# Stirling Engine Regenerator Analysis Essay

## Introduction

The Stirling engine was invented by Rev. Robert Stirling some 200 years ago, at the time the engine received some attention and saw commercial use (Stirling, 1816). However, at the time the rapid development of the internal combustion engine quickly overshadowed the Stirling engine. In recent years, there has been renewed interest in the Stirling engine, especially with the rise in interest in renewable energy technology. Stirling engines are poised to play a pivotal role in this industry as they are quiet, have multi-fuel capabilities, produce little pollution and are efficient (Thombare and Verma, 2008). Furthermore, the Stirling engine is categorized as a Reitlinger class cycle, which means it can theoretically achieve Carnot Efficiency (Senft, 1998). The current uncertainty over the future availability of fossil fuels and the imminent threat of climate change means that new sources of energy need to be utilized (Ellabban et al., 2014). The Stirling engine is a prime candidate for use with renewable sources of energy as the engine operates through a closed thermodynamic cycle that can utilize any heat source. The Stirling engine has also been cited as being the most economically viable solar converter in the range of 5–100 kWe (Kongtragool and Wongwises, 2003).

There are a variety of different approaches to Stirling engine modeling and there exist several different orders of models (Dyson et al., 2004). These models are of varying complexity and there have been several studies that analyze, compare, and discuss the different models available. Initially, Stirling engines were modeled with isothermal working spaces; however, in recent times the working spaces have been modeled as having finite rates of heat transfer, or as being adiabatic. The analysis conducted by Finkelstein was the first of its kind and represented a major advancement in Stirling engine analysis when he considered non-isothermal working spaces (Finkelstein, 1960). Subsequently, Urieli and Berchowitz developed the ideal adiabatic model. This model more accurately predicts performance as in real Stirling engines the cylinders are not designed for heat transfer (Berchowitz and Urieli, 1984). In these models, a variety of losses are quantified, and included in the analysis to predict the output of a real engine operating at specified conditions (Walker, 1980). In the most recent study, a model with polytropic working spaces has been developed (Babaelahi and Sayyaadi, 2015). A significant portion of Stirling cycle optimization studies have moved toward numerical simulations that in the past were too computationally expensive to use for optimization purposes. However, with the extensive work carried out by Berchowitz and Urieli (1984), the models have been significantly sped up, and more than 30 years later computing speed has also drastically increased. A Stirling engine simulation that originally took 10 min to run now takes seconds. There have been several studies that have aimed to predict and optimize engine performance using numerical models. The study conducted by Timoumi et al. (2008) presented a new model of the Stirling cycle. The model presented was used to optimize the GPU-3 Stirling engine and a medium temperature difference solarpowered engine (Tlili et al., 2008). The study conducted by Campos et al. (2012), used a similar model to maximize the non-dimensional engine efficiency at specified conditions. The effect that changing parameters had on performance was also analyzed and it was found that the engine performance is robust to changes in some parameters. Senft used the classical Schmidt analysis and conducted an analysis that looked to find the optimum engine geometry of a Stirling engine (Senft, 2002). Other studies have also looked to analyze the second-order non-ideal adiabatic model using a multi-objective optimization approach (Toghyani et al., 2014). The analysis optimized the engine in terms efficiency and pressure drop, a Pareto frontier of optimal solutions is presented.

There have been several studies that have applied the exergy analysis methodology to ideal Stirling cycle models. The study conducted by Martaj et al. (2006), applied the exergetic, energetic and entropic analysis techniques to the Stirling cycle to optimize the performance. The same authors in a separate study analyzed and optimized a low temperature difference Stirling engine at steady state operation (Martaj et al., 2007). de Boer (2003) showed the importance of including pressure drop in his analysis of the Stirling engine regenerator, where he proved the maximum achievable efficiency for a Stirling cycle engine is half of Carnot efficiency. The analysis by Wu et al. (1998) formulated criteria to optimize the heat transfer area in the heater and cooler. Similarly, Costea and Feidt (1998) conducted an analysis looking at the irreversibility and the effects that the heat exchanger area had on Stirling engine performance. These analyses and optimizations usually begin with highly idealized models of the Stirling cycle which is problematic as they do not necessarily accurately model the Stirling cycle. There have been several studies that have looked at Stirling engine power density. Also, the dead-volume ratio has been seen to have a significant effect on performance, especially when adiabatic working spaces are assumed (Wills and Bello-Ochende, 2016). The effects of dead-volume have also been mentioned by several authors as negatively affecting the power output of Stirling engines (Kongtragool and Wongwises, 2006; Puech and Tishkova, 2011). These studies have analyzed how the dead-volume negatively affects the cycle efficiency and power output. Several studies have looked to optimize the power density of the Stirling engine as the engine size is of economic interest (Erbay and Yavuz, 1997, 1999). These analyses optimized the device in terms of maximum power density as this results in engines with good power density and high efficiency.

