CROSSREFERENCES TO RELATED APPLICATIONS

The present application is a continuation of and claims priority to provisional patent application Nos. 61,180,913 filed May 25, 2009 titled Thermally Isolated Counter Flowing Heat Exchanger; 61,178,881 filed May 15, 2009 titled Thermal Energy Transfer and Storage System; 61,178,828 filed May 15, 2009 titled Energy System Using Thermally Isolated Cold and Hot Transfer Areas, each of which is hereby incorporated by reference in its entirety.
FIELD OF THE INVENTION

The present invention relates generally to thermal energy transfer and specifically to transfer of thermal energy between isolated counter flowing fluids.
BACKGROUND OF THE INVENTION

Heat transfer is the transition of thermal energy from a heated item to a cooler item. When an object or fluid is at a different temperature than its surroundings or another object, transfer of thermal energy, also known as heat transfer, or heat exchange, occurs in such a way that the body and the surroundings reach thermal equilibrium. Heat transfer always occurs from a hot particle to a cold particle, a result of the second law of thermodynamics. Therefore, where there is a temperature difference between objects in proximity, heat transfer between them can never be stopped; it can only be slowed.

Thermal energy is conducted well by metals such as copper, platinum, gold, iron, etc. Energy can also be transferred through the walls of other materials such as polymers e.g. polyethylene, polypropylene, nylon, if there is sufficient time or an increased rate of flow of fluids relative to surface area on either side of the material—thereby utilizing forced convection. The increased surface area coupled with an increased length of time for the heat transfer increases the heat transfer efficiency for polymer materials

Heat Reclamation Introduction—a Model of a Simple Heat Exchanger

A simple fluid heat exchanger might be thought of as two straight pipes with fluid flow, which are thermally connected. Let the pipes be of equal length L, carrying fluids with heat capacity C_{i }(energy per unit mass per unit change in temperature) and let the mass flow rate of the fluids through the pipes be j_{i }(mass per unit time), where the subscript i applies to pipe 1 or pipe 2.

The temperature profiles for the pipes are T_{1}(x) and T_{2}(x) where x is the distance along the pipe. Assume a steady state, so that the incoming temperature profiles are not functions of time. Assume also that the only transfer of heat from a small volume of fluid in one pipe is to the fluid element in the other pipe at the same position. There will be no transfer of heat along a pipe due to temperature differences in that pipe. By Newton's law of cooling, see below, the rate of change in energy of a small volume of fluid is proportional to the difference in temperatures between it and the corresponding element in the other pipe:

$\frac{\uf74c{u}_{1}}{\uf74ct}=\gamma \ue8a0\left({T}_{2}{T}_{1}\right)$
$\frac{\uf74c{u}_{2}}{\uf74ct}=\gamma \ue8a0\left({T}_{1}{T}_{2}\right)$

Where u_{i}(x) is the thermal energy per unit length and γ is the thermal connection constant per unit length between the two pipes. This change in internal energy results in a change in the temperature of the fluid element. The time rate of change for the fluid element being carried along by the flow is:

$\frac{\uf74c{u}_{1}}{\uf74ct}={J}_{1}\ue89e\frac{\uf74c{T}_{1}}{\uf74cx}$
$\frac{\uf74c{u}_{2}}{\uf74ct}={J}_{2}\ue89e\frac{\uf74c{T}_{2}}{\uf74cx}$

Where J_{i}=C_{iji }is the “thermal mass flow rate”. The differential equations governing the heat exchanger may now be written as:

${J}_{1}\ue89e\frac{\partial {T}_{1}}{\partial x}=\gamma \ue8a0\left({T}_{2}{T}_{1}\right)$
${J}_{2}\ue89e\frac{\partial {T}_{2}}{\partial x}=\gamma \ue8a0\left({T}_{1}{T}_{2}\right)$

Note that, since the system is in a steady state, there are no partial derivatives of temperature with respect to time, and since there is no heat transfer along the pipe, there are no second derivatives in x as is found in the heat equation. These two coupled firstorder differential equations may be solved to yield:

${T}_{1}=A\frac{{\mathrm{Bk}}_{1}}{k}\ue89e{\uf74d}^{k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ex}$
${T}_{2}=A+\frac{{\mathrm{Bk}}_{2}}{k}\ue89e{\uf74d}^{k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ex}$

Where k1=γ/J_{1}, k_{2}=γ/J_{2}, k=k_{1}+k_{2 }and A and B are two as yet undetermined constants of integration. Let T_{10 }and T_{20 }be the temperatures at x=0 and let T_{1L }and T_{2L }be the temperatures at the end of the pipe at x=L, where L is the pipe length. Define the average temperatures in each pipe as:

${\stackrel{\_}{T}}_{1}=\frac{1}{L}\ue89e{\int}_{0}^{L}\ue89e{T}_{1}\ue8a0\left(x\right)\ue89e\uf74cx$
${\stackrel{\_}{T}}_{2}=\frac{1}{L}\ue89e{\int}_{0}^{L}\ue89e{T}_{2}\ue8a0\left(x\right)\ue89e\uf74cx$

Using the solutions above, these temperatures are:

${T}_{10}=A\frac{{\mathrm{Bk}}_{1}}{k}$
${T}_{20}=A+\frac{{\mathrm{Bk}}_{2}}{k}$
${T}_{1\ue89eL}=A\frac{{\mathrm{Bk}}_{1}}{k}\ue89e{\uf74d}^{\mathrm{kL}}$
${T}_{2\ue89eL}=A+\frac{{\mathrm{Bk}}_{2}}{k}\ue89e{\uf74d}^{k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eL}$
${\stackrel{\_}{T}}_{1}=A\frac{{\mathrm{Bk}}_{1}}{{k}^{2}\ue89eL}\ue89e\left(1{\uf74d}^{k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eL}\right)$
${\stackrel{\_}{T}}_{2}=A+\frac{{\mathrm{Bk}}_{2}}{{k}^{2}\ue89eL}\ue89e\left(1{\uf74d}^{k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eL}\right)$

Choosing any two of the above temperatures will allow the constants of integration to be eliminated, and that will allow the other four temperatures to be found. The total energy transferred is found by integrating the expressions for the time rate of change of internal energy per unit length:

$\frac{\uf74c{U}_{1}}{\uf74ct}={\int}_{0}^{L}\ue89e\frac{\uf74c{u}_{1}}{\uf74ct}\ue89e\uf74cx={J}_{1}\ue8a0\left({T}_{1\ue89eL}{T}_{10}\right)=\gamma \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eL\ue8a0\left({\stackrel{\_}{T}}_{2}{\stackrel{\_}{T}}_{1}\right)$
$\frac{\uf74c{U}_{2}}{\uf74ct}={\int}_{0}^{L}\ue89e\frac{\uf74c{u}_{2}}{\uf74ct}\ue89e\uf74cx={J}_{2}\ue8a0\left({T}_{2\ue89eL}{T}_{20}\right)=\gamma \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eL\ue8a0\left({\stackrel{\_}{T}}_{1}{\stackrel{\_}{T}}_{2}\right)$

By the conservation of energy, the sum of the two energies is zero. The quantity T _{2}− T _{1 }is known as the log mean temperature difference and is a measure of the effectiveness of the heat exchanger in transferring heat energy.

Log Mean Temperature Difference

The log mean temperature difference (LMTD) is used to determine the temperature driving force for heat transfer in flow systems (most notably in heat exchangers). The LMTD is a logarithmic average of the temperature difference between the hot and cold streams at each end of the exchanger. The use of the LMTD arises straightforwardly from the analysis of a heat exchanger with constant flow rate and fluid thermal properties.

For countercurrent flow (i.e. where the hot stream, liquid or gas, goes from left to right, and the cold stream, again liquid or gas goes from right to left), is given by the following equation:

$\mathrm{LMTD}=\frac{\left({T}_{1}{t}_{2}\right)\left({T}_{2}{t}_{1}\right)}{\mathrm{ln}\ue8a0\left(\frac{{T}_{1}{t}_{2}}{{T}_{2}{t}_{1}}\right)}$

And for parallel flow (i.e. where the hot stream, liquid or gas, goes from left to right, and so does the cold stream), is given by the following equation:

$\mathrm{LMTD}=\frac{\left({T}_{1}{t}_{1}\right)\left({T}_{2}{t}_{2}\right)}{\mathrm{ln}\ue8a0\left(\frac{{T}_{1}{t}_{1}}{{T}_{2}{t}_{2}}\right)}$

T_{1}=Hot Stream Inlet Temp.

T_{2}=Hot Stream Outlet Temp.

t_{1}=Cold Stream Inlet Temp.

t_{2}=Cold Stream Outlet Temp.

It makes no difference which temperature differential is 1 or 2 as long as the nomenclature is consistent. The larger the LMTD, the more heat is transferred based on a larger heat difference.

Yet a third type of unit is a crossflow exchanger, in which one system (usually the heat sink) has the same nominal temperature at all points on the heat transfer surface. This follows similar mathematics, in its dependence on the LMTD, except that a correction factor F often needs to be included in the heat transfer relationship.

Derivation

Assume heat transfer is occurring between two fluids (T_{1 }and T_{2}) with a temperature difference of ΔT(A) at point A and ΔT(B) at point B (where ΔT(z)=T_{2}(z)−T_{1}(z)). The direction of fluid flow does not need to be considered. Since LMTD is the average temperature difference of the two streams between points A and B the following formula defines LMTD:

$\mathrm{LMTD}=\frac{{\int}_{A}^{B}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(z\right)\ue89e\uf74cz}{{\int}_{A}^{B}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74cz}$

Where z is the distance parallel to the motion of the two fluids.

The rate of change of the temperature of the two fluids, T_{1 }and T_{2 }respectively, is proportional to the temperature difference between the two fluids:

$\frac{\uf74c{T}_{1}}{\uf74cz}={k}_{a}\ue8a0\left({T}_{1}\ue8a0\left(z\right){T}_{2}\ue8a0\left(z\right)\right)={k}_{a}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(z\right)$
$\frac{\uf74c{T}_{2}}{\uf74cz}={k}_{b}\ue8a0\left({T}_{2}\ue8a0\left(z\right){T}_{1}\ue8a0\left(z\right)\right)={k}_{b}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(z\right)$

Therefore:

$\frac{\uf74c\left(\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\right)}{\uf74cz}=\frac{\uf74c\left({T}_{2}{T}_{1}\right)}{\uf74cz}=\frac{\uf74c{T}_{2}}{\uf74cz}\frac{\uf74c{T}_{1}}{\uf74cz}=K\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(z\right)$

Where K=k_{a}+k_{b}.

This gives a value for dz:

$\mathrm{dz}=\frac{1}{K}\ue89e\frac{\uf74c\left(\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\right)}{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT}$

Substituting back into the formula for LMTD:

$\mathrm{LMTD}=\frac{{\int}_{A}^{B}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(z\right)\ue89e\uf74cz}{{\int}_{A}^{B}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74cz}=\frac{{\int}_{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\left(A\right)}^{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)}\ue89e\frac{1}{K}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\left(\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\right)}{{\int}_{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(A\right)}^{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)}\ue89e\frac{1}{K}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\frac{\uf74c\left(\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\right)}{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT}}$

Integrating gives:

$\mathrm{LMTD}=\frac{{\int}_{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(A\right)}^{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)}\ue89e\frac{1}{K}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\left(\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\right)}{{\int}_{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(A\right)}^{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)}\ue89e\frac{1}{K}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\frac{\uf74c\left(\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\right)}{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT}}=\frac{{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT]}_{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(A\right)}^{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)}}{{\mathrm{ln}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT]}_{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(A\right)}^{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)}}=\frac{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(A\right)}{\mathrm{ln}\ue8a0\left(\frac{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(B\right)}{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(A\right)}\right)}$

Trivial case:

$\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(z\right)=C\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{for}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{all}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89ez$
$\mathrm{LMTD}=\frac{{\int}_{A}^{B}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT\ue8a0\left(z\right)\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74cz}{{\int}_{A}^{B}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74cz}=\frac{C\ue89e{\int}_{A}^{B}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74cz}{{\int}_{A}^{B}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74cz}=C$

Newton's Law of Cooling

A related principle, Newton's law of cooling, states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings, or environment. The law is

$\frac{\uf74cQ}{\uf74ct}=h\xb7A\ue8a0\left({T}_{0}{T}_{\mathrm{env}}\right)$

Q=Thermal energy in Joules

h=Heat transfer coefficient

A=Surface area of the heat being transferred

T_{0}=Temperature of the object's surface

T_{env}=Temperature of the environment

This form of heat loss principle is sometimes not very precise; an accurate formulation may require analysis of heat flow, based on the (transient) heat transfer equation in a nonhomogeneous, or else poorly conductive, medium. The following simplification may be applied so long the surface conductance related to the interior thermal conductivity in a body is uniform (Biot number). This is the case with a body of water as it has relatively high internal conductivity, such that (to good approximation) the entire body is at the same uniform temperature as it is cooled from the outside, by the environment. If this is the case, then it is easy to derive from these conditions the behavior of exponential decay of temperature of a body. In such cases, the entire body is treated as lumped capacitance heat reservoir, with total heat content which is proportional to simple total heat capacity Q=mcT, and the temperature of the body. If T(t) is the temperature of such a body at time t, and T_{env }is the temperature of the environment around the body, then

$\frac{\uf74cT\ue8a0\left(t\right)}{\uf74ct}=r\ue8a0\left(T{T}_{\mathrm{env}}\right)$

Where r is a positive constant characteristic of the system, which must be in units of 1/time, and is therefore sometimes expressed in terms of a time constant: r=1/t_{0}.

