Experimental Investigation and Theoretical Modelling of a High-Pressure Pneumatic Catapult Considering Dynamic Leakage and Convection

Article Experimental Investigation and Theoretical Modelling of a High-Pressure Pneumatic Catapult Considering Dynamic Leakage and Convection

Jie Ren 1,2,* , Jianlin Zhong 1, Lin Yao 3 and Zhongwei Guan 2

1 School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China; [email protected] 2 School of Engineering, University of Liverpool, Liverpool L69 3GQ, UK; [email protected] 3 Nanjing Research Institute on Simulation Technique, Nanjing 210016, China; [email protected] * Correspondence: [email protected]; Tel.: +44-0151-794-5210

Received: 6 July 2020; Accepted: 8 September 2020; Published: 10 September 2020

Abstract: A high-pressure pneumatic catapult works under extreme boundaries such as high-pressure and rapid change of pressure and temperature, with the features of nonlinearity and gas-solid convection. In the thermodynamics processes, the pressure is much larger than the critical pressure, and the compressibility factor can deviate from the Zeno line significantly. Therefore, the pneumatic performance and thermo-physical properties need to be described with the real gas hypothesis instead of the ideal gas one. It is found that the analytical results based on the ideal gas model overestimate the performance of the catapult, in comparison to the test data. To obtain a theoretical model with dynamic leakage compensation, leakage tests are carried out, and the relationship among the leakage rate, pressure and stroke is fitted. The compressibility factor library of the equation of state for compressed air is established and evaluated by referring it to the Nelson-Obert generalized compressibility charts. Based on the Peng–Robinson equation, a theoretical model of the high-pressure pneumatic catapult is developed, in which the effects of dynamic leakage and the forced convective heat transfer between the gas and the metal wall are taken into account. The results from the theoretical model are consistent with the data from ejection tests. This research presents an approach to study the performance of a high-pressure pneumatic catapult with high precision.

Keywords: thermodynamics; mathematical modelling; dynamic leakage; convective heat transfer; compressibility factor; pneumatic catapult

1. Introduction High-pressure pneumatic catapult works with high-pressure air, which has the advantages of no pollution, inexpensive nature, recycling, high power density and stable performance [1]. On top of high-pressure, it is also characterized by high-speed and heavy load. It is widely adopted in the industrial field of automation and robot driving. It can also be applied to aerospace and marine [2–4]. A high-pressure pneumatic catapult has the characteristics of high nonlinearity, rapid change of gas-solid convective heat transfer and strong real gas effects. In thermodynamics processes, the pressure is much higher than the critical pressure, and the compressibility factors deviate from the Zeno line significantly. In this situation, the analytical results from an ideal gas model have a large deviation; furthermore, as a result of this, the accuracy of the theoretical model based on the ideal gas hypothesis is not suitable for theoretical analysis and practical implementation [5,6]. Therefore, the pneumatic performance and thermo-physical properties need to be described with the real gas hypothesis. However, some previous studies are based on the hypothesis of the ideal gas and

Entropy 2020, 22, 1010; doi:10.3390/e22091010 www.mdpi.com/journal/entropy Entropy 2020, 22, x FOR PEER REVIEW 2 of 17

hypothesis. However, some previous studies are based on the hypothesis of the ideal gas and adiabatic, and some are considered on the real gas effects without considering the convective heat Entropytransfer2020 [7–9]., 22, 1010 Few studies take the real gas effects and the convective heat transfer2 ofinto 16 consideration at the same time. Furthermore, the research on the theoretical modelling of a high-pressure catapult with a consideration of dynamic leakage and convective heat transfer is adiabatic, and some are considered on the real gas effects without considering the convective heat rarely reported, nor is the experimental work on the dynamic leakage of a catapult. Here, a novel transfer [7–9]. Few studies take the real gas effects and the convective heat transfer into consideration high-pressure pneumatic catapult with an open-type cylinder is adopted to drive a heavy object at the same time. Furthermore, the research on the theoretical modelling of a high-pressure catapult with a high velocity, in which the compressed air with strong instantaneous expansibility is chosen with a consideration of dynamic leakage and convective heat transfer is rarely reported, nor is the for the pneumatic catapult [10,11]. Accurate theoretical modelling of a high-pressure pneumatic experimental work on the dynamic leakage of a catapult. Here, a novel high-pressure pneumatic catapult is a prerequisite condition for the design and implementation of such systems. Many catapult with an open-type cylinder is adopted to drive a heavy object with a high velocity, in which equations have been proposed to describe the properties of real gases, including the van der Waals the compressed air with strong instantaneous expansibility is chosen for the pneumatic catapult [10,11]. equation [12,13], the virial equation [14], the Redlich–Kwong equation [15], the Accurate theoretical modelling of a high-pressure pneumatic catapult is a prerequisite condition for Benedict–Webb–Rubin equation [16], the Soave–Redlich–Kwong equation [17], and the the design and implementation of such systems. Many equations have been proposed to describe Peng–Robinson equation of state [18,19], which are applicable to gases within a certain pressure the properties of real gases, including the van der Waals equation [12,13], the virial equation [14], and temperature range. the Redlich–Kwong equation [15], the Benedict–Webb–Rubin equation [16], the Soave–Redlich–Kwong To date, most of the studies on theoretical modelling of pneumatic catapult have been based on equation [17], and the Peng–Robinson equation of state [18,19], which are applicable to gases within a the assumption of the ideal gas and adiabatic or based on real gas assumptions, but without certain pressure and temperature range. considering convective heat transfer and dynamic leakage, with few researchers taking the real gas To date, most of the studies on theoretical modelling of pneumatic catapult have been based on the effects, convective heat transfer, and leakage into account all in one theoretical model. Besides, the assumption of the ideal gas and adiabatic or based on real gas assumptions, but without considering accuracy of theoretical models is not fully evaluated by experimental work. In this study, ejection convective heat transfer and dynamic leakage, with few researchers taking the real gas effects, tests of an experimental catapult prototype were first carried out, in which test results were used to convective heat transfer, and leakage into account all in one theoretical model. Besides, the accuracy compare with the theoretical model of the ideal gas state. Then, leakage tests of the pneumatic of theoretical models is not fully evaluated by experimental work. In this study, ejection tests of an catapult were carried out to obtain a precise model for dynamic leakage, by fitting the relationship experimental catapult prototype were first carried out, in which test results were used to compare among leakage rate, pressure and stroke. Finally, by considering the dynamic leakage and the with the theoretical model of the ideal gas state. Then, leakage tests of the pneumatic catapult were forced convective heat transfer between the working medium and the metal wall in the real gas carried out to obtain a precise model for dynamic leakage, by fitting the relationship among leakage state, an accurate theoretical model of a high-pressure pneumatic catapult was established, which rate, pressure and stroke. Finally, by considering the dynamic leakage and the forced convective heat was verified by the experimental data. This research provides support for designing and optimizing transfer between the working medium and the metal wall in the real gas state, an accurate theoretical high-pressure pneumatic catapults. model of a high-pressure pneumatic catapult was established, which was verified by the experimental data. This research provides support for designing and optimizing high-pressure pneumatic catapults. 2. Experimental Study on Ejection of a High-Pressure Pneumatic Catapult 2. Experimental Study on Ejection of a High-Pressure Pneumatic Catapult 2.1. The Working Mechanism of a High-Pressure Pneumatic Catapult 2.1. TheFigure Working 1 shows Mechanism a workflow of a High-Pressure chart of a Pneumatichigh-pressure Catapult pneumatic catapult with an open-type cylinder,Figure which1 shows is acomprised workflow of chart a controller, of a high-pressure servo valve-controlled pneumatic catapult module with (including an open-type a hydraulic cylinder, whichcylinder is comprisedand a high-precision of a controller, servo servo valve), valve-controlled an open-type cylinder, module (includingand a high-pressure a hydraulic air cylinder source. andThe aservo high-precision valve-controlled servo valve), module an open-typeis used for cylinder, setting andthe amovement high-pressure of the air valve source. core The for servo the valve-controlledthrottle control moduleof high-pressure is used for air setting to meet the movement pressure change. of the valve The core working for the mechanism throttle control of the of high-pressurehigh-pressure airpneumatic to meet pressure catapult change. can be The described working as mechanism follows, i.e., of the the high-pressure servo valve-controlled pneumatic catapultmodule canstarts be describedto work with as follows, the control i.e., the command, servo valve-controlled and the gas moduleflows into starts the to chamber work with of thean controlopen-type command, cylinder and from the the gas gas flows source into theafter chamber opening of anthe open-type high-precision cylinder servo from valve. the gas Then, source the afteracceleration opening of the analogue high-precision load attached servo valve. to the Then, pist theon accelerationis executed by of analoguethe high-pressure load attached gas toin thethe pistoncylinder. is executed When ejection by the is high-pressure finished, the gasgas inin thethe cylinder.cylinder is When relieved ejection and isthe finished, velocity the of gasthe inpiston the cylinderis reduced is relievedto zero by and a buffering the velocity cylinder. of thepiston is reduced to zero by a buffering cylinder.