There have been many studies that have optimized power cycles using exergy analysis methodology. Where exergy is defined as the energy that is available to do work. The Gouy–Stodola theorem, which describes the relationship between reversible work , irreversible work , entropy generation and environmental temperature *T*_{0} (Bejan, 1996), can be seen below as Eq. 1.

The development of this equation was a major advancement in the thermodynamics of the time, and the expression shows that the rate of entropy generation is directly proportional to the rate at which work is destroyed. While using this methodology, it has been emphasized that it is crucial to optimize the system in its entirety, rather than as individual components (Bejan, 2006). This is done, as spreading the irreversibility over the entire system rather than minimizing it in individual components results in a truly optimized system.

This paper presents a novel approach to modeling the losses and optimizing the alpha type Stirling engine, which involves the application of exergy analysis methodology to the ideal adiabatic model of the Stirling cycle. The model incorporates the irreversibility due to heat transfer through a finite temperature difference, pressure drops and conductive thermal bridging loss. The model presented is used with the implicit filtering algorithm to optimize a 1,000 cm^{3} Stirling engine for maximum power production with four different regenerator mesh types and a fixed energy input. In the analysis, the working fluid is assumed to be an ideal gas and a finite heat capacity rate is assumed in the heater and cooler, the number of heater and cooler tubes are also fixed.

## Physical Model

The methodology used to optimize the 1,000 cm^{3} alpha type Stirling engine for maximum work output with a fixed energy input is presented in this section. The working fluid is assumed to be pressurized air which behaves as an ideal gas. The heater and cooler external fluids are assumed to have finite heat capacity rates, and four different regenerator mesh types are used in this analysis.

### Model Description

A diagram of the alpha type Stirling engine used in the analysis can be seen as Figure 1. In this diagram, the operating frequency *f*, volume *V*, and heater, cooler, and regenerator lengths *L* are shown.

**Figure 1**. Diagram of the alpha type Stirling engine.

The expressions for the compression and expansion space volumes are Eqs 2 and 3. Eq. 2 calculates the volume in the compression space *V*_{c} using the clearance volume *V*_{ccl}, swept volume *V*_{c,swept}, and the crank angle θ.

Similarly, Eq. 3 calculates the volume in the expansion space using the clearance volume *V*_{ecl}, swept volume *V*_{e,swept}, crank angle θ, and phase difference α.

Equations are required to determine the volumes from the geometric variables for the heat exchangers. Equations 4–9 are the equations for the volume *V* and area *A* of the cooler, heater and regenerator, respectively.

Equations 4 and 5 define the volume *V _{k}* and surface area

*A*in the cooler. These equations use the number of cooler tubes

_{k}*N*, cooler length

_{k}*L*and cooler tube diameter

_{k}*D*.

_{k}Equations 6 and 7 define the volume *V _{h}* and surface area

*A*of the heater. These equations use the number of heater tubes

_{h}*N*, heater length

_{h}*L*and heater tube diameter

_{h}*D*.

_{h}Equations 8 and 9 define the volume *V _{r}* and surface area

*A*in the regenerator. These equations use the mesh porosity

_{r}*ε*, regenerator length

*L*, regenerator diameter

_{r}*D*, and hydraulic diameter

_{r}*d*

_{hyd}.

Figure 1 shows the different engine variables. To model the engine, some of the variables are fixed, and Table 1 gives these variables and their values. The values are chosen based on an idealized model of the Stirling engine, and the number of heater and cooler tubes was chosen based on real Stirling engines (Timoumi et al., 2008). The number of heater tubes is far lower than the number of cooler tubes; this is because decreasing the temperature of the working fluid in the cooler has a greater effect on engine performance than increasing the temperature of the working fluid in the heater. Another reason there are fewer tubes in the heater is that in real Stirling engines the heater may have combustion products flowing through it. Depending on the fuel source this may foul the heat exchanger, and therefore the spacing of the tubes is important to facilitate cleaning. The cooler usually has water flowing through it which means it would not foul nearly as quickly as the heater.

**Table 1**. Table of fixed parameters.

Along with these fixed parameters, four different mesh types are used in the optimization. The mesh types and their properties can be seen in Table 2.

## Mathematical Model

The following section presents and describes the equations used to model the alpha type Stirling engine which is optimized in this study. First, the model and the ideal adiabatic model of the Stirling cycle are presented. Following this the equations that describe the heat transfer, flow friction and thermal bridging loss are presented and explained. Finally, the exergy and rate of entropy generation equations are introduced and the method of solution is described.