The solution of this differential equation, by standard methods of integration and substitution of boundary conditions, gives:

T(t)=T _{env}+(T)(0)−T _{env})e ^{−rt}.

Here, T(t) is the temperature at time t, and T(0) is the initial temperature at zero time, or t=0.

If: ΔT(t) is defined as: T(t)−T_{env}, where ΔT(0) is the initial temperature difference at time 0, then the Newtonian solution is written as:

ΔT(t)=ΔT(0)e ^{−rt}.

Uses

For example, simplified climate models may use Newtonian cooling instead of a full (and computationally expensive) radiation code to maintain atmospheric temperatures that are greater than the heat of vaporization.

Assumptions

It has been assumed that the rate of change for the temperature of both fluids is proportional to the temperature and volume difference; this assumption is valid for fluids with a constant specific heat, which is a good description of fluids changing temperature over a relatively small range, i.e. within the temperatures of boiling and solidifying. However, if the specific heat changes, the LMTD approach will no longer be accurate. Two particular cases where the LMTD is not applicable are condensers and reboilers, where the latent heat associated to phase change makes the hypothesis invalid. The present invention operates within the assumed parameters.

Incorporated by reference herein are:

Kay J M & Nedderman R M (1985) Fluid Mechanics and Transfer Processes, Cambridge University Press

S. S. Kutateladze and V. M. Borishanskii, “A Concise Encyclopedia of Heat Transfer”, Pergamon Press, 1966.

Thermal Energy Storage (TES)

The standard enthalpy of fusion (symbol: ΔHfus), also known as the heat of fusion or specific melting heat, is the amount of thermal energy which must be absorbed or evolved for 1 mole of a substance to change states from a solid to a liquid or vice versa. It is also called the latent heat of fusion or the enthalpy change of fusion, and the temperature at which it occurs is called the melting point.

When thermal energy is withdraw from a liquid or solid, the temperature falls. Conversely, when heat energy is added, the temperature rises. However, at the transition point between solid and liquid (the melting point), extra energy is required called the heat of fusion. To go from liquid to solid, the molecules of a substance must become more ordered. For them to maintain the order of a solid, extra heat must be withdrawn. In the other direction, to create the disorder from the solid crystal to liquid, extra heat must be added.

The heat of fusion can be observed if the temperature of water is measure as it freezes. If a closed container of room temperature water is plunged into a very cold environment (about −20° C.), the temperature will fall steadily until it drops just below the freezing point (0° C.). The temperature then rebounds and holds steady while the water crystallizes Once completely frozen, the temperature will fall steadily again. The temperature stops falling at (or just below) the freezing point due to the heat of fusion. The energy of the heat of fusion must be withdrawn (the liquid must turn to solid) before the temperature can continue to fall.

To heat one kilogram (about 1 liter) of water from 10° C. to 30° C. requires 20 kcal. However, to melt ice and raise the resulting water temperature 20° C. requires extra energy. To heat ice from 0° C. to water at 20° C. requires:

(1) 80 cal/g (heat of fusion of ice)=80 kcal for 1 kg

PLUS

(2) 1 cal/(g·° C.)=20 kcal for 1 kg to go up 20° C.

=100 kcal−or approximately 5 times the energy with the heat of fusion

High peak summertime loads drive the capital expenditures of the electricity power generation industry. However, the peak demand is not sustained throughout the day. As a result, power companies must supplement the sustained load with additional power generation that can be brought on to provide peak load but turns off when demand subsides. The power industry meets these peak loads with lowefficiency peaking power plants, usually gas turbines, which have lower capital costs but higher fuel costs. At night, the baseline demand uses less expensive sources of electricity like nuclear power. A kilowatthour of electricity consumed at night can be produced at much lower marginal cost. In order to reduce peak demand and to increase offpeak usage, utilities have begun to pass these lower costs to consumers, in the form of Time of Use (TOU) rates, or Real Time Pricing (RTP) Rates.

Thermal energy is cheaper than any other energy source. Thermal energy storage is made practical by the high heat of fusion of water. One metric ton of water, just one cubic meter, can store 334 MJ (317 k BTUs, 93 kWh or 26.4 tonhours). In fact, ice was originally transported from mountains to cities for use as a coolant, and the original definition of a “ton” of cooling capacity (heat flow) was the heat to melt one ton of ice every 24 hours. This is the heat flow one would expect in a 3,000squarefoot (280 m2) house in Boston in the summer. This definition has since been replaced by less archaic units: one ton HVAC capacity=12,000 BTU/hour. Either way, a relatively small storage facility can hold enough ice to cool a large building for a day or a week, whether that ice is produced by anhydrous ammonia chillers or hauled in by horsedrawn carts.

Use in Air Conditioning

The most widely used form of thermal energy storage technology is in large building or campuswide air conditioning or chilled water systems. Air conditioning systems, especially in commercial buildings, are the most significant contributors to the peak electrical loads on hot summer days. In this application a relatively standard chiller is run at night to produce a pile of ice. During the day, water is circulated through the ice storage system to produce chilled water that would normally be produced by the chillers.

Ice storage systems are often classified as partial or full storage. A partial storage system minimizes capital investment by running the chillers 24 hours a day. At night they produce ice for storage, and during the day they chill water for the air conditioning system, their production augmented by water circulating through the melting ice. A conventional ice storage system usually runs in icemaking mode for 16 to 18 hours a day, and in icemelting mode for 6 to 8 hours a day. Capital expenditures can be reduced because the chillers can be sized smaller to accommodate a smaller need.

A full storage system provides the complete need for cooling during a peak demand. This minimizes the cost of energy to run the system by shutting off the chillers entirely during peak load hours. Such a system requires chillers larger than a partial storage system, and a larger ice storage system, so the capital cost is higher. However, ice storage systems are inexpensive enough that full storage systems are often competitive with conventional air conditioning designs.

There are advantages to air conditioning thermal storage. The fuel used at night to produce electricity is a domestic resource in most countries, so that less imported fuel is used. This process also has been shown in studies to significantly reduce the emissions associated with producing the power for air conditioners, since inefficient “peaker” plants are replaced by low emission base load facilities in the evening. The plants that produce this power are often more efficient than the gas turbines that provide peaking power during the day. And, because the load factor on the plants is higher, fewer plants are needed to service the load.

A new twist on ice storage technology uses ice as a condensing medium for refrigerant. In this case, regular refrigerant is pumped to ice storage units where it is used. However, instead of needing a compressor to convert it back in to a liquid, the low temperature of the ice is used to chill the refrigerant back in to a liquid. This type of system allows existing refrigerant based HVAC equipment to be converted to Thermal Energy Storage systems, something that could not previously be easily done with chilled water technology. In addition, unlike watercooled chilled water systems that do not experience a tremendous difference in efficiency from day to night, this new class of equipment typically displaces daytime operation of air cooled condensing units. In areas where there is a significant difference between peak daytime temperatures and off peak temperatures, this type of unit is typically more energy efficient than the equipment it is replacing.

Further information regarding thermal storage systems is found at Thermal Storage and Deregulation by Brian Silvetti, PE, and Mark MacCracken, PE. 1998, and Thermal Storage System Achieves Operating and FirstCost Savings by Edward O'Neal, PE 1999, each of which are incorporated herein by reference.

Air conditioning during summer daytime hours is the largest single contributor to a building's energy cost. The present invention significantly reduces energy cost by enabling energyintensive cooling equipment to be operated during offpeak hours when electricity rates are the least expensive, while still providing a cool and very comfortable environment for occupants.

The thermal energy storage system of the present invention enables facilities operators to shift the largest portion of a building's electrical load from HighCost Peak Hours to LowCost OffPeak Hours. Other benefits of the present invention include those to an owner:

 Reduces cooling costs by approximately 40 percent by shifting a building's energy demand from onpeak to offpeak electric times.
 With a partial storage OffPeak Cooling system, engineers can specify chillers at about 50 to 60 percent of the previous size, reducing capital outlays.
 The size and cost of air handlers, motors, ducts, and pumps can be reduced by about 20 to 40 percent.
 OPC lowers the relative humidity within a building and, as a result, occupants feel comfortable even if the thermostat is set at a higher, more costsaving setting.
 Increases a building's load factor so lessexpensive energy rates can be negotiated with energy providers.
 The equipment required for the present invention can be added to an existing chiller system to increase cooling capacity and reduce cooling costs.
 The present invention provides a quantifiable return on investment.
 Hypothetically, on a 600 ton, 300,000 squarefoot building, where the cost of a kW is $10, saving 600 kW per month translates to a savings of $6,000.
 Being constructed of noncorrosive materials, the present invention lasts longer than traditional heat exchangers.

Other benefits of the present invention include those to the environment and society:

 Because the chiller used within the present invention is about 50 to 60 percent smaller, less refrigerant is needed. Because refrigerant escapes from a chiller system over time, the smaller chiller of the present minimizing the use of refrigerant benefits the environment.
 Reduces sourceenergy consumption by about 8 to 34 percent, which means that energy providers will generate fewer polluting emissions.
 Reduces emissions and use of dirtiest power plants.
 Increases load factor of generation by about 25 percent.
 Delays the need for additional power plants.

Other benefits of the present invention include those to energy providers:

 The present invention reduces peak electrical demand, allowing energy providers to produce more electricity at increased efficiencies and avoid costly expansion.
 In numerous studies, it has been proven that electricity is produced and delivered much more efficiently during offpeak hours than during onpeak periods. For every kilowatthour of energy that is shifted from onpeak usage to offpeak, there is a reduction in the source fuel needed to generate it. While the exact amount of savings varies, studies show a range from 8 to 30 percent for two of the major utilities studied. The reduction in source fuel normally results in a reduction of greenhousegas emissions produced by the power plant. (Source Energy & Environmental Impacts of Thermal Energy Storage, California Energy Commission (CEC), P50095005, February 1996)
 Increases a utility's load factor.

In conventional air conditioning system design, cooling loads are measured in terms of “Tons of Refrigeration” (or kW's) required, or more simply “Tons.” TES systems, however, are measured by the term “TonHours” (or kWh). Realistically, no building air conditioning system operates at 100% capacity for the entire daily cooling cycle. Air conditioning loads peak in the afternoon—generally from 2 to 4 PM—when ambient temperatures are highest. In a typical 100ton chiller capacity building air conditioning load profile during the peak design day, the full capacity is needed for only two hours in the cooling cycle. For the other eight hours, less than the total chiller capacity is required. The total actual TonHr. needed is only 750. A 100ton chiller must be specified, however, to handle the peak 100ton cooling load.

“Diversity Factor” is defined as the ratio of the actual cooling load to the total potential chiller capacity, or: Diversity Factor (%)=Actual TonHr./Total Potential TonHr.=750/1000. This chiller, then, has a Diversity Factor of 75 percent. It is capable of providing 1000 tonhours when only 750 tonhours are required. If the Diversity Factor is low, the system's cost efficiency is also low. (The lower the Diversity Factor, the greater the potential benefit from a TES system.) Dividing the total tonhours of the building by the number of hours the chiller is in operation gives the building's average load throughout the cooling period. If the air conditioning load could be shifted to the offpeak hours or leveled to the average load, less chiller capacity would be needed, 100 percent diversity would be achieved, and better cost efficiency would result.

Full Versus Partial Thermal Energy Storage

There are any number of control strategies that can be utilized to take advantage of the benefit of TES, however, there are two basic approaches that define the common limits of the system design. The electric rates will determine which control strategies are best for the project. When electric rates justify a complete shifting of airconditioning loads, a conventionally sized chiller can be used with enough energy storage to shift the entire load into offpeak hours. This is called a Full Storage system and is used most often in retrofit applications using existing chiller capacity.

Ideal Isothermal Analysis

The invention of the Stirling engine in 1816 was well in advance of all pertinent scientific knowledge of that time. The first attempt at an analysis of the cycle was published in 1871 by Gustav Schmidt. Much as the Otto cycle has become the classic Air standard cycle to describe the spark ignition engine, the cycle described by Schmidt has become the classic ideal Stirling cycle. This is unfortunately mainly because the Schmidt analysis yields a closed form solution rather than its ability to predict the real cycle; however, we use it as a starting point to guide us ultimately to a more realistic approach.

Consider the Ideal Isothermal model of a Stirling engine as shown in FIG. 4.

The principal assumption of the analysis is that the gas in the expansion space and the heater is at the constant upper source temperature and the gas in the compression space and the cooler is at the constant lower sink temperature. This isothermal assumption makes it possible to generate a simple expression for the working gas pressure as a function of the volume variations. This expression may then be used to investigate how different drive mechanisms affect the output power. To obtain closed form solutions, Schmidt assumed that the volumes of the working spaces vary sinusoidally.

The assumption of isothermal working spaces and heat exchangers implies that the heat exchangers (including the regenerator) are perfectly effective, with a spatial temperature distribution as indicated in the figure above. The engine is considered as a five component serially connected model, consisting respectively of a compression space c, cooler k, regenerator r, heater h and expansion space e. Each component is considered as a homogeneous entity or cell, the gas therein being represented by its instantaneous mass m, absolute temperature T, volume V and pressure p, with the suffix c, k, r, h, and e identifying the specific cell.

