Thermodynamic and dynamic analysis of an alpha type Stirling engine and numerical treatment

Energy Conversion and Management 169 (2018) 34–44

Contents lists available at ScienceDirect

Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Thermodynamic and dynamic analysis of an alpha type Stirling engine and numerical treatment

T

⁎

Duygu Ipci , Halit Karabulut Gazi University, Technology Faculty, Automotive Engineering Department, Teknikokullar, Ankara, Turkey

A R T I C LE I N FO

A B S T R A C T

Keywords: Numerical analysis Alpha type Stirling engine Stirling engine with Scotch-yoke mechanism Thermodynamic and dynamic analysis of Stirling engines

In this study, the nodal thermodynamic and dynamic analysis of an alpha type Stirling engine driven by Scotchyoke mechanism is presented. The nodal thermodynamic section of the analysis is performed via 15 nodal volumes. The temperature variations in nodal volumes are calculated by means of the first law of the thermodynamics given for the open systems. The pressures in all of the nodal volumes are assumed to be equal and calculated via Schmidt relation. The momentary masses in nodal volumes are calculated via the perfect gas relation. The dynamic section of the analysis involves the motion equations of pistons and crankshaft. The motion equations are derived by means of the Newton method. In the derivation of the motion equations of pistons, the working fluid forces and friction forces are considered beside the inertia forces. In the derivation of motion equation of the crankshaft, moments of working fluid forces, moments of friction forces, the moment of external load and the moment of starter motor are considered as well as mass inertia moments. It is estimated that an engine having 1.8 L swept volume, 1000 K hot end temperature, 400 K cold end temperature, 3000 cm2 total inner heat transfer area, 5.1 bar charge pressure and 2000 W/m2 K inner heat transfer coefficient is capable of producing a shaft power above 2 kW. For these inputs and shaft power; the speed, speed fluctuation and torque are optimized as 138 rad/s, 16% and 14.9 N m respectively. The presented analysis is useful for engine development studies.

1. Introduction Stirling engines are externally heated, closed cycle, piston type energy conversion machines invented in 1816 by Robert Stirling. The ideal theoretical cycle of Stirling engines consists of a constant temperature compression process, a constant volume heating process, a constant temperature expansion process and a constant volume cooling process [1]. The cycle is a regenerative cycle and its thermal efficiency is equal to the efficiency of the Carnot cycle [2]. The cycle of practical Stirling engines exhibits a considerable amount of deviations from the ideal theoretical cycle and as the result of these deviations, performance values of practical Stirling engines such as thermal efficiency, cyclic work generation, running speed, power output, specific power and torque become uncompetitive with Internal Combustion Engines. Stirling engines have too much application fields. Due to external heating property, Stirling engine enables the conversion of clean and renewable energies into useful energy forms. Among the clean and renewable energies, the solar energy comes first. There are some opinions that a hybrid engine with a higher thermal efficiency may be developed by integrating the Stirling and Internal Combustion Engines [3]. Beside

⁎

Corresponding author. E-mail address: [email protected] (D. Ipci).

https://doi.org/10.1016/j.enconman.2018.05.044 Received 20 December 2017; Received in revised form 4 May 2018; Accepted 13 May 2018 Available online 26 May 2018 0196-8904/ © 2018 Elsevier Ltd. All rights reserved.

thermal efficiency, the hybrid engines are considered to have better exhaust emissions. Stirling engines have also importance in space investigations as well [4]. The domestic Combined Heat and Power cogeneration systems [5]and, the Combined Cooling Heating and Power cogeneration systems [6] are also considered as application fields for Stirling engines. For the current situation, the development level of Stirling engine is not appropriate for commercial use but, too much investigations are undergoing to improve its development level. In the following paragraphs some of studies conducted within last decades are presented. Costea et al. [7] developed an analytical model of estimating the performance of Stirling engines based on the first and second laws of the thermodynamics. Authors named this model as Finite Speed Thermodynamic analysis. From some aspects, the analysis resembles the Finite Time Thermodynamic analysis developed in 1975 by Curzon and Ahlborn [8]. The model presented by Costea et al. [7] directly connects the irreversibilities to the operation speed of the cycle. Beside the heat transfer irreversibilities, flow and mechanical frictions were taken into account. As the result of this study, authors stated that, the practical efficiency of Stirling cycle engines is about the half of the ideal Stirling

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

Nomenclature

Ai Ap Acc Ahc Acs Ahs Cv Cp Cd Cl Cm

D Fh Fc F∞ he ho

Hie Hio Icr p pd pch Qi QH ΔQR QC R Rm R Lp Lm m Δmi mi mi f me mo

mp Mh Mc Ms Mq Δt ΔTi

heat transfer area of nodal volumes (m2) top area of pistons (m2) cold cylinder heat transfer area (m2) hot cylinder heat transfer area (m2) area augmentation in cold cylinder by slotting (m2) area augmentation in hot cylinder by slotting (m2) specific heats at constant volume (J/kg K) specific heats at constant pressure (J/kg K) damping constant of piston due to lubrication (N s/m) damping constant of crankshaft due to external loading (N m s/rad) damping constant of crankshaft bearings due to lubrication (N m s/rad) cylinder diameter (m) total force exerting on hot piston (N) total force exerting on cold piston (N) dry friction on the piston surface (N) specific enthalpy of entering fluid into a nodal volume (J/ kg) specific enthalpy of outgoing fluid from a nodal volume (J/kg) enthalpy flow into a nodal volume within the time step Δt (J) enthalpy flow out of a nodal volume within the time step Δt (J) inertia moment of crankshaft and flywheel (m2 kg) working volume pressure (Pa) engine block pressure (Pa) charge pressure (Pa) the heat exchange of the working fluid with solid boundaries of the nodal volume i within a time step of Δt (J) heat exchange of working fluid with heater during Δt (J) heat exchange of working fluid with regenerator matrix during Δt (J) heat exchange of working fluid with cooler during Δt (J) crank radius (m) radius of crank journals (m) gas constant (J/kg K) piston length (m) length of crank journals (m) total value of working fluid mass (kg) mass variation in a nodal volume within the time step (kg) gas mass in nodal volumes (kg) mass of working fluid in nodal volume i at pervious time step (kg) mass flow into the nodal volume i during Δt (kg) mass flow out of the nodal volume i during Δt (kg)

Ti Ti f

Tiw U

(ΔU )i

ΔVi Vcc Vhc Vi Vw Wi yh

yc Z

piston mass (kg) moment of hot piston force (N m) moment of cold piston force (N m) starter moment (N m) external load (N m) length of time interval (s) temperature variation in a nodal volume within the time step Δt (K) gas temperature in nodal volume i within the current time step (K) gas temperature in nodal volume i within the previous time step (K) wall temperature of a nodal volume within the time step Δt (K) distance between cylinder top and crankshaft center, (U = R + Z + ε ) (m) internal energy variation in a nodal volume within the time step Δt (J) variation of a nodal volume within the time step Δt volume of cold cylinder, Fig. 1 (m3) volume of hot cylinder, Fig. 1 (m3) value of a nodal volume (m3) total value of nodal volume (m3) work generation in the nodal volume i within the time step of Δt (J) distance between coordinate origin and the top of hot piston, Fig. 2 (m) distance between coordinate origin and the top of cold piston, Fig. 2 (m) distance between piston top and slot, Fig. 2 (m)

