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Article Development of a Beta-Type Moderate-- Differential Based on Computational and Experimental Methods

Chin-Hsiang Cheng * and Jhen-Syuan Huang

Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan; [email protected] * Correspondence: [email protected]; Tel.: +886-6-2757575 (ext. 63627)

 Received: 30 October 2020; Accepted: 16 November 2020; Published: 18 November 2020 

Abstract: is a favorable technique in the application of recovery or system. This paper aims at developing a beta-type Stirling engine which is operated at moderate heating temperature (773–973 K). mechanism is utilized to make coaxial motion of displacer and . Based on the proposed dimensions, a theoretical model combining thermodynamic and dynamic analysis is built to predict the performance of the Stirling engine. Thermodynamic analysis deals with variations of properties in each chamber while dynamic analysis handles the resultant shaft torque produced by the Stirling engine. Furthermore, a prototype engine is manufactured, and experimental test is carried out to validate the simulated results in this research. Under heating temperature of 973 K, charged of 8 bar, rotation speed of 1944 rpm, shaft power of 68 W is obtained from the 1 prototype Stirling engine. is calculated to be 1.889 W-c.c.− by theoretical prediction 1 and 1.725 W-c.c.− by tested result. The impact of the geometrical dimensions is investigated to survey the optimal piston diameter which is related to and swept volume.

Keywords: Stirling engine; moderate temperature difference; theoretical model; experimental validation

1. Introduction As an external , the Stirling engine features availability to most heat sources, such as solar, , geothermal, nuclear, and . In recent years, capacity and efficiency of Stirling have been remarkably lifted and lots of effort were paid to implement the practical utilization of this technology [1]. Atmosphere like air, , , and can be adopted as working medium in Stirling engines [2,3]. Three main kinds of structure are exploited in Stirling engines, which are alpha-, beta-, and gamma-type. According to first-order analysis on design parameters, beta-type Stirling engine has the highest dimensionless output [4]. Coaxial movement of piston and displacer is the characteristic of beta-type Stirling engine, which results in a more compact configuration and better power density. Cinar et al. [5] designed an air-charged beta-type Stirling engine with crank mechanism. At hot-source temperature of 1000 ◦C generated from a radioactive heater, the test engine can generate 5.98 W at 208 rpm. Aksoy et al. [6] adopted a slotted lever to implement the displacement of displacer and piston. The engine was tested with halogen lamp and power of 127.17 W was achieved at heating temperature of 873 K. In contrast to crank mechanism, Meijer [7] demonstrated a rhombic drive mechanism firstly which successfully operated to achieve the scale of 3 W–8.95 kW. The advantage of rhombic drive mechanism is that piston and displacer bear less lateral force due to the symmetrical geometry, and hence vibration and noise are remarkably reduced. Erol and Caliskan [8] compared performance of four mechanisms of beta-type Stirling engines with isothermal and kinematic analysis. At the same swept volume of 365 cm3 and charged mass of 7.16 10 4 kg, net × −

Energies 2020, 13, 6029; doi:10.3390/en13226029 www.mdpi.com/journal/energies Energies 2020, 13, 6029 2 of 14 output works obtained by bell crank, slide crank, rhombic, and scotch yoke drive mechanisms are 12.85, 12.09, 15.49, and 12.91 J, respectively. Rhombic drive mechanism performed roughly 20 % better than others. Hirata et al. [9] designed a beta-type Stirling engine whose swept volume is 81.4 c.c. and the output power obtained was 60 W with nitrogen being the working gas. Ni et al. [10] demonstrated a beta-type Stirling engine with rhombic drive mechanism operated at 16–30 bar pressure. The maximum powers for the working gas of helium and nitrogen are 69.44 and 36.78 W, respectively. One of the attractive applications of the Stirling engine is waste heat recovery [11–13]. Since Stirling engines can be integrated with different forms of thermal energy without contaminating their , it is possible to harvest waste heat and generate power. Durcansky, Nosek, and Jandacka [14] evaluated the possibility of utilizing waste energy between 300 and 800 ◦C. A commercial Cleanergy Stirling engine operated between 15 and 90 bar was shown to obtain 400–4900 W. Aladayleh and Alahmer [15] studied the potentiality of recycling the waste heat of an automobile engine with Stirling engine. Exhaust temperature of a combustion engine may reach 200–700 ◦C, which is in moderate level for Stirling engines. Wang et al. [16] reviewed Stirling engines that are operated at moderate temperature differences ranging from 0.5 to 375 ◦C. Power density of Stirling engine under moderate temperature locates between 0.005 1 and 2 W-c.c.− [17]. Kropiwnicki and Furmanek [18] developed an alpha-type Stirling engine prototype which is driven by the waste of a combustion engine. The designed swept volume is 730 c.c. and the working gas is air. The 114 W power is achieved at temperature of 350 ◦C and pressure of 6 bar. With optimization on the heater and cooler size, engine power was elevated to 369 W. Combination of nonideal adiabatic model and simplified conjugate-gradient method was used to optimize a four-cylinder double-acting Stirling engine by Cheng and Tan [19]. The approach was validated with robustness and the indicated power can be improved by 56%. Three-dimensional simulation based on CFD software has been used to investigate the distribution of temperature and pressure inside a beta-type Stirling engine [20,21]. Compared to high-temperature Stirling engines, moderate-temperature ones produce lower power and efficiency. Nevertheless, the moderate-temperature-differential Stirling engines feature higher reliability and smaller volume. The aim of the present study is to develop a beta-type Stirling engine with rhombic drive mechanism that is operated at moderate heating temperature. In order to predict engine performance, a theoretical model that deals with the thermodynamic and dynamic process is established. Moreover, a prototype Stirling engine based on the analyzed dimensions is designed and manufactured. Experimental apparatus for measuring engine performance is constructed and running tests are carried out for model validation. Effects of charged pressure and heating temperature are investigated by means of numerical simulation and experimental measurement in parallel. Parametric study on clearance length and size of compression chamber is also conducted to find out the optimal design parameters.