The starting point of the analysis is that the total mass of gas in the machine is constant, thus:

M=mc+mk+mr+mh+me

Substituting the ideal gas law given by

m=p V/RT

we obtain

M=p(Vc/Tk+Vk/Tk+Vr/Tr+Vh/Th+Ve/Th)/R

For the assumed linear temperature distribution in the regenerator we can show that the effective regenerator temperature Tr is given by

Tr=(Th−Tk)/ln(Th/Tk)

Thus given the volume variations Vc and Ve the above equation is solved for pressure p as a function of Vc and Ve.

$p=\mathrm{MR}/\left(\frac{\mathrm{Vc}}{\mathrm{Tk}}+\frac{\mathrm{Vk}}{\mathrm{Tk}}+\frac{\mathrm{Vr}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{ln}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\left(\mathrm{Th}/\mathrm{Tk}\right)}{\left(\mathrm{Th}\mathrm{Tk}\right)}+\frac{\mathrm{Vh}}{\mathrm{Th}}+\frac{\mathrm{Ve}}{\mathrm{Th}}\right)$

The work done by the system over a complete cycle is given respectively by the cyclic integral of p dV:

$W=\mathrm{We}+\mathrm{Wc}=\oint p\ue89e\uf74c\mathrm{Vc}+\oint p\ue89e\uf74c\mathrm{Ve}=\oint p\ue8a0\left(\frac{\uf74c\mathrm{Vc}}{\uf74c\theta}+\frac{\uf74c\mathrm{Ve}}{\uf74c\theta}\right)\ue89e\uf74c\theta $

On evaluating the heat transferred over a complete cycle to the various cells we find that the cyclic heat transferred to all three heat exchanger cells is zero. Thus:

Qc=We

Qe=We

Qk=0

Qh=0

Qr=0

This rather startling result implies that all the heat exchangers in the ideal Stirling engine are redundant since all the external heat transfer occurs across the boundaries of the compression and expansion spaces. This apparent paradox is a direct result of the definition of the Ideal Isothermal model in which the compression and expansion spaces are maintained at the respective cooler and heater temperatures. Obviously this cannot be correct, since the cylinder walls are not designed for heat transfer. In real machines the compression and expansion spaces will tend to be adiabatic rather than isothermal, which implies that the net heat transferred over the cycle must be provided by the heat exchangers. This will be resolved when we consider the Ideal Adiabatic model in the next section.

The set of pertinent equations is shown in the following table.

$p=\mathrm{MR}/\left(\frac{\mathrm{Vc}}{\mathrm{Tk}}+\frac{\mathrm{Vk}}{\mathrm{Tk}}+\frac{\mathrm{Vr}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{ln}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\left(\mathrm{Th}/\mathrm{Tk}\right)}{\left(\mathrm{Th}\mathrm{Tk}\right)}+\frac{\mathrm{Vh}}{\mathrm{Th}}+\frac{\mathrm{Ve}}{\mathrm{Th}}\right)$
$\mathrm{Pressure}$
$\mathrm{Qe}=\mathrm{We}=\oint p\ue8a0\left(\frac{\uf74c\mathrm{Ve}}{\uf74c\theta}\right)\ue89e\uf74c\theta $
$\mathrm{Heat}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{transferred}$
$\mathrm{Qc}=\mathrm{Wc}=\oint p\ue8a0\left(\frac{\uf74c\mathrm{Vc}}{\uf74c\theta}\right)\ue89e\uf74c\theta $
$\mathrm{Work}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{done}$
$W=\mathrm{Wc}+\mathrm{We}$
$\mathrm{Efficiency}$
$\eta =W/\mathrm{Qe}$

In order to solve these equations, one must specify the working space volume variations Vc and Ve as well as the respective volume derivatives dVc and dVe with respect to crank angle □. One of the case studies of this course is the Ross Yokedrive engine for which we have analyzed the volume variations, thus the above equation set can be solved by numerical integration. In 1871 Gustav Schmidt published an analysis in which he obtained closed form solutions for the above equation set for the special case of sinusoidal volume variations. We continue now with the Schmidt analysis.

The Schmidt Analysis

The Ideal Isothermal model, is shown in the following table:

$p=\mathrm{MR}/\left(\frac{\mathrm{Vc}}{\mathrm{Tk}}+\frac{\mathrm{Vk}}{\mathrm{Tk}}+\frac{\mathrm{Vr}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{ln}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\left(\mathrm{Th}/\mathrm{Tk}\right)}{\left(\mathrm{Th}\mathrm{Tk}\right)}+\frac{\mathrm{Vh}}{\mathrm{Th}}+\frac{\mathrm{Ve}}{\mathrm{Th}}\right)$
$\mathrm{Pressure}$
$\mathrm{Qe}=\mathrm{We}=\oint p\ue8a0\left(\frac{\uf74c\mathrm{Ve}}{\uf74c\theta}\right)\ue89e\uf74c\theta $
$\mathrm{Heat}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{transferred}$
$\mathrm{Qc}=\mathrm{Wc}=\oint p\ue8a0\left(\frac{\uf74c\mathrm{Vc}}{\uf74c\theta}\right)\ue89e\uf74c\theta $
$\mathrm{Work}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{done}$
$W=\mathrm{Wc}+\mathrm{We}$
$\mathrm{Efficiency}$
$\eta =W/\mathrm{Qe}$

Gustav Schmidt of the German Polytechnic Institute of Prague Published an analysis in 1871 in which he obtained closed form solutions of these equations for the special case of sinusoidal volume variations of the working spaces with respect to the cycle angle □. This analysis was published in detail in the appendix of the book by Urieli & Berchowitz, “Stirling Cycle Machine Analysis”, Adam Hilger 1984, incorporated herein by reference.

Consider the diagram in FIG. 5 showing the volume variations of the compression and expansion spaces (Vc and Ve) over a single cycle. Notice the phase advance angle □ of the expansion space volume variation with respect to the compression space volume variation:

The sinusoidal volume variations of the compression and expansion spaces are respectively as follows:

Vc=Vclc+Vswc(1+cos □)/2

Ve=Vclc+Vswe(1+cos(□+□))/2

Where Vcl and Vsw represent respectively clearance and swept volumes, and □ is the cycle angle. Substituting for Vc and Ve in the pressure equation above and simplifying we obtain:

$p=\mathrm{MR}/\left[s+\left(\frac{\mathrm{Vswe}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\alpha}{2\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{Th}}+\frac{\mathrm{Vswc}}{2\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{Tk}}\right)\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \left(\frac{\mathrm{Vswe}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\alpha}{2\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{Th}}\right)\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \right]$
$\mathrm{where}$
$s=\left[\frac{\mathrm{Vswc}}{2\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{Tk}}+\frac{\mathrm{Vclc}}{\mathrm{Tk}}+\frac{\mathrm{Vk}}{\mathrm{Tk}}+\frac{\mathrm{Vr}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{ln}\ue8a0\left(\mathrm{Th}/\mathrm{Tk}\right)}{\left(\mathrm{Th}\mathrm{Tk}\right)}+\frac{\mathrm{Vh}}{\mathrm{Th}}+\frac{\mathrm{Vswe}}{2\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{Th}}+\frac{\mathrm{Vcle}}{\mathrm{Te}}\right]$

In order to simplify the pressure equation one must consider a trigonometric substitution of □ and c as defined by the following rightangled triangle:


Substituting for β and c in the pressure equation above and simplifying

$p=\frac{\mathrm{MR}}{s\ue8a0\left(1+b\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\phi \right)}$
$\mathrm{where}$
$\phi =\theta +\beta $
$b=c/s$

The maximum and minimum values of pressure can now be evaluated for the extreme values of cos φ:

${p}_{\mathrm{min}}=\frac{\mathrm{MR}}{s\ue8a0\left(1+b\right)}$
${p}_{\mathrm{max}}=\frac{\mathrm{MR}}{s\ue8a0\left(1b\right)}$

The average pressure over the cycle is given by:

${p}_{\mathrm{mean}}=\frac{1}{2\ue89e\pi}\ue89e{\int}_{0}^{2\ue89e\pi}\ue89ep\ue89e\uf74c\phi $
${p}_{\mathrm{mean}}=\frac{\mathrm{MR}}{2\ue89e\pi \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89es}\ue89e{\int}_{0}^{2\ue89e\pi}\ue89e\frac{1}{\left(1+b\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\phi \right)}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\phi $

From tables of integrals, this reduces to:

p _{mean}=^{M R}/(s√{square root over (1−b^{2})})

This equation is the most convenient way of relating the total mass of working gas in the cycle to the more conveniently specified mean operating pressure.

The net work done by the engine is the sum of the work done by the compression and expansion spaces. Over a complete cycle:

$\mathrm{Qe}=\mathrm{We}={\int}_{0}^{2\ue89e\pi}\ue89e\left(p\ue89e\frac{\uf74c\mathrm{Ve}}{\uf74c\theta}\right)\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\theta $
$\mathrm{Qc}=\mathrm{Wc}={\int}_{0}^{2\ue89e\pi}\ue89e\left(p\ue89e\frac{\uf74c\mathrm{Vc}}{\uf74c\theta}\right)\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\theta $
$W=\mathrm{Wc}+\mathrm{We}$

The volume derivatives are obtained by differentiating Vc and Ve above:

$\frac{\uf74c\mathrm{Vc}}{\uf74c\theta}=\frac{1}{2}\ue89e\mathrm{Vswc}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta $
$\frac{\uf74c\mathrm{Ve}}{\uf74c\theta}=\frac{1}{2}\ue89e\mathrm{Vswe}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{sin}\ue8a0\left(\theta +\alpha \right)$

Substituting these and the pressure equation into the equations for We and We:

$\mathrm{Wc}=\frac{\mathrm{Vswc}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{MR}}{2\ue89es}\ue89e{\int}_{0}^{2\ue89e\pi}\ue89e\frac{\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}{1+b\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue8a0\left(\beta +\theta \right)}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\theta $
$\mathrm{We}=\frac{\mathrm{Vswe}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{MR}}{2\ue89es}\ue89e{\int}_{0}^{2\ue89e\pi}\ue89e\frac{\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\left(\theta +\alpha \right)}{1+b\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue8a0\left(\beta +\theta \right)}\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\theta $

The solution of these integrals requires the judicious use of tables of integrals and is done in the appendix of “Stirling Cycle Machine Analysis.”

Wc=πVswc p _{mean }sin β(√{square root over (1−b^{2})}−1)/b

We=πVswe p _{mean }sin(β−α) (√{square root over (1−b^{2})}−1)/b

Recently Siegfried “Zig” Herzog from Pennsylvania State University presented an alternative derivation of the Schmidt Analysis (http://mac6.ma.psu.edu/stirling/; incorporated herein by reference), which parallels and complements the above analysis in that it goes into much more detail concerning the solution of the above integrals.

Schmidt Analysis—Equation Summary

The Schmidt analysis is done specifically for an Alpha type engine. For Beta or Gamma type engines we examine the equivalent sinusoidal volume variations (see FIG. 5) to determine the effective Vclc, Vswc, Vcle, Vswe, and α required for this analysis.

$\mathrm{Vc}=\mathrm{Vclc}+\mathrm{Vswc}\ue8a0\left(1+\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \right)/2$
$\mathrm{Ve}=\mathrm{Vcle}+\mathrm{Vswe}\ue8a0\left[1+\mathrm{cos}\ue8a0\left(\theta +\alpha \right)\right]/2$
$\mathrm{Wc}=\mathrm{Qc}=\pi \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{Vswc}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{p}_{\mathrm{mean}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\beta \ue8a0\left(\sqrt{1{b}^{2}}1\right)/b$
$\mathrm{We}=\mathrm{Qe}=\pi \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{Vswe}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{p}_{\mathrm{mean}}\ue89e\mathrm{sin}\ue8a0\left(\beta \alpha \right)\ue89e\left(\sqrt{1{b}^{2}}1\right)/b$
$W=\mathrm{Wc}+\mathrm{We}$
$\eta =W/\mathrm{Qe}=1\mathrm{Tk}/\mathrm{Th}$
$\mathrm{where}$
$\mathrm{tan}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\beta =\left(\frac{\mathrm{Vswe}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\alpha /\mathrm{Th}}{\mathrm{Vswe}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\alpha /\mathrm{Th}+\mathrm{Vswc}/\mathrm{Tk}}\right)$
$c=\frac{1}{2}\ue89e\sqrt{{\left(\frac{\mathrm{Vswe}}{\mathrm{Th}}\right)}^{2}+2\ue89e\frac{\mathrm{Vswe}}{\mathrm{TH}}\ue89e\frac{\mathrm{Vswc}}{\mathrm{Tk}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\alpha +{\left(\frac{\mathrm{Vswc}}{\mathrm{Tk}}\right)}^{2}}$
$s=\left[\frac{\mathrm{Vswc}}{2\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{Tk}}+\frac{\mathrm{Vclc}}{\mathrm{Tk}}+\frac{\mathrm{Vk}}{\mathrm{Tk}}+\frac{\mathrm{Vr}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{ln}\ue8a0\left(\mathrm{Th}/\mathrm{Tk}\right)}{\left(\mathrm{Th}\mathrm{Tk}\right)}+\frac{\mathrm{Vh}}{\mathrm{Th}}+\frac{\mathrm{Vswe}}{2\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{Th}}+\frac{\mathrm{Vcle}}{\mathrm{Te}}\right]$
$b=c/s$
${p}_{\mathrm{mean}}=\mathrm{MR}/\left(s\ue89e\sqrt{1{b}^{2}}\right)$

Heater and Cooler Simple Analysis

One can determine an effectiveness of the heater or cooler heat exchanger in a similar way to that of the regenerator in terms of the following equation:

ε=1−e^{−NTU }

where ε is the effectiveness of the heat exchanger and NTU is the “Number of

Transfer Units” (Refer to “Compact Heat Exchangers”, Kays & London, Krieger Pub Co, 1997; incorporated herein by reference). Both concepts are described herein at Regenerator Simple analysis. Unfortunately, one cannot determine a simple relation between the heater and cooler effectiveness and the engine efficiency, as with the regenerator. Referring to the temperature profile diagram in FIG. 6 observes that the nonideal heater results in the mean effective temperature of the gas in the heater space (Th) being lower than that of the heater wall (Twh). Similarly the nonideal cooler results in the mean effective temperature of the gas in the cooler space (Tk) being higher than that of the cooler wall (Twk). This has a significant effect on the engine performance, since it is effectively operating between lower temperature limits than those of the heater and cooler walls. Thus the Simple analysis of the heater and cooler iteratively determines these temperature differences using the convective heat transfer equations, the values of Qh and Qk being evaluated by the Ideal Adiabatic analysis.