The model outlined assumes finite heat capacity rates in the heater and cooler. The compartment temperature diagram shows the different thermodynamic properties and temperature in each compartment, seen as Figure 2.

**Figure 2**. Serially connected component and temperature diagram of the Stirling cycle.

Figure 2 is used as a means of graphically showing and defining different thermodynamic properties in the engine compartments and energy flows. The properties shown are pressure *P*, temperature *T*, and volume *V*. The compartments are the compression space *c*, cooler *k*, regenerator *r*, heater *h*, and expansion space *e*. The compartment interfaces are the compression space to cooler interface cb, the cooler to regenerator interface, the regenerator to heater interface and the heater to expansion space interface. All the equations presented in this section can be understood using the diagram and the knowledge that *R* is the ideal gas constant, *C _{p}* is the constant pressure specific heat,

*C*is the constant volume specific heat and γ is the ratio of specific heats.

_{V}### Ideal Adiabatic Model

The ideal adiabatic model was developed by Urieli and Berchowitz as a means of more accurately modeling the real Stirling cycle. At the time of the development of these models the iterative schemes took too long to solve, to make the model useful in the optimization of Stirling engine geometry. However, due to advances in computing and better models the solutions are arrived at in seconds rather than minutes, making these numerical models suitable for optimization purposes.

The full derivation of the equations is not presented but the equations are listed and briefly explained. To see the complete derivation of the equations, see the book by Urieli and Berchowitz (Berchowitz and Urieli, 1984), or the online resources maintained by Urieli (2017).

The ideal adiabatic model assumes that there is negligible pressure variation throughout the engine. Therefore, Eq. 10 describes the pressure *P* in all the engine compartments.

Assuming that the total mass of working fluid in the device is the sum of the masses of working fluid in each component yields Eq. 11.

Assuming, the mass of working fluid remains constant, yields Eq. 12.

For the cooler, regenerator and heater the volume and temperature are assumed to be constant. Therefore, the mass differential is defined as Eqs 13–15.

Substituting Eqs 13–15 into Eq. 12 and rearranging, yields Eq. 16.

Applying the mathematical expression for the first law to a generalized cell of working space yields Eq. 17.

Rearranging Eq. 17 to give the change in mass in the compression and expansion spaces yields Eqs 18 and 19.

Substituting Eqs 18 and 19 into Eq. 16 and rearranging yields Eq. 20.

Defining the temperature differentials in the compression and expansion spaces, yields Eqs 21 and 22.

The mass flows through the compartment boundaries are defined as Eqs 23–26.

The conditional temperatures which depend on the direction of fluid flow in the heater and cooler are Eqs 27 and 28.

The energy equations that describe the heat absorbed and rejected in the cooler, regenerator, and heater are Eqs 29–31.

The energy equations which describe the work output of the cycle are Eqs 32 and 33.

### Flow Losses

When calculating the pressure drop Δ*P* in the heat exchangers, the Reynolds friction factor *f _{r}* approach is used as this results in the change of pressure drop sign with change in flow direction (Berchowitz and Urieli, 1984). The Reynolds friction factor is defined as the Darcy friction factor

*f*multiplied by the Reynolds number Re.

_{D}Equation 34 is used to calculate the pressure drop Δ*P*, where viscosity μ, gas velocity *u*, volume *V*, flow area *A*_{flow} and hydraulic diameter *d*_{hyd} are used.

Equation 35 is the Darcy friction factor *f _{D}* in the regenerator and is calculated using the maximum Reynolds number Re

_{max}(Tanaka et al., 1990).

In the case of the cooler and the heater unidirectional smooth pipe flow relations are used to calculate the Darcy friction factor *f _{D}* (Joseph and Yang, 2010). These relations are for flow in the turbulent regime, seen as Eqs 36 and 37.

### Heat Transfer Relations

Equations 38 and 39 are for the Nusselt number Nu and effectiveness *ε* of the regenerator (Tanaka et al., 1990).

Equation 40 is the Gnielinski relation which is used to calculate the Nusselt number in the heater and the cooler (Gnielinski, 1975). This is then in turn used to calculate the heat transfer coefficient *h* in the heater and the cooler.

Renewable Energy Department, Energy and Environment Research Center, Niroo Research Institute, Ministry of Energy, P.O. Box 14665 517, Tehran, Iran

Copyright © 2012 A. Asnaghi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper provides numerical simulation and thermodynamic analysis of SOLO 161 Solar Stirling engine. Some imperfect working conditions, pistons' dead volumes, and work losses are considered in the simulation process. Considering an imperfect regeneration, an isothermal model is developed to calculate heat transfer. Hot and cold pistons dead volumes are accounted in the work diagram calculations. Regenerator effectiveness, heater and cooler temperatures, working gas, phase difference, average engine pressure, and dead volumes are considered as effective parameters. By variations in the effective parameters, Stirling engine performance is estimated. Results of this study indicate that the increase in the heater and cooler temperature difference and the decrease in the dead volumes will lead to an increase in thermal efficiency. Moreover, net work has its maximum value when the angle between two pistons shaft equal to 90 degrees while efficiency is maximum in 110 degrees.