2. Theoretical Model In beta-type Stirling engine, a displacer and a power piston are used to generate volume variations in expansion and compression chamber with a certain phase angle. Expansion and compression chamber are set at high and low temperature regions, respectively, and working gas will shuttle between these two chambers. Figure1 shows the schematic of studied beta-type Stirling engine where three heat exchangers are also incorporated. A heater and cooler are introduced to enhance the in hot and cold ends. They are composed of numerous tubes and immersed in the heat sinks. Between the heater and cooler, regenerator is an internal that serves as thermal inertia and provides a great amount of heat transfer area. The regenerator is usually composed of porous material and benefits maintaining the axial temperature gradient. While high temperature working gas passes through the regenerator, its internal energy will be preserved inside the material. The energy is afterward recycled as working gas returns from the compression chamber. (SS304) mesh that is composed of crossed metal wire is employed as regenerator for the present Stirling engine. Below power piston, the back chamber functions as a buffer for the working zone and it is taken isolated in the present model. Rhombic drive mechanism is adopted to drive a pair of gears where the Energies 2020, 13, x FOR PEER REVIEW 3 of 14 theEnergies present2020, 13Stirling, 6029 engine. Below power piston, the back chamber functions as a buffer for3 ofthe 14 working zone and it is taken isolated in the present model. Rhombic drive mechanism is adopted to drive a pair of gears where the right gear serves as the main rotor to endure shaft load. In the figure, right gear serves as the main rotor to endure shaft load. In the figure, yp and yd are displacements y p and y d are displacements of the piston and displacer while θ is the crank angle about the of the piston and displacer while θ is the crank angle about the main shaft. Dd, Dp, and Db are the main shaft. D d , Dp , and D b are the diameters of displacer, piston, and back chamber rod, diameters of displacer, piston, and back chamber rod, respectively. Lce and Lcc are clearance lengths of respectively. L ce and L cc are clearance lengths of expansion and compression chamber. expansion and compression chamber.

Dd Lce

D p Lcc

y p

Db θ

yd

Figure 1. Schematic of beta-type Stirling engine with rhombic drive mechanism. Figure 1. Schematic of beta-type Stirling engine with rhombic drive mechanism. In this paper,a theoretical model is used to simulate the transient process via thermodynamic analysis for In this paper, a theoretical model is used to simulate the transient process via thermodynamic working gas and dynamic analysis for mechanical components. As indicated on the schematic, the working analysis for working gas and dynamic analysis for mechanical components. As indicated on the zone consists of expansion, heater, regenerator, cooler, and compression chambers. In thermodynamic schematic, the working zone consists of expansion, heater, regenerator, cooler, and compression analysis, properties including volume, mass, pressure, and temperature in each chamber are calculated by chambers. In thermodynamic analysis, properties including volume, mass, pressure, and ideal-gas, continuity, momentum, and energy equations. Volume variations of expansion and compression temperature in each chamber are calculated by ideal-gas, continuity, momentum, and energy chamber are calculated once the displacements of piston and displacer are updated with crank angle. equations. Volume variations of expansion and compression chamber are calculated once the After that, average pressure inside engine pave is obtained and mass in each chamber can be evaluated as displacements of piston and displacer are updated with crank angle. After that, average pressure inside engine p ave is obtained and mass in each chamber can be evaluated as mi = paveVi/RconstTi, i = e, h, r, k, c (1) mpVRTiaveiconsti= / , i = e, h, r, k, c (1) In the above equation, mi, Vi, and Ti are mass, volume, and temperature in each chamber. Rconst is In the above equation, mi, Vi, and1 Ti are1 mass, volume, and temperature in each chamber. Rconst gas constant whose value is 2077 J-kg− -K− in the present case that helium is used. Subscripts e, h, r, k, −1 −1 isc standgas constant for expansion, whose value heater, is regenerator,2077 J-kg -K cooler, in the and present compression case that chamber. helium is used. Subscripts e, h, r, k, c Massstand flowfor expansion, rate across heater, the boundaries regenerator, can cooler, be evaluated and compression by continuity chamber. equation. As working gas flowsMass through flow heater,rate across regenerator, the boundaries and cooler, can be it suevffaluateders from by viscous continuity equation. on the As channel working surface. gas flowsTherefore, through pressure heater, drops regenerator, in these and three cooler, elements it suffe shouldrs from be considered.viscous friction Friction on the factors channelf regarding surface. f Therefore,Reynolds numberpressure can drops be described in these as three [22] elements should be considered. Friction factors regarding Reynolds number can be described as [22] fi = 64/Rei, i = h, k (2) fii= 64 / Re , i = h, k (2) 0.103 fi = 129/Rei + 2.91Re− , i = r (3) =+i−0.103 f ii129 / Re 2.91Re i , i = r (3) where Reynolds number is determined in terms of the hydraulic diameter. As a result, pressure drop whereacross Reynolds three heat number exchangers is determined can be estimated in terms with of the the hydraulic help of Darcy–Weisbach diameter. As a result, equation. pressure In present drop acrossmodel, three average heat pressure exchangers is set can at thebe regenerator,estimated with and the the help of Da inrcy–Weisbach other chambers equation. are determined In present as model, average pressure is set at the regenerator, and the pressures in other chambers are determined as pe = ph + ∆ph/2 (4)

ph = pr + (∆ph + ∆pr)/2 (5)