From the basic equation for convective heat transfer we obtain:

{dot over (Q)}=h Awg(Tw−T)

where {dot over (Q)} (watts) is the heat transfer power, h is the convective heat transfer coefficient, Awg refers to the wall/gas, or “wetted” area of the heat exchanger surface, Tw is the wall temperature, and T the gas temperature. In order to reduce the units of this equation to the net heat transferred over a single cycle Q (joules/cycle) we divide both sides by the frequency of operation (freq), thus:

Qk=hk Awgk(Twk−Tk)/freq

Qh=hh Awgh(Twh−Th)/freq

Where, as shown in the diagram above, the suffix h refers to the heater, and the suffix k refers to the cooler. We now rewrite these equations to evaluate the respective gas temperatures Tk and Th:

Tk=Twk−Qk freq/(hk Awgk)

Th=Twh−Qh freq/(hh Awgh)

The Simple solution algorithm requires iterative invoking of the Ideal Adiabatic simulation, each time with new values of Tk and Th, until convergence is attained. After each simulation run values of Qk and Qh are available. The mass flow rates through the heater and cooler are used to determine the average Reynolds numbers and thus the heat transfer coefficients in accordance with the methods in the section on Scaling Parameters. Substituting these values in the above equations yields Tk and Th, and convergence is attained when their successive values are essentially equal.

The Simple simulation of the D90 Ross Yokedrive engine case study results in the temperature distribution as shown below. The mean temperature of the gas in the heater space is 59 degrees below that of the heater wall, and similarly the mean temperature of the gas in the cooler space is 15 degrees above that of the cooler wall. This lower temperature range of operation reduced the output power from 178 W to 147 W.

Regenerator Simple Analysis

The first mathematical theories to describe regenerator operation were published in the late 1920s, more than 100 years after its invention by Robert Stirling. Significantly, these and subsequent theories of regenerator operation are based on assumptions which are neither relevant nor applicable to Stirling engine regenerators. In the book “The Regenerator and the Stirling Engine” by Allan Organ (John Wiley & Sons 1997; incorporated herein by reference) a significant step towards “bridging the gap” is made between Hausen's celebrated regenerator analysis, widely used in the analysis of gas turbine engines, and the unique conditions that apply to Stirling engines.

By definition a regenerator is a cyclic device. On the first part of the cycle the hot gas flows through the regenerator from the heater to the cooler, and in so doing transfers heat to the regenerator matrix. This is referred to as a “single blow”. Subsequently during the second part of the cycle the cold gas flows in the reverse direction, absorbing the heat that was previously stored in the matrix. Thus at steady state the net heat transfer per cycle between the working gas and the regenerator matrix is zero.

The regenerator quality is usually defined on an enthalpy basis in terms of a regenerator effectiveness E as follows:

$\varepsilon \equiv \frac{\begin{array}{c}\mathrm{actual}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{enthalpy}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{change}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{gas}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{during}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89ea\\ \mathrm{single}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{blow}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{through}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{regenerator}\end{array}}{\begin{array}{c}\mathrm{equivalent}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{maximum}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{theoretical}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{enthalpy}\\ \mathrm{change}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{in}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{an}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{ideal}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{regenerator}\end{array}}$

However this definition is not amenable to usage in Stirling engines. We use an equivalent definition in the context of the Ideal Adiabatic model (http://www.ent.ohiou.edu/˜urieli/stirling/simple/htx_simple.html; incorporated herein by reference), which represents the limiting maximum performance measure, as follows:

$\varepsilon \equiv \frac{\left(\begin{array}{c}\mathrm{amount}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{heat}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{transferred}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{from}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{matrix}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{to}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{gas}\\ \mathrm{during}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89ea\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{single}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{blow}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{through}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{regenerator}\end{array}\right)}{\left(\begin{array}{c}\mathrm{equivalent}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{amount}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{heat}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{transferred}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{in}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\\ \mathrm{regenerator}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Ideal}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Adiabatic}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{model}\end{array}\right)}$

The regenerator effectiveness ε thus varies from 1 for an ideal regenerator (as defined in the Ideal Adiabatic model) to 0 for no regenerative action. For example, FIG. 7 shows the cyclic energytheta diagram of the Ideal Adiabatic analysis of the Ross D90 engine.

The thermal efficiency of the Ideal Adiabatic cycle (suffix “i”) is given in terms of the energy values accumulated at the end of the cycle by:

ηi=Wi/Qhi=(Qhi+Qki)/Qhi

Note from the diagram that Qki is a negative quantity, thus ηi=0.627 for the D90 engine as shown. Notice also the significant amount of heat transferred during a single blow of the regenerator given b y Qrî. Thus for the D90 engine as shown the ratio Qrî/Qhi=5.66.

Now for a system having a nonideal regenerator, during the single blow when the working gas flows from the cooler to the heater, on exit from the regenerator it will have a temperature somewhat lower than that of the heater. This will result in more heat being supplied externally over the cycle by the heater in increasing the temperature of the gas to that of heater and can be written quantitatively as follows:

Qh=Qhi 30 Qrî(1−ε)

Similarly, when the working gas flows from the heater to the cooler, then an extra cooling load will be burdened on the cooler, as follows:

Qk=Qki−Qrî(1−ε)

The thermal efficiency of the nonideal engine (without the suffix “i”) is given by:

h=W/Qh=(Qh+Qk)/Qh

Substituting for Qh, Qk, and ηi from the above equations:

$\eta =\frac{\eta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ei}{\left[1+\left(\frac{\mathrm{Qri}^}{\mathrm{Qhi}}\right)\ue89e\left(1\varepsilon \right)\right]}$

The diagram in FIG. 8 shows a plot of the above equation for the specific case of the D90 engine and shows the effect of regenerator effectiveness E on thermal efficiency η.

Notice that as ε varies from 1 for an ideal regenerator (Ideal Adiabatic cycle) to 0 for no regenerative action, the thermal efficiency η drops from more than 60% to less than 10%. Furthermore, differentiating the efficiency equation η with respect to E, and substituting ε=1:

$\begin{array}{c}\frac{\uf74c\eta}{\uf74c\varepsilon}\ue89e{}_{\varepsilon =1}=\eta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ei\ue89e\frac{\mathrm{Qri}^}{\mathrm{Qhi}}\\ =5.66\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\eta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ei\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{for}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eD\ue89e\text{}\ue89e90\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Ross}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Yoke}\ue89e\text{}\ue89e\mathrm{drive}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{engine}.\end{array}$

Thus, for highly effective regenerators (close to ε=1) a 1% reduction in regenerator effectiveness results in a more than 5% reduction in thermal efficiency η. Furthermore, if one has a regenerator that has an effectiveness of 0.8, the thermal efficiency has dropped by half to around 30%. This not only means a significantly less efficient machine, but one that has to have a significantly larger cooler. Obviously we need to have a means of determining the actual regenerator effectiveness in any specific machine.

Evaluating the Regenerator Effectiveness ε

We now consider the regenerator effectiveness in terms of the temperature profile of the ‘hot’ and ‘cold’ gas streams with respect to the regenerator matrix. We assume an equal difference in temperature Δ T on the hot and the cold sides, and linear temperature profiles, leading to the definition of regenerator effectiveness ε in terms of temperatures, as shown in FIG. 9.

Combining the two equations in the figure, we obtain:

$\varepsilon =\frac{1}{\left(1+\frac{2\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eT}{\left({\mathrm{Th}}_{1}{\mathrm{Th}}_{2}\right)}\right)}$

Now from energy balance considerations of the hot stream, the change in enthalpy of the hot stream is equal to the heat transfer from the hot stream to the matrix, and subsequently from the matrix to the cold stream, thus:

{dot over (Q)}=cp {dot over (m)}(Th1−Th2)=2 h Awg ΔT

Where {dot over (Q)}(watts) is the heat transfer power, h is the overall heat transfer coefficient (hot stream/matrix/cold stream), Awg refers to the wall/gas, or “wetted” area of the heat exchanger surface, cp the specific heat capacity at constant pressure, and {dot over (m)} (kg/s) the mass flow rate through the regenerator. Substituting in the effectiveness equation we obtain:

$\varepsilon =\frac{1}{\left(1+\frac{\mathrm{cp}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\stackrel{.}{m}}{\mathrm{hAwg}}\right)}$

We now introduce the concept of Number of Transfer Units (NTU) which is a well known measure of heat exchanger effectiveness, and is defined in the section on Scaling Parameters.

$\mathrm{NTU}\equiv \frac{\mathrm{hAwg}}{\mathrm{cpm}}$
Thus:

$\begin{array}{c}\varepsilon \ue89e\phantom{\rule{0.3em}{0.3ex}}=\phantom{\rule{0.3em}{0.3ex}}\ue89e\frac{\mathrm{NTU}}{\left(1+\mathrm{NTU}\right)}\end{array}$

Notice that the NTU value is a function of the type of heat exchanger as well as its physical size. It includes the actual wetted area Awg, as well as the actual mass flow through the regenerator {dot over (m)} (kg/s). In heat exchanger analysis it is more usual to evaluate local heat exchanger parameters in terms of fluid property values which are independent of size. Thus we define a Stanton number (refer to the section on Scaling Parameters) as follows:

NST=h/(ρ u cp)

where ρ is the fluid density, and u is the fluid velocity, thus:

{dot over (m)}=ρ u A

Where A is the free flow area through the matrix. Tables and graphs of empirical values of Stanton number vs. Reynolds number are available from heat exchanger texts for various heat exchanger types. We use two types of matrices in our regenerator analysis—woven mesh and coiled annular foil (however there is some renewed interest in Spherical Bed matrices, consisting of randomly stacked spherical pebbles). The NTU value can then be obtained in terms of the Stanton number as follows:

NTU=NST (Awg/A)/2

The factor 2 in this equation is unusual, and stems from the fact that the Stanton number is usually defined for the transfer of heat from the gas stream to the matrix alone, whereas the NTU usage in this section is for overall transfer of heat from the hot stream to the regenerator matrix, and subsequently to the cold stream.

Pumping Loss Simple Analysis

Throughout this analysis we have assumed that at any instant the pressure is constant throughout the engine. However we find that the high heat fluxes required in the heat exchangers in turn requires a large wall/gas, or wetted area Awg. This requirement together with the conflicting requirement of a low void volume will result in heat exchangers with many small diameter passages in parallel. The fluid friction associated with the flow through the heat exchangers will in fact result in a pressure drop across all the heat exchangers which has the effect of reducing the power output of the engine. This is referred to as the “Pumping Loss” and in this section we attempt to quantify this power loss. We first evaluate the pressure drop across all three heat exchangers with respect to the compression space. Subsequently we can determine the new value of work done by integrating over the complete cycle, and isolate the Pumping Loss term as follows:

$\mathrm{thus}\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eW=\mathrm{We}+\mathrm{Wc}=\oint p\ue89e\uf74c\mathrm{Vc}+\oint \left(p\mathrm{\Sigma \Delta}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ep\right)\ue89e\uf74c\mathrm{Ve}$
$\mathrm{where}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{summation}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{\Sigma \Delta}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ep\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{is}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{taken}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{over}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e3\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{heat}$
$\mathrm{exchangers}$
$\mathrm{thus}\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eW=\oint p\ue8a0\left(\uf74c\mathrm{Vc}+\uf74c\mathrm{Ve}\right)\oint \mathrm{\Sigma \Delta}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ep\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\uf74c\mathrm{Ve}={W}_{1}\xb7\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eW$
$\mathrm{where}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Wi}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{is}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Ideal}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Adiabatic}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{work}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{done}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{per}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cycle}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{and}$
$\phantom{\rule{4.2em}{4.2ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eW\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{is}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{pressure}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{drop}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{loss}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{or}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{pumping}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{loss}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{per}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cycle}$
$\mathrm{thus}\ue89e\text{:}\ue89e\phantom{\rule{1.1em}{1.1ex}}\ue89e\begin{array}{c}\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eW={\int}_{0}^{2\ue89e\pi}\ue89e\left(\sum _{i=1}^{3}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{p}_{i}\ue89e\frac{\uf74c\mathrm{Ve}}{\uf74c\theta}\right)\ue89e\phantom{\rule{0.2em}{0.2ex}}\ue89e\uf74c\theta \end{array}$

The pressure drop Δp is due to fluid friction as it flows through the heat exchanger sections. Our model has assumed onedimensional flow throughout, however the fundamental concepts of fluid friction paradoxically break down under onedimensional flow. Newton's law of viscosity states that the shear stress τ between adjacent layers of fluid is proportional to the velocity gradient (du/dz) in these layers normal to the flow direction, as shown in FIG. 10.

