This exercise concerns the ideal performance
of the EG1000 Stirling engine (developed by Sunpower,
Inc) which is gas fired and has been designed to generate
electricity as well as to provide hot water for a private home.
This engine is shown in the figure below together with a simplified
schematic diagram. Notice that there are two pistons  a power
piston which allows compression and expansion of the working gas
(helium) and a displacer piston which shuttles the working gas
between the hot expansion space V_{E}
and the cold compression space V_{C},
through the series connected heater, regenerator and cooler. Conceptually
the Stirling engine is the simplest of all heat engines. The working
gas is sealed within its cylinder by the power piston. The displacer
piston shuttles the gas such that the gas will compress while
it is mainly in the cool compression space and expand while in
the hot expansion space. Since the gas is at a higher temperature,
and therefore pressure, during its expansion than during its compression,
more power is produced during expansion than is reabsorbed during
compression, and this net excess power is the useful output of
the engine. Note that there are no valves or intermittent combustion,
which is the major source of noise in an internal combustion engine.
The same working gas is used over and over again, making the Stirling
engine a sealed, closed cycle system. All that is added to the
system is steady high temperature heat, and all that is removed
from the system is low temperature heat and mechanical power.
The linear electrical generator (not shown in the above schematic) is comprised of powerful rareearth magnets in the piston cutting a magnetic circuit and coils in the cylinder. This produces 240 Volts at 50 Herz  designed for operation in Europe, and is capable of producing more than one kilowatt of electrical power output at an efficiency of around 90%.

The working gas used is helium, which has the advantage of having a low molecular weight and high thermal conductivity compared to air, allowing a high efficiency system. Process (1)(2) is the isothermal compression of the helium at temperature T_{C} = 50°C, during which heat Q_{C} is rejected to the cooling water. Process (2)(3) is the constant volume displacement process during which heat Q_{R} is absorbed from the regenerator matrix. Process (3)(4) is the power producing isothermal expansion process at temperature T_{E} = 500°C, during which heat Q_{E} is absorbed from the gas burner, and finally process (4)(1) is the constant volume displacement process during which heat Q_{R} is lost to the regenerator matrix. Thus the ideal Stirling cycle consists of four distinct processes, each one of which can be separately analysed in accordance with the methods that are described in Chapter 3b. State (1) is defined at a maximum volume of 650 cu.cm and a pressure of 10 bar, and State (2) is defined at a minimum volume of 550 cu.cm. This large minimum volume is the dead space due to the unswept volumes including the heater, regenerator and cooler spaces. (Note that the values presented here are not actual values of the EG1000, however were devised by your instructor for purposes of this exercise only).
Figure 2. The ideal Stirling cycle engine PV
diagram
Since the Stirling cycle is a closed cycle, we can consider each process separately. Thus the work done for each process can be determined by integration. This is equivalent to evaluating the area under the Pv curve, as follows:
The working fluid is helium which is an ideal gas, we use the ideal gas equation of state throughout. Thus P V = m R T, where R = 2.077 kJ/kg K, and = C_{V} , where C_{V} = 3.116 kJ/kg K. (refer: Ideal Gas Properties)
1. From the given conditions at state 1 (P = 10 bar = 1000 kPa, V = 650 cc, T = 50°C) determine the mass of working gas (helium) used in the cycle. [m = 0.00097 kg (close to 1 gm)]
2. Determine the net work done per cycle (kiloJoules): W_{E} + W_{C} (Note that the compression work WC is always negative). At a frequency of 50 Herz (cycles per second) determine the power output produced by the engine. [Wnet = 0.151 kJ/cycle, Power = 7.55 kW]
3. Determine the heat absorbed in the expansion space Q_{E} during the expansion process (3)(4). [Q_{E} = 0.260 kJ]
4. Evaluate the Thermal Efficiency _{th}, defined as: _{th} = (W_{E} + W_{C}) / Q_{E}. (Net mechanical work done divided by the heat supplied externally by the gas burner). [58 %]
5. Determine the amount of thermal power rejected to the cooling water. Note that at a temperature of 50°C this is suitable for providing hot water for the home, as well as providing home space heating capability. [Q_{C} = 0.109 kJ/cycle, Thermal power to cooling water = 5.45 kW]
6. Determine the amount of heat transferred to the working fluid Q_{R} as it passes through the regenerator during process (2)(3). [1.36 kJ] If this heat were to be supplied externally by the gas burner, (i.e. no regenerator) what would be the new value of thermal efficiency _{th}? [9.3%]
In this photograph we see the Sunpower EG1000 being demonstrated using sawdust pellets as the fuel, and generating more than 1000W of electricity to a light panel. This was done at the Sustainability Fair in the Fairgrounds of Athens Ohio, 2001. A closeup photograph of the basic system is shown. Notice the closed cycle radiator and vibration pump used in the water cooling system. 
Update (Nov. 2013): Sunpower has recently introduced a 1 kW Stirling Developers Kit based on the EG1000 free piston Stirling engine fired by Propane or natural gas.
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Engineering Thermodynamics by Israel Urieli is licensed under a
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