Figure 1. A workﬂow chart of the high-pressure pneumatic catapult.

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Entropy 2020, 22, x FOR FigurePEER REVIEW 1. A workflow chart of the high-pressure pneumatic catapult. 3 of 17 Entropy 2020, 22, 1010 3 of 16 Figure 1. A workflow chart of the high-pressure pneumatic catapult. The open-type cylinder, shown in Figure 2, is a key component in the high-pressure pneumatic catapult, which mainly includes an open cylinder, piston, seal belt, guide rail, load pushing table TheThe open-type open-type cylinder, cylinder, shown shown inin FigureFigure2 2,, isis aa keykey component in in the the high-pressure high-pressure pneumatic pneumatic and buffering cylinder. The load pushing table is a loading platform, which is attached to piston for catapult,catapult, which which mainly mainly includes includes anan openopen cylinder,cylinder, piston, seal seal belt, belt, guide guide rail, rail, load load pushing pushing table table transferring movement. The buffering cylinder is designed to provide a shelter for the load pushing andand bu bufferingﬀering cylinder. cylinder. The The load load pushingpushing tabletable is a loading platform, platform, which which is isattached attached to topiston piston for for table and piston. A set of holes are set to relieve the pressure in the cylinder when ejection is transferringtransferring movement. movement. The The bubufferingﬀering cylindercylinder is designed to to provide provide a ashelter shelter for for the the load load pushing pushing completed. tabletable and and piston. piston. A setA ofset holes of holes are setare to set relieve to relieve the pressure the pressure in the cylinderin the cylinder when ejection when isejection completed. is completed.

FigureFigure 2. 2.Schematic Schematic of of an an open-type open-type cylinder. cylinder. Figure 2. Schematic of an open-type cylinder. 2.2.2.2. The The Ejection Ejection Test Test System System of of a a High-Pressure High-Pressure Pneumatic Pneumatic Catapult Catapult 2.2. The Ejection Test System of a High-Pressure Pneumatic Catapult TheThe ejection ejection test test system system of of a a high-pressure high-pressure pneumatic pneumatic catapult catapult is is shown shown in in Figure Figure3. 3. Figure Figure3a 3a is a prototypeThe ejection and test system system. of Figure a high-pressure3b–e show pneu the gasmatic source, catapult air is compressor, shown in Figure hydraulic 3. Figure module, 3a isis a aprototype prototype andand testtest system. FigureFigure 3b–e 3b–e show show the the gas gas source, source, air air compressor, compressor, hydraulic hydraulic module, module, and pneumatic valve, respectively. Pressure sensors with 0.075% F. S. accuracy and 50 MPa range andand pneumatic pneumatic valve,valve, respectively. PressurePressure sens sensorsors with with 0.075% 0.075% F. F.S. S.ac curacyaccuracy and and 50 MPa50 MPa range range are installed on the cylinder to measure the pressure, and a wireless linear acceleration sensor with areare installed installed on on thethe cylindercylinder to measuremeasure the the pressure, pressure, and and a awireless wireless linear linear acceleration acceleration sensor sensor with with 20 g range and voltage sensitivity 100 mv/g is used to measure the ejection acceleration, as shown in 2020 g g range range and and voltagevoltage sensitivity 100100 mv/g mv/g is is used used to to measure measure the the ejection ejection acceleration, acceleration, as shown as shown in in Figure3f,g. Figure3h,i show the laser displacement sensor with 50 m range and 0.12% F. S. accuracy FigureFigure 3f,g.3f,g. FigureFigure 3h,i3h,i show thethe laserlaser displacementdisplacement sensor sensor with with 50 50m mrange range± and and ± 0.12% ± 0.12% F. S.F. S. trackedaccuracyaccuracy the trackedtracked piston displacementthethe piston displacement during the during during ejection. the the Twoejection. ejection. photoelectric Two Two photoele photoele sensorsctricctric withsensors sensors 450 with mm with 450 range 450 andmmmm 0.5 rangerange ms accuracy andand 0.50.5 are ms used accuracy in the terminal areare usedused velocity inin thethe subsystem terminal terminal tovelocity recordvelocity thesubsystem subsystem instantaneous to torecord velocityrecord the the at theinstantaneousinstantaneous end of the stroke, velocityvelocity as at shown the endend in Figureofof thethe stroke, 3stroke,j. as as shown shown in in Figure Figure 3j. 3j.

(a) Prototype and test system (b)Gas source (c) Air compressor (a) Prototype and test system (b)Gas source (c) Air compressor

(g) Linear (e)Pneumatic (d) Hydraulic module (f) Pressure sensor acceleration(g) Linear (e)PneumaticValve (d) Hydraulic module (f) Pressure sensor sensoracceleration Valve sensor Figure 3. Cont.