From the equation we see that a Newtonian fluid cannot sustain a shear stress unless the flow is two dimensional. This paradox is bypassed by stating that the flow is not strictly onedimensional, but rather represented by its mean bulk mass flow rate. The dynamic viscosity μ is basically a measure of the internal friction which occurs when the molecules of the fluid in one layer collide with molecules in adjacent layers traveling at different speeds, and in so doing transfer their momentum.

Over the pressure range of interest the dynamic viscosity μ is independent of pressure. Its temperature dependence for the gasses of interest is obtained as in the diagram depicted in FIG. 11. (Refer: Bretsznajder, A, 1971, “Prediction of the transport and Other Physical Properties of Fluids”, International Series of Monographs in Chemical Engineering, II, Oxford: Pergamon, incorporated herein by reference)

The frictional drag force F is related to the shear stress τ as follows:

F=τ Awg

Where Awg is the wall/gas, or wetted area of the heat exchanger.

In setting up the working expressions to describe pumping loss we introduce the concept of a “hydraulic diameter” d, which describes the ratio of the two important variables of a heat exchanger—the void volume V and the wetted area Awg:

d=4V/Awg

The factor 4 is included for convenience. For flow in a circular pipe (or a homogeneous bundle of circular pipes) the hydraulic diameter thus becomes equal to the pipe internal diameter. Substituting in the force equation above:

F=4τV/d

We now define a Coefficient of Friction Cf as the ratio of the shear stress τ to the “dynamic head” (see “Compact Heat Exchangers”, Kays & London):

$\mathrm{Cf}\equiv \frac{\tau}{\frac{1}{2}\ue89e\rho \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{u}^{2}}$

Where ρ is the fluid density and u is the fluid bulk velocity. Thus substituting for τ in the force equation we obtain the frictional drag force in terms of the Coefficient of Friction:

F=2 Cf ρ u ^{2 } V/d

Under the quasisteady flow assumption (no acceleration or deceleration forces) the frictional drag force is equal and opposite to the pressure drop force, thus:

F+Δ p A=0

Where A is the cross sectional (free flow) area. Substituting for F, the pressure drop Δ0 p is given by:

Δp+2 Cf ρ u ^{2 } V/(d A)=0

Note that Δ p can be positive or negative, depending on the direction of flow. However the second term in this equation is always positive, and thus the equation violates the momentum conservation principle in the case of reversing flow. We resolve this by defining a “Reynolds Friction Coefficient” (Cref) by multiplying the Reynolds Number by the Coefficient of Friction as follows:

Cref=Nre Cf

Where Nre=ρ u d/μ is the Reynolds Number, defined and discussed in the section Scaling Parameters. By definition, the Reynolds Number is always positive, independent of the direction of flow. Thus finally:

$\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ep=\frac{2\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{Cref}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mu \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eu\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eV}{{d}^{2}\ue89eA}$

This equation satisfies the momentum conservation principle for both positive and reversed flow, since the sign of Δ p is always correctly related to the sign of the velocity u. Since all current empirical data on the Coefficient of Friction is presented as a function of Reynolds Number, it is a simple matter to convert that data to the required Reynolds Friction Coefficient. For example, the Coefficient of Friction vs. Nre curves for circular pipes (Moody Diagram) have been in widespread use over the past half a century. These curves have been simplified and rearranged in terms of the Reynolds Friction Coefficient Cref as depicted in FIG. 12.

Similar formulations can be done for the various heat exchanger and regenerator types of interest. (Refer to “Compact Heat Exchangers”, Kays & London). Refer to the Spherical Bed type heat exchanger consisting of randomly stacked spherical pebbles, currently under consideration for regenerators.

The Simple simulation of the D90 Ross Yokedrive engine case study results in the pressure vs. crank angle plots depicted in FIG. 13. The first plot shows the pressure drop across the three heat exchangers. Note the relative magnitude (as well as the phase) of the regenerator pressure drop with respect to those of the heater and cooler.

The plot depicted in FIG. 14 shows the expansion and compression space pressures vs. crank angle. Under these conditions the pumping loss is 10.3 W, or about 7.5% of the net output power.

Scaling Parameters

Forced convection heat transfer is fundamental to Stirling engine operation. Heat is transferred from the external heat source to the working fluid in the heater section, cyclically stored and recovered in the regenerator, and rejected by the working fluid to the external heat sink in the cooler section. All of this is done in compact heat exchangers (large wetted area to void volume ratio) so as to limit the “dead space” an acceptable value and thus allow for a reasonable specific power output of the engine. We find that effective heat exchange comes at a price of increased flow friction, resulting in the socalled “pumping loss”. This loss refers to the mechanical power required to “pump” the working fluid through the heat exchangers, and thus reducing the net power output of the engine.

When we try to design a machine for a specific performance, we find that there are a large number of parameters involved which affect the performancein non intuitive ways. The Scaling Parameter approach uses dimensional analysis to reduce the number of parameters to a basic set of dimensionless scaling groups, and thus allow experimental data to be used in various contexts. This is a standard technique in forced convection heat transfer analysis and can be reviewed in many heat transfer texts. We have based our analysis on the book “Compact Heat Exchangers” by Kays & London, both the 1955 and 1964 editions. An extremely lucid discussion of the basic scaling parameters involved and their applicability to the oscillating flow conditions of Stirling engines is found in the book “The Regenerator and the Stirling Engine” by Allan Organ (1997), and in particular Chapter 3: “Heat Transfer—and the Price”.

Standard dimensionless scaling parameters for each heat exchanger section in the machine, based on a basic (exhaustive) set of variables is described as follows (http://www.sesusa.org/DrIz/scaling.html; incorporated herein by reference):
 d—the Hydraulic Diameter (m).

This variable represents ratio of the two important size parameters of a heat exchanger—the void volume V and the wetted area Awg. It is defined by:

$d\equiv \frac{4\ue89eV}{\mathrm{Awg}}$

The factor 4 is included for convenience. For flow in a circular pipe (or a homogeneous bundle of circular pipes) the Hydraulic Diameter thus becomes equal to the pipe internal diameter. Note that some researchers (e.g. Allan Organ) use the socalled Hydraulic Radius (rh) as their scaling parameter. This is simply defined as rh=V/Awg, thus d=4 rh.
 μ—the working gas dynamic viscosity (Pa s). This is defined in terms of Newtons Law of Viscosity in the section on Pumping Loss.
 u—the mean bulk velocity of the flowing fluid (m/s)
 ρ—the density of the working gas (kg/cu.m)
 h—the convective heat transfer coefficient (W/sq.m K). This is defined in the section on Heater and Cooler Simple analysis.
 k—the working gas thermal conductivity (W/m K)
 cp—the working gas specific heat capacity at constant pressure (J/kg K)

Reynolds Number (Nre)

$\mathrm{Nre}\equiv \frac{\rho \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{ud}}{\mu}$

This grouping is obtained by considering the ratio of the inertial forces to the viscous forces. The value of Nre determines the flow regime, whether laminar or turbulent. Both the friction factor and the heat transfer coefficient are strongly dependent on the flow regime, thus Nre is invariably used as the independent variable in the presentation of flowfriction and heat transfer data. Note that by definition Nre is always positive, independent of the direction of fluid flow.

Stanton Number (Nst)

$\mathrm{Nst}\equiv \frac{h}{\rho \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eu\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cp}}$

This grouping is one of the two standard methods for the presentation of heat transfer by convection. The physical significance of Nst is that it can be related to the ratio of the convective heat transfer to the thermal capacity of the flowing fluid. It has found favor because of the ease in which it can be obtained from experimental data. Thus from an energy balance of a heated (or cooled) fluid flowing through a heat exchanger:


Where:
 {dot over (Q)} is the rate of heat transfer
 {dot over (m)} is the mass flow rate
 Awg is the wall/gas, or wetted area
 A is the free flow area (normal to the direction onf flow)
 Tw, T are the respective wall and bulk fluid temperatures
 Ti, To are the respective inlet and outlet fluid temperatures

Substituting for Nst above we obtain:

$\mathrm{Nst}=\frac{A}{\mathrm{Awg}}\ue89e\left(\frac{\mathrm{To}\mathrm{Ti}}{\mathrm{Tw}T}\right)$

Thus the value of Nst can be obtained directly from the heat exchanger dimensions and temperature measurement without reference to the fluid properties.

“Number of Transfer Units”, or NTU, can be defined from the above energy balance equation as follows:

$\mathrm{NTU}\equiv \frac{h\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{Awg}}{\rho \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eu\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cp}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89eA}=\mathrm{Nst}\ue89e\frac{\mathrm{Awg}}{A}$

NTU is a function of the heat exchanger dimensions and as such is not considered a fundamental heat transfer grouping in the classical sense. However formulation in terms of NTU allows a solution in terms of Nst, and thus avoids the tedium of extracting the heat transfer coefficient h. This approach is used to advantage in the section on Regenerator Simple analysis.

Prandtl Number (Npr)

$\mathrm{Npr}\equiv \frac{\mathrm{cp}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mu}{k}$

This grouping is obtained from the ratio of the kinematic viscosity μ=□/□ (sq.m/s) (also known as the momentum diffusivity) to the thermal diffusivity □=k/□ cp (sq.m/s), and thus represents the ratio of the viscous to the thermal boundary layers. Thus for fluids having a value of Npr close to unity the classic Reynolds Analogy can be used to relate simply between flowfriction and heat transfer data. It involves three fluid properties and is thus itself a property of the fluid, and frequently appears in heat transfer data. For the range of working gases used in Stirling engines and for the temperature range of interest (about 300 to 1000 K), Npr is approximately constant at around a value of 0.7.

Developing the Reynolds Analogy, Organ, in Chapter 3 of “The Regenerator and the Stirling Engine”, shows that one can relate the Stanton Number to the friction factor Cf as follows:

Nst=Cf/2

According to Organ, “this vital result is not quantitatively exact, but serves a more valuable purpose than any precise formula by confirming the inevitable tie between friction factor and Stanton number. It warns against unrealistic expectation of increasing heat transfer without penalty of increased pumping power.”

Nusselt Number (Nnu)

$\mathrm{Nnu}\equiv \frac{h\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ed}{k}$

This grouping is often used as an alternative to the Stanton Number for the presentation of heat transfer data, and is usually presented in graphical form in terms of the Prandtl and Reynolds numbers (e.g. Kays & London, “Compact Heat Exchangers”). It is not an independent grouping, and can be defined as a function of the other three dimensionless groups as follows:

Nnu=Nst Npr Nre

In the foregoing, the influence of temperature on the fluid properties is not expressed. Both dynamic viscosity μ and thermal conductivity k vary significantly with temperature. However, since the specific heat capacity cp and Prandtl Number Npr are approximately constant over the temperature range of interest we see from the definition of Npr that it is sufficient to consider the temperature dependence of the dynamic viscosity μ.

Ideal Adiabatic Analysis

An ideal Stirling engine model is one in which the compression and expansion spaces were maintained at the respective cooler and heater temperatures. This leads to the paradoxical situation that neither the heater nor the cooler contributed any net heat transfer over the cycle and hence were redundant. All the required heat transfer occurred across the boundaries of the isothermal working spaces. Obviously this cannot be correct, since the cylinder walls are not designed for heat transfer. In real machines the working spaces will tend to be adiabatic rather than isothermal, which implies that the net heat transferred over the cycle must be provided by the heat exchangers. We thus consider an alternative ideal model for Stirling cycle engines, the Ideal Adiabatic model depicted in FIG. 15.

As before the engine is configured as a five component serially connected model having perfectly effective heat exchangers (including the regenerator) and in this respect is similar to the Ideal Isothermal model defined previously. However both the compression and expansion spaces are adiabatic, in which no heat is transferred to the surroundings. In the diagram depicted in FIG. 16, we define the Ideal Adiabatic model nomenclature. Thus we have a single suffix (c, k, r, h, e) representing the five cells, and a double suffix (ck, kr, rh, he) representing the four interfaces between the cells. Enthalpy is transported across the interfaces in terms of a mass flow rate m′ and an upstream temperature T. The arrows on the interfaces represent the positive direction of flow, arbitrarily defined from the compression space to the expansion space.

Notice from the temperature distribution diagram that the temperature in the compression and expansion spaces (Tc and Te) are not constant, but vary over the cycle in accordance with the adiabatic compression and expansion occurring in the working spaces. Thus the enthalpies flowing across the interfaces ck and he carry the respective adjacent upstream cell temperatures, hence temperatures Tck and The are conditional on the direction of flow and are defined algorithmically as follows:

if mck′>0 then Tck=Tc else Tck=Tk

if mhe′>0 then The=Th else The=Te

In the ideal model there is no gas leakage, the total mass of gas M in the system is constant, and there is no pressure drop, hence p is not suffixed and represents the instantaneous pressure throughout the system.