Greek symbols

ε δ

μ ω ω Ω θ λi

minimum distance between piston top and cylinder top (m) thickness of lubricant layer between piston and cylinder, or journal and bearing (m) dynamic viscosity of working fluid (N s/m2) average speed (rad/s) average crankshaft speed (rad/s) a dummy variable to avoid divergence crankshaft angle (rad) heat transfer coefficient at a nodal volume (W/m2 K)

Subscripts

i n

counter for nodal volumes counter for time steps

volume to displacer swept volume ratio and phase angle were examined. The optimal value of the piston to displacer swept volume ratio was found to be about 1. The optimal value of the phase angle was found to be in the range of 83°-91°. Tanaka et al. [12] conducted an experimental study and determined friction and heat transfer characteristic of three different wire screens made of stainless steel which were named as WN50, WN100, WN150 and WN200. Here WN50, WN100, WN150, etc. indicates 50, 100 and 150 wires per inch of screen. For WN50 wire screen, the heat transfer coefficient range was determined as 1000 < h < 3000 W/m2 K while the fluid velocity is ranging from 1 m/s to 6 m/s. Rogdakis et al. [13] developed thermodynamic and dynamic analysis of the beta type free piston Stirling engine. The dynamic analysis involves three motion equations; one of them is for displacer, other is

cycle. Kaushik and Kumar[9] conducted a finite time thermodynamic analysis of an endoreversible Stirling engine. The study intended to maximize the power output and corresponding thermal efficiency. For the case of 100% regenerator efficiency, the power output and thermal efficiency of the endoreversible Stirling engine is found to be equal to the power output and thermal efficiency of the Carnot cycle. Finkelstein and Organ [10] made a comprehensive examination of studies conducted before the year 2000 and published as a book. The book presents most of the Stirling driving mechanisms used before the year 2000 as well as theoretical analysis. By using Schmidt approximation, Senft [11] optimized the geometry of an alpha type Stirling engine with respect to the indicated and shaft works. The influences of cold to hot space temperature ratio, piston to displacer swept volume ratio, dead 35

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

which are motion equation of the free piston and the free displacer. The piston was coupled with a linear alternator for electricity generation. When deriving the motion equations, gas forces, hydrodynamic frictions due to lubrication and magnetic forces were considered. Via the developed thermodynamic-dynamic model, the stable working conditions of the engine were examined. The stable working of the engine was demonstrated for some working conditions. Grosu et al. [21] analyzed the performance of Stirling engines with finite-size and finite speed thermodynamics. In the analysis, instead of working gas mass, some new constrains such as maximum pressure, maximum volume, extreme temperature and total heat conductance were used. Analytical expressions were obtained for power, work, efficiency and speed. Karabulut and coworkers [22] developed a thermodynamic dynamic model of a gamma type free piston Stirling engine. The dynamic model involves momentum equations of piston and displacer. In the derivation of momentum equations, beside gas forces, the hydrodynamic frictions due to lubrication and the magnetic force due to electricity generation were considered. The thermodynamic model is a third order approximation involving the first law of thermodynamics, perfect gas relation and pressure equation of Schmidt. The thermodynamic and dynamic equations were solved simultaneously. Via the simulation program, stable working conditions of the engine were investigated as well as thermodynamic performance predictions. In order to maximize the shaft work of the rhombic drive Stirling engine, Cheng and Yang [23] conducted a theoretical and experimental study. In the study, the gas pressure in the engine was calculated with Schmidt formula and converted to a dimensionless form. The dimensionless pressure was used in dimensionless forced work formula and dimensionless indicated work formulas of Senft. The dimensionless shaft work was calculated via taking the difference of dimensionless indicated work and forced work. The results of the theoretical study were compared with experimental findings and a 5.2% deviation was determined between them. Hooshang and coworkers [24] developed a dynamic model for an alpha type Stirling engine with V form and, examined speed variation of its crankshaft. In this dynamic model, a motion equation was derived for the crankshaft. The equation contains equivalent torques generated by gas forces, and inertia forces of piston and displacer. The masses of connecting rods were divided into two parts and shared between piston and crankshaft and, displacer and crankshaft. In the analysis, equations related to the thermodynamic analysis were probably solved prior to the motion equation of crankshaft. As the result of this simulation, authors presented some working gas pressure variations and crankshaft speed variations according to the time. Ben-Mansour et al. [25] conducted a CFD analysis of the rhombic drive engine. In the analysis, the software of ANSYS fluent 14.5 was used. The heat transfer between the working fluid flowing through the regenerative channel between the displacer and its cylinder was analyzed. Authors concerned rather with the determination of radiation heat transfer between working fluid and solid surfaces. For air, Helium, and Hydrogen, the ratios of radiation to the total heat transfer were determined as 9–12%, 7–%9 and less than 2% respectively. Altin et al. [26] developed a thermodynamic and dynamic model for an alpha type Stirling engine with Scotch-yoke driving mechanism. In the thermodynamic model, the gas pressure was calculated with an isothermal approximation. The power of the engine was maximized with respect to the phase angle and the area ratio of the hot and the cold pistons. The highest power was obtained at 77° phase angle while the area ratio of the hot piston to the cold piston was 1.73. Ahmadi et al. [27] demonstrated a finite speed approach for the thermodynamic analysis of Stirling engines. The analysis is based on the first law of the thermodynamics. On the engine performance, as well as the heat source temperatures, effects of the regenerator efficiency, volumetric ratios, piston stroke, engine speed and flow losses were considered. The output power in optimal speed was stated as an equation. In alpha type Stirling engines, whose pistons are driven by Scotchyoke mechanisms, the thrust force between the piston side surface and the cylinder surface is expected to be relatively lower than that of the

for piston and the other is for casing of the engine. The thermodynamic calculations were conducted with Schmidt isothermal analysis. Equations were solved in terms of exponential function. Regarding the roots of the characteristic equation, authors determined running conditions of the engine. Authors presented the stable running conditions of the engine as a surface graphic. Boucher and coworkers [14] conducted the dynamic analysis of a double acting free piston beta type Stirling engine. The engine has two free displacers and a free dual piston which is coupled with a linear alternator. In the derivation of motion equations of dual piston, the magnetic force was described as a damping force. Via this dynamic analysis, authors examined the stable working conditions of the free piston engine. Authors stated that the stable working of the engine may be obtained by regulating the geometric, dynamic and thermodynamic variables. Thombare and Verma [15] carried out a literature review on Stirling engine studies conducted before the year 2006. The document presented by Thombare and Verma contains Schmidt’s theory, isothermal analysis, components of Stirling engines, engine configurations, working fluids, power and speed control, factors governing the engine performance, operational characteristics and so on. Snyman and coworkers [16] predicted the performance parameters of a Heinrici Stirling engine by preparing a five volume nodal analysis which was first developed by Berchowitz and Urieli in 1984. Beside nodal analysis, authors predicted the same parameters with second order adiabatic and with Schmidt analyses as well. Authors made also some nodal pressure and temperature measurements on the same Stirling engine and presented the results as pressure-volume, pressure-crank angle, nodal mass-crank angle and nodal temperature-crank angle diagrams. Timoumi and coworkers [17] developed a thermodynamic model and used in the performance prediction of General Motor GPU-3 engine. The model takes into account most of the heat losses and flow frictions in the engine. Via the model, the power and thermal efficiency of GPU-3 were predicted with a high degree of approximation to practical measurements. Karabulut et al. [18] designed and manufactured a beta type Stirling engine which works at relatively lower temperatures. To perform a better approach to the theoretical Stirling cycle, the motion of displacer was governed by a lever. At the ambient pressure testing, the engine started to run at 93 °C hot-end and 27 °C cold-end temperatures. To increase the heat transfer area, the inner surface of the displacer cylinder was augmented by means of growing span wise slots. By comparing the experimental cyclic shaft work of the engine with the work of nodal thermodynamic analysis, the overall heat transfer coefficient of the engine was determined as 2392 W/m2 K when Helium was used as working fluid. Via developing a combined thermodynamic and dynamic model, Cheng and Yu [19] conducted the performance calculations of a beta type Stirling engine. Moving components of the engine are a piston, a displacer, two connecting rods and a flywheel. The dynamic model was established by deriving the angular motion equation of the flywheel on which all moments generated by gas forces and inertia forces of moving components were exerted. In the derivation of angular motion equation of flywheel, the hydrodynamic frictions were also considered. The thermodynamic model incorporated with the dynamic model has been presented in a previous publication of the authors which is a third order nodal approximation. In the analysis, the flow between displacer and its cylinder was assumed to be a fully developed flow and the flow friction of the working gas and heat transfer between gas and solid surfaces were predicted. The most appropriate value of the gap between the regenerative displacer and its cylinder was determined as 0.4 mm. For maximum power output and thermal efficiency, the phase angle between piston and displacer was optimized as 70°. Karabulut [20] developed the thermodynamic and dynamic model of a beta type free piston Stirling engine working with closed and open thermodynamic cycles. The thermodynamic cycle of the engine was modeled with an isothermal nodal analysis. The dynamic model involves two equations 36