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(j) Photoelectric (h) Laser displacement sensor (i) Laser test point (j) Photoelectric (h) Laser displacement sensor (i) Laser test point speedometer(j) Photoelectric (h) Laser displacement sensor (i) Laser test pointspeedometer speedometer Figure 3. The ejection test system of a high-pressure pneumatic catapult. Figure 3. The ejection test system of a high-pressure pneumatic catapult. Figure 3. The ejection test system of a high-pressure pneumatic catapult. 2.3. Comparison of the EjectionFigure Testing 3. The and ejection the Theoretical test system Model of a high-pressurein the Ideal Gas pneumatic State catapult. 2.3.2.3. Comparison of the Ejection Testing andand thethe TheoreticalTheoretical ModelModel inin thethe IdealIdeal GasGas StateState Ejection2.3. Comparisonstudies were of thecarried Ejection out Testing based andon thethe Theoreticalcatapult prototype, Model in the with Ideal the Gas test State results being comparedEjectionEjection with studiesstudies the analytical were carriedcarried results outout of basedbased the theoreti onon thethecal catapultca modeltapult in prototype,prototype, the ideal withwithgas state, thethe test testand resultsresults the accuracy beingbeing Ejection studies were carried out based on the catapult prototype, with the test results being comparedcomparedof the ideal withwith gas themodelthe analytical analytical being resultsassessed. results of of the the theoretical theoretical model model in in the the ideal ideal gas gas state, state, and and the the accuracy accuracy of compared with the analytical results of the theoretical model in the ideal gas state, and the accuracy theof the idealHere, ideal gas an gas model ejection model being beingtest assessed. case assessed. is shown in Figure 4. In this example, the initial pressure of the gas of the ideal gas model being assessed. sourceHere,Here, is set anan to ejectionejection 30 MPa; testtest Figure casecase is4ais shown shownshows inina catapult FigureFigure4 4. .moment InIn thisthis example, example,in the ejection thethe initialinitial and Figure pressurepressure 4b ofshowsof thethe gas gasthe Here, an ejection test case is shown in Figure 4. In this example, the initial pressure of the gas sourcesourcegas discharges isis setset toto 3030from MPa;MPa; vents. FigureFigure Comparing4 4aa shows shows a athe catapult catapult analytic moment momental results in in the thein ejection theejection ideal and and gas Figure Figure state4 b4bwith shows shows the the testthe source is set to 30 MPa; Figure 4a shows a catapult moment in the ejection and Figure 4b shows the gasgasresults dischargesdischarges under the from from same vents. vents. conditions, Comparing Comparing the the contraststhe analytical analytic ar resultseal shown results in thein in Figure idealthe ideal gas 5. It state gasis found withstate the withthat, test withoutthe results test gas discharges from vents. Comparing the analytical results in the ideal gas state with the test underresultsconsidering the under same the the conditions,leakage same conditions,and the convective contrasts the contrasts areheat shown transf ar iner,e Figureshownthe maximum5 .in It Figure is found deviation 5. that,It is without foundon the that,piston considering without stroke results under the same conditions, the contrasts are shown in Figure 5. It is found that, without theconsideringand leakage the gas and thepressure leakage convective in andthe heat convectivecylinder transfer, are heat the16.7 transf maximum and er,24.5%, the deviation maximumrespectively. on deviation the Moreover, piston on stroke the the piston trend and the strokeof gasthe considering the leakage and convective heat transfer, the maximum deviation on the piston stroke pressureanddeviation the gas in is the pressuregetting cylinder inbigger arethe 16.7cylinderwith and time. 24.5%,are 16.7As respectively. aand re sult,24.5%, the respectively. Moreover, precision the ofMoreover, trendthe ideal of the gas deviationtrend model of the isis and the gas pressure in the cylinder are 16.7 and 24.5%, respectively. Moreover, the trend of the gettingdeviationunacceptable bigger is gettingfor with practical time. bigger As engineering a result,with time. the applications. precision As a re ofsult, the idealthe precision gas model of is unacceptablethe ideal gas for model practical is engineeringunacceptabledeviation applications. for practical is getting engineering bigger applications.with time. As a result, the precision of the ideal gas model is unacceptable for practical engineering applications.

(a)A moment in the ejection (b)Completion moment, gas discharges from vents (a)A moment in the ejection (b)Completion moment, gas discharges from vents Figure(a)A moment4. Ejection in test the of ejection the high-pressure pneumatic(b)Completion catapult moment, at 30 MPa. gas discharges from vents FigureFigure 4.4. Ejection test of the high-pressure pneumaticpneumatic catapultcatapult atat 3030 MPa.MPa. Figure 4. Ejection test of the high-pressure pneumatic catapult at 30 MPa. 15 25 15 25 Ideal gas state Ideal gas state 25 Ideal gas state 12 15 20 Ejection test IdealEjection gas teststate Ideal gas state 12 20 Ejection test Ejection test Ideal gas state 12 15 20 Ejection test 9 Ejection test 9 15 15 in cylinder (MPa) 10 6 9

in cylinder (MPa) 10 6 in cylinder (MPa) Piston stroke(m) 10 3 6 5 Pressure Piston stroke(m) 3 5 Pressure Piston stroke(m) 3 0 5 0 0.00 0.15Pressure 0.30 0.45 0.60 0.75 0.90 0.00 0.15 0.30 0.45 0.60 0.75 0.90 0 0 0.00 0.15 0.30Time 0.45 (s) 0.60 0.75 0.90 0.00 0.15 0.30Time 0.45 (s) 0.60 0.75 0.90 0 0 0.00Time 0.15 (s) 0.30 0.45 0.60 0.75 0.90 0.00 0.15Time (s) 0.30 0.45 0.60 0.75 0.90 Time (s) (a)The pressure of working mediumTime (s) in cylinder (b)The piston stroke (a)The pressure of working medium in cylinder (b)The piston stroke (FigureFigurea)The 5.5.pressure ContrastsContrasts of ofof working thethe testtest results resultsmedium andand in analyticalanalytical cylinder resultsresults ofof thethe idealideal(b)The gasgas model. model.piston stroke Figure 5. Contrasts of the test results and analytical results of the ideal gas model. Figure 5. Contrasts of the test results and analytical results of the ideal gas model.

Entropy 2020, 22, x FOR PEER REVIEW 5 of 17 Entropy 2020, 22, 1010 5 of 16 3. Experimental Investigation Leakage of the High-Pressure Pneumatic Catapult 3. ExperimentalA high-pressure Investigation pneumatic Leakage catapult of is the equipped High-Pressure with an Pneumatic open-type Catapultcylinder, which indicates that thereA high-pressure will be leakage. pneumatic The catapultleakage rate is equipped is closely with linked an open-type to the gas cylinder, flow and which pressure indicates in thatthe cylinder,there will which be leakage. affects The the leakageaccuracy rate of the is closely theoreti linkedcal model to the of gas the flow pneumatic and pressure catapult in the[20–22]. cylinder, As thewhich piston affects moves the accuracyfast during of thethe theoretical ejection, the model leak ofage the area pneumatic increases catapult accordingly, [20–22 so]. Asthat the leakage piston ratemoves changes fast during significantly. the ejection, The dynamic the leakage model area of increases the leakage accordingly, rate and socoefficients that leakage can ratebe fitted changes by leakagesignificantly. testing. The dynamic model of the leakage rate and coefficients can be fitted by leakage testing.