#### 1. Introduction

The urgent need to preserve fossil fuels and use renewable energies has led to the use of Stirling engines, which have excellent theoretical efficiency, equivalent to the related Carnot cycle. They can consume any source of thermal energy (combustion energy, solar energy, etc.) and they make less pollution than the traditional engines, [1].

A Stirling cycle machine is a device, which operates in a closed regenerative thermodynamic cycle. In the ideal cycle of the Stirling engine, the working fluid is compressed at constant temperature, heated at constant volume, expanded at constant temperature, and cooled at constant volume. The flow is regulated by volume changes so there is a net conversion of heat to work or vice versa.

The Stirling engines are frequently called by other names, including hot-air or hot-gas engines, or one of a number of designations reserved for particular engine arrangement.

Numerous applications of Stirling engines were raised during the 19th century and at the beginning of the 20th century. Stirling engine in the 19th century was confined largely by the metallurgical possibilities and problems of the time. By these reasons, the engine was finally pushed back by newly developed internal combustion engines. The Stirling engine was almost forgotten until the 1920s.

It was only in 1938, a small Stirling engine with an output of 200 W by Mr. N. V. Philips which stimulated interest in this engine type again. Development in material production technologies that took place in the 1950s opened new perspectives as well for the Stirling engine.

In the course of 1969-1970, Philips developed a drive unit with a rhombic mechanism for a municipal bus. A detailed calculation finally showed that, with a batch of 10000 pieces annually, the price would still be 2.5 times higher than that of a compression ignition engine of the same output due to the substantial complexity of the engine.

In the 1970s United Stirling worked hard on the development of a drive unit for passenger cars. One of the following V4X35 types was fitted in the Ford Taurus car in 1974. Despite the satisfactory results of a driving test covering 10,000 km, series production was never commenced due to the price of the drive unit.

High heat efficiency, low-noise operation, and ability of Stirling engines to use many fuels meet the demand of the effective use of energy and environmental security. Stirling engine-based units are considered best among the most effective low-power range solar thermal conversion units [2].

On the other side, the main disadvantages of Stirling engines are their large volume and weight, low compression ratio, and leakage of working fluid from the engine inner volume. To increase the specific power of Stirling engines, a number of methods have been developed, such as using hydrogen or helium as working fluid at high charge pressure, increasing the temperature difference between hot and cold sources, increasing the internal heat transfer coefficient and heat transfer surface and using simple mechanical arrangements, for example, a free piston Stirling engine [3].

The Stirling engine performance depends on geometrical and physical characteristics of the engine and on the working fluid gas properties such as regenerator efficiency and porosity, dead volume, swept volume, temperature of sources, pressure drop losses, and shuttle losses.

A machine that utilizes the Stirling cycle could function as an engine that converts heat energy from an appropriate heat source into kinetic energy, or, by employing the reverse cycle, as a refrigerator that can achieve low temperatures or provide a heat-absorbing effect by the injection of kinetic energy from an electric motor. The Stirling engine has also been proposed as a driver for electricity generators and heat pumps, and its practical applications have been realized. Conventionally, fossil fuels and solar energy have been considered as potential heat sources; more recently, however, the practical applications of engines that utilize biomass or the waste heat generated from diesel engines as fuel have been discovered [4].

The thermal limit for the operation of a Stirling engine depends on the material used for its construction. In most instances, the engines operate with a heater and cooler temperature of 923 and 338 K, respectively. Engine efficiency ranges from about 30 to 40% resulting from a typical temperature range of 923–1073 K, and normal operating speed range from 2000 to 4000 rpm [5].

Numerical investigations of Stirling engines have been performed by many researches. Thermodynamics analysis is the basic of almost all of these researches. Some of these researches focus on a specific part of engine, for example, regenerator [6]. Some of the others focus on a specific type of Stirling engine, for example, free piston Stirling engine [7]. Considering specific operating conditions, for example, thermoacoustic conditions, are objective in many of these researches [8]. In all of these researches, the main objective is to calculate the optimum operating condition. In the current study, maximum output power and maximum performance efficiency are considered as main objectives. Numerical model is developed using thermodynamics analysis to generate performance variations comparing different parts’ characteristics and conditions.