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

adiabatic case rather than isothermal. The other reason is that some of the working fluid is compressed in other nodal volumes having different solid surface temperatures than cooler. During the second 90° rotation of the crankshaft, the crankpin driving the cold piston moves from B to C. The pin driving the hot piston moves from C′ to D′. While the cold cylinder volume is decreasing, the hot cylinder volume increases. The decrease of the cold cylinder volume is equal to the increase of the hot cylinder volume and therefore, the total volume of the working fluid remains approximately constant. The working fluid in the cold cylinder is removed to the hot cylinder. During the movement of the working fluid, it flows through the cooler, regenerator and heater and its temperature increases. This process is assumed as a heating process at constant volume. When the second 90° rotation of the crankshaft is completed, the volume of the cold cylinder becomes almost zero while the volume of the hot cylinder is the half of its total volume. During this process, the great amount of the working fluid is subjected to a heating process. This process is enoughly similar to the ideal heating process of the standard Stirling cycle. There is, however, a little deviation from the ideal heating process of the standard Stirling cycle because of that the temperature of the fluid masses retained in cooler, regenerator and flow passages are lower than the heater temperature. During the third 90° rotation of the crankshaft, the crank pin driving the cold piston moves from C to D. The crankpin driving the hot piston moves from D′ to A′. Volumes of both the cold and hot cylinder increase. This process is assumed as an expansion period at the constant temperature. However, this process deviates from the isothermal expansion process of the standard Stirling cycle. One of the reasons of the deviation is that the inner heat transfer surface of the hot cylinder is limited and therefore, the temperature of the working fluid in the hot cylinder decreases during the expansion period. The other reason is that, at beginning of this period, the volume of the cold cylinder is almost zero and during the period it increases. This results in a mass flow from hot cylinder towards the cold cylinder through the heater, regenerator, and cooler. Due to this phenomenon, the working fluid loses some heat. Due to limited heat transfer surface, the expansion in the cold cylinder is closer to an adiabatic expansion rather than isothermal. In summary, this process is significantly different from the isothermal expansion process of the standard Stirling cycle. The process is a polytrophic expansion process closer to the adiabatic case rather than isothermal. During the fourth 90° rotation of the crankshaft, the crankpin driving the cold piston moves from D to A. The crankpin driving the hot piston moves from A′ to B′. While the cold cylinder volume is increasing, the volume of hot cylinder decreases. The increase of the

other engines. As the result of this feature, the engine is expected to have lower frictional losses. Except for crankpins, the other moving components of the engine makes no lateral motions and generates no lateral inertia forces. Therefore, the vibrational concerns of the engine may be less than the other engines as well. Despite that the alpha type Stirling engine with Scotch yoke driving mechanism promises a better performance, its thermodynamic and dynamic aspects have not been investigated in details. This study concerns with the thermodynamic and dynamic analysis of the Stirling engine having a Scotch yoke driving mechanism. 2. Physical system and mathematical model Fig. 1 indicates the working space of the engine and nodal volumes. The working space has been divided into 15 nodal volumes. The nodal volumes indicated by Vhc , Vh , Vc and Vcc are the hot cylinder volume, the heater volume, the cooler volume and the cold cylinder volume respectively. The others are regenerator volumes. Fig. 1 also indicates the solid surface temperature of the nodal volumes. Fig. 2 illustrates the schematic view of the engine. The cylinders on the left and on the right are the cold and the hot cylinders respectively. In both cylinders, there exists a piston. Both pistons are equipped with a slot bearing holder. The holder has a tail whose surface is machined at a super finish quality. The tail of the holder operates in a bore and prevents tilting of the piston motion. The slot bearings are connected to crankpins making circular motion. Crankpins may take place on the same crankshaft. If the cylinders of the engine will be situated side by side, the crankpins are situated on the crankshaft with 90° phase angle. By this, the motions of pistons are synchronized so as to generate thermodynamic processes to form a cycle. The origin of the coordinate system used in the analysis was situated at the center of crankshaft. The crankshaft angle θ was measured from negative y-axis as shown in Fig. 2. During the first 90° rotation of the crankshaft, while the crankpin driving the cold piston is moving from A to B, the crankpin driving the hot piston moves from B′ to C′. During this period of crankshaft rotation, the volumes of both cylinders decrease. The working fluid taking place in the hot cylinder is removed to the other volumes as well as being compressed simultaneously. During this process, the larger percent of working fluid mass is in the cold cylinder. This process is assumed as the compression of the working fluid at cooler temperature. However, the process deviates from the isothermal compression process of the standard Stirling cycle. One of the reasons of the deviation is that the heat transfer surface of the cold cylinder is limited and therefore, the compression occurs at a polytrophic situation which is closer to the

Fig. 1. Nodal volumes and their wall temperature. 37

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

Fig. 2. Schematic view of mechanism.

Variable heat transfer areas inside the hot and cold cylinders may be stated as

volume of the cold cylinder is equal to the decrease of the volume of the hot cylinder. Therefore, the total volume of the working fluid remains constant. Some of the working fluid taking place in the hot cylinder is removed to the cold cylinder through the heater, regenerator end cooler. The temperature of the fluid flowing through the heater, regenerator and cooler decreases about to the cooler temperature. This process is assumed as a cooling process at constant volume. However, it deviates from the constant-volume cooling process of the standard Stirling cycle. The principal cause of the deviation is that a significant amount of working fluid remains in the hot cylinder and no cooling process is applied to it. This process is significantly different from the constant-volume cooling process of the standard Stirling cycle. When the crankpin of the cold cylinder arrived at A, or the crankpin of the hot cylinder arrived at B′, the thermodynamic cycle is completed. According to the coordinate system seen in Fig. 2, the locations of the cold and hot pistons may be stated as

yc = −Rcosθ + Z ,

(1)

yh = −Rcos(θ + π /2) + Z .

(2)

Acc = (U + Rcosθ−Z )πD + Acs Ahc = [U + Rcos(θ + π/2)−Z ]πD + Ahs .