3.1.3.1. Schematic Schematic Diagram Diagram of of the the Test Test System on Leakage Rate FigureFigure 66 shows shows a schematica schematic diagram diagram of a leakageof a leakage testing testing system, system, which consists which ofconsists an open-type of an open-typecylinder, position cylinder, limiter, position high-pressure limiter, high-pressure gas source, gas servo source, valve, servo gate valve, valve, gate relief valve, valve, relief pressure valve, pressuresensors, temperaturesensors, temperature sensor, air sensor, compressor, air compressor, dryer, and dryer, data and acquisition. data acquisition. In this work, In this the work, leakage the leakagemeasurement measurement adopts positive adopts pressurepositive pressure detection dete method,ction bymethod, taking by several taking fixed several positions fixed withinpositions the withinstroke ofthe the stroke piston, of loadingthe piston, diff loadingerent pressures different respectively, pressures respectively, as well as measuring as well as and measuring recording and the recordingdata of gas the flow, data pressure of gas andflow, temperature. pressure and The temperature. state parameters The state in the parameters gas source in and the thegas cylinder source andare measuredthe cylinder in are a relatively measured stable in a relatively state, which stable can state, be applied which tocan the be ideal applied gas to law. the With ideal these gas law. test Withdata, thethese leakages test data, and the leakage leakages rate ofand the leakage working rate medium of the duringworking the medium valve opening during process the valve can openingbe inferred. process can be inferred.

FigureFigure 6. 6. SchematicSchematic diagram diagram of of leakage leakage tests. tests.

3.2.3.2. The The Principle Principle of of Leakage Leakage Rate Rate Test Test and Test Procedure InIn orderorder toto build build the the leakage leakage rate rate model model of high-pressure of high-pressure air in theair ejectionin the ejection process, process, it is necessary it is necessaryto estimate to the estimate leakages the with leakages different with strokes different and st pressures.rokes and In pressures. leakage tests, In leakage there is tests, a large there internal is a largeforce internal between force the piston between and the the piston cylinder and due the to cylinder the piston due being to the constrained piston being by theconstrained position by limiter. the positionMaintaining limiter. a high-pressure Maintaining fora high-pressure a long time may for a cause long structuraltime may deformation.cause structural To guaranteedeformation. safety, To guaranteevarious low safety, pressures various are low set pressures for the leakage are set test.for th Fore leakage the floating test. For seal the structure floating seal in an structure open-type in ancylinder, open-type the concept cylinder, of designthe concept is that of the design greater is thethat pressure, the greater the smallerthe pressure, the seal the gap, smaller and the the better seal gap,the sealing and the performance. better the sealing Therefore, performance. the leakage Therefor in a lowe, pressurethe leakage is greater in a low than pressure that in ais high greater pressure. than thatA testing in a high pressure pressure. up to 4A MPatesting can pressure be applied upto to evaluate 4 MPa thecan outerbe applied envelope to evaluate of the leakage the outer rate, envelopewith an unexaggerated of the leakage performance rate, with an of theunexaggerated system. In addition, performance if necessary, of the thesystem. correlation In addition, between if necessary,leakage and the pressure correlation diff erencebetween can leakage be further and pressure calibrated difference by the relationship can be further between calibrated leakage by andthe relationshippressure [23 –between25]. leakage and pressure [23–25]. TheThe procedures ofof the the leaking leaking test test are are as follows. as follows Firstly,. Firstly, adjust adjust the piston the inpiston the specified in the specified positions positionswith a limiter. with a Secondly, limiter. Secondly, supplement supplement high-pressure high-pressure air for the air gas for source the gas using source an air using compressor. an air compressor.Thirdly, open Thirdly, the gate open valve the and gate the valve pneumatic and the servopneumatic valve servo to allow valve the to high-pressure allow the high-pressure air to flow airinto to the flow open into cylinder the open from cylinder the gas from source, the gas and so keepurce, the and valve keep opening the valve until opening the system until reachesthe system the reachessteady-state. the steady-state. Finally, record Finally, the pressure record changesthe pres ofsure the gaschanges in the of gas the source gas in and the open-type gas source cylinder and open-typeby the data cylinder acquisition, by the then data close acquisition, the valve then and relieveclose the the valve pressure. and relieve To make the the pressure. data reliable, To make the themeasurements data reliable, were the repeatedmeasurements three timeswere repeat and aned average three times value and was an taken. average value was taken. InIn these these tests, tests, the piston positionspositions werewere setset to to six six di difffferenterent values, values, i.e., i.e., 0, 0, 1.1, 1.1, 2.2, 2.2, 3.3, 3.3, 4.2, 4.2, and and 5.2 5.2 m. m.Similarly, Similarly, the the pressures pressures of workingof working medium medium in-cylinder in-cylinder were were set set as 2.0,as 2.0, 2.75, 2.75, and and 3.8 MPa.3.8 MPa. Keep Keep the thevalve valve open open for about for about 10 s for 10 retainings for retaining enough enough time to time complete to complete the test. Thethe test. density The and density the remaining and the