Energies 2020, 13, 6029 4 of 14

pr = pave (6)

p = pr (∆ pr + ∆p )/2 (7) k − k pc = p ∆p /2 (8) k − k where ∆ph, ∆pr, and ∆pk are pressure of heater, regenerator, and cooler chamber, respectively. For the back chamber, it is assumed that there is no mass leakage in the chamber and its temperature remains constant. Thus, the pressure in the back chamber (pb) can be evaluated with ideal-gas equation as

pbVb = constant (9) where back chamber volume Vb is related to the position of piston. In order to determine temperature variation of the working gas, the energy equation of each chamber is used. That is

d(micvTi) . . . . = Q W + m cpT m cpT , i = e, h, r, k, c (10) dt i − i in,i in,i − out,i out,i . . In the above equation, Q is the heat transfer input and W is the work output. Work output is evaluated by the boundary work in the expansion and compression chambers. Since volumes of heater, . . regenerator, and cooler chamber remain unchanged, no boundary work is exerted. min and mout are mass flow rate into and out of the chamber. Tin and Tout are inlet and outlet . Heat transfer input depends on thermal resistance between the working gas and the . For the working gas flowing in the channel of heater, regenerator, and cooler, convection heat transfer dominates. The convective heat transfer coefficient can be determined from Nusselt number [23] as

Nui = 3.66, i = h, k (11)

0.67 Nui = 0.33Rei , i = r (12) In thermodynamic analysis, volume, mass, pressure, and temperature are calculated. Subsequently, pressure of working zone is brought into dynamic analysis for predicting the resultant shaft torque of the engine. Yang, et al. [24] presented the dynamic relations of rhombic drive mechanism for a 1-kW Stirling engine, in which friction of each joint is considered. The friction caused by the ball bearing is relatively small compared to the dry friction at the piston–cylinder interface and the resistance of the shaft seal. Besides, gravitational and inertial energies of components are demonstrated to be trivial compared to the driving power as well. Cheng et al. [25] conducted experimental measurements for the mechanical loss of rhombic drive mechanism. It is found that the overall friction power can be represented in polynomial form as a function of the rotation speed. In this study, the analysis in [24] is simplified by neglecting the effects of joint friction, gravitational, and inertial energies. All the frictions involved in the engine are summarized in terms of angular speed. The angular acceleration α of the main shaft can be evaluated in the following equation once the pressures in the expansion, compression, and back chambers are determined. . . . α = (W W W )/ωI (13) dri − f ri − sh f where . h i W = (pc p ) (Ap A )vp + A v (pe pc)A v (14) dri − − b − b b d − − d d . 2 W f ri = c1ω + c2ω (15) . Wsh = τsh ω (16) . In the above equations, Wdri is the driving power induced by the gas pressures of expansion, . compression, and back chambers; W f ri is the frictional power represented in a second-order polynomial Energies 2020, 13, x FOR PEER REVIEW 5 of 14

2 Wcc fri =+12ωω (15)

 =τω Wsh sh (16)

Energies 2020In ,the13, 6029above equations, Wdri is the driving power induced by the gas pressures of expansion,5 of 14 compression, and back chambers; W fri is the frictional power represented in a second-order .  τ Wsh sh form;polynomialW is the form; shaft power is the defined shaft by power the product defined of by exerted the product shaft torque of exertedτ and shaft angular torque speed andω. sh ω sh angular speed . v p and v d are linear velocities of the piston and displacer. In addition, c1 and vp and vd are linear velocities of the piston and displacer. In addition, c1 and c2 are two coefficients c2 are two coefficients determined according to the experimental data. For the present1 case, c1=2.025 determined according to the experimental data. For the present case, c1 = 2.025 10− W-s, and c2 = −1 −4 2 × 1.125× 10 10 W-s,4 W-s and2. c2=1.125 × 10 W-s . × − FigureFigure2 shows 2 shows the the flow flow chart chart of theof the computation computation with with the the proposed proposed theoretical theoretical model. model. An An initial initial rotationrotation speed speed is applied is applied and and after after a starting a starting period period the engine the engine is able is toable reach to reach the steady the steady operation operation state. At eachstate. time At each step, time thermodynamic step, thermodynamic and dynamic andanalyses dynamic are analyses performed are performed simultaneously simultaneously and iteratively and iteratively until the properties, m, p, T, and α converge to the solutions. Table 1 lists the design until the properties, m, p, T, and α converge to the solutions. Table1 lists the design parameters of the parameters of the studied Stirling engine. studied Stirling engine.

j = 0 τ shsh=×j Δτ

1 θθωΔαΔkkk++112=+tt + k 2

mpT , , , α converged ?

Output Ω , W sh

Figure 2. Calculation flow chart of theoretical model. Figure 2. Calculation flow chart of theoretical model. Table 1. Design parameters of the Stirling engine. Table 1. Design parameters of the Stirling engine. Displacer Bore 0.07 0.0186 m × × PistonDisplacer Bore BoreStroke × Stroke 0.070.05 × 0.01860.02 m m Piston Bore× × Stroke 0.05× × 0.02 m Expansion chamber clearance length, Lce 0.003 m Expansion chamber clearance length, Lce 0.003 m CompressionCompression chamber chamber clearance clearance length, length,Lcc Lcc 0.01650.0165 m m

DiameterDiameter of backof back chamber chamber rod, rod,Db Db 0.0040.004 m m Number of heater channels 24 Number of heater channels 24 Length of heater channel 0.2345 m Length of heater channel 0.2345 m Cross-sectional area of heater channel 8.067 × 10−6 m2 6 2 Cross-sectionalNumber area of cooler of heater channels channel 8.067 9010 − m × NumberLength of coolerof cooler channels channel 0.03 90 m Cross-sectional area of cooler channel 1.647 × 10−6 m2 Length of cooler channel 0.03 m Volume of regenerator, Vr 1.679 × 10−5 m3 Cross-sectional area of cooler channel 1.647 10 6 m2 × − 5 3 Volume of regenerator, Vr 1.679 10 m × − 3 2 inertial, If 7.123 10 kg-m × − Compression ratio, Γ 1.242