#### 2. Stirling Engine Cycle

##### 2.1. Description

Stirling engines are mechanical devices working theoretically on the Stirling cycle in which compressible fluids, such as air, hydrogen, helium, nitrogen, or even water vapor, are used as working fluids. The Stirling engine offers possibility for having high-efficiency engine with less exhaust emissions in comparison with the internal combustion engine. The Stirling engine is an external combustion engine. Therefore, most sources of heat can power it, including combustion of any combustive material, field waste, rice husk, or the like, biomass methane and solar energy. In principle, the Stirling engine is simple in design and construction and can be operated without difficulty [9].

##### 2.2. Application

The Stirling engine could be used in many applications and is suitable where(1)multifueled characteristic is required;(2)a very good cooling source is available; (3)quiet operation is required; (4)relatively low speed operation is permitted; (5)constant power output operation is permitted; (6)slow changing of engine power output is permitted; (7)A long warmup period is permitted.

##### 2.3. Ideal Stirling Cycle

The engine cycle is represented on PV and TS diagrams in Figure 1. The ideal cycle of the Stirling engine is formed by two isochoric processes and two isothermal processes. Consider a cylinder containing two opposed pistons with a regenerator between the pistons as shown in Figure 2. The regenerator is like a thermal sponge alternatively absorbing and releasing heat which is a matrix of finely divided metal in the form of wires or strips. The volume between regenerator and the right side piston is expansion volume and between regenerator and left side piston is compression volume. Expansion volume is maintained at heater temperature which is called hot temperature, and compression volume is maintained at cooler temperature which is called cold temperature [2].

**Figure 1: ***P-V *and *T-S* diagrams of Stirling engine.

**Figure 2: **Stirling engine cyclic pistons’ arrangements [10].

To start with Stirling cycle, we assume that the compression space piston is at outer dead point (at extreme right side) and the expansion space piston is at inner dead point close to regenerator. The compression volume is at maximum and the pressure and temperature are at their minimum values represented by point 1 on PV and TS diagrams of Figure 1.

Four processes of the Stirling cycle are [11] as follows.

###### 2.3.1. Process 1-2, Isothermal Compression Process

During compression process from 1 to 2, compression piston moves towards regenerator while the expansion piston remains stationery. The working fluid is compressed in the compression space and the pressure increases from P1 to P2. The temperature is maintained constant due to heat flow from cold space to surrounding. Work is done on the working fluid equal in magnitude to the heat rejected from working gas. There is no change in internal energy and there is a decrease in entropy.

###### 2.3.2. Process 2-3, Constant Volume Regenerative Transfer Process

In the process 2-3, both pistons move simultaneously, that is, compression piston towards regenerator and expansion piston away from regenerator, so that the volume between pistons remains constant. The working fluid is transferred from compression volume to expansion volume through porous media regenerator. Temperature of working fluid increased from to by heat transfer from regenerator matrix to working fluid. The gradual increase in temperature of working fluid while passing through regenerator causes increase in pressure. No work is done and there is an increase in the entropy and internal energy of the working fluid.

###### 2.3.3. Process 3-4, Isothermal Expansion Process

In the expansion process 3-4, the expansion piston continues to move away from the regenerator towards outer dead piston while compression piston remains stationery at inner dead point adjacent to regenerator. As the expansion proceeds, the pressure decreases as volume increases. The temperature maintained constant by adding heat to the system from external source at . Work is done by the working fluid on piston equal in the magnitude to the heat supplied. There is no change in the internal energy, but an increase in the entropy of the working fluid.

###### 2.3.4. Process 4-1, Constant Volume Regenerative Transfer Process.

In the process 4-1, both pistons move simultaneously to transfer working fluid from expansion space to compression space through regenerator at constant volume. During the flow of working fluid through regenerator, the heat is transferred from the working fluid to the regenerator matrix reducing the temperature of working fluid to . No work is done; there is a decrease in the internal energy and the entropy of the working fluid.

The Stirling cycle is highly idealized thermodynamic cycle, which consists of two isothermal and two constant volume processes and the cycle is thermodynamically reversible. The first assumptions of isothermal working and heat exchange imply that the heat exchangers are required to be perfectly effective, and to do so, infinite rate of heat transfer is required between cylinder wall and working fluid. The second assumption requires zero heat transfer between walls and working fluid, both assumptions remain invalid in actual engine operation [2].

##### 2.4. Different Mechanical Configurations

With respect to their mechanical arrangements, Stirling engines are classified into three groups: alpha, beta, and gamma. Each configuration has the same thermodynamic cycle but has different mechanical design characteristics, see Figure 3.

**Figure 3: **The basic mechanical configurations for Stirling engine.

In the alpha-configuration, two pistons, called the hot and cold pistons, are used on either side of the heater, regenerator, and cooler. In the alpha type of mechanical arrangement, the thermodynamic cycle is performed by means of two pistons working in separate cylinders: one is held at the hot temperature and the other at the cold temperature.