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remainingEntropy 2020, 22mass, x FOR of PEER the REVIEW working medium can be calculated from the measured pressure6 of 17 and Entropy 2020 22 temperature, , 1010 after opening the valve according to the standard atmospheric conditions, and6 of 16the leakageremaining can mass be obtained. of the working medium can be calculated from the measured pressure and masstemperature of the working after opening medium the can valve be calculated according fromto the the standard measured atmospheric pressure andconditions, temperature and the after opening3.3.leakage Leakage the can valve Testsbe obtained. according and Fitting to on the Dynamic standard Leakage atmospheric Rate Model conditions, and the leakage can be obtained. A leak test is shown in Figure 7; in this test case, the pressure sensor with 8 MPa range and 3.3.3.3. Leakage Leakage Tests Tests and and Fitting Fitting on on Dynamic Dynamic LeakageLeakage RateRate Model 0.25% F. S. accuracy and the temperature sensor with 0.2% F. S. accuracy and −50 °C–150 °C range. TheA pressureA leak leak test test of is isa shown shownworking in in Figure mediumFigure7 ;7; in inin this thethis gas test test source case,case, thethe is 4.2 pressurepressure MPa, thesens sensor pistonor with with fixed 8 8MPa MPa position range range andis and 2.2 m, 0.25% F. S. accuracy and the temperature sensor with 0.2% F. S. accuracy and −50 °C–150 °C range. 0.25%and F.the S. pressure accuracy of and relief the valve temperature is set to sensor4.0 MPa with to guarantee 0.2% F. S. safety. accuracy At andthe time50 of◦C–150 6 s, all◦ Cvalves range. are The pressure of a working medium in the gas source is 4.2 MPa, the piston fixed− position is 2.2 m, Thefully pressure opened of and a working kept open medium for about in the 10 gass. The source pressure is 4.2 in MPa, cylinder the piston reaches ﬁxed a maximum position issoon 2.2 m,after and the pressure of relief valve is set to 4.0 MPa to guarantee safety. At the time of 6 s, all valves are andthe the high-pressure pressure ofrelief airflow valve into is the set cylinder. to 4.0 MPa Due to guaranteeto the leakage, safety. the At pressure the time drops of 6 s, after all valves reaches are the fully opened and kept open for about 10 s. The pressure in cylinder reaches a maximum soon after fullypeak, opened as shown and keptin Figure open 8. for The about high-pressure 10 s. The pressure air leaked in cylinderforms a reacheswhite mist, a maximum which can soon be seen after in the high-pressure airflow into the cylinder. Due to the leakage, the pressure drops after reaches the theFigure high-pressure 7. From Figure airﬂow 7a,b, into it the can cylinder. be seen that Due th toe theleakage leakage, occurs the when pressure all valves drops afteropen, reaches then leakage the peak, as shown in Figure 8. The high-pressure air leaked forms a white mist, which can be seen in peak,isFigure increasing, as shown7. From in asFigure Figureshown 7a,b,8 .in Theit Figure can high-pressure be 7c.seen The that open-t th aire leakage leakedype cylinder occurs formswhen acannot white all mist,valvesmaintain which open, high-pressure canthen be leakage seen infor a Figurelongis increasing,7 .time, From as Figure as it shown may7a,b, in itbe can Figuredamaged be seen7c. The thatand open-t thecause leakageype an cylinder occursaccident. cannot when Therefore, allmaintain valves the open,high-pressure leakage then leakage duration for a is is increasing,maintainedlong time, as as shownabout it may 10 in s Figure beto meetdamaged7c. the The testand open-type requirements cause cylinderan accident., then cannot the Therefore, servo maintain valve the high-pressure isleakage closed, duration while for athe longis relief time,valvemaintained as itis may open, about be damagedas 10 shown s to meet and in causetheFigure test an requirements7d. accident. Taking Therefore, the, then test the thewith servo leakage 3.8 valve MPa duration is closed,pressure is maintainedwhile in cylinderthe relief about as a 10representativevalve s to meet is open, the testascase, shown requirements, the testin Figuredata then of 7d. the theTaking pressu servo there valve andtest istemperaturewith closed, 3.8 whileMPa are pressure the shown relief in in valve cylinder Figures is open, as8 anda as 9. shownWithrepresentative inthese Figure test7 case,d. data, Taking the the test the leakage data test withof rate the 3.8 canpressu MPa be re pressureinferred and temperature infrom cylinder the equationare as ashown representative of in state Figures for case, a8 realand the gas,9. test as datashownWith of thethese in pressure Table test data,1. and the temperature leakage rate are can shown be inferred in Figures from8 andthe equation9. With these of state test for data, a real the gas, leakage as rateshown can be in inferredTable 1. from the equation of state for a real gas, as shown in Table1.

(b) All valves open, t = (a) Initial state, t = 0 s (c) Leakage, t = 10 s (d) Relief state, t = 18 s (a) Initial state, t = 0 s (b) All valves open,6 s t = 6 s (c) Leakage, t = 10 s (d) Relief state, t = 18 s

Figure 7. An example of leakage test at maximum pressure 3.8 MPa in cylinder. FigureFigure 7. 7.An An example example ofof leakageleakage test at maximum pressure pressure 3.8 3.8 MPa MPa in in cylinder. cylinder.

6 4.8 6 lpiston=0 m 4.8 lpiston=0 m lpiston=0 m l =1.1 m lpiston=0 m piston 5 lpiston=1.1 m lpiston=1.1 m 5 lpiston=1.1 m lpiston=2.2 m 4.4 lpiston=2.2 m 4.4 lpiston=2.2 m lpiston=2.2 m lpiston=3.3 m 4 lpiston=3.3 m lpiston=3.3 m 4 lpiston=3.3 m l =4.2 m lpiston=4.2piston m 4.0 l lpiston=4.2=4.2 m m piston l =5.2 lpiston m=5.2 m l =5.2 m 3 piston lpistonpiston=5.2 m 3 3.6 2 2

3.2 1 1 Pressure in-cylinder (MPa) Pressure in-cylinder (MPa) Pressure in the gas source (MPa) Pressure in the gas source (MPa) 2.8 0 0 00 4 4 8 8 121620 12162000 4 4 8 8 12162024 12162024 Time (s) TimeTime (s) (s) Time (s) ((a)) TheThe pressurepressure in in the the gas gas source source (b) (Theb) The pressure pressure in the in cylinderthe cylinder Figure 8. Test data of the pressure at the maximum pressure 3.8 MPa in cylinder. FigureFigure 8. 8.Test Test data data of of the the pressure pressure at theat the maximum maximum pressure pressure 3.8 3.8 MPa MPa in cylinder.in cylinder.

Entropy 2020, 22, 1010 7 of 16 Entropy 2020, 22, x FOR PEER REVIEW 7 of 17

lp= 0 m 308 lp= 2.2 m

lp= 5.2 m 304

Entropy 2020, 22, x FOR PEER REVIEW 7 of 17 300

lp= 0 m 296308 lp= 2.2 m

Temperaturecylinder in (K) lp= 5.2 m 292304 0 4 8 121620 Time (s) 300 Figure 9.9. The temperature at the maximum pressure 3.83.8 MPaMPa inin cylinder.cylinder.

Table 1.1. 296TheThe leakageleakage ratesrates atat didifferentfferent pressurepressure andand positions.positions. Temperaturecylinder in (K) PistonPiston Position Position lplp [m][m] The LeakageThe Leakage Rate η292 Rateleakage ηleakage[%/s] [%/s] 0 4 80.00 1216200.00 1.101.10 2.20 2.20 3.30 3.30 4.20 4.205.20 5.20 Time (s) 2.00 2.77 2.85 2.95 3.05 3.12 3.27 2.00 2.77 2.85 2.95 3.05 3.12 3.27 Maximum pressure in-cylinder p2 [MPa] 2.75 2.62 2.77 2.87 3.00 3.06 3.13 Maximum pressureFigure in-cylinder 9. The temperaturep2 [MPa] at the2.75 maximum 2.62 pressure 2.77 3.8 2.87MPa in cylinder. 3.00 3.06 3.13 3.803.80 2.41 2.41 2.572.57 2.72 2.72 2.82 2.82 2.89 2.892.95 2.95 Table 1. The leakage rates at different pressure and positions. According to Table 1 and Figure 10, the least-squares method is used to fit the dynamic According to Table1 and Figure 10, the least-squares methodPiston is used Position to fit lp the [m] dynamic leakage leakage rate model,The which Leakage is Ratea quadratic ηleakage [%/s] one. The relationship of the leakage rate ηleak and the rate model, which is a quadratic one. The relationship of0.00 the leakage1.10 2.20 rate η3.30 and4.20 the5.20 pressure and pressure and piston position is fitted as follows. leak piston position is fitted as follows. 2.00 2.77 2.85 2.95 3.05 3.12 3.27 Maximum pressure in-cylinder p2 [MPa] 2.75 2.62 2.77 2.87 3.00 3.06 3.13 2 2