Charged pressure, pch 4–8 bar

Heating temperature, TH 773–973 K Energies 2020, 13, x FOR PEER REVIEW 6 of 14

Flywheel inertial, If 7.123 × 10−3 kg-m2 Compression ratio, Γ 1.242 Charged pressure, pch 4–8 bar Heating temperature, TH 773–973 K

3. Prototype Engine and Experimental Setup In this paper, a prototype beta-type Stirling engine based on the above-mentioned parameters was manufactured to validate the raised model. Figure 3a shows the picture of the designed Stirling Energies 2020, 13, 6029 6 of 14 engine prototype. Components of the Stirling engine require high precision machining and accurate assembly. The piston and cylinder were made with fine surface roughness and matched in tight tolerance.3. Prototype Hermetic Engine seals and Experimentalwere applied Setup at the junctions of parts to avoid gas leakage. Copper pipe was used as the heat exchanger in the heating zone for its good . In this paper, a prototype beta-type Stirling engine based on the above-mentioned parameters wasFor manufactured the purpose to validate of acquiring the raised performance model. Figure of3 athe shows designed the picture Stirling of the engine, designed a Stirlingsetup of experimentalengine prototype. apparatus Components shown of in the Figure Stirling 3b engine was constructed. require high precisionKuehl [26] machining put forward and accurateempirical equationsassembly. for The estimating piston and cylinderthe transport were madeproperti withes fineof surfacegases. roughnessDynamic and matched and in thermal tight conductivitytolerance. Hermetic of the available seals were working applied atgases the junctionsin Stirling of engine parts to are avoid listed gas in leakage. Table 2, Copper where pipethe values was wereused calculated as the heat based exchanger on the in theequations heating in zone [26] for and its the good conditions thermal conductivity.were set at 873 K and 6 bar based on TableFor the1. It purpose can be offound acquiring that dynamic performance viscosity of the of designed hydrogen Stirling is roughly engine, half a setup of air, of experimental nitrogen, and helium.apparatus In shownterms inof Figurethermal3b wasconducti constructed.vity, helium Kuehl [and26] puthydrogen forward perform empirical 6 equations–13 times forbetter estimating than air andthe transportnitrogen. propertiesDespite the of fact gases. that Dynamic conductivity viscosity of helium and thermal is 44 conductivity% of hydrogen, of the helium available was working chosen as thegases working in Stirling gas enginedue to areits availability listed in Table and2, wheresafety. theInterior values air were should calculated be drawn based out on firstly the equations by vacuum pumpin [26] and thehigh conditions purity (99.999%) were set at helium 873 K and is then 6 bar basedfilled oninto Table the1 .engine. It can be Charged found that pressure dynamic was controlledviscosity of by hydrogen a pressure is roughlyregulator half whose of air, operational nitrogen, and range helium. is 0–10 In bar terms and of resolution thermal conductivity, is 0.01 bar. In experiments,helium and hydrogen an electric perform heater 6–13 was times used better to supp thanly air thermal and nitrogen. energy Despite for the the engine, fact that and conductivity the heating temperatureof helium is 44was % ofadjusted hydrogen, with helium a PID was controller. chosen as The the workingengine shaft gas duewas to connected its availability to a andtorque safety. and speedInterior sensor air should and a be hysteresis drawn out brake. firstly byIn this vacuum manner, the and shaft high power purity (99.999%)can be determined helium is thenin terms filled of theinto torque the engine. and Chargedrotation pressurespeed measured. was controlled Variatio by an pressureof heating regulator temperature, whose operational engine torque, range isand rotation0–10 bar speed and resolution were collected is 0.01 bar.and In reco experiments,rded by a andata electric acquisition heater wassystem. used to supply thermal energy for theThe engine, major andparameters the heating measured temperature in the wasexperiment adjusted were with charged a PID controller. pressure, The heating engine temperature, shaft was shaftconnected torque, to and a torque rotation and speed. speed Uncertainty sensor and a analysis hysteresis proposed brake. In by this Moffat manner, [27] theis performed shaft power in canTable 3be where determined the parameters in terms of are the estimated torque and with rotation 95% speedconfidence. measured. Relative Variation uncertainty of heating of shaft temperature, power is calculatedengine torque, to be and 3.7%. rotation speed were collected and recorded by a data acquisition system.

(a) (b)

FigureFigure 3. PrototypePrototype Stirling engine andand experimentalexperimental apparatusapparatus built built in in this this study: study: (a ()a picture) picture of of studiedstudied Stirling Stirling engine prototype;prototype; ((bb)) experimentalexperimental apparatus. apparatus.

Table 2. Transport properties of gas at 873 K and 6 bar [26].

Gas Dynamic Viscosity Thermal Conductivity Air 3.918 10 5 Pa-s 6.099 10 2 W-m 1-K 1 × − × − − − Nitrogen 3.703 10 5 Pa-s 5.844 10 2 W-m 1-K 1 × − × − − − Heliuim 4.073 10 5 Pa-s 3.236 10 1 W-m 1-K 1 × − × − − − Hydrogen 1.837 10 5 Pa-s 7.272 10 1 W-m 1-K 1 × − × − − − Energies 2020, 13, x FOR PEER REVIEW 7 of 14

Table 2. Transport properties of gas at 873 K and 6 bar [26].

Energies 2020, 13, 6029 Gas Dynamic Viscosity Thermal Conductivity 7 of 14 Air 3.918 × 10−5 Pa-s 6.099 × 10−2 W-m−1-K−1 Nitrogen 3.703 × 10−5 Pa-s 5.844 × 10−2 W-m−1-K−1 The major parametersHeliuim measured 4.073 in the × experiment10−5 Pa-s were 3.236 charged × 10−1 W-m pressure,−1-K−1 heating temperature, shaft torque, and rotationHydrogen speed. Uncertainty 1.837 × 10 analysis−5 Pa-s proposed 7.272 × by 10− Mo1 W-mffat− [1-K27−]1 is performed in Table3 where the parameters are estimated with 95% confidence. Relative uncertainty of shaft power is calculated to be 3.7%. Table 3. Uncertainty analysis for measured parameters.