15

Vw =

yḣ = Rsin(θ + π /2) θ.̇

(4)

(5)

y¨h = Rcos(θ + π /2) (θ)̇ 2 + Rsin(θ + π /2) θ¨.

(6)

Vi .

(11)

By assuming that the pressures of all nodal volumes are equal to each other, the working gas pressure in the engine may be stated as

mR 15

∑i = 1 (Vi / Ti )

.

Qi−Wi = (mo ·ho)i−(me ·he )i + (ΔU )i.

(7)

Vhc = [U + Rcos(θ + π/2)−Z ] Ap .

(8)

(13)

In the last equation Qi , Wi , (ΔU )i , (mo ·ho)i and (me ·he )i indicate respectively the heat exchange of the working fluid with solid boundaries of the nodal volume, the work generated in the nodal volume, the internal energy variations in the nodal volume, the enthalpy conveyed out of the nodal volume and the enthalpy conveyed into the nodal volume. In Eq. Wi = p ΔVi , Qi = λ iAi (Tiw−Ti )Δt , (13) by substituting

Volumes of the cold and hot cylinders are determined as

Vcc = (U + Rcosθ−Z ) Ap

(12)

For the calculation of temperature variation in nodal volumes, the first law of the thermodynamics given for unsteady open systems may be used. By ignoring the kinetic energy and potential energy, the first law of the thermodynamics for a nodal volume designated by i may be stated as

Accelerations of the cold and hot pistons become as

y¨c = Rcosθ (θ)̇ 2 + Rsinθ θ¨,

∑ i=1

p= (3)

(10)

In last equations Acs and Ahs are additional heat transfer areas inside the hot end the cold cylinders obtained by growing slots on piston top and cylinder top surfaces. With respect to the order of nodal volumes seen in Fig. 1, from left to right by indicating the hot cylinder and heater volumes with i = 1 and i = 2 ;regenerator volumes with i = 3,4,…,13; cooler and cold cylinder volumes with i = 14 and i = 15, the total volume of the working gas in the working space of the engine may be stated as

Speeds of the cold and hot pistons become as

yċ = Rsinθ θ,̇

(9)

38

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

(ΔU )i = mi f Cv ΔTi + Δmi Cv Ti f , (mo ·ho)i = Hio , and (me ·he )i = Hie ; the equation λ iAi (Tiw−Ti )Δt −p ΔVi = Hio−Hie + mi f Cv ΔTi + Δmi Cv Ti f

(14)

is obtained. Into the both sides of the last equation, by adding the term ΩΔTi and then solving for ΔTi results in

ΔTi =

λ iAi (Tiw−Ti )Δt −p ΔVi + ΩΔTi + (Hie−Hio)−Δmi Cv Ti f (Ω + mi f Cv )

.

(15)

In this equation addition of ΩΔTi avoids dividing by zero. The value of Ω may be any number such as Ω = 1. In last equation, mi f and Ti f are the mass and temperature of the working fluid in the nodal volume i at previous time step. The enthalpy flow in last equation can be calculated as follows

(Hie−Hio ) = −Cp − Cp

Ti − 1 + Ti (Δm1 2

Ti + Ti + 1 (Δmi + 1 2

+ Δmi + 2 +⋯+Δm15 )

+ Δm2 +⋯+Δmi − 1).

(16)

The subscripts (1, 2, 3, i, i + 1, i + 2, etc.) used in Eq. (16) are indicated in Fig. 3. The mass, mass variation and temperature of the nodal volume i are calculated respectively as

mi = pVi /(RTi ),

(17)

Δmi = (mi−mi f )

(18)

Ti = Ti f + ΔTi.

(19)

Fig. 4. Forces exerted on cold piston.

When the starter motor spins, the crankpin rotates in anticlockwise direction and the cold piston moves in positive y direction with a positive acceleration. While this operation is progressing, the forces exerting on the cold piston are; the pressure force of the gas in the working space of the engine, the pressure force of the gas in the crankcase of the engine, the hydrodynamic friction at side surface of the piston, the dry friction at the side surface of the piston, the hydrodynamic friction at the tail of the slot holder and the dry friction at the tail of the slot holder. In Fig. 4, the forces exerting on the cold piston are illustrated. The dry friction force at the side surface of the piston and the dry friction force at the tail of slot holder may be assumed as a unique force. Similarly, the hydrodynamic friction at the piston side surface and the hydrodynamic friction at the tail surface may also be unified. By Newton law, the motion equation of the cold piston may then be stated as

Fc j + pd Ap j−F∞sgn(yċ ) j−Cd yċ j−pAp j = mp y¨c j.

Fh j + pd Ap j−F∞sgn(yḣ ) j−Cd yḣ j−pAp j = mp y¨h j.

From the last equation, the force exerted by the hot piston to the crankpin is stated as

⎯→ ⎯ Fh = −[mp y¨ h + (p−pd ) Ap + F∞sgn(yḣ ) + Cd y ̇ h ] j.

(24)

The crank radiuses of crankpins of cold and hot pistons may be stated as respectively

⎯→ ⎯ R c = Rsinθ i−Rcosθj,

(25)

⎯→ ⎯ Rh = Rsin(θ + π /2) i−Rcos(θ + π /2) j.

(26)

The moments generated by forces defined with Eqs. (22) and (24) may be calculated as

i j k Mc = Rsinθ −Rcosθ 0 0 −[mp y¨c + (p−pd ) Ap + F∞sgn(yċ ) + Cd yċ ] 0

(20)

From this equation, the crankpin force exerted onto the piston is solved as

= −Rsinθ [mp y¨c + (p−pd ) Ap + F∞sgn(yċ ) + Cd yċ ],

⎯→ ⎯ Fc = [mp y¨c + (p−pd ) Ap + F∞sgn(yċ ) + Cd yċ ] j.

(21) ⎯→ ⎯ The force exerted by the piston to the crankpin is equal to Fc as magnitude but it exerts in opposite direction. So, the force exerted by the cold piston to the crankpin becomes

⎯→ ⎯ Fc = −[mp y¨c + (p−pd ) Ap + F∞sgn(yċ ) + Cd yċ ] j.

(23)

i

(

Rsin θ + Mh =

j π 2

)

(

−Rcos θ +

(27)

k π 2

)

0

−[mp y¨ h + (p−pd ) Ap + F∞sgn(yḣ ) 0

0

+ Cd y ̇ h ] (22)

(

= −Rsin θ +

As seen in Fig. 2, while the engine is being accelerated by the starter motor, the hot piston moves also at positive y direction with a positive acceleration. Therefore, forces exerting onto the hot piston are similar to those exerting onto the cold piston. Under this circumstances, the motion equation of the hot piston becomes

π 2

) [m y¨

p h

+ (p−pd ) Ap + F∞sgn(yḣ ) + Cd y ̇ h ].

(28)

Beside moments given with Eqs. (27) and (28), by considering the moment of bearing frictions, the moment of starter motor and the external load; the motion equation of crankshaft may be written as follows Fig. 3. Nomination of nodal volumes with subscripts.