ηleak = k1 + kp2 + k3lp3.80+ k 4p2.412lp +2.57k5p2 +2.72k6l p 2.82 2.89 2.95 (1) )

3.2

k k k k k k

whereAccording1, 2, 3, to4, Table5, and 1 6andare( 3.0Figure the ﬁtted 10, coe theﬃ cientsleast-squares of the leakage method rate is model.used to Forfit thisthe catapult,dynamic

g 2 4 5 a the coeﬃcients with 95% conﬁdencek bounds are k = 2.864 10 , k = 9.549 10 , k =2.202leak 10 ,

leakage rate model, which is aa quadratic one. 1The relationship− 2of the leakage− rate3 η and the− e l 2.8 × × ×

5 ? 5 4

kpressure4 = 4.465 and 10piston− , k 5position= 9.998 is 10fittede − , andas follows.k6 = 3.228 10− , respectively. t

− × × a − × r

2.6

a k

a 2.4

L )

3.2 s 4.0

0 ) / 3.6

1 a Pi 2 3.2 P % st M o ( ( 3.0 n 3 2.8 p 2

o p e sit 4 2.4 e g io r a n 5 u k l 2.0 s a p s e ( e l 2.8 m 6 r

? ) P

a r

2.6

e Figure 10. The leakage rate.

a k

a 2.4 e

L 24.0 2 η =0 + + + + + ) k1 kp k lp k p lp k 3.6p a k l (1) leak P 1 2 2 3 4 2 3.25 2 P 6 p isto M n p 3 2.8 ( os 4 p 2 itio 2.4 re 1 2 3 4 5 6 n 5 u where k , k , k , k , k , and k are the fitted coefficientsl of 2.0the leakagess rate model. For this catapult, the p ( m 6 re ) P coefficients with 95% confidence bounds are k1 = 2.864 × 10−2, k2 = 9.549 × 10−4, k3 =2.202 × 10−5, k4 = −4.465 × 10−5, k5 = 9.998 × 10−5, and k6 = Figure−Figure3.228 10.10.× 10 TheThe−4, leakagerespectively.leakage rate.rate.

4. Modelling of the High-Pressure Pneumatic Catapult Based on Real Gas Consideration 4. Modelling of the High-Pressure Pneumatic Catapult Based on2 Real Gas2 Consideration η = + + + + + leak k1 kp2 k3lp k4 p2lp k5 p2 k6lp (1) 4.1.4.1. The Theoretical ModellingModelling FlowFlow ofof thethe High-PressureHigh-Pressure PneumaticPneumatic CatapultCatapult where k1, k2, k3, k4, k5, and k6 are the fitted coefficients of the leakage rate model. For this catapult, the A ﬂow chart for modelling the high-pressure pneumatic catapult is given in Figure 11. Based on coefficientsA flow withchart 95% for modellingconfidence the bounds high-pressure are k1 = 2.864 pneu ×matic 10−2, catapultk2 = 9.549 is × given 10−4, kin3 =2.202Figure × 11. 10 −Based5, k4 = the Peng–Robinson equation of state for real gases, by considering the eﬀects of dynamic leakage and on−4.465 the Peng–Robinson× 10−5, k5 = 9.998 ×equation 10−5, and of k 6state = −3.228 for real × 10 ga−4,ses, respectively. by considering the effects of dynamic leakage takingand taking into accountinto account the convective the convective heat transfer heat transfer between between the working the working medium medium and the and metal the wall, metal a wall,4. Modelling a high-precision of the High-Pressure theoretical model Pneumatic of the Catapulthigh-pressure Based pneumatic on Real Gas catapult Consideration is developed. The

4.1. The Theoretical Modelling Flow of the High-Pressure Pneumatic Catapult

A flow chart for modelling the high-pressure pneumatic catapult is given in Figure 11. Based on the Peng–Robinson equation of state for real gases, by considering the effects of dynamic leakage and taking into account the convective heat transfer between the working medium and the metal wall, a high-precision theoretical model of the high-pressure pneumatic catapult is developed. The

Entropy 2020, 22, 1010 8 of 16

Entropy 2020, 22, x FOR PEER REVIEW 8 of 17 high-precision theoretical model of the high-pressure pneumatic catapult is developed. The simulated transientsimulated gas transient pressure gas and pressure temperature and temperature are veriﬁed are withverified the with test datathe test of thedata ejection of the ejection test of thetest high-pressureof the high-pressure pneumatic pneumatic catapult. catapult.

FigureFigure 11.11. A modeling flowflow chartchart ofof thethe high-pressurehigh-pressure pneumaticpneumatic catapult.catapult. 4.2. The Convective Heat Transfer between the Working Medium and the Metal Wall 4.2. The Convective Heat Transfer between the Working Medium and the Metal Wall For the working medium inside a high-pressure gas source, such a source is a large heat one with For the working medium inside a high-pressure gas source, such a source is a large heat one high heat capacity. It can be assumed that the temperature of the gas source body stays constant during with high heat capacity. It can be assumed that the temperature of the gas source body stays work. Therefore, for the gas source, the heat convection between the outer surface of the gas source constant during work. Therefore, for the gas source, the heat convection between the outer surface and external air can be ignored, and only that between the working medium and the inner wall surface of the gas source and external air can be ignored, and only that between the working medium and of the gas source needs to be considered, as does that in the cylinder. the inner wall surface of the gas source needs to be considered, as does that in the cylinder. The convective heat transfer includes forced convection heat one and natural convection heat one. The convective heat transfer includes forced convection heat one and natural convection heat When the catapult ejects, the working medium flows into the cylinder from the gas source, which is one. When the catapult ejects, the working medium flows into the cylinder from the gas source, generated by the external force. Thus, it is the forced convection heat transfer between the working which is generated by the external force. Thus, it is the forced convection heat transfer between the medium and the metal wall, and the heat flux can be calculated by Newton’s law of cooling. working medium and the metal wall, and the heat flux can be calculated by Newton’s law of When the working medium is heated, the rate of heat increase can be expressed as below. cooling. When the working medium isd Qheated, the rate of heat increase can be expressed as below. = αA∆T(t) = hA Tw T (2) dt − f dQ = α Δ = ()− A T (t) hA Tw T f (2) When the working medium is cooled,dt the rate of heat loss is shown as follows.

