may cause container fatigue and constitute a safety concern.

 

A second alternative, which yields the same efficiency equation as the Ericsson Cycle, and Variation I is presented here.  The only short coming of this variation is that a container produces slightly less output power per volume.  Hence, for the same power output, a slightly larger container is required.

 

In this variation, every thing is the same as in Variation I, except that the container is only maintained at To, rather than steadily increased to Tc.  Interestingly the efficiency equation does ratify to the same result, and the ability to use heat regeneration is still highly available to not only preheat the packet charge, but to maintain the container temperature at To.

 

 

Summary

Described here in is a Thermodynamic Heat Engine process called the Galt Cycle, based on an adaptation of the Ericsson Cycle, consisting of four major phases.

 

Reiterated they are:

1.)                  Disassociated Isothermal Compression

1’.)          Working Packet Extraction

2.)                  Working Packet Heat Addition – Isobaric Heating

3.)                  Working Packet Expansion – Polytropic Expansion (Ideally n=1 Isothermal Expansion)

4.)                  Packet Heat Rejection – Regeneration and Isobaric Cooling

 

By separating in time, the compression of air, and the action of Heat Addition, Working Expansion and Cooling, the drawbacks of the Ericsson Cycle can be avoided.  This method allows for the slow Isothermal Compression of air into a storage container, which can then be used to power a heat engine based on the Ericsson Cycle, until the reservoir of stored air is exhausted.  The compression can then be conducted where there is sufficient time to dissipate the accumulated heat, and an available heat sink to receive the heat.

 

Note that the new drawback is now operational time.  The engine can only operate while there is available air in the storage container.  When the container is exhausted, it will require recharging.

 

There are, however major applications for a heat engine which are accommodating to this constraint, such as Solar Energy, and Hybrid Automobiles.

 

As a calculation for efficiency, the PV diagram depicting the full Ericsson Cycle will be used.  It will be for a follow up paper to present the derivations of Variation I and Variation II to illustrate the equivalence.

 

Efficiency

The Net Work performed by the Galt Cycle is visualized by the bounded area within the PV Diagram of Fig 2 from paths (1), (2), (3) and (4).

 

Calculations for Net Thermal Efficiency are as follows:

 

Thermal Efficiency = Net Work Out  / Net Heat Added  which follows:

 

Effic        =              W(out)

                                Q(in)

 

 

=              W(2-3) + W(3-4) – W(4-1) – W(1-2)

                                Q(2-3) + Q(3-4) – Rg Q(4-1)

 

                =              Pc [V(3) – V(2)] + R Tc ln (Rc) - Pa [V(4)-V(1)] + R Ta ln (Rc)

                                778 Cp (Tc – Ta) + R Tc ln (Rc) – Rg 778 Cp (Tc – Ta)

 

                =              Pc V(3) – Pc V(2) + R ln (Rc) (Tc – Ta) - Pa V(4) + Pa V(1)

                                (1 – Rg) 778 Cp (Tc – Ta) + R Tc ln (Rc)

 

                =              R Tc – R Ta + R ln (Rc) (Tc – Ta) - R Tc + R Ta

                                (1-Rg) 778 Cp (Tc – Ta) + R Tc ln (Rc)

 

                =              R ln (Rc) (Tc – Ta)

                                (1-Rg) 778 Cp (Tc – Ta) + R Tc ln (Rc)

 

For Rg ~ 1 =>

 

Effic        =              R ln (Rc) (Tc - Ta)                =              Tc – Ta                  ( Carnot Efficiency)

                                R Tc ln (Rc)                                           Tc

 

 

Higlights

 

1.)                  A mechanical thermodynamic heat engine apparatus can be built which has the ability to execute the Galt Cycle and realize a functioning system that closely approaches  the ideal, theoretical, Heat Engine defined by the Carnot Cycle.

2.)                  By using a Polytropic Compression process designed to yield an Isothermal profile, maximum compression ratios can be obtained while maintaining lower ceiling temperatures for improved power density of the heat engine.

3.)                  Greater Thermal Efficiency can be obtained during the working expansion phase by using a Polytropic Expansion process to yield an Isothermal Expansion profile thus creating the opportunity to add more heat during expansion from the Heat Source for a greater conversion of heat to work output.

4.)                  Having an Exhaust Temperature which is at or near the ceiling operating temperature provides an opportunity for useful thermal regeneration with a large overall efficiency gain.

5.)                  Higher efficiency at lower ceiling temperatures and greater power density yield a heat engine suited for low grade energy sources such as solar.

6.)                  Disassociated Compression eliminates the run time loss of power for compression and the draining container makes a ready heat sink for exhaust heat regeneration.