39

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

M M C M Mq θ¨ = h + c − m θ ̇ + s − . Icr Icr Icr Icr Icr

3. Numerical procedure (29) In the solution of equations, the Taylor series method has been used which is given above by Eqs. (30) and (31). For the solution of equations, a FORTRAN program has been developed. The unknowns of the analysis are θ , θ ,̇ θ¨ , yc , ẏc , y¨c , yh , ẏh , y¨h , Vcc , Vhc , Vw , Ahc , Acc , mi , Δmi , Ti , ΔTi , (Hie−Hio) , Mh , Mc and p . For the calculation of these unknowns, equations to be used respectively are (30), (31), (29), (1), (3), (5), (2), (4), (6), (7), (8), (11), (10), (9), (17), (18), (19), (15), (16), (28), (27) and (12). The solution of the dynamic model is carried out similar to the operation of the engine. The solution process is started by estimating the initial value of θ¨ via Eq. (29). The initial value of θ¨ is indicated by θ¨0 . When estimating θ¨0 , the only input to be used in Eq. (29) is starting motor moment Ms . The other parameter taking part in Eq. (29) are not known or zero. After estimating θ¨0 , the initial values of yc , ẏc , y¨c , yh , ẏh , y¨h , Vcc , Vhc , Vw , Ahc , Acc , mi , Δmi , Ti , ΔTi , (Hie−Hio), Mh , Mc and p are calculated via Eqs. (1), (3), (5), (2), (4), (6), (7), (8), (11), (10), (9), (17), (18), (19), (15), (16), (28), (27) and (12). And then, θ¨0 is updated one time more. By iterating these calculations, a number of times, the correct initial values of unknowns are determined. The iteration number is determined by inspecting the convergence of the calculated values. After finishing the calculations of the initial values, the solution operation is progressed into the second time step. The new time step is numbered by 1. Via Eqs. (30) and (31), θ1 and θ1̇ are calculated. Then yc , ẏc , y¨c , yh , ẏh , y¨h , Vcc , Vhc , Vw , Ahc , Acc , mi , Δmi , Ti , ΔTi , (Hie−Hio) , Mh , Mc and p are calculated via using the relevant equations indicated above. Finally, θ¨1 is calculated and the first iteration of the time step numbered by 1 is completed. Then, by iterating the same calculations an adequate number of times, the precise values of the unknowns are determined for the time step numbered by 1. The rest of the numerical simulation is the repetition of the same operations. The starter moment (Ms ) in Eq. (29) is not a permanent term. It is kept until the crankshaft rotates one revolution and then removed. The magnitude of the starter moment is dependent on the charge pressure or the mass of working fluid. Its appropriate value is determined by a trial and error process. If the starter moment is inadequate, the crankshaft spins but not revolves. In order to obtain reliable results, the simulation program should be run by using appropriate values for time step (Δt ). The appropriate

In order to exert an external load onto the engine, Mq can be manually adjusted. If Mq is set to a value higher than the torque generation ability of the engine, then the engine tends to stop. If Mq is set to a value lower than the torque generation ability of the engine, the speed of the engine continues to increase until being restricted by friction forces and moC ments: F∞sgn(yḣ ) , Cd ẏ h , F∞sgn(yċ ) , Cd ẏc and I m θ .̇ In order to exert an cr external load onto the engine, Mq can also be applied via a hydrodynamic loading device such as a torque converter or an air fan. In this case the external load in Eq. (29) is stated as Mq = Cl θ .̇ This type of loading may be called as hydrodynamic loading. As can be understood from the equation Mq = Cl θ ,̇ the external load on the engine is regulated by the speed of a torque converter or air fan and, engine does not tend to stop. The speed of the engine increases until the torque generated by the engine is being balanced by the reaction torque of the hydrodynamic device. In order to adjust the nominal speed of the crankshaft to the certain value, Cl should be increased or decreased. By considering that initially, the engine is in stop, the boundary condition of Eq. (29) may be determined. In order to determine the boundary conditions, the lowest position of the cold piston may be regarded. While the cold piston is at the lowest position, as illustrated in Fig. 2, the crank pin of the cold piston is at A and the crankshaft angle is zero. So, the boundary conditions may be stated as t = 0, θ = 0, θ ̇ = 0 . The numerical solution of Eq. (29) may be carried out via the relations,

θn = θn − 1 +

θṅ − 1 θ¨ Δt + n − 1 Δt 2 1! 2!

(30)

θ¨n − 1 Δt . 1!

(31)

and

θṅ = θṅ − 1 +

The step counter taking part in last equations varies as n = 1,2,3,…. The dry friction forces at the side surface of pistons and at the tail of slot holders may be caused by metal to metal contacts due to lateral forces. For the current situation, there is not any practical data about the magnitudes of the dry friction forces acting on pistons and slot holders. On the other hand, in a Stirling engine with Scotch-yoke driving mechanism, the lateral forces exerting on pistons are thought to be small enough. In simulation program, therefore, the dry friction forces (F∞) in motion equations of pistons were taken to be zero. The magnitudes of hydrodynamic friction coefficients of pistons and μ crankshaft bearings may be estimated via equations, Cd = δ πDLp and

Table 1 Specific values of the engine.

2πL μR 3

m m Cm = which are derived from Couette flow. As long as the δ thickness of lubricant layer (δ ) is correctly known, these equations are capable of providing reasonable values. The damping constant of pistons(Cd ) and the total damping constant of crankshaft bearings (Cm ) given in Table 1 are determined from these equations. Equations given so far are ones required for thermodynamic and dynamic simulation of the engine. The heat transfer in the heater, cooler and regenerator are dominantly convective. The equations

ΔQH = λ1 A1 (T1w−T1)Δt + λ2 A2 (T2w−T2)Δt ,

(32)

Parameters

Specifications

Engine type For both pistons, bore × stroke (mm) Piston top area (cm2) Minimum distance between piston top and cylinder top (mm) Piston mass (kg) Damping constants of both pistons (N s/m) Total damping constant of crankshaft bearings (N m s/rad) Mass inertia moment of crankshaft and flywheel (m2 kg) Heat transfer coefficient (W/m2 K) Volumes of nodal spaces (cm3)

Alpha, α 112.8 × 80 100 2

13

ΔQR =

∑

λi Ai (Tiw−Ti )Δt ,

i=3 w w ΔQC = λ14 A14 (T14 −T14 )Δt + λ15 A15 (T15 −T15)Δt

Heat transfer area (cm2)

(33) (34)

enables the calculation of heat transfer between heater and working fluid, regenerator and working fluid and, cooler and working fluid respectively for the differential time interval Δt .

Working fluid

40

4 2 0.005 0.1

1500 ⩽ λ ⩽ 3000 Heater Regenerator Cooler Average of hot cylinder Heater Regenerator Cooler Average of cold cylinder Helium