When the working medium isdQ cooled, the rate of heat loss is shown as follows. = αA∆T(t) = αA T Tw (3) dt dQ f − = αAΔT (t) = αA()T − T (3) dt f w where α is the heat transfer coeﬃcient, Q the heat, A the heat transfer surface area, Tw and Tf are the temperaturewhere α is the of heat the wall transfer and workingcoefficient, medium Q the heat, respectively. A the heat transfer surface area, Tw and Tf are the temperature of the wall and working medium respectively. When Reynolds number Re ≤ 2300, the convective heat transfer is a laminar heat transfer, and the Nusselt number is calculated by the Sieder–Tate equation. 1 3 η 0.14  d   f  Nu = 1.86 Re ⋅ Pr ⋅   f  f  η  (4)  L   w 

Entropy 2020, 22, 1010 9 of 16

When Reynolds number Re 2300, the convective heat transfer is a laminar heat transfer, and the ≤ Nusselt number is calculated by the Sieder–Tate equation.

!1/3 !0.14 d η f Nu f = 1.86 Re f Pr (4) · · L ηw where Pr is the Prandtl number, d the characteristic inner diameter of the ﬂow channel, L the characteristic length, ηf and ηw are the viscosity when the qualitative temperature is the temperature of the working meduim and the wall. If the relationship between gas viscosity and temperature is not considered, ηf/ηw = 1. When Re > 2300, it is a turbulent heat transfer, and the Nusselt number needs to be calculated by the Dittus–Boelter correction equation.

0.8 0.4 Nu = 0.023Re Pr ε εR εt (5) f f · f · l · · where εl, εR and εt are the tube length, bending and temperature correction factor, respectively. When the ejection starts, all the valves are fully opened, the working medium flows into the cylinder, and the pressure and the temperature of the working medium rise rapidly in a small enclosed space. Then the piston is pushed to move fast, while the internal volume of the cylinder increases, resulting in continually decreasing the temperature and the pressure of the working medium. The gas source is in the state of deflation and the temperature of the working medium in the gas source decreases further. Consequently, the working medium is heated by the inner wall surface of the gas source. According to Newton’s law of cooling [12], the heat flux between the working medium and the inner wall of the gas source is as follows.

 1/3 0.14  d η f λ  1.86 Re f Pr f L η D A1 Tw1 T f 1 , Re f 2300 δQ = α A Tw T = · · w · 1 · − ≤ (6) 1 1 1 f  0.8 0.4 λ −  0.023Re Pr εl εR εt A1 Tw1 T f 1 , Re f > 2300 f · f · · · · D1 · − and the heat ﬂux between the working medium and the inner wall surface of the cylinder is

 1/3 0.14  d η f λ  1.86 Re f Pr f L η D A2 T f 2 Tw2 , Re f 2300 δQ = α A T Tw = · · w · 2 · − ≤ (7) 2 2 2 f  0.8 0.3 λ −  0.023Re Pr εl εR εt A2 T f 2 Tw2 , Re f > 2300 f · f · · · · D2 · − The heat transfer coeﬃcient α can be calculated by Nusselt number, as shown in Equation (8).

α = Nu f λ/d (8) where λ is the thermal conductivity of ﬂuid, d the characteristic inner diameter of round tube.

4.3. Compressibility Factor and Thermodynamic Variables of a Real Gas

4.3.1. Compressibility Factor of a Real Gas Compressibility factor is a function of the corresponding pressure and temperature, and it can be used as a basis to select the equation of state. As the working medium is dry air, the virial equation, Soave–Redlich–Kwong equation and Peng–Robinson (P-R) equation are all applicable to the high-pressure pneumatic system. From our previous research, the P-R equation of state is more suitable for an open-type catapult. The expression of the compressibility factor is shown as follows.

pVm Z = (9) RT Entropy 2020, 22, 1010 10 of 16

where Z is the compressibility factor, Vm the molar volume, R the universal gas constant, p the pressure, T the temperature. The P-R equation is an improvement on the van der Waals equation, as follows.

RT a(T) p = (10) Vm b − Vm(Vm + b) + b(Vm b) − − where a(T) and b are van der Waals constants. The cubic equation for the molar volume Vm can be obtained after solve the P-R equation of state. Together with Equations (9) and (10), the compression factor at a special point can be obtained.

RT a(T) Z = ( )Vm/RT (11) Vm b − Vm(Vm + b) + b(Vm b) − − For dry air, when the pressure is less than half of the critical pressure or the temperature is greater than 5 times of the critical temperature, the P-Z curves can be approximately considered as linear and the compressibility factor Z is nearly 1. When the pressure is greater than half of the critical or the temperature is less than 5 times of the critical temperature, a reliable real gas equation of state needs to be derived to ﬁt the data. The Nelson-Obert (N-O) generalized compressibility charts proposed by Nelson-Obert are the most precise ones, with pressures of pr = 0~1 MPa (meiobar range), pr = 1~10 MPa (middle-pressure range) and pr = 10~100 MPa (high-pressure range). From the ideal gas model, it can be known that the temperature range of gas in a cylinder is from 220 K to 463.5 K. Although it is larger than the actual range, it still can be a basis for selection of temperature. The comparison of compressibility factors for air under the speciﬁed pressure and temperature with the N-O compressibility charts and P-R equation of state is shown in Table2.

Table 2. Compressibility factors of air under the speciﬁed pressure and temperature.

Pressure [MPa] Compressibility Factors Z 0.101 1 6 12 18 24 30 35 P-R equation 0.9982 0.9793 0.8960 0.8521 0.8647 0.9322 1.0969 1.1259 220 (T = 1.6) r N-O value 0.9981 0.9938 0.9068 0.8421 0.8666 0.9425 1.0388 1.1310 P-R equation 0.9990 0.9903 0.9550 0.9421 0.9555 0.9903 1.0326 1.0762 265 (Tr = 2) Temperature N-O value 0.9988 0.9928 0.9683 0.9588 0.9761 1.0039 1.0152 1.0276 [K] P-R equation 0.9997 0.9975 0.9920 0.9989 1.0323 1.0564 1.0914 1.1137 331 (T = 2.5) r N-O value 0.9969 0.9975 0.9996 1.0101 1.0429 1.0753 1.1034 1.1328 P-R equation 1.00027 1.0038 1.0146 1.0336 1.0662 1.0929 1.1086 1.1321 463.5 (T = 3.5) r N-O value 1.0033 1.0042 1.0181 1.0405 1.0709 1.1067 1.1236 1.1527

As noted in Table2, when the pressure is relatively low, compressibility factors are close to 1.0 within the temperature range of the table. The deviation of the compressibility factors corresponding to the P-R equation and N-O values is small, which means the ideal gas model can be applied when the pressure is low. With the increase of the pressure and temperature, the compressibility factor gradually deviates from 1.0. The diﬀerence in pressure and temperature between real gas and ideal gas cannot be neglected in such situations. Within the changing range of temperature and pressure of the ejection, the P-R equation has a high accuracy.