In the beta-configuration, a displacer and a power piston are incorporated in the same cylinder. The displacer moves working fluid between the hot space and the cold space of the cylinder through the heater, regenerator, and cooler. The power piston, located at the cold space of the cylinder, compresses the working fluid when the working fluid is in the cold space and expands the working fluid when the working fluid is moved into the hot space.

The gamma-configuration uses separated cylinders for the displacer and the power pistons, with the power cylinder connected to the displacer cylinder. The displacer moves working fluid between the hot space and the cold space of the displacer cylinder through the heater, regenerator, and cooler. In this configuration, the power piston both compresses and expands the working fluid. The gamma-configuration with double acting piston arrangement has theoretically the highest possible mechanical efficiency. This configuration also shows good self-pressurization. However, the engine cylinder should be designed in vertical type rather than horizontal in order to reduce bushing friction [9]. It should be noted that, in this study, the thermodynamic equations and relations for alpha type has been presented. However, the presented algorithm can be extended to two other types of Stirling engines. To do so, basic thermodynamic equations for Beta and Gamma engines arrangements can be found in [11].

##### 2.5. Dead Volumes and Regenerator Deficiency

In the ideal Stirling cycle, it is assumed that the total heat rejected during process 4-1 is absorbed by the regenerator and then released to the working fluid during the process 2-3. In reality, we cannot find the ideal regenerator and all of the regenerators due to their structure and used materials have deficiency. So, an imperfect regenerator cannot absorb the total heat released during process 4-1, and consequently cannot provide the total required heat of process 2-3. For this study, the temperatures of working fluid at exit of the imperfect regenerator are noted as and . In Figure 1, the positions of these two temperatures in *P-V* and *T-S* diagram are presented. Although regenerator effectiveness values of 95, 98-99, and 99.09% have been reported, engine developers who do not have efficient-regenerator technology in hand should take into account the regenerator effectiveness, and then an analysis with imperfect regeneration should be made [5].

Total dead volume is defined as the sum of Stirling engine void volumes. The dead volumes are considered for regenerator, cold and hot pistons. It is evidenced that a real Stirling engine must have some unavoidable dead volume. In normal Stirling engine design practice, the total dead volume is approximately 58% of the total volume. Although many researchers have analyzed Stirling engines, there still remains room for further development. One can use the Schmidt equations to consider dead volumes on his/her analysis. However, ideal regeneration is assumed in the Schmidt analysis [11]. Considering Figure 2, the dead volumes contributions of Stirling engine are presented in Figure 4.

**Figure 4: **Stirling engine volumes contributions.

Another important issue which should be considered is the temperature of the regenerator. Correct estimating of regenerator temperature will have direct effects on the final results. In our approach, it is assumed that half of the regenerator dead volume is at , and the other half is at . Following, the effective temperature of the regenerator can be calculated by arithmetic mean or log mean of these two temperatures.

#### 3. Thermodynamics Equations

The basic assumptions for the Stirling engine are as follows [11].(i)Temperature in each gas space (cold and hot) is known and constant.(ii)There is no pressure difference between the gas spaces.(iii)Ideal gas law can be used for the working fluid.(iv)There is no leakage into or out of the working fluid space.

##### 3.1. Regenerator Effectiveness

Effectiveness of a regenerator for hot and cold sides is defined as

For ideal regenerator, the effectiveness is equal to one. To consider the Stirling engines which do not use regenerator, the effectiveness should be set equal to zero. The temperatures at the exit of the regenerator are defined as

If we consider that , the regenerator should have same effectiveness for cold and hot sides which means .

As it is discussed previously, estimation of the regenerator effective temperature is important. Three main approaches for estimation of regenerator effective temperature are as follows.(i)Arithmetic mean approach: (ii)Logarithmic mean approach: (iii)Half hot space-half cold space approach:

By substitution of (2) into (3), the regenerator effective temperature will be As it is clear in (6), in the arithmetic mean approach the regenerator effective temperature isn’t depended on the effectiveness of the regenerator. However, in the two other approaches, the effective temperature of regenerator is dependent on the regenerator effectiveness.

##### 3.2. Volumes

It is considered that pistons have simple harmonic movements. Therefore, volumes of pistons are defined as follows: In these equations, is the angle of crank shaft, and ALPH is the phase angle difference between hot and cold pistons, respectively.

As it is shown in Figure 4, the total dead volumes can be calculated as follows:

The dimensionless total dead volume is presented as follows:

It should be noted that the utilized algorithm is independent from volume variations definitions. However, in this study simple harmonic motion is used.