40 10 × 11 40 500 150 150 × 11 150 500

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

examination are practically possible values [18]. Tanaka et al. [12] have also presented similar heat transfer coefficient for wire screen regenerators. The numerical data used in drawing of the resting figures were obtained for λ = 2000 W/m2 K heat transfer coefficient. The influence of charge pressure on the thermodynamic and dynamic performance of the engine was examined by obtaining results for different values of the charge pressure. Fig. 6 indicates variations of the indicated work, effective work, effective thermal efficiency, heat from the heater and heat to the cooler with charge pressure. The charge pressure in the engine block is equal to the average of the maximum and minimum values of the pressure profiles seen on the p-V diagram. From the sealing point of view of the working fluid, a charge pressure below 6 bars is applicable to a prototype of Stirling engine having an open ended crankshaft. As seen from Fig. 6, the indicated work is slightly higher than the effective work. The difference of indicated work and effective work is caused by hydrodynamic frictions on piston surface and in crankshaft bearings. Up to 10 bar of charge pressure, which corresponds to 0.55 g working fluid mass, the indicated and effective works increase. Beyond 10 bar, both of works decrease. The decrease of indicated work is sharper than the effective work which may be caused by numerical errors. At 10 bar, the effective and indicated works are about 138 and 141 J per cycle. When the charge pressure was increased to 13 bar, the effective and indicated works decrease to 133.7 J and 134.3 J respectively. While the charge pressure increasing, the decrease of effective and indicated works is caused by inadequate variation of the working fluid temperature during the heating and cooling processes of the cycle. In Stirling engines, there is a critical equilibrium between the charge pressure and heat transfer coefficient λ , or the inner heat transfer area of the engine. In order to be able to further increase the charge pressure, the inner heat transfer area or the heat transfer coefficient of the engine should be increased. At 1.84 bar charge pressure, the effective thermal efficiency of the engine is about 52%. When the charge pressure is increased to 10 bar, the effective thermal efficiency decreases to 19.4%. The decrease of thermal efficiency with charge pressure is also caused by inadequate temperature variation of the working fluid during the heating and cooling processes of the cycle. As seen from Fig. 6, the speed fluctuation of the crankshaft (ωmax −ωmin )/ ω increases with charge pressure as well. At moderate loadings, the allowable speed fluctuation of an engine should be about 15%. As seen from Fig. 6, even the lowest value of the speed fluctuation is above the acceptable limit. In these examinations, the speed of the engine varied between limits 47 rad/s and 102 rad/s. All of the data used in Fig. 6 were obtained for Cl = 0.2 N m s/rad which is an extreme loading. At lower values of Cl , the speed fluctuation becomes acceptable as indicated in Fig. 7. In order to examine the thermodynamic and dynamic responses of

value of Δt is determined by a trial and error procedure. For this purpose, the simulation program is run for two different values of Δt and the obtained results are compared. Comparison is made via p-V and θ−ω diagrams. Inherently, the appropriate value of Δt is dependent on the other inputs such as working fluid mass or charge pressure, external load, heat transfer coefficient, inertia moment of flywheel. When inputs given in Table 1 are used and the charge pressure is increased gradually up to 0.5 g Helium, which corresponds to 9 bar charge pressure, a time step of 0.00015 s becomes appropriate for obtaining accurate results. If the working fluid mass increased to 0.7 g Helium, the time step requires to be reduced to about 0.00005 s. To obtain reliable results from a nodal analysis, the working space of the engine should be divided into an adequate number of nodal volumes. The heater, cooler, hot cylinder and cold cylinder are considered as single nodal volumes. The regenerator should be divided into an adequate number of nodal volumes. While the number of nodal volumes in the regenerator are increasing, the accuracy of results increases as well. The number of nodal volumes is also relevant to the temperature difference between the hot and cold ends of the regenerator. In this study, the temperature difference between the two ends of the regenerator is considered to be 600 K. In order to determine the number of nodal volumes required in the regenerator, the cyclic works and p-V diagrams of two cycles with 6 and 11 nodal volumes were compared. The mass of working fluids in both cycles was 0.3 g Helium which corresponds to 5.5 bar charge pressure. The dimensions and other physical properties of regenerators divided into 6 and 11 nodal volumes were the same. The simulation program with 6 nodal volumes in regenerator provided 122.8 J work, while the other was providing 123.6 J work. Despite such proximity of works, the shapes of p-V diagrams were a little differing. Therefore, 6 nodal volumes in regenerator were found to be a bit poor. 4. Results and discussion Table 1 indicates inputs of the simulation program and specific values of the conceptual engine. The total inner heat transfer area of the engine is 3000 cm2. The wall temperatures of nodal volumes are indicated in Fig. 1. The inner heat transfer coefficient was assumed to be uniform everywhere in the engine. In order to examine the influence of the internal heat transfer coefficient on the thermodynamic performance of the engine, results were obtained for different values of heat transfer coefficients and compared. Fig. 5 indicates the comparison of p-V diagrams obtained for 1500, 2000, 2500 and 3000 W/m2 Kheat transfer coefficients as well as isothermal situation. The gas mass in the working volume of the engine was 0.15 g Helium which corresponds to 2.75 bar charge pressure. For 1500, 2000, 2500 and 3000 W/m2 K heat transfer coefficients, the indicated work generations per cycle are 72.81, 83.75, 89.03 and 93.3 J respectively. While the heat transfer coefficient was ranging from 1500 W/m2 K to 3000 W/m2 K, the speed of the engine ranged from 47.43 rad/s to 53.58 rad/s. So, the speed may be assumed as approximately constant. The torque ranged from 12.8 N m to 14.46 N m. At isothermal situation, where the gas temperatures in nodal volumes were assumed to be equal to the wall temperature of the nodal volume, the work generation reaches to 102.9 J/cycle. As can be seen from Fig. 5, the compression ratio of the engine is about 3.43. A standard Stirling cycle working between the same temperature limits and having the same compression ratio provides 231 J/cycle indicated work. The ratio of the isothermal work to the work of the standard cycles is about 0.44 which is a moderate value compared to the other practical Stirling cycles that have been examined by authors before. As seen from the numerical results presented in Fig. 5, the cyclic work generation displays a decelerating increase with heat transfer coefficient. The difference of works obtained for 2000 W/m2 K heat transfer coefficient and the isothermal situation is about 19%. According to the former studies of authors, the values of heat transfer coefficient used in this

5,0

h= 1500 W/m K, W i= 72.81 J/cyc

4,5

h= 2000 W/m K, W i= 83.75 J/cyc

2 2 2

h= 2500 W/m K, W i= 89.03 J/cyc

Pressure (bar)

4,0

2

h= 3000 W/m K, W i= 93.3 J/cyc isothermal, W i=103.9 J/cyc

3,5 3,0 2,5 2,0 1,5 1,0 0,4

0,6

0,8

1,0 1,2 Volume (Lit)

Fig. 5. p-V diagram. 41

1,4

1,6

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

charge pressure may be assumed as a constant value. The maximum effective power appears at about 200 rad/s speed as 2166 W. The maximum indicated power appears at about 225 rad/s as 2322 W. While the speed is increasing, the deviation between the indicated power and shaft power increases. This deviation is caused by frictional losses. At 200 rad/s speed, the mechanical efficiency of the engine is about 93%. If the effective power, the indicated power, and the torque were made dimensionless by dividing with the maximum values of themselves and then the results were presented as a graph, it is observed that the power curves and torque curve intersects each other at about 125 rad/s speed. This speed may be chosen as the optimum running speed of the engine. At this speed, as seen from Fig. 8, the torque and effective power of the engine are about 14.15 N m and 1.95 kW respectively. In order to develop an engine appropriate for different applications, its mass and volume should be kept at minimum level. The mass and shape of the flywheel are significant factors to satisfy this requirement. The minimization of the speed fluctuation is accomplished via changing the mass inertia moment of the flywheel. Fig. 9 indicates the speed fluctuation and transient behavior of the engine. For an acceptable value of the speed fluctuation, the appropriate value of the mass inertia moment is determined with a trial and error process by inspecting Fig. 9. In this study, the appropriate value of the mass inertia moment has been determined as Icr = 0.1 m2kg corresponding to 0.1 m2kg mass inertia moment and 16% speed fluctuation. The data used in Fig. 9 were obtained for 0.28 g working fluid mass, which corresponds to 5.115 bar charge pressure, and 2000 W/m2 K heat transfer coefficient. The other inputs are given in Table 1. At steady running of the engine, the cyclic average of the speed is 138 rad/s, the effective work per cycle is 93.64 J, the effective power is 2057 W, the indicated work per cycle is 96.93 J, the indicated power is 2129 W, frictional power loss is 72 W, the torque is 14.9 N m, the effective thermal efficiency is 24.7%, and indicated thermal efficiency is 25.26%. The speed fluctuation may be reduced below 16% via reducing the torque below 14.9 N m if acceptable. The total working space volume of this engine is about 1790 cm3. The engine provides about 1 kW power per liter working space volume. Results presented in this paragraph are optimized final values of this engine. Fig. 10 indicates the variation of the cumulative heat exchange in the regenerator. Regenerator is a tube filled with a porous material named as regenerator matrix. At constant volume cooling period of the working fluid, the working fluid flows from the hot cylinder to the cold cylinder through the regenerator (which is called as hot blow) and the excess heat on the working fluid is transferred to the regenerator matrix. At constant volume heating period of working fluid, the working fluid flows from the cold cylinder to the hot cylinder (which is called as cold blow) and the heat that has been transferred to the regenerator

Fig. 6. Variation of heats, cyclic work, thermal efficiency and speed fluctuation with charge pressure.

