Literature Review

This is on-going work and will be updated regularly.

The Stirling Engine

Abstract —This review attempts to provide an investigation into the thermodynamic analysis of a Stirling engine.

Nomenclature
CV specific heat at constant volume (J/kg K)
e  Stirling engine thermal efficiency (%)
n mole number 
p absolute pressure (N/m²)
Q heat
Qh heat furnished by the heater (J)
Qk heat furnished by the cooler (J)
R gas constant (J/kg K)
U internal energy (J)
T temperature (K)
Th heater temperature (K)
Tk cooler temperature (K)
V volume (m3)
Vh heater volume (m3)
Vk cooler volume (m3)
Vm minimum volume considering Vh and Vk only (m3)

VM maximum volume considering Vh and Vk only (m3)
W work (J)
Wh work for the heater piston (J)
Wk work for the cooler piston (J)
Wt engine net work (J)

I.     INTRODUCTION

 

The Stirling engine was invented in 1816 by Robert Stirling (Stirling 1816). The engines’ working principles are based on the laws of thermodynamics and ability of volume expansion of ideal gases at different temperatures. Although, further development of the engine was mitigated by the quick advancement of internal-combustion and the electric motor, later years saw lengthy amounts of investigations related to the further development of the Stirling engine  (Senft, Senft 1993, Thombare, Verma 2008). In contrast to the modern combustion engine there is no explosion inside the cylinder during operation. Instead, the engine uses an external heat source which results in chamber gas expanding and compressing cyclically and repeatedly to produce motion. As a result, a wide variety of heat sources such as solar, combustion of any fuel, nuclear and even geothermal, can be utilised and the compressible fluid gas can range from air, hydrogen, helium, nitrogen or vapour (Abuelyamen, Ben-Mansour 2018, Hachem, Gheith et al. 2015). In certain applications the engines displays high performance and its’ service can be desirable in situations where a long warm-up period is warranted, quiet operation is required, multi-fuelled characteristics is required or slow changing of engine output power is warranted (Kongtragool, Wongwises 2003).

Figure 1 modern Stirling Engine

Geometry & Operation

Stirling engines are commonly found in three different configurations. They have the same thermodynamic cycle but differ in mechanical design characteristics. These consist of the alpha, beta and gamma. The alpha type consist of twin power pistons separate cylinders (Fig. 2.). One cylinder is exposed to a heat source while the other is subjected to a cold temperature source. 

Figure 2 alpha type configuration

Both cylinders are connected to each other by a connecting pipe which is usually fitted with regenerative material. The purpose of the regenerator is to improve the thermal efficiency. The beta type vary's from the alpha by incorporating only one cylinder. Inside this cylinder holds one sealed piston and a displacer (Fig. 3.).

Figure 3 beta type configuration

Heat is applied to the top end whilst the bottom end is cooled. The displacers function is for transferring gas from the hot end to the cool end during the compression and expansion process. The power piston is fixed at ninety degrees out of phase with the displacer piston (Abuelyamen, Ben-Mansour 2018). The gamma-type engine is similar to the beta-type. Main difference being the cooling chamber is shifted to a small cooing chamber (Fig. 4.). 

figure 4 gamma type configuration

The theoretical Stirling cycle consists of four transformations of an ideal gas between two constant temperature heat sources and separated by a heat exchanger called a regenerator (Hachem et al. 2015). It consists of two isothermal and two isochoric processes (Fig. 5.).

figure 5 Stirling cycle

Heat is applied to the hot side which subsequently results in increased gas temperature. This in turn creates a rise in pressure which pushes the work piston down. Next the displacer piston pushes cold air into the hot end. During this phase, the cooler air creates decreased pressure, causing a contraction in the gas, thereby pulling the power piston back up. Stored energy in the rotating flywheel also assists with this phase shift (Urieli, Berchowitz 1984).

Prior Knowledge

 Many studies have been carried out about modeling and analysis of the Stirling engine (Chahartaghi, Sheykhi 2018). More than twenty sets of parameters like pressure, temperature, engine working frequency, power piston and displacer geometries, heat-exchanger geometries have direct effect on engine output power and efficiency and they add complication to the design procedure (Hooshang, Askari Moghadam et al. 2015). To analyze the conversion of energy from one form to another, the availability of energy to do work, and how heat will flow, it is important to understand the laws and applications of thermodynamics.  In terms of the Stirling engine, the emphasis is on engineering thermodynamics as it involves the process of transforming useful work from heat (Lorentzen 1981). 