Parameters TypicalTable 3. Value,Uncertainty x Uncertainty analysis for a measured, δx Relative parameters. Uncertainty, δx/x (%) pch 4–8 bar 0.12 bar 2 a δ δ ParametersTH Typical773–973 Value, K x Uncertainty 6.5 K , x Relative Uncertainty, 0.83 x/x (%)

pτchsh 0.02–0.284–8 bar N-m 0.004 0.12 N-m bar 2.7 2 T 773–973 K 6.5 K 0.83 ΩH 350–2700 rpm 20 rpm 1.3 τsh 0.02–0.28 N-m 0.004 N-m 2.7  b WΩ sh 350–27001–68 W rpm 20 rpm- 3.7 1.3 .  22221/2b W 1–68 WδW/Wsh sh=+++[(δpp - ch / ch ) (δTT H / H )3.7 (δτ sh / τ sh ) ( δ Ω / Ω) ] a All estimated withsh 95% confidence. b . a b 2 2 2 2 1/2 All estimated with 95% confidence. δ W.sh/ W.sh = [(δpch/pch) + (δTH/TH) + (δτsh/τsh) + (δ Ω / Ω ) ] . 4. Results and Discussion 4. Results and Discussion In the computation, once the steady operation was reached, an increment of shaft torque was added.In The the shaft computation, torque was once increased the steady to alter operation the ro wastation reached, speed as an well increment as the ofshaft shaft power. torque Figure was 4 displaysadded. the The dynamic shaft torque simulation was increased by plotting to alter the the vari rotationations speedin the asaverage well as rotation the shaft speed power. and Figure the shaft4 powerdisplays in response the dynamic to shaft simulation torque by change. plotting The the charged variations pressure in the average is 4 bar rotationand the speedheating and temperature the shaft is power723 K.in As response shown toin shaftFigure torque 4a, the change. Stirling The engine charged is pressure accelerated is 4 bar from and the the initial heating condition temperature without is shaft723 K.torque As shown and it in takes Figure about4a, the 64 Stirling s to attain engine th ise acceleratedsteady operation from the regime. initial conditionWhile the without resisting shaft shaft torque and it takes about 64 s to attain the steady operation regime. While the resisting shaft torque is torque is applied, engine speed will be gradually declined to another level. The variation in the shaft applied, engine speed will be gradually declined to another level. The variation in the shaft power power is plotted in Figure 4b. In this figure, data of the friction power, driving power, and net heat is plotted in Figure4b. In this figure, data of the friction power, driving power, and net heat input input are also provided. The net heat input of the Stirling engine,. Q net , is the sum of heat transfer are also provided. The net heat input of the Stirling engine, Qnet, is the sum of heat transfer rates in ratesall chambers.in all chambers. It is clearly It is observedclearly observed that larger that heat larger input heat is required input is intorequired the engine into the in the engine first fewin the firstseconds. few seconds. At each At steady each steady level, net level, heat net input heat rateinput is rate balanced is balanced by the by driving the driving power. power. Note that Note the that thedriving driving power power is consumed is consumed by the by friction the friction power po andwer the and shaft the power shaft inpower the steady in the operation steady operation regime regimewith α with= 0, α as = indicated 0, as indicated in Equation in Equation (13). (13).

0.5 3000 τ _sh Ω 0.4 2500

2000 0.3

1500 (rpm) (N-m) 0.2 sh _ Ω τ 1000 0.1 500 0 0 0 50 100 150 200

Time (s) (a)

Figure 4. Cont.

Energies 2020, 13, x FOR PEER REVIEW 8 of 14 EnergiesEnergies2020 2020, 13, 13, 6029, x FOR PEER REVIEW 8 8of of 14 14

. 120 .Qnet Q . 120 .Wnet W. dri 100 . dri Wfri

(W) 100 Wfri. (W) .W . sh W sh sh 80 sh . W 80 , W , . fri fri 60 . W 60 , W , dri

. dri 40 . W 40 W , , net

. net 20 . Q 20 Q

0 0 5050 100100 150150 200200 Time (s) Time (s) ((bb))

FigureFigureFigure 4. 4. 4.Dynamic DynamicDynamic simulation simulation of of engine engine behavior behaviorbehavior inin in responseresponse response to to variation variation in in shaft shaft shaft torque: torque: torque: (a ( a)( a)average) average average rotationrotationrotation speed; speed;speed; ( b ((b)b) shaft) shaftshaft power, power, friction frictionfriction power, power,power, driving driving driving power, power, power, and and and net net net heatheat heat input.input. input.