Work W is done on the surroundings by virtue of the varying volumes of the working spaces Vc and Ve, and heat Qk and Qh is transferred from the external environment to the working gas in the cooler and heater cells, respectively. The regenerator is externally adiabatic, heat Qr being transferred internally from the regenerator matrix to the gas flowing through the regenerator void volume Vr.

Development of the Equation Set

The general approach for deriving the equation set is to apply the equations of energy and state to each of the cells. The resulting equations are linked by applying the continuity equation across the entire system. Consider first the energy equation applied to a generalized cell which may either be reduced to a working space cell or a heat exchanger cell. As depicted in FIG. 17, enthalpy is transported into the cell by means of mass flow mi′ and temperature Ti, and out of the cell by means of mass flow mo′ and temperature To. The derivative operator is denoted by d, thus for example dm refers to the mass derivative dm/dQ, where Q is the cycle angle.

The word statement of the energy equation for the working gas in the generalized cell is:

$\begin{array}{c}\mathrm{rate}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{heat}\\ \mathrm{transfer}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{into}\\ \mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cell}\end{array}+\begin{array}{c}\mathrm{net}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{enthalpy}\\ \mathrm{convected}\\ \mathrm{into}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cell}\end{array}=\begin{array}{c}\mathrm{rate}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{work}\\ \mathrm{done}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{on}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\\ \mathrm{surroundings}\end{array}+\begin{array}{c}\mathrm{rate}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{increase}\\ \mathrm{of}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{internal}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{energy}\\ \mathrm{in}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{the}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cell}\end{array}$

Mathematically, this word statement becomes

dQ+(cp Ti mi′−cp To mo′)=dW+cv d(m T)

where cp and cv are the specific heat capacities of the gas at constant pressure and constant volume respectively. This equation is the well known classical form of the energy equation for non steady flow in which kinetic and potential energy terms have been neglected.

We assume that the working gas is ideal. This is a reasonable assumption for Stirling engines since the working gas processes are far removed from the gas critical point. The equation of state for each cell is presented in both its standard and differential form as follows:

p V=m R T

dP/p+dV/V=dm/m+dT/T

The starting point of the analysis is that the total mass of gas in the machine is constant, thus:

mc+mk+mr+mh+me=M

Substituting for the mass in each cell from the ideal gas law above

p(Vc/Tc+Vk/Tk+Vr/Tr+Vh/Th+Ve/Te)/R=M

Where for the assumed linear temperature profile in the regenerator the mean effective temperature Tr is equal to the log mean temperature difference Tr=(Th−Tk)/ln(Th/Tk).

Solving the above equation for pressure:

p=M R/(Vc/Tc+Vk/Tk+Vr/Tr+Vh/Th+Ve/Te)

Differentiating the equation for mass above:

dmc+dmk+dmr+dmh+dme=0

For all the heat exchanger cells, since the respective volumes and temperatures are constant, the differential form of the equation of state reduces to:

dm/m=dp/p

dm=dp m/p=(dp/R)V/T

Substituting in the mass equation above:

dmc+dme+(dp/R) (Vk/Tk+Vr/Tr+Vh/Th)=0

We wish to eliminate dmc and dme in the above equation so as to obtain an explicit equation in dp. Consider the adiabatic compression space (dQc=0) depicted in FIG. 18.

Applying the above energy equation to this space we obtain

−cp Tck mck′=dWc+cv d(mc Tc)

From continuity considerations the rate of accumulation of gas dmc is equal to the mass inflow of gas given by −mck′, and the work done by dWc is given by p dVc, thus

cp Tck dmc=p dVc+cv d(mc Tc)

Substituting the ideal gas relations p Vc=mc R Tc, cp−cv=R, and cp/cv=□, and simplifying

dmc=(p dVc+Vc dp/□)/(R Tck)

Similarly for the expansion space

dme=(p dVe+Ve dp/□)/(R The)

Substituting for dmc and dme above and simplifying

$\mathrm{dp}=\frac{\gamma \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ep\ue8a0\left(\mathrm{dVc}/\mathrm{Tck}+\mathrm{dVe}/\mathrm{The}\right)}{\left[\mathrm{Vc}/\mathrm{Tck}+\gamma \ue8a0\left(\mathrm{Vk}/\mathrm{Tk}+\mathrm{Vr}/\mathrm{Tr}+\mathrm{Vh}/\mathrm{Th}\right)+\mathrm{Ve}/\mathrm{The}\right]}$

From the differential form of the equation of state above we obtain relations dTc and dTe

dTc=Tc(dp/p+dVc/Vc−dmc/mc)

dTe=Te(dp/p+dVe/Ve−dme/me)

Applying the energy equation above to each of the heat exchanger cells (dW=0, T constant) and substituting for the equation of state for a heat exchanger cell (dm=dp m/p=(dp/R)V/T):

dQ+(cp Ti mi′−cp To mo′)=cv T d =V dp cv/R

Thus for the three heat exchanger cells we obtain

dQk=Vk dp cv/R−cp(Tck mck′−Tkr mkr′)

dQr=Vr dp cv/R−cp(Tkr mkr′−Trh mrh′)

dQh=Vh dp cv/R−cp(Trh mrh′−The mhe′)

We note that since the heat exchangers are isothermal and the regenerator is ideal, Tkr=Tk and Trh=Th.

Finally the work done in the compression and expansion cells is given by

W=Wc+We

dW=dWc+dWe

dWc=p dVc

dWe=p dVe

With reference to FIG. 16, the final set of pertinent differential and algebraic equations required for solution is gathered in the Equation Summary following.

Ideal Adiabatic analysisequation summary & method of solution
p = M R / (Vc / Tc + Vk / Tk + Vr / Tr + Vh / Th + Ve / Te
Pressure
$\mathrm{dp}=\frac{\gamma \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ep\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\left(\mathrm{dVc}/\mathrm{Tck}+\mathrm{dVe}/\mathrm{The}\right)}{\left[\mathrm{Vc}/\mathrm{Tck}+\gamma \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\left(\mathrm{Vk}/\mathrm{Tk}+\mathrm{Vr}/\mathrm{Tr}+\mathrm{Vh}/\mathrm{Th}\right)+\mathrm{Ve}/\mathrm{The}\right]}$
mc = p Vc / (R Tc)
Masses
mk = p Vk / (R Tk)
mr = p Vr / (R Tr)
mh = p Vh / (R Th)
me = p Ve / (R Te)
dmc = (p dVc + Vc dp / γ) / (R Tck)
Mass
dme = (p dVe + Ve dp / γ) / (R The)
Accumula
dmk = mk dp / p
tions
dmr = mr dp /p
dmh = mh dp / p
mck′ = −dmc
Mass Flow
mkr′ = mck′ − dmk
mhe′ = dme
mrh′ = mhe′ + dmh
if mck′ > 0 then Tck = Tc else Tck = Tk
Conditional
if mhe′ > 0 then The = Th else The = Te
Temperatures
dTc = Tc (dp / p + dVc / Vc − dmc / mc)
Temperatures
dTe = Te (dp / p + dVe / Ve − dme / me)
dQk = Vk dp cv / R − cp (Tck mck′ − Tk mkr′)
Energy
dQr = Vr dp cv /R − cp (Tk mkr′ − Th mrh′)
dQh = Vh dp cv / R − cp (Th mrh′ − The mhe′)
dWc = p dVc
dWe = p dVe
dW = dWc + dWe
W = Wc + We

The Method of Solution

We now consider the solution of the equation set above. Because of the nonlinear nature of the equations (in particular with regards to the Conditional Temperatures) we have to resort to a numerical solution of specific configurations and operating conditions.

The specific engine configuration and geometry defines Vc, Ve, dVc, and dVe as analytic functions of the crankangle θ, and the heat exchanger geometry defines the void volumes Vk, Vr, Vh. The choice of working gas (typically air, helium or hydrogen) specifies R, cp, cv, and γ. The operating conditions specify Tk and Th, and thus the mean effective temperature Tr=(Th−Tk)/ln(Th/Tk). Specifying the total mass of working gas M is a problem, since this is not normally a known parameter. The approach we use is to specify the mean operating pressure pmean and then use the Schmidt Analysis to evaluate M. Even though the Ideal Adiabatic model is independent of operating frequency, we nevertheless specify it in order to evaluate power and other time related effects (such as thermal conduction loss in the regenerator housing.)

We notice that apart from the constant parameters specified above, there are 22 variables and 16 derivatives in the equation set, to be solved over a complete cycle (θ=[0, 2 π]):
 Tc, Te, Qk, Qr, Qh, Wc, We—seven derivatives to be integrated numerically
 W, p, Vc, Ve, mc, mk, mr, mh, me—nine analytical variables and derivatives
 Tck, The, mck′, mkr′, mrh′, mhe′—six conditional and mass flow variables (derivatives undefined)

We treat this as a “quasi steadyflow” system, thus over each integration interval the four mass flow variables mck′, mkr′, mrh′, and mhe′ remain constant and there are no acceleration effects. Thus we consider the problem as that of solving a set of seven simultaneous ordinary differential equations.

The simplest approach to solving a set of ordinary differential equations is to formulate it as an initialvalue problem, in which the initial values of all the variables are known and the equations are integrated from that initial state over a complete cycle. The initial value problem can be stated in simple terms. Let the vector Y collectively represent the seven unknown variables, thus y[Tc] is the compression space temperature, y[We] is the work done by the expansion space, and so on. Given an initial condition Y(θ=0)=Y0 and the corresponding set of differential equations dY=F(θ, Y), evaluate the unknown functions Y(θ) that satisfy both the differential equations and the initial conditions. A numerical solution to this problem is accomplished by by first computing the values of the derivatives at θ 0 and proceeding in small increments of θ to a new point θ 1=θ 0+Δθ. Thus the solution is composed of a series of short straightline segments that approximate the true curves Y(θ). Among the vast number of methods available for solving initialvalue problems, the classical fourthorder RungeKutta method is probably the most frequently used.

In order to develop our specific method of solution of the initialvalue problem, we have presented a case study in the MATLAB language involving a largeangle pendulum. We do not use the MATLAB builtin functions for solving ordinary differential equations, since our method requires overloading features not available in these builtin functions.

Unfortunately the Ideal Adiabatic model is not an initialvalue problem, but is instead a boundaryvalue problem. We do not know the initial values of the working space gas temperatures Tc and Te, which result from the adiabatic compression and expansion processes as well as enthalpy flow processes. The only guidance that we have to their correct choice is that their values at the end of the steadystate cycle should be equal to their respective values at the beginning of the cycle.

However, because of its cyclic nature, the system can be formed as an initial value problem by assigning arbitrary initial conditions, and integrating the equations through several complete cycles until a cyclic steady state has been attained. This is equivalent to the transient “warmup” operation of an actual machine. Experience has shown that the most sensitive measure of convergence to cyclic steady state is the residual regenerator heat Qr at the end of the cycle, which should be zero.

The compression and expansion space temperatures are thus initially specified at Tk and Th respectively. The system of equations can then be solved through as many cycles as necessary in order to attain cyclic steady state. For most configurations, between five and ten cycles will be sufficient for convergence.

In December 2005, Siegfried “Zig” Herzog from Pennsylvania State University presented an Ideal Adiabatic Analysis (http://mac6.ma.psu.edu/stirling/simulations/IdealAdiabatic/; incorporated herein by reference) that essentially parallels the above analysis. A unique feature of this presentation is that since the Ideal Adiabatic Analysis of necessity requires computer analysis, the simulation program that Herzog presents can also be executed remotely on the web.

Ideal Adiabatic Solution—The D90 Ross Yokedrive Engine

The 90 cc D90 engine is fully described in “Making Stirling Engines” Andy Ross (1993, incorporated herein by reference). Performing an Ideal Adiabatic simulation of the D90 engine under specific typical operating conditions as follows

 Mean operating pressure—pmean=2 bar.
 (The crankcase is sealed, and the output shaft power is obtained by a magnetic coupling. Andy typically pressurizes the crankcase with a bicycle pump to about 2 bar.)
 Cooler temperature 27 degrees Celsius (300 K), and heater temperature 650 degrees Celsius (923 K)
 Operating frequency 50 Hz. (Note that the Ideal Adiabatic model is independent of operating speed—all results are presented per cycle)

In order to simulate the engine by means of the Ideal Adiabatic model equation set given previously, we require the equations for the Yokedrive volume variations and derivatives Vc, Ve, dVc and dVe (all functions of crank angle θ), as well as the void volumes of the heat exchangers Vk, Vr, and Vh.

The cyclic convergence behavior of the Ideal Adiabatic model is extremely good, and using 360 increments over the cycle, the system effectively converges within 5 cycles. The convergence criterion chosen is that after a complete cycle both variable temperatures Te and Tc must be within one degree Kelvin of their initial values. We now consider the solution of the temperature variables Tc and Te, the heat energy variables Qk, Qr, Qh, and the work energy variables Wc, We, and the net work done W. These results are presented as plots showing the variation of these parameters with the crank angle θ in FIG. 19.

In the temperaturetheta diagram depicted in FIG. 20, we observe a large cyclic temperature variation of the gas in the expansion space (>100 K), its mean value being less than that of the heater temperature of 923 K. Similarly the mean gas temperature in the compression space is higher than the cooler temperature. This suggests that the adiabatic working spaces effectively reduce the temperature limits of operation, thus reducing the thermal efficiency to less than that of the Carnot efficiency.