##### 3.3. Pressure Equation

It is assumed that the working gas is an ideal gas and, therefore, the ideal gas state equation can be used for it. Total mass of the working fluid is sum of hot piston volume, hot piston dead volume, regenerator dead volume, cold piston dead volume, and cold piston volume. Therefore,

Each space follows the ideal gas state equation:

It is assumed that the pressure in these three spaces is equal. Substituting (11) into (10), the pressure relation will be obtained as follow:

##### 3.4. Heat and Work Values during Four Steps of the Stirling Cycle

As it is mentioned previously, the Stirling cycle consists of four steps: isothermal compression process (based on Figure 1, point 1 to 2), isochoric heating process (point 2 to 3), and isothermal expansion process (point 3 to 4), isochoric cooling process (point 4 to 1).

Having the volumes and working gas mass, one can calculate the pressure using (12). As volumes are relative to crank shaft angle (7), pressure is also proportional to crank shaft angle.

###### 3.4.1. Work Diagram

By changing the crank shaft angle from zero to 360 degree, it would be possible to obtain *P-V* diagram. Integrating areas of this diagram, expansion work, compression work, and net work can be calculated. For this reason, *P-V* diagram is called as work diagram. The expansion work which theoretically occurs in process 3-4 of Figure 1 is equal to added heat. Compression work which theoretically occurs in process 1-2 of Figure 1 is equal to absorbed heat during the cycle. Therefore, by having the work diagram, added heat and absorbed heat can be calculated. -

###### 3.4.2. Isochoric Heating Process

Regenerator will provide required heat from 2-3′. Therefore, the remaining heat to warm the working gas from 3′-3 should be provided by heater. By considering that the process occurs in constant volume, the heat added during the isochoric heating process 3′-3 is given by In (13), all parameters except the mass are known. In some literatures like [5], this mass is considered equal to total mass of working fluid. In this assumption, the mass of working gas in cold dead volume is neglected.

##### 3.5. Thermal Efficiency

Thermal efficiency of Stirling engine including dead volumes and regenerator deficiency is given by

#### 4. Solution Procedure

In what follows, the numerical solution algorithm is presented, see Figure 5. First, the values of engine dimensions and working conditions are specified. In Table 1, the basic used values are presented.

**Table 1: **Values of basic engine parameters.

**Figure 5: **Stirling engine performance estimation procedure diagram.

It should be noted that the conditions presented in Table 1 are the same as SOLO 161 Solar Stirling Unit which is under experimental tests in NRI.

#### 5. Results

##### 5.1. Basic Engine Performance

In Figure 6, the *P-V* diagram of the basic engine is presented. Because the integration of area inside the *P-V* curve indicates the net work, this diagram is called as the work diagram. In Table 2, the performance characteristics of the basic engine are provided.

**Table 2: **Performance characteristics of the basic engine.

**Figure 6: ***P-V* (work) diagram for the basic engine.

##### 5.2. Effects of Hot Temperature

###### 5.2.1. Effects of Regenerator Effectiveness

Theoretically, in the *P-V* diagram of Stirling engine (Figure 1), the hot temperature indicates the locations of point 3 and 4. Having constant volume and mass, increase in hot temperature will results in increase of pressure of these two points. Therefore, the area of below line 3-4 will increase which means the increase in total input heat. As it is shown in Figure 7, the total input heat increases with increase in hot temperature. It should be noted that the amount of increase for lower values of regenerator effectiveness is more. The added heat during process 3-4 is equal for all regenerator effectiveness values and is equal to values. The difference is added heat during process 3′-3. In lower values of regenerator effectiveness, this value is higher and, therefore, more heat should be added to the engine.

**Figure 7: **Variations of total input heat against hot temperature for different regenerator effectiveness values.

Increase in hot temperature will result in increase of expansion work, the area below the line 3-4 in Figure 1. While the compression work is constant, this will lead to increase in net work. Because the regenerator effectiveness does not have effect on process 3-4 and 1-2, it does not have effect on net work. In Figure 8, variations of net work with increase in hot temperature are presented.

**Figure 8: **Variations of net work against hot temperature.

As it is stated before, increase in hot temperature will result in increase in total input heat and net work. When the difference between hot temperature and cold temperature is low, hot temperature from 400–600 K, in Figure 9, an increase in hot temperature has more effects on net work than total input heat which will lead to increase in thermal efficiency. In higher values of hot temperature, hot temperature higher than 600 K in Figure 9, the increase rate in net work and total input heat is approximately equal, and, therefore, thermal efficiency tends to a limited value. Similar results are observed by experimental tests [12].