PerformancePerformancePerformance curves curvescurves of theof the designed designed Stirling StirlingStirling engine engineengine under underunder different different different charged charged charged pressures pressures pressures and heating and and temperaturesheatingheating temperatures temperatures are illustrated are illustrated in Figure 5inin, where FigureFigure both 5,5, wherewhere numerical both both numerical andnumerical experimental and and experimental experimental results are results provided.results are are Forprovided.provided. the present For For thethe engine presentpresent the engine heating the temperature heatingheating temperaturetemperature varied from varied varied 500 from from to 700500 500 ◦toC to 700 (i.e., 700 °C ° 773–973 C(i.e., (i.e., 773–973 773–973 K). Thus, K). K). theThus,Thus, diff erencethe the differencedifference between betweenbetween heating heating and cooling andand temperaturescoolingcooling tetemperaturesmperatures ranges withinranges ranges 473–673within within 473–673 K.473–673 For 4-barK. K. For For charged 4-bar 4-bar pressurechargedcharged shownpressure pressure in shownshown Figure 5ina, Figure the engine 5a, thethe can engineengine be actuated ca cann be be actuated actuated at heating at at heating heating temperature temperature temperature of 773 of K of 773 and 773 K peakKand and shaftpeakpeak power shaft shaft power ofpower the prototypeofof thethe prototype is 16 W is at 1616 1108 WW atat rpm. 11081108 As rpm.rpm. the As heatingAs the the heating heating temperature temperature temperature is increased is isincreased increased to 973 to K, to shaft973973 K, powerK, shaftshaft is powerpower lifted toisis 47lifted W into numerical47 W inin numericanumerica simulationll simulationsimulation and 44 W and and in measurement.44 44 W W in in measurement. measurement. For 6-bar For For charged 6-bar 6-bar pressure,chargedcharged thepressure, pressure, operated thethe heating operatedoperated temperature heating temperaturetemperature is between is is 823between between and 973 823 823 K and and as displayed973 973 K K as as displayed displayed in Figure in5 b.in Figure ShaftFigure power5b.5b. Shaft Shaft ranges powerpower from rangesranges 36 to 64from W in36 theoreticalto 64 WW inin model theoreticaltheoretical and frommodel model 29 and toand 60 from from W in29 29 experimental to to 60 60 W W in in experimental data.experimental In case ofdata.data. 8-bar In In charged case case of of 8-bar pressure,8-bar chargedcharged the lowest pressure, heating thethe lowestlowest temperature heating heating for temperature temperature starting engine for for starting starting is observed engine engine to is be isobserved observed 873 K in testing.toto be be 873 873 It can KK inin be testing.testing. found It thatIt can the be lowest found heatingthatthat thethe temperaturelowestlowest heating heating depends temperature temperature on the depends chargeddepends on pressureon the the charged charged of the designedpressurepressure engine. ofof thethe designeddesigned Since more engine. working Since gas moremore is filled workinworkin ing theg gasgas engine is is filled filled at in larger in the the engine charged engine at pressure,atlarger larger charged charged higher heatingpressure,pressure, temperature higherhigher heatingheating is required temperature to supply isis su requiredrequiredfficient heat toto supply energysupply tosufficient sufficient actuate theheat heat engine. energy energy At to 8-barto actuate actuate charged the the pressureengine.engine. andAt At 8-bar8-bar 973-K chargedcharged heating pressure temperature, and 973-K shaft973-K power heatingheating is temperature, predictedtemperature, to beshaft shaft 74 Wpower power in simulation is is predicted predicted and toverified to be be 74 74 W in simulation and verified to be 68 W in experiment. toW be in 68 simulation W in experiment. and verified to be 68 W in experiment.

60 60 TH = 973 K (Num.) TH = 973 K (Num.) 50 TH = 973 K (Exp.) 50 TH = 973 K (Exp.) TH = 873 K (Num.) TH = 873 K (Num.) 40 TH = 873 K (Exp.) 40 TH = 873 K (Exp.) TH = 773 K (Num.) TH = 773 K (Num.) (W) 30 TH = 773 K (Exp.) sh . (W) 30 TH = 773 K (Exp.) W . sh W 20 20 10 10 0 0 1000 2000 3000 4000 5000 6000 7000 0 _ 0 1000 2000 3000Ω_ (rpm)4000 5000 6000 7000 Ω (rpm) (a) (a) Figure 5. Cont.

Energies 2020, 13, 6029 9 of 14 Energies 2020, 13, x FOR PEER REVIEW 9 of 14

80 T = 973 K (Num.) 70 H TH = 973 K (Exp.)

60 TH = 873 K (Num.)

TH = 873 K (Exp.) 50 TH = 823 K (Num.)

(W) 40 TH = 823 K (Exp.) . sh W 30

20

10

0 0 1000 2000 3000_ 4000 5000 6000 7000 Ω (rpm)

(b) 100

90 TH =973K(Num.) T = 973 K (Exp.) 80 H T =873K(Num.) 70 H TH = 873 K (Exp.) 60

(W) 50 . sh W 40 30 20 10 0 0 1000 2000 3000_ 4000 5000 6000 7000 Ω (rpm)

(c)

FigureFigure 5.5. NumericalNumerical andand experimentalexperimental performance curve of thethe deigneddeigned StirlingStirling engineengine atat chargedcharged pressurepressure ofof 4–8 4–8 bar bar and and heating heating temperature temperature of 773–973 of 773–973 K: (a )K: 4-bar (a) charged4-bar charged pressure; pressure; (b) 6-bar (b charged) 6-bar pressure;charged pressure; (c) 8-bar charged(c) 8-bar pressure.charged pressure.

ClearanceClearance lengthlength isis defineddefined to describe the least distance that causes dead volume for a a chamber. chamber. InIn thethe presentpresent designdesign ofof beta-typebeta-type StirlingStirling engine,engine, clearanceclearance lengthslengths ofof expansionexpansion andand compressioncompression chamberschambers areare 0.0030.003 and and 0.0165 0.0165 m, m, respectively. respectively. If If the the dead dead volume volume of of the the compression compression chamber chamber can can be reduced,be reduced, the the compression compression ratio ratio is increased is increased and engineand engine performance performa cannce be can improved be improved due to due the to more the emorefficient efficient compression compression of working of working gas. Figure gas. Figure6 shows 6 theshows e ff ectsthe effects of the clearanceof the clearance length andlength piston and diameterpiston diameter on the engineon the performance engine performance under 8-bar under charged 8-bar pressure charged and pressure 973-K heating and 973-K temperature. heating Intemperature. Figure6a, clearance In Figure length 6a, cleara of thence compression length of the chamber compression is decreased chamber from is 0.0165decreased to 0.0005 from m, 0.0165 which to makes0.0005 anm, increasewhich makes of compression an increase ratio of compression from 1.242 to ratio 1.300. from As 1.242 a result, to 1.300. shaft As power a result, is raised shaft from power 74 tois raised 82 W, indicatingfrom 74 to 11 82 % W, enhancement. indicating 11 % enhancement. InfluenceInfluence of the piston diameter diameter is is displayed displayed in in Figure Figure 6b.6b. Compression Compression ratio ratio will will be be elevated elevated to to1.463 1.463 while while the the diameter diameter is increased is increased to 0.07 to 0.07 m. m.It is Itnoted is noted that thatperformance performance curve curve is narrowed, is narrowed, and andthe maximum the maximum rotation rotation speed speed is lowered is lowered as the as thediamet diameterer gets gets larger. larger. Stirling Stirling engine engine is accelerated is accelerated by bythe the driving driving torque torque which which comes comes from from the the normal normal pres pressuresure force force on on the the piston. piston. However, However, pressure ofof backback chamberchamber whichwhich alsoalso exertsexerts onon thethe crosscross areaarea ofof thethe pistonpiston willwill balancebalance thethe resultantresultant forceforce onon thethe piston. As the cross section is larger, the force induced by the back chamber is increased under the same charged pressure, which drops the maximum speed of the engine. It can be seen that shaft