The energytheta diagram shows the accumulated heat transferred and work done over the cycle. Notice that the work done W starts with the (positive slope) expansion process then the compression process, and again returning to the expansion process, Thus the total work excursion is almost 15 joules, however the net work done at the end of the cycle is only 3 joules. The most significant aspect of the energytheta diagram is the considerable amount of heat transferred in the regenerator over the cycle, almost ten times that of the net work done per cycle. This tends to indicate that the engine performance depends critically on the regenerator effectiveness and its ability to accommodate high heat fluxes.

Significantly the energy rejected by the gas to the regenerator matrix in the first half of the cycle is equal to the energy absorbed by the gas from the matrix in the second half of the cycle, thus the net heat transfer to the regenerator over a cycle is zero. It is for this reason that the importance of the regenerator was not understood for about 100 years after Stirling's original patent describing the function and importance of the regenerator. The Lehmann machine on which Schmidt did his analysis was apparently not fitted with a regenerator, and it is conceivable that Schmidt did not appreciate its importance, He refers to the textbook by Zeuner as containing a “complete, simple and clear theory” of air engines, but in the same textbook Zeuner decries the use of regenerators for air engines (Finkelstein, T., 1959, Air Engines in The Engineer part 1, 27 March, incorporated herein by reference).

It is of interest to examine the two components, Wc and We, which added together gives the net work done W. These are shown as dashed lines in the diagram depicted in FIG. 21.

Notice in particular that the expansion space work done (We) undergoes a vastly different process from that of heat transferred to the heater (Qh), however at the end of the cycle they have equal values (Qh=We). Similarly for the compression space work done (Wc) and the heat transferred to the cooler (Qk). In retrospect this must be so in order to retain an energy balance, however it did catch us unawares and surprised us when we first noticed this. The ideal regenerator thus behaves as the perfect isolator, isolating the energy balance of the heater and expansion space from that of the cooler and compression space. Thus for the Ideal Adiabatic model over a complete cycle

Qh=We; (Qe=0)

Qk=Wc; (Qc=0)

W=Wc+We

Recall that for the Ideal Isothermal model

Qe=We; (Qh=0)

Qc=Wc; (Qk=0)

W=Wc+We

Furthermore the Ideal Adiabatic model in itself does not give results which are significantly different from those of the Ideal Isothermal model. The pressurevolume diagram is of similar form, and the power output and efficiency are quantitatively similar (albeit the efficiency of the Ideal Adiabatic model is about 10% lower for reasons described above). However the behavior of the Ideal Adiabatic model is more realistic, in that the various results are consistent with the expected limiting behavior of real machines. Thus the heat exchangers become necessary components without which the engine will not function. The required differential equation approach to solution reveals the considerable amount of heat transferred in the regenerator, indicating its importance in the cycle, and provides a natural basis for extending the analysis to include nonideal heat exchangers. Thus the solution of the Ideal Adiabatic model equations is equivalent to a simulation of the engine behavior in all respects, from setting up the initial conditions until convergence to cyclic steady state is attained. Throughout this process all the variables of the system are available as byproducts of the simulation and can be used for extending the analysis. Thus for example the mass flow rates through all the heat exchangers can be used in order to evaluate the heat transfer and flow friction effects over the cycle. The work of Kay J M & Nedderman R M (1985) Fluid Mechanics and Transfer Processes, Cambridge University Press and S. S. Kutateladze and V. M. Borishanskii, “A Concise Encyclopedia of Heat Transfer”, Pergamon Press, 1966 is also incorporated herein by reference.
BRIEF DESCRIPTION OF THE INVENTION

The present invention is a thermally isolated counter flowing heat exchanger comprising two isolated fluids having different energy levels. Each fluid flows in a contained system, such as a container, such as coiled tubing, having an inlet and an outlet. The isolated systems comprise thermally isolated cells. The isolated fluids flow in opposite directions. The cells are separated from each other by fluid heat trap passages. The isolated fluid systems are in contact with each other such that energy is transferred between the isolated fluids. In an embodiment, gates control which cell initially receives the wastewater by a signal from a temperature gauge.

The two or more fluids in a unit of the present invention that are totally isolated and exchanging heat through counter flowing through thermally isolated cells can be almost any fluid, e.g. liquid or gas, which remains inert to polymer tubes. Therefore, embodiments exist for other fluids than those referenced here.

Each thermally isolated cell in the present invention reaches a thermal equilibrium independent from adjacent cells; therefore, as the hot and cold fluids counter flow from cell to cell the heat transfer efficiency can continue to increase and is only limited by the number of cells and the effectiveness of the thermal isolation/insulation of the cells.

Unlike traditional heat recovery devices, the device of the present invention is also an integrated thermal storage device; therefore, heat can be extracted from it even when a hot fluid is not currently flowing into it.

In an embodiment with a hot fluid, e.g. hot wastewater or any hot liquid or gas, flowing through multiple thermally isolated cells that contain polymer tubes with a counter flowing colder fluid is a more efficient heat recovery device to operate for both intermittent and continuous fluid flows than other traditional heat recovery devices.

In an embodiment with a hot fluid flowing through multiple thermally isolated cells that contain polymer tubes with a counter flowing colder fluid, rather than using more costly metal tubes or plates, is a more costeffective heat recovery device to manufacture for both intermittent and continuous fluid flows than other traditional heat recovery devices.

Due to the circular shape of each insulated cell of the present invention, and the intercellular thermal isolation passages for the flow of the hot fluid between cells, the hot fluid is quickly and evenly distributed in a circular flow over the coils of tubing and thereby quickly transfers its heat to the counter flowing colder fluid in the coils. The direction of flow can be from top to bottom or vice versa; however, flowing waste fluid from the bottom creates less pressure.

Unlike the traditional heat reclamation devices, e.g. shell and tube and plate and frame, which rely on the transfer of heat from continuously flowing fluids through a continuously thermally connected device containing a typically large metallic surface area over which the two separated fluids flow, the present invention allows the fluids to flow either continuously or intermittently through thermally isolated and insulated cells—little heat will be lost and heat transfer between fluids will continue even while there is no fluid flow.

Traditional shell and tube or plate and frame heat exchangers are typically between 60 and 70 percent efficient, whereas, the simplest version of the present invention is over 70 percent efficient with more sophisticated fluid temperature balancing embodiments over 90 percent efficient. High efficiency is accomplished by directing a source fluid to a specific cell based upon its current temperature; thereby, making the overall heat transfer more efficient.

The coils of polymer tubing can be manifolded so that one or more fluids are flowing through the coils isolated from one another and may even counter flow to one another. This allows the present invention to transfer thermal energy from multiple fluids within one device. Thermal energy can be transferred efficiently through the walls of materials other than metal, such as polymer materials e.g. polyethylene, polypropylene, nylon, etc. tubing, if there is sufficient time or an increased rate of flow of fluids relative to surface area on either side of the material. The increased surface area coupled with an increased length of time for the heat transfer increases the heat transfer coefficient for a material such as polyethylene as well as other polymers. Thereby, utilizing the cellular approach and forced convection, the thermal energy transfer rate of the present invention exceeds that of a copper or other metal based traditional heat reclamation device.

The present invention is a system for transferring and storing thermal energy comprising a refrigerant circulating in a tank. The refrigerant flowing into the tank at a series of jets located along an exterior wall of the tank and exiting the tank at a drain located at a center of the tank. The flow from the jets creates a vortex circulation of the refrigerant in the tank. The vortex flow of the refrigerant is in contact with multiple spaced tubing located inside the tank such that energy from a fluid flowing inside the tubing is transferred to the refrigerant causing the fluid to freeze to a solid state and energy from the refrigerant is transferred to the frozen fluid causing the frozen fluid to return to a liquid state. The tubing comprises a volume expansion space. In an embodiment, the flow of the refrigerant in reversed. In an embodiment, the tank is insulated.

The vortex of the system causes the refrigerant to flow faster over the tubing such that the energy is transferred more quickly. The vortex speed may be varied by the rate of introduction of the refrigerant to the tank. The refrigerant rate may be varied by a pump speed.

In an embodiment, the cooled refrigerant is introduced into the tank via jets located at the wall of the tank to create a vortex motion across tubing inside the tank that contains a fluid until the fluid is transformed to a solid from a transfer of energy from the fluid to the refrigerant. At a given time, a noncooled refrigerant is introduced into the tank in the vortex motion across the tubing until the solid fluid (such as ice) is transformed to a liquid from a transfer of energy from the refrigerant to the solid. The refrigerant cooled by the solid is used for further energy transfer.

A system for generating energy comprising a hot regenerator and a cold regenerator, each independent, dedicated and thermally isolated. The regenerators each connected to counter cycling hot expansion pistons that utilize compression of exhausting hot gas as it flows into the cold regeneration area to create a suction effect on the exhausting hot gas that adds power to the compression stroke of the piston, said pistons connected to a scotch yolk and a crankshaft, said crankshaft turned from the movement of the pistons and providing energy to a kinetic drive.

As used herein, “approximately” means within plus or minus 25% of the term it qualifies. The term “about” means between ½ and 2 times the term it qualifies.

The compositions and methods of the present invention can comprise, consist of, or consist essentially of the essential elements and limitations of the invention described herein, as well as any additional or optional ingredients, components, or limitations described herein or otherwise useful in compositions and methods of the general type as described herein.

Numerical ranges as used herein are intended to include every number and subset of numbers contained within that range, whether specifically disclosed or not. Further, these numerical ranges should be construed as providing support for a claim directed to any number or subset of numbers in that range or to be limited to the exact conversion to a different measuring system, such, but not limited to, as between inches and millimeters.

All references to singular characteristics or limitations of the present invention shall include the corresponding plural characteristic or limitation, and vice versa, unless otherwise specified or clearly implied to the contrary by the context in which the reference is made.

All combinations of method or process steps as used herein can be performed in any order, unless otherwise specified or clearly implied to the contrary by the context in which the referenced combination is made.

Terms such as “top,” “bottom,” “right,” “left,” “above”, “under”, “side” and the like are words of convenience and are not to be construed as limiting.
BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is diagrammatical view of an embodiment of the invention.

FIG. 2 is diagrammatical view of an embodiment of the invention.

FIG. 3 is diagrammatical view of an embodiment of the invention. FIGS. 421 are diagrams and charts used to illustrate the workings of embodiments of the invention.
DETAILED DESCRIPTION OF THE INVENTION

As shown in FIG. 1, the present invention is a system of thermally isolated “cells” where temperature equilibrium can occur within each cell through both forced convection of moving fluids and with passive convection of submerged coils in liquid. This volume adjusted temperature difference defines the work to be performed on the other fluid volume and temperature mix. Unlike the traditional heat reclamation devices, i.e. shell and tube and plate and frame, which rely on the transfer of heat from continuously flowing fluids through a continuously thermally connected device containing a large surface area (typically metallic) over which the two separated fluids flow. The present invention allows the fluids to flow either continuously or intermittently through thermally isolated and insulated cells—heat transfer will continue while there is no flow.

As described in the energy transfer equations above, each cell of the present invention achieves a thermal equilibrium as the cold fluid moves up in coils through the cells and the hot fluid moves down. The equilibrium is impacted by the difference of the volume sizes for each of the fluids in the cell and by the rate that each fluid flows through the cell. The transfer efficiency can be improved by directing the hot waste fluid to a specific cell based on its temperature.

The device of the present invention takes advantage of the above mentioned laws of thermodynamics and increases its heat transfer efficiency by creating multiple thermally isolated and insulated heat exchange cells within the device. Each cell is thermally insulated and allows the hot fluid to pass from cell to cell through thermal isolation passages that close after the hot fluid passes creating a heat isolation chamber. Each cell has hundreds of feet of low cost polymer tubes that provide a high surface area for heat exchange for the fluid that is counterflowing through those tubes. Due to the circular shape of the cell and its hot fluid isolation passages, the hot fluid is quickly and evenly distributed over the coils of tubing which transfers its heat to the colder fluid traveling inside the coils.

Each cell is thermally isolated from adjacent cells and from the outside environment. The present invention can thereby support either intermittent or continuous flows of fluids. Hot fluids can have temperatures that vary from anything above the incoming cold fluid side temperature to just below boiling. More sophisticated embodiments of the present invention direct the hot fluid to the appropriate cell based upon its temperature. Traditional shell and tube or plate and frame heat exchangers are typically between 60 and 70 percent efficient, whereas, a basic embodiment of the present invention is up to 80 percent with more sophisticated embodiments above 90 percent efficient. Key claims are that the present invention with a hot fluid, i.e. wastewater, flowing through multiple thermally isolated cells that contain polymer tubes with counter flowing cold fluid is both a more efficient and a more costeffective heat recovery device for both intermittent and continuous wastewater flow than traditional wastewater heat recovery devices.

As shown in FIG. 2, the present invention is a system for transferring and storing thermal energy comprising a refrigerant 10 circulating in a tank 20. The refrigerant flows into the tank at a series of jets 30 located along an exterior wall of the tank and exiting the tank at a drain 40 located at a center of the tank. The flow from the jets creates a vortex 50 circulation of the refrigerant in the tank. In an embodiment, the flow of the refrigerant in reversed. The vortex flow of the refrigerant is in contact with multiple spaced tubing 60 located inside the tank such that energy from a fluid 70 flowing inside the tubing is transferred to the refrigerant causing the fluid to freeze to a solid state and energy from the refrigerant is transferred to the frozen fluid causing the frozen fluid to return to a liquid state. The tubing comprises a volume expansion space 80. The refrigerant is circulated to the tank and cooled by a chiller/heat pump 90. In an embodiment, the tank is insulated.