Fig. 7. Interactions of thermodynamic and dynamic behaviors.

the engine against the external loading, results were obtained for two different values of the external load and compared. As explained when deriving Eq. (29), the external load is adjusted via changing Cl in equation Mq = Cl θ .̇ Fig. 7 indicates the comparison of two p-V diagrams obtained for Cl = 0.2 and Cl = 0,12 N ms/rad. Both of p-V diagrams were obtained for 2.76 bar charge pressure. The other inputs are ones given in Table 1. As seen from Fig. 7, the p-V diagrams obtained for Cl = 0.2 N ms/rad and Cl = 0.12 N sm/rad are slightly different from each other. At high load (Cl = 0.2 ), the speed of the engine is low. Therefore, the heat exchange between the gas and surrounding solid surfaces becomes better. As the result of this, the work becomes a bit higher. Fig. 7 indicates also the speed and speed fluctuations of the engine. At high load, the nominal speed is 69.43 rad/s, the speed fluctuation is 28%. At low load, the nominal speed is 106.23 rad/s, the speed fluctuation is 11.8%. Via the equation Mq = Cl θ ,̇ the external loads, are calculated as 13.88 and 12.75 N m corresponding to 69.43 and 106.23 rad/s speeds. For high load, the effective and indicated powers are determined as 964 and 1017 W. For low load, the effective and indicated powers are determined as 1354 and 1420 W. In order to investigate the influence of engine speed on the torque, effective power and indicated power, results were obtained for different values of engine speed. Fig. 8 indicates the variation of torque and power with engine speed. While obtaining the data used in Fig. 8, the charge pressure ranged within the interval 4.27–4.43 bar. So, the

Fig. 8. Variation of torque and power with speed. 42

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

of differential heats such as (QR = ΔQR1 + ΔQR2 + ΔQR3 + ⋯) . In this equation the suffixes 1, 2, 3, etc. indicate the time intervals. The variation of the cumulative heat with crankshaft angle is as shown in Fig. 10 and it is seen that, over a cyclic period, the heat given to the matrix by the working fluid is not equal to the heat received by the working fluid from the matrix. This is caused by unequal temperature difference between the working fluid and matrix during the hot and cold blows, unequal contact period between the matrix and working fluid during the hot and cold blows and, unequal heat transfer coefficient between the matrix and working fluid during hot and cold blows. In this analysis however, the heat transfer coefficients at hot and cold blows are taken to be equal. If the working fluid temperature was in equilibrium with matrix temperature, which corresponds to the case of ideal regenerator, the heat given to the matrix and the heat received from the matrix by the working fluid would be equal. In Fig. 10, each of waves corresponds to a cycle. The difference between two peaks indicates the net heat flow from the working fluid to the regenerator per cycle. If the regenerator was an ideal regenerator, the cumulative heat profile would be a periodic wave function and the net heat would be zero. Despite that the cumulative heat profile illustrated in Fig. 10 has been obtained from the current theoretical analysis, the same phenomena appear in the practice as well. That means, in practice, the regenerator may perform heating or cooling duties as well as heat regeneration. Due to this, the hot and cold ends of the regenerator should be connected to the heater and cooler respectively. If not, the axial temperature distribution in the regenerator will not have an appropriate form providing better regenerator efficiency. The numerical data used in Fig. 10 were obtained for λ = 2000 W/m2 K. The net heat flow from the working gas to the regenerator is 45.678 J per cycle. If the heat transfer coefficient was taken to be λ = 4000 W/ m2 K, the net heat flow from the working gas to the regenerator decreases to 26.14 J per cycle thereby becoming closer to the ideal regenerator. For a robust calculation of engine efficiency, there is need for regenerator net heat. As mentioned above, the regenerator performs heating or cooling duties beside regeneration. In this analysis, the thermal efficiency illustrated in Fig. 6 was calculated by taking into account the net heat of regenerator. Fig. 11 indicates the theoretical p-V diagram of the Scotch-yoke alpha engine and theoretical and experimental p-V diagrams of Rhombicdrive beta engine comparatively. The experimental data were obtained by Aksoy et al. [28]. The specific values of the test engine and experimental conditions are shown in Table 2. The hot and cold volumes of the test engine are internally connected via the annular channel between displacer and its cylinder. The width of the channel is about 1.2 mm. This engine has no regenerator except from the annular

Fig. 9. Variation of crankshaft speed.

Fig. 10. Heat exchange with regenerator.

matrix is given back to the working fluid. During the flow of working fluid from the hot cylinder to the cold cylinder or from the cold cylinder to the hot cylinder, if the temperature difference between the gas and matrix is infinitely small at everywhere in the regenerator, then the regenerator is said to be ideal. For the ideal regenerator to exist, the matrix temperature in it should be linearly varying between the cooler and heater temperatures. Besides this, the matrix heat transfer area should be infinitely large or the heat transfer coefficient between gas and matrix should be infinitely large. In practice, for transfer of heat between working fluid and matrix, a certain amount of temperature difference is imperative. The temperature difference between matrix and working fluid varies from volume to volume as well as varying with time in the same volume. Therefore, the ideal regenerator is practically not available. In the current analysis, the matrix temperature in the regenerator is assumed to be in Fig. 1. According to this profile, at each nodal volume of the regenerator, there is a uniform matrix temperature. This temperature does not vary within time. For a differential interval of time, the differential heat exchange of working fluid with matrix all over the regenerator is calculated via Eq. (33). In a nodal volume, the heat transfer between matrix and working fluid may be a positive or a negative magnitude according to the temperature of working fluid in that nodal volume. In some of time intervals, the differential heat calculated via Eq. (33) is a negative magnitude while it is positive in some of time intervals. The cumulative heat exchange of working fluid with regenerator matrix over any time period is determined by taking the sum

Fig. 11. Comparison of theoretical and experimental results. 43

Energy Conversion and Management 169 (2018) 34–44

D. Ipci, H. Karabulut

138 rad/s speed and 5.1 bar charge pressure; the torque, effective power, speed fluctuation, and effective thermal efficiency of the engine were determined as 14.9 N m, 2057 W, 16%, 24.7%. The ratio of the thermal efficiency of this engine to the Carnot efficiency is 41%.