By observing the first law of thermodynamics, we examine two situations. First, consider adding heat to a system of fixed volume. Then the internal energy increases by an amount equal to heat added:

The second case to consider is expansion of a gas which pushes a piston. In this case if no heat is added to or removed from the system, the internal energy of the system decreases by an amount equal to the work done by the system:

Considering both cases, the internal energy can be written as:

This is the first law of thermodynamics. From this heat can be written as:

This means that the internal energy of the system is related to its temperature. Furthermore, heat can be transferred into or out of the system both by a temperature gradient and by work done by the system (Halliday, Resnick et al. 2013, Young, Freedman et al. 2010).

Theoretically, the thermal efficiencies of the Stirling engine are close to the Carnot cycle, therefore they have higher values of thermal efficiency than other heat engines (Batmaz, Üstün et al. 2008). The Carnot cycle (Fig. 6.), founded by French physicist Sadi Carnot in 1824, is a glorified representation of the operation of the steam engine. The process is such that an ideal gas is restricted in a cylinder by a piston, and is allowed to absorb heat from a reservoir held at Th or to reject heat to a reservoir held at T_k.

figure 6 Ideal Carnot cycle

The gas is led through a cyclical path which, overall converts heat into useful work (Kerker 1960). During the isothermal compression (segment 4-1 in figure 6 and 1-2 in figure 5), the cold piston moves towards the regenerator. In the following step (segment 1-2 figure in 6 and 2-3 figure in 5) both pistons move similarly so that the volume remains constant and the working fluid is transferred through the regenerator. This step is followed by the isothermal expansion process (segment 2-3 in figure 6 and 3-4 in figure 5), with only the hot piston moving. In the step (segment 3-4 in figure 6 and 4-1 in figure 5) both pistons move keeping the volume constant and transferring the working fluid through the regenerator. This is an ideal cycle assuming the perfect heat transfer and does not take into account heat losses during the regeneration (Puech, Tishkova 2011). For each cyclic process, heat Q_h  is added from the hot reservoir T_h  to the engine (Fig. 7.) and the engine does work W by using that heat. The heat that is not converted to work, Qk = Qh - W, leaves the engine and is dumped into the cold reservoir at Tk (Young et al, 2010, Halliday, et al, 2013).

                                                                                 figure 7 The heat engine

Calculating Efficiency

Data gathered from the Stirling engine cycle can also be displayed in a pressure p versus volume V diagram (Fig. 8.).

 figure 8 The P V diagram

Here the product of pressure and volume represents a quantity of work. This is the area enclosed by the four curves of the PV diagram. As mentioned earlier, referring to the second law of thermodynamics it is possible to show that no heat engine can be more efficient than a reversible heat engine working between two fixed-temperature limits (Lorentzen 1981).

In order to calculate the efficiency of an ideal Stirling engine, the author will assume there is perfect regeneration with no regenerator dead volume. Dead volume is the total void volume in the engine and is generally referred to as the volume of working fluid contained in the total dead space in the engine (Kongtragool, Wongwises 2006).

The following (Table 1) shows energy calculations without dead volume and with perfect regeneration during the phases of the PV diagram (Fig. 8.).

Table 1 Energy quantities without dead volume

During the isochore processes, pressure is given by:

With the pressure obtained we can calculate the work done during both isothermal transformations which is:

This can be displayed to show the work for cold and hot space (Table 2). The work for a transition is W = W_h + W_k.

Table 2 Work for cold and hot space

Thus, the internal efficiency of the ideal Stirling cycle will be:

Improving Efficiency

Various research and inventions have been undertaken into improving the efficiency of Stirling engines for both motion and power generation. One such example is the United States patent number 3,513,659 which is for an invention for a Stirling Cycle Amplifying Machine (Fig. 9.). This invention aims to produce a large amount of output energy from a small amount of input energy.

The conventional Stirling engine has the displacer and power piston mechanically linked by a crank shaft and phased 90 degrees from each other. However, the Martini concept assumes control of the displacer which is independently oscillated in a reciprocating motion. The motion is controlled by any suitable means such as an electrical motor. With the regenerator being able to be controlled independently, this also allows for instantaneous control of the power output of the engine by varying either amplitude or the phase angle of the regenerator motion (Martini 1970). This concept of the engine is called the Martini Type Stirling Engine.

The free piston Stirling engine is another configuration which aims to improve the efficiency by having no fluid leakage. It was invented by William Beale in 1964 (Walker, Senft 1985) (Beale 1969). The engine consists of a few numbers of components such as a piston, displacer, springs and casing. Variations of this engine for creating electrical power also includes a linear alternator. This model (Fig. 10.) is a beta-type configuration and includes two linear alternators for generating electricity (Mou, Li et al. 2016).