**Figure 9: **Variations of thermal efficiency against hot temperature for different regenerator effectiveness values.

###### 5.2.2. Working Gas Effects

In this study, we have fixed engine volumes and average pressure, and then the mass of working fluid is calculated. Because volumes and pressure are constant, net work is independent from working gas type. Therefore, thermal efficiency is just proportional to total input heat. Four different gases are considered, air, helium, hydrogen and nitrogen. As it is presented in Figure 10, by increase in hot temperature, helium needs lower total input heat in similar conditions. Because the net work is equal, the thermal efficiency of helium is more than other working gases which is presented in Figure 11.

**Figure 10: **Variations of total input heat against hot temperature for different working gas types.

**Figure 11: **Variations of thermal efficiency against hot temperature for different working gas types.

##### 5.3. Effects of Cold Temperature

###### 5.3.1. Effects of Regenerator Effectiveness

Compared to hot temperature effects, increase in cold temperature has reverse effects on total input heat, net work, and thermal efficiency. Although increase in cold temperature will result in decrease in total input heat which is desired, it will reduce net work which is undesired. The variations of total input heat and net work against cold temperature are presented in Figures 12 and 13.

**Figure 12: **Variations of total input heat against cold temperature for different regenerator effectiveness values.

**Figure 13: **Variations of net work against cold temperature.

Because the decrease in net work is more than decrease amount of total input heat, thermal efficiency decreases with increase in cold temperature. The amount of decrease in higher values of regenerator effectiveness is more which is shown in Figure 14.

**Figure 14: **Variations of thermal efficiency against cold temperature for different regenerator effectiveness values.

###### 5.3.2. Working Gas Effects

In lower values of cold temperature, helium needs lower amount of total input heat compared to air, hydrogen, and nitrogen, see Figure 15. By increase in cold temperature and reduction in difference between the cold and hot temperatures, the total input heat for all gases tend to a constant value. Therefore, considering that net work is equal, helium has higher thermal efficiency which reduces by increase in cold temperature, see Figure 16.

**Figure 15: **Variations of total input heat against cold temperature for different working gas types.

**Figure 16: **Variations of thermal efficiency against cold temperature for different working gas types.

##### 5.4. Effects of Regenerator Effectiveness

Effectiveness of regenerator indicates the required heat during process 3′-3 of Figure 1. Therefore, having regenerator with higher effectiveness will reduce total input heat. In Figure 17, variations of total input heat against regenerator effectiveness for different working fluid are presented. As it is shown, regenerator effectiveness has more effects on air, hydrogen, and nitrogen than helium. For example, the total input heat of nitrogen decreases from 9000 Joule to about 3000 Joule, approximately 66 percent reduction.

**Figure 17: **Variations of total input heat against regenerator effectiveness for different working gas types.

It should be considered that, even in these cases, thermal efficiency of helium is much more than other considered working gases. By an increase in regenerator effectiveness, thermal efficiency of all working gases leads to a constant value, see Figure 18.

**Figure 18: **Variations of thermal efficiency against regenerator effectiveness for different working gas types.

##### 5.5. Effects of Phase Angle Difference

###### 5.5.1. Comparison of Performance Characteristics Variations

In the considered piston arrangement for Stirling engine, the difference angle between hot piston and cold piston is called phase angle difference. In this study, variations of volumes are considered harmonically based on the Simple Harmonic Motion theory [11]. Therefore, phase angle difference has straight effects on volumes variation and, therefore, engine performance. In this section, we are going to find the best phase angle difference to optimize the net work and thermal efficiency. Having appropriate comparable values, we have to normalize net work, total input heat, and thermal efficiency. Therefore, the maximum value is considered in each case as a reference value.

As it is shown in Figure 19, the maximum value of thermal efficiency can be obtained in ALPH = 110 degree while the maximum value of net work can be obtained in ALPH = 80 degree. Being able to follow, the details of normalized net work, total input heat, and thermal efficiency around ALPH = 90 degree are presented in Table (3). For phase angle difference bigger than 180 degree, the net work and consequently thermal efficiency are negative. This means that engine will use work instead of generating it.

**Figure 19: **Variations of normalized net work, efficiency, and total input heat against difference phase angle.

###### 5.5.2. Effects of Regenerator Effectiveness

To find the optimized value for phase change difference, variations of thermal efficiency in different regenerator effectiveness are shown in Figure 20. As it is presented, optimized value is depended on regenerator effectiveness. For , ALPH = 90–100 degree is the optimum, while for =0.9–0.95, ALPH = 150–170 degree is the optimum point. For , the optimum point is approximately ALPH = 175 degree. It should be noted that in reality it is impossible to find ideal regenerator. Ordinary regenerators have effectiveness around 0.5–0.8 which means that the optimum point is approximately ALPH = 110–130 degree.

**Figure 20: **Variations of thermal efficiency against difference phase angle for different regenerator effectiveness values.

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