Table 2 Specific values of the test engine [28]. Parameters

Specifications

Drive mechanism Bore (mm) × stroke (mm) Working fluid Swept volume (cm3) Compression ratio Cylinder length (mm) Displacer length (mm) Volume of annular channel (cm3) Total heat transfer area (cm2) Rhombus side length (mm)

Rhombic 87 × 72.47 Helium 431 2.51 655 285 115 1500 80

References [1] Walker G. Stirling engines. Oxford: Oxford University Press; 1980. [2] Singh O. Applied thermodynamics, 3rd ed. New Age International Ltd; 2009. [3] Saxena S, Ahmed M. Automobile exhaust gas heat energy recovery using Stirling engine: thermodynamic model. SAE technical paper 2017; no. 2017-26-0029. [4] Fan S, Li M, Li S, Zhou T, Hu Y, Wu S. Thermodynamic analysis and optimization of a Stirling cycle for lunar surface nuclear power system. Appl Therm Eng 2017;111:60–7. [5] Balcombe P, Rigby D, Azapagic A. Energy self-sufficiency, grid demand variability and consumer costs: integrating solar PV, Stirling engine CHP and battery storage. Appl Energy 2015;155:393–408. [6] Kong XQ, Wang RZ, Huang XH. Energy efficiency and economic feasibility of CCHP driven by Stirling engine. Energy Convers Manage 2004;45:1433–42. [7] Costea M, Petrescu S, Harman C. The effect of irreversibilities on solar Stirling engine cycle performance. Energy Convers Manage 1999;40:1723–31. [8] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power output. Am J Phys 1975;43(1):22–4. [9] Kaushik SC, Kumar S. Finite time thermodynamic analysis of endoreversible Stirling heat engine with regenerative losses. Energy 2000;25(10):989–1003. [10] Finkelstein T, Organ AJ. Air engines. New York: The American Society of Mechanical Engineers; 2001. [11] Senft JR. Optimum Stirling engine geometry. Int J Energy Res 2002;26(12):1087–101. [12] Tanaka M, Yamashita I, Chisak F. Flow and heat transfer characteristics of the Stirling engine regenerator in an oscillating flow. JSME Int J. Ser 2, Fluids Eng, Heat Transf, Power, Combust, Thermophys Prop 1990;33(2):283–9. [13] Rogdakis ED, Bormpilas NA, Koniakos IK. A thermodynamic study for the optimization of stable operation of free piston Stirling engine. Energy Convers Manage 2004;45:575–93. [14] Boucher J, Lanzetta F, Nika P. Optimization of a dual free piston Stirling engine. Appl Therm Eng 2007;27:802–11. [15] Thombare DG, Verma SK. Technological development in the Stirling cycle engines. Renew Sustain Energy Rev 2008;12:1–38. [16] Snyman H, Harms TM, Strauss JM. Design analysis methods for Stirling engines. J Energy Southern Africa 2008;19(3):4–19. [17] Timoumi Y, Tlili I, Nasrallah SB. Design and performance optimization of GPU-3 Stirling engines. Energy 2008;33(7):1100–14. [18] Karabulut H, Yucesu HS, Cınar C, Aksoy F. An experimental study on the development of a b-type Stirling engine for low and moderate temperature heat sources. Appl Energy 2009;86:68–73. [19] Cheng CH, Yu Y. Dynamic simulation of a beta-type Stirling engine with cam-drive mechanism via the combination of the thermodynamic and dynamic model. Renew Energy 2011;36:714–25. [20] Karabulut H. Dynamic analysis of a free piston Stirling engine working with closed and open thermodynamic cycles. Renew Energy 2011;36:1704–9. [21] Grosu L, Rochelle P, Martaj N. An engineer-oriented optimization of Stirling engine cycle with Finite-size finite-speed of revolution thermodynamics. Int J Exergy 2012;11(2):191–204. [22] Karabulut H, Solmaz H, Okur M, Şahin F. Gamma tipi serbest pistonlu bir Stirling Motorunun dinamik ve termodinamik analizi. J Fac Eng Architecture Gazi Univ 2013;28(2):213–65. [23] Cheng CH, Yang HS. Optimization of rhombic drive mechanism used in beta-type Stirling engine based on dimensionless analysis. Energy 2014;64:970–8. [24] Hooshang M, Moghadam RA, AlizadehNia S. Dynamic response simulation and experiment for gamma-type Stirling engine. Renew Energy 2016;86:192–205. [25] Ben-Mansour R, Abuelyamen A, Mokheimer EMA. CFD analysis of radiation impact on Stirling engine performance. Energy Convers Manage 2017;152:354–65. [26] Altin M, Okur M, Ipci D, Halis S, Karabulut H. Thermodynamic and dynamic analysis of an alpha type Stirling engine with Scotch Yoke mechanism. Energy 2018;148:855–65. [27] Ahmadi MH, Ahmadi MA, Pourfayaz F, Bidi M, Hosseinzade H, Feidt M. Optimization of powered Stirling heat engine with finite speed thermodynamics. Energy Convers Manage 2016;108:96–105. [28] Aksoy F, Solmaz H, Cinar C, Karabulut H. 1.2 kW beta type Stirling engine with rhombic drive mechanism. Int J Energy Res 2017;41(91):310–1321. [29] Aksoy F, Solmaz H, Karabulut H, Cinar C, Ozgoren YO, Polat S. A thermodynamic approach to compare the performance of rhombic-drive and crank-drive mechanisms for a beta-type Stirling engine. Appl Therm Eng 2016;93:359–67.

channel between displacer and its cylinder. The channel between displacer and its cylinder performs the functions of heater, regenerator, and cooler. The heat transfer coefficient of this channel is estimated to be about 500 W/m2 K [29]. The theoretical p-V diagrams of both engines were obtained by using the specific values of the experimental engine in simulation programs. As seen from Fig. 11, the experimental work of Rhombic drive is 32 J/cyc, the theoretical works of Scotch-yoke and Rhombic drive are 69 J/cyc and 45 J/cyc. There is a considerable amount of difference between experimental and theoretical results. The difference is caused by the inadequate heat transfer coefficient in the annular channel. 5. Conclusion An engine concept based on Scotch yoke mechanism has been depicted. The engine is expected to have relatively lower friction and vibration. The thermodynamic and dynamic features of the engine have been investigated theoretically by developing a simulation program. As working fluid, the Helium was used. Some of conducted examinations and obtained results are as follows. 1. By introducing 1500, 2000, 2500 and 3000 W/m2 K heat transfer coefficients; 72.81, 83.75, 89.03 and 93.3 J cyclic works were obtained. In this examination 0.15 g working fluid mass and 3000 cm2 total inner heat transfer area were used. In this examination, the speed of the engine varied within the limits 47.43–53.58 rad/s. 2. For the charge pressure range 1.83–13 bar, the thermodynamic and dynamic behavior of the engine has been examined under high external loads. Up to 10 bar, the effective and indicated work increased. The thermodynamic efficiency exhibited a steady decrease between 52% and 19.4%. The speed fluctuations exhibited a steady increase between 24% and 122%. At 10 bar, the effective and indicated works were about 138 and 141 J per cycle. In these examinations, the lowest and highest limits of the speed were 47 rad/s and 102 rad/s. 3. The variations of the torque, indicated power and effective power with engine speed has been examined by varying the speed within the range 60–276 rad/s. In this examination, the charge pressure varied between 4.27 and 4.43 bars which may be assumed as a constant value. The maximum effective and indicated powers were determined as 2166 W and 2322 W at 200 rad/s and 225 rad/s speeds respectively. By considering dimensionless forms of the power and torque curves, the optimum running speed of the engine was determined as 138 rad/s where the torque and effective power are 14.15 N m and 1.95 kW respectively. 4. As the result of this study, for steady running of the engine at

44