Energies 2020, 13, 6029 10 of 14 piston.Energies 2020 As, the13, xcross FOR PEER section REVIEW is larger, the force induced by the back chamber is increased under10 of the 14 same charged pressure, which drops the maximum speed of the engine. It can be seen that shaft power ispower lifted is to lifted 85 W to in 85 the W case in the of 0.06case m of piston 0.06 m diameter, piston diameter, where 15 where % power 15 % improvement power improvement is obtained. is Forobtained. 0.065 mFor piston 0.065 diameter,m piston shaftdiameter, power shaft will power slightly will decrease slightly with decrease a lower with speed. a lower When speed. the pistonWhen diameterthe piston is diameter 0.07 m, shaft is 0.07 power m, shaft declines power to declines 80 W. to 80 W.

180

Lcc = 0.0165 m 160 Lcc = 0.0125 m

140 Lcc = 0.0085 m L = 0.0045 m 120 cc Lcc = 0.0005 m 100 (W)

. sh 80 W 60

40

20

0 0 1000 _2000 3000 4000 Ω (rpm)

(a) 180 D =0.05m 160 p Dp =0.055m 140 Dp =0.06m D =0.065m 120 p

Dp =0.07m 100 (W)

. sh 80 W 60

40

20

0 0 1000 _2000 3000 4000 Ω (rpm)

(b)

Figure 6. Parametric analysis of clearance length of compression chamber and piston diameter under 8-bar charged pressure and 973-K heating temperature: ( a) eeffectffect of clearance length of compressioncompression chamber; (b)) eeffectffect ofof pistonpiston diameter.diameter.

Figure7 7 shows shows thethe effecteffect ofof pistonpiston diameterdiameter onon thethe compressioncompression ratioratio andand powerpower densitydensity underunder 8-bar chargedcharged pressure pressure and and 973-K 973-K heating heating temperature. temperatur Compressione. Compression ratio ratio is increased is increased with larger with pistonlarger diameter,piston diameter, as plotted as inplotted the figure. in the Power figure. density Power is density defined is as defined the ratio as of the peak ratio shaft of peak power shaft and power swept 1 volumeand swept of thevolume piston. of Inthe the piston. proposed In the Stirling proposed engine, Stirling power engine, density power is predicted density to is be predicted 1.889 W-c.c. to be− 1 and1.889 tested W-c.c. to−1 beand 1.725 tested W-c.c. to be− .1.725 On the W-c.c. other−1. hand,On the it other is noticed hand, from it is thenoticed figure from that the an optimalfigure that value an 1 ofoptimal power value density of exists.power Highestdensity powerexists. Highest density ispower 2.020 W-c.c.density− isin 2.020 the case W-c.c. of 0.042−1 in the m pistoncase of diameter. 0.042 m Itpiston can be diameter. deduced It that can piston be deduced diameter that of piston 0.06 m diam leadseter to theof 0.06 largest m leads shaft to power the largest and piston shaft diameterpower and of 0.042piston m diameter yields the of best 0.042 power m yields density. the Forbest the power purpose density. of more Forcomprehensive the purpose ofanalysis more comprehensive of the Stirling analysis of the Stirling engine, design parameters including piston stroke, displacer diameter, displacer stroke, and phase angle should be studied simultaneously and optimized through optimization method.

Energies 2020, 13, 6029 11 of 14

engine,Energies 2020 design, 13, x parameters FOR PEER REVIEW including piston stroke, displacer diameter, displacer stroke, and phase11 angle of 14 should be studied simultaneously and optimized through optimization method.

1.6 2.6 Γ 2.4

λ )

1.5 -1 2.2 Γ io, t 1.4 2 (W-c.c. a r 1.8 λ 1.3

ession 1.6 r

1.2 1.4 Comp

1.2 Power density, 1.1 1 0.03 0.04 0.05 0.06 0.07 D (m) p

Figure 7.7. EEffectffect of pistonpiston diameterdiameter on thethe compressioncompression ratio andand powerpower densitydensity under 8-bar charged pressurepressure andand 973-K973-K heatingheating temperature.temperature. A comparison of developed beta-type Stirling is shown in Table4. It is noted that the present Stirling A comparison of developed beta-type Stirling is shown in Table 4. It is noted that the present engine yields higher power density than those presented in [6] and [9] at nearly equal pressure and Stirling engine yields higher power density than those presented in [6] and [9] at nearly equal temperature. In comparison to a high-pressure system (29.6 bar) demonstrated in [10], the present engine pressure and temperature. In comparison to a high-pressure system (29.6 bar) demonstrated in [10], is lower only by 0.075 W-c.c. 1. the present engine is lower −only by 0.075 W-c.c.−1. Note that the present study aimed to develop a moderate-temperature-differential Stirling Table 4. Comparisons of developed beta-type Stirling engines. engine. From design point of view, the β-type Stirling engines are well suitable for development of the compactAuthor engines. Therefore, Working Gas both Chargedmodeling Pressure and prototyping Heating Temperaturephases were combined Power Density in this study 1 so as to fulfillPresent the aim. The Helium compact property 8 bar of the engine is a criterion 973 K to be competitive1.725 W-c.c. in −personal 1 powerAksoy generation et al. [6 ]applications. Helium 5 bar 873 K 0.553 W-c.c.− 1 Hirata et al [9] Nitrogen 8 bar 923 K 0.737 W-c.c.− 1 Ni et al. [10]Table Helium 4. Comparisons 29.6 of developed bar beta-type Stirling 808 K engines. 1.800 W-c.c.−