4.3.2. Thermodynamic Variables The real gas thermodynamic property variables can be obtained by subtracting the residual function from the ideal value. The deﬁnition of residual function can be expressed as

Fre = F* F (12) − Entropy 2020, 22, 1010 11 of 16

where Fre denotes the residual of a arbitrary extensive properties or speciﬁc properties, F* and F are the properties of ideal gas and real gas. The residual speciﬁc thermodynamic energy for real gas is as follows.

Z v" # Z v" # ∂p 2 ∂(p/T) ur = p T( ) dv = T dv (13) − ∂T − ∂T v ∞ ∞ The speciﬁc enthalpy for real gas and ideal gas are deﬁned as

h = u + pv (14)

h∗ = u∗ + RgT (15) where h is the speciﬁc enthalpy, u speciﬁc thermodynamic energy, Rg is gas constant, p the pressure, T the temperature, superscripts * denotes ideal gas. Combination of Equations (12)–(15) will lead to

Z v" # 2 ∂(p/T) hr = T dv pv + RgT (16) − ∂T v − ∞ Therefore, by introducing the P-R equation, the expressions of the specific thermodynamic energy can be derived as follows.     r2 + r V + 0.0778 1 √2 RTc    m Pc  u = C T 2.0778R T  (r + 1)2 + T0.5 ln −  (17) v g c 0.5   RT  − − Tc  · V + 0.0778 1 + √2 c  m Pc where R is the Molar gas constant, Tc the critical temperature, Pc the critical pressure, Vm the molar volume, v the specific volume and v = Vm/M, r the characteristic constant determined by material types, Cv the specific heat at constant volume of ideal gas state. The experimental results show that the specific heat capacity of an ideal gas is a complex function of temperature, which increases with temperature. The formula of air is in a cubic relationship between the specific constant pressure heat capacity Cp and temperature [26,27]. By combining with the Meyer formula, Cp can be obtained by the equation below.

2 3 Cp = Cv + Rg = C0 + C1θ + C2θ + C3θ (18) where θ = (T) /1000. To dry air, there are C = 1.05, C = 0.365, C = 0.85, C = 0.39. k 0 1 − 2 3 − 4.4. The Accurate Theoretical Model Considering Dynamic Leakage and Convective Heat Transfer In the procedure of ejection, the mass ﬂow equation of the pneumatic valve can be written as

 s " #  2 k+1 k  p1Av 2k p2 k p2 k 2 k 1 p2  µx k 1 p p , k+1 − < p < 1  √RgT1 1 − 1 1 Gc =  − (19)  k+1 k  2 2(k 1) p p2 2 k 1  µx √k − p1Av/ RgTc, − k+1 p1 ≤ k+1 where Gc is the mass flow at the throttle, subscripts 1 and 2 indicate gas source and cylinder, respectively, µx the flow correction factor, Av the equivalent flow area of a large flow pneumatic valve, k the adiabatic index, Rg the gas constant, Tc the critical temperature of a working medium, T1 the working temperature of a working medium in gas source, p1 and p2 the pressures of working medium in gas source and cylinder. Entropy 2020, 22, 1010 12 of 16

The mass equation and the energy equation of a working medium for a high-pressure gas source can be expressed in Equations (20) and (21) below, respectively.

d (ρ V ) = Gc (20) dt 1 1 − d (ρ V u ) = Gch (21) dt 1 1 1 − 1 The equation for the motion of a piston can be expressed as:

dvp p2S (ma + mt)g f = − − (22) dt (ma + mt) where vp is the velocity of piston, S the eﬀective pushing area of piston, g the acceleration of gravity, f is the friction force, ma the mass of analogue load, mt the total mass of piston and load pushing table, p2 the pressure of working medium in cylinder. The energy equation of a working medium for the cylinder is shown.

d(m2u2) dlp = Gch Sp (23) dt 1 − 2 dt where m2 is the mass of the working medium in the cylinder. The pressure and temperature in the cylinder rise firstly and then decrease, which vary greatly during the ejection. Based on the conservation laws of mass and energy, mass and flow equations, P-R state equations, coupled dynamic leakage models and convective heat transfer models, an accurate theoretical model of the high-pressure pneumatic catapult is established and can be expressed as the closed-form equations. There are seven independent variables, i.e., ρ1, T1, m2, T2, lp, vp and Av. Let X1 = ρ1, X2 = T1, X3 = m2, X4 = T2, X5 = lp, X6 = vp and X7 = Av. The closed-form equations can be calculated by five-stage fourth-order Runge–Kutta method.

 .   X1V1 + Gc = 0  . . .  X1V1(h1 u1)u1+δQ1  X2W1 X1 Y1 − = 0  . − · − X1   X3 Gc ηleak = 0  . − · . (24)  MS2X3X6 M(V01+S2X5)X3  X4X3W2 + Gcu2 + S2p2X6 0.85Gch1 Y2 − + δQ2 = 0  − − · X3  .  X5 X6 = 0  . −  S2p2 (ma+mt)g f  X6 − − = 0 − (ma+mt)

p1 where u1 = Cv X2 Y1, u2 = Cv2X4 Y2, h1 = u1 + , 1 − − X1 2 2 ηleak = k1 + k2p2 + k3X5 + k4p2X5 + k5p2 + k6X5

2 3 X2 X2 X2 Cv = C + C + C + C Rg, 1 0 1 1000 2 1000 3 1000 − 2 3 X4 X4 X4 Cv = C + C + C + C Rg 2 0 1 1000 2 1000 3 1000 −  s " #  2 k+1 k  p1X7 2k p2 k p2 k 2 k 1 p2  µx k 1 p p k+1 − < p < 1  √RgT1 1 − 1 1 Gc =  −  k+1 k  2 2(k 1) p p2 2 k 1  µx √k − p1X7/ RgTc − k+1 p1 ≤ k+1 Entropy 2020, 22, 1010 13 of 16

  RTc    Vm1 + 0.0778(1 √2)   C1 C2 2 C3 3 0.5 2 0.5  − pc  W1 =  C0 2X2 + 3X2 + 4X2 Rg 1.0389RgTc (r + r)X2− ln   − 103 · 106 · 109 · − −  V + 0.0778(1 + √2) RTc  m1 pc

  RT  2 √ c  C C C (r + r)  Vm2 + 0.0778(1 2) p  =  1 + 2 2 + 3 3 0.5  − c  W2  C0 2X4 3X4 4X4 Rg 1.0389RgTc X4− ln   − 103 · 106 · 109 · − − T0.5  V + 0.0778(1 + √2) RTc  c m2 pc

  ( 2 + )   2 r r 0.5  1 1  Y1 = 2.0778RgTc[ (r + 1) + X ]   0.5 2  RTc RTc  − Tc ·  Vm1 + 0.0778(1 √2) − Vm1 + 0.0778(1 + √2)  − pc pc

  ( 2