The present invention comprises a thermal heat exchanger and thermal storage unit contained in an insulated nonmetallic tank that circulates a refrigerant solution over fluid filled polymer tubes using jets located along the exterior walls and the center of the tank to a return near or at the center of the tank, and vice versa, to and from one or more heat pumps. The tank may be fabricated from any noncorrosive metal or nonmetallic substance, such as a glass, a polymer, a plastic and the like. Polymer substances for the tubes and the tank include polyethylene, polypropylene, nylon and the like. The refrigerant may be any typically refrigerant, such as but not limited to water, an alcohol, a glycol, and mixtures thereof, such as but not limited to water containing 25% ethylene glycol. In an embodiment, the refrigerant is ethylene glycol and water. Because of the flow of the refrigerant from the jets, the refrigerant flows in a vortex in the tank. The fluid in the tubes is any fluid that can freeze, such as water and water mixtures. As the refrigerant vortex flows over the tubes, the fluid inside the tubes transitions to a solid state, such as ice. The tubes comprise pressure relief air spaces at the top of the tubes to absorb the expansion and contraction of the fluid before and after the phase change to and from a fluid to a solid.

Key to the invention is the vortex flow of the refrigerant over the tubes of fluid/solid. The vortex provides an advantage over traditional flow of refrigerants through tubes in a tank of water to provide a more uniform and efficient cooling of the fluid when cooling the fluid to a solid, and a more uniform and efficient cooling of the refrigerant when transferring heat to the solid, such as ice. The uniform delivery of cooling tonnage allows for the system to deliver a stable quantity of cooling capacity over time rather than drop off significantly in time as with traditional ice storage systems.

Another advantage of the invention is the more viscous refrigerant is flowing through the tank rather than in the small tubes, such that the dynamic head pressure of the invention is significantly lower than the traditional approach causing a large reduction in the energy required to circulate the refrigerant. The lower pressure allows for more fluid to be circulated per tank ice volume than in traditional ice storage systems.

The invention can be easily scaled in size and can be ganged together to increase capacity linearly. In an embodiment, a single unit of the invention ranges in size from 2.5 (2′×3′ tank) to over 26 tonhours (3′×7′ tank).

During offpeak rate times and/or at night when outside temperatures are lower, a refrigerant is cooled by a chiller/heat pump and is circulated through the tank extracting heat until essentially all of the fluid in the tubes is frozen solid. In an embodiment where the fluid is water, ice is built uniformly throughout the tubes by the vortex flow of the refrigerant in the tank flowing over the water filled heat exchanger tubes of water/ice. The temperature remains uniform throughout the tank and water in the tubes does not become surrounded by ice during the freezing process and can move freely to expand while still in the liquid state before it transitions to ice, thereby preventing damage to the tubes or the tank. The following day, the stored ice cools the circulating refrigerant from approximately 52° F. to approximately 34° F. This cooled refrigerant can then be used either partially or fully for heat pump cooling or direct cooling of air through coils in air handlers or other cooling needs.

A Full Storage system chiller is used in the present invention to store ice during the night. The 32° F. energy stored in the ice then provides the required 750 tonhours of cooling during the day. The average load has been lowered to 53.6 tons (750 tonhours÷14=53.6). The chiller does not run at all during the day, which results in significantly reduced demand charges. In new construction, a Partial Storage system is usually the most practical and costeffective load management strategy. In this case, a much smaller chiller is allowed to run any hour of the day. It charges the ice storage tanks at night and cools the load during the day with help from stored cooling. Extending the hours of operation from 14 to 24 results in the lowest possible average load (750 tonhours÷24=31.25). Demand charges are greatly reduced and chiller capacity can often be decreased by 50 to 60 percent or more. Note that although the building's average 24hour load is 31.25 tons, the chiller's actual capacity is slightly higher during the day and lower at night. This is because of the chiller's 30 to 35 percent derated capacity for ice making (not to be mistaken for an efficiency derating).

An example of the system designed for residential and small commercial applications uses water in coiled tubing inside a polyethylene container. A refrigerant is cooled by a low temperature heat pump using the less expensive energy provided at night, and circulated in the tank over the tubing to create ice. The ice is melted during the next day's peak cooling loads to assist in the cooling requirements during the expensive daytime electric billing. The present invention allows for more efficient pumping as the refrigerant is circulated through the tank, not through the tubing. The tubing contain the water allows for more controlled rate of discharge of the stored potential ice energy.

The vortex flow of the refrigerant over the tubes containing fluid is opposite the traditional flow of the refrigerant through tubes in a tank of water resulting in a more uniform, faster and efficient cooling of the fluid (such as water to ice) because the faster flow of refrigerant over the larger surface area of the water/ice storage tubes will extract heat from the water more quickly. The vortex flow of the refrigerant over the tubing causes a more uniform, faster and efficient cooling of the refrigerant when extracting cold from the ice because the faster flow of refrigerant over the larger surface area of the water/ice storage tubes transfers heat to the water/ice in the tubes more quickly that that of traditional flow of the refrigerant through tubes in a tank of water. The more viscous refrigerant flowing through the tank rather than the small tubing significantly lowers the dynamic head pressure of the refrigerant causing a larger volume of the refrigerant to be circulated while reducing the energy required to circulate the refrigerant. The more viscous refrigerant flows through the tank rather than the small tubes; therefore, the dynamic head pressure of the refrigerant is significantly lower than with the traditional approach causing the volume of refrigerant being circulated over the tubing to be easily varied to allow for a quick and efficient increase or decrease the rate of thermal energy transfer. The significantly lower dynamic head pressure of the refrigerant causes a larger volume of the refrigerant to be circulated thereby reducing the number of ice storage units required to support the refrigerant flow demanded by the chiller/heat pumps. As the thermal energy is transferred from the refrigerant to the frozen fluid, the solid state fluid remaining in the tubing as the fluid returns to a fluid state floats to the top of the tubing, but is still in contact with the tubing walls which are in contact with the refrigerant circulating over them in the vortex flow causing a more uniform and faster transfer of heat to the frozen fluid than in traditional ice storage systems. As the thermal energy is transferred from the fluid to the refrigerant in the solid forming cycle (for example, water to ice), the ice that forms in the tubing as they cool will float to the top of the tubes and will allow the remaining fluid that has not yet solidified to be in contact with the tubing walls which are in contact with the refrigerant circulating around them in the vortex flow causing a more uniform and faster transfer of heat from the ice/water to the refrigerant than in traditional ice storage systems. A major departure and improvement from the traditional ice storage system is that the present system approach requires much fewer tubing connections and those that exist can be less costly mechanical connections rather than thermal welds because the connections are located in the refrigerant or above it where those environments never experience the extreme cycles of continuous freezing and thawing.

The energy required to form the solid and the energy required to melt the solid is less than with traditional ice storage systems making the present invention a more efficient and therefore less costly ice storage system to operate. The lower cost of materials, the lower cost of pumps, fewer connections and simplicity to manufacture the present invention ice storage system makes it a less costly ice storage system to produce than traditional ice storage systems.

As shown in FIG. 3, the present invention is a system of thermally isolated cold thermal transfer areas and hot thermal transfer areas where temperature equilibrium can occur within each cell through both forced convection of moving fluids and with passive convection of submerged coils in liquid. This volume adjusted temperature difference defines the work to be performed on the other fluid volume and temperature mix. Traditional heat motor devices, (i.e. Carnot cycle and various forms of Stirling engines) rely on the transfer of heat from continuously flowing fluids through a continuously thermally connected regeneration device containing a large surface area (typically metallic) over which the fluid flows as it moves back and forth between the cold area and the hot area. The present invention allows the fluids to flow either continuously or intermittently through thermally isolated and insulated areas that do not share the same regeneration surface; thereby, increasing the effectiveness of the heat transfer. Therefore, the heat transfer will be twice as efficient and the heat transfer will continue while there is no flow.

As described in the energy transfer equations, each hot and cold heat transfer area of the present invention approaches a thermal equilibrium. As hot gas is exhausted from the associated piston's compression, the gas goes through atomizers and moves up over the cold tubes transfer area causing the gas to cool and contract. Then, as the cold gas continues to move up again through atomizers into the hot tubes transfer area, the hot water surrounding the gas acts as a one way valve causing it to expand into the piston and produces pressure on the piston powering it to go up. This cycle is continuously repeated as the two expansion pistons cycle. The cycle is controlled by air release valves that hold the air pressure in the piston unit it reaches the extended position and then the air is released until the piston reaches its lowest position and then the valve is once again closed. The pistons cycle up and down on a scotch yolk as hot gases flow from one piston to the other.

The power produced is a direct function of the size of the pistons and the temperature difference between the cold and hot heat transfer areas. The pistons transfer their kinetic energy to a crankshaft utilizing a transfer device, such as a scotch yoke, piston rods, and the like. The crankshaft turns to deliver power to either direct drive devices, such as pumps, propulsion drives, etc., or indirect devices, such as hydraulic drive systems, electrical generators, etc. The hot fluid can be supplied by solar thermal panel(s), waste heat fluid(s), geothermal, other hot fluid sources and/or any combination thereof. The cold fluid can be supplied by geothermal fluid(s), cold waste fluid(s), other cold fluid sources and/or any combination thereof.

Thermal energy is conducted well by metals such as copper, platinum, gold, iron, etc. Energy can also be transferred through the walls of other materials such as polymers, such as polyethylene, polypropylene, nylon, etc., if there is sufficient time or an increased rate of flow of fluids relative to surface area on either side of the material, thereby utilizing forced convection. The increased surface area coupled with an increased length of time for the heat transfer increases the heat transfer efficiency for polymer materials.

The present invention takes advantage of the laws of thermodynamics and increases heat transfer efficiency by creating multiple thermally isolated and insulated heat exchange areas within the system. Each cold heat transfer area is thermally insulated and allows the hot gas to pass from the piston through a thermally isolated water filled cold tube area where the cold fluid passing through the tubes may come from a ground source, such as a lake, pond, ground water, well water, etc. The water filled area acts as a one way valve forcing the gas to go up through a thermally isolated (heat trapped) tube to the next heat transfer area. Each hot heat transfer area is thermally insulated and allows the cooled gas to pass from the piston through a thermally isolated water filled hot tube area where the hot fluid passing through the tubes may come from a hot fluid source, such as solar thermal panel(s), hot wastewater fluid(s), etc. Each heat transfer area has hundreds of feet of nonmetallic tubes that provide a high surface area for heat exchange for the fluid that is flowing through the tubes. Due to the circular shape of the cell, its isolation and the thousands of small holes that distribute the gas over the coils of tubes, the gas is quickly and evenly distributed over the coils of tubing thereby transferring its heat to and from the fluid traveling inside the coils.

The present invention is more efficient than a traditional Stirling engine by both separating the regenerator into two independent, dedicated and thermally isolated hot and cold regenerators and by utilizing two counter cycling hot expansion pistons rather than using one cold compression piston and one hot expansion piston.

As described in the Ideal Adiabatic analysis and Simple Regenerator analysis sections, the present invention is more efficient than a traditional Stirling engine by separating the regenerator into two independent, dedicated and thermally isolated hot and cold regenerators.

The present invention is more efficient than a traditional Stirling engine by utilizing two counter cycling hot expansion pistons rather than using one cold compression piston and one hot expansion piston.

The solar/geothermal embodiment of the present invention utilizes no fuels other than the sun and earth's ground water to generate electricity or deliver kinetic energy. Other embodiments can utilize waste hot and cold fluids to generate power.

The present invention utilizes materials that are both inert and highly durable that coupled with the fact that it can generate enough power to run simple and reliable pumps makes it capable of operating for years without any maintenance.

The present invention can vary in size to accommodate various thermal differences of fluid sources or various flow rates of fluids to deliver a required power.

The present invention can be integrated in sets in parallel or in series to more efficiently utilize available thermal variations or fluid flow rate variations to deliver the required power.

The present invention utilizes the compression of the exhausting hot gas as it flows into the cold regeneration area to create a suction effect on the exhausting hot gas that will add power to the compression stroke of the piston.

The present invention system can be utilized in one or more sets of cycling pairs of expansion pistons as indicated in FIG. 1 and be used in conjunction with heating and cooling systems such as those described above to deliver a complete energy package for private or commercial applications.

In an embodiment with a buried and/or insulated hot fluid storage container can allow the system to continue to operate when the sun is not shining, e.g. during the night or in cloudy weather. The size of the storage container will directly determine the length of the sunless operation.

The present invention regenerators contain atomizing gas diffusers in the bottom of each heat transfer area such that as the atomized gas flows over the large surface area of the hot fluid coils or the cold fluid coils, there is a more efficient transfer of thermal energy to and from the gas than is realized in a traditional Stirling engine regenerator.

While the forms of the invention herein disclosed constitute presently preferred embodiments, many others are possible. It is not intended herein to mention all of the possible equivalent forms or ramifications of the invention. It is to be understood that the terms used herein are merely descriptive, rather than limiting, and that various changes may be made without departing from the spirit of the scope of the invention.