Author Working Gas Charged Pressure Heating Temperature Power Density NotePresent that the present Helium study aimed to develop 8 bar a moderate-temperature-di 973 K fferential Stirling 1.725 W-c.c. engine.−1 FromAksoy design et al. [6] point of view, Helium the β-type Stirling 5 enginesbar are well suitable 873 for K development of 0.553 the W-c.c. compact−1 engines.Hirata et Therefore,al [9] both Nitrogen modeling and prototyping 8 bar phases were 923 combined K in this study 0.737 W-c.c. so as−1 to fulfillNi et the al. aim.[10] The compact Helium property of the 29.6 engine bar is a criterion to be 808 competitive K in personal 1.800 W-c.c. power−1 generation applications. 5. Conclusions 5. Conclusions In this study, a beta-type Stirling engine for heating temperature from 773 to 973 K was developed.In this study,Dynamic a beta-type and thermodynamic Stirling engine analyses for heating were temperature built to simulate from 773 the to 973 characteristic K was developed. of the Dynamicengine behavior. and thermodynamic A prototype analyses Stirling were engine built was to simulate manufactured, the characteristic and experiments of the engine of behavior.different Acharged prototype pressure Stirling and engine heating was temperature manufactured, were performed. and experiments Conclusions of different obtained charged from pressure this research and heatingare listed temperature as below. were performed. Conclusions obtained from this research are listed as below.

(1)(1) Dynamic behavior ofof thethe Stirling Stirling engine engine in in response response to to varied varied shaft shaft torque torque was was demonstrated demonstrated in the in thepaper. paper. Time-dependent Time-dependent variations variations of rotation of speedrotation and speed energies and were energies illustrated. were Conservation illustrated. Conservationof energy was of also energy satisfied was toalso certify satisfied the presented to certify the method. presented method. (2)(2) At 4-bar charged pressure, the prototype Stirling engine generates 16 16-44-44 W shaft power in thethe temperaturetemperature span of 773–973 K. Meanwhile, numerical results according to the proposed modelmodel showshow 17–47 W within the same temperature. As the working conditions are elevated to 8-bar8-bar chargedcharged pressure and 973-K heating temperature, thethe shaft power of the designed engine is able to achieve 68 W while predicted power is 74 W. Close agreements between the experimental results and the numerical predictions were found. (3) By means of parametric study on the compression chamber, shaft power is increased with smaller clearance. It was observed that optimal piston diameters corresponding to maximum

Energies 2020, 13, 6029 12 of 14

to achieve 68 W while predicted power is 74 W. Close agreements between the experimental results and the numerical predictions were found. (3) By means of parametric study on the compression chamber, shaft power is increased with smaller clearance. It was observed that optimal piston diameters corresponding to maximum shaft power and maximum power density are 0.06 and 0.042 m, respectively. It was deduced that the power 1 density of the moderate-temperature-differential Stirling engine may achieve 2 W-c.c.− . (4) The present study successfully developed a moderate-temperature-differential Stirling engine which is well suitable for development of the compact engines. The compact property of the engine is a criterion to be competitive in engineering applications.

Author Contributions: Conceptualization, C.-H.C.; methodology, C.-H.C. and J.-S.H.; validation, C.-H.C. and J.-S.H.; formal analysis, C.-H.C.; investigation, C.-H.C. and J.-S.H.; resources, C.-H.C.; data curation, J.-S.H.; writing—original draft preparation, J.-S.H.; writing—review and editing, C.-H.C.; supervision, C.-H.C.; project administration, C.-H.C.; funding acquisition, C,-H.C. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Ministry of Science and Technology, Taiwan, under Grant MOST 106-2221-E-006 -125 -MY3. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

Variables Abbreviations A area (m2) c1, c2 constants of friction power equation 1 1 cp, cv constant-pressure and constant-volume specific heat (J-kg− -K− ) D diameter (m) f friction factor 2 If moment inertial of flywheel (kg-m ) Lcc,Lce clearance length of compression and expansion chamber (m) m mass (kg) Nu Nusselt number p pressure (Pa) pave average pressure of working zone (Pa) pch charged pressure (Pa) ∆p pressure drop (Pa) . Q heat transfer input (W) . Qnet sum of heat transfer rate in all chambers (W) 1 1 Rconst gas constant (J-kg− -K− ) Re Reynolds number T temperature (K) TH heating temperature of Stirling engine (K) V volume (m3) 1 v velocity (m-s− ) . . . Wdri, W f ri, Wsh driving, friction and shaft power (W) y displacement (m) Greek Symbols Abbreviations 2 α angular acceleration (rad-s− ) Γ compression ratio 1 λ power density (W-c.c.− ) θ crank angle (rad) τsh shaft torque (N-m) 1 ω angular velocity (rad-s− ) Ω rotation speed (rpm) Energies 2020, 13, 6029 13 of 14

Subscript Abbreviations b, c, e back, compression, and expansion chamber d, p displacer and piston h, k, r heater, cooler, and regenerator chamber in, out inlet and outlet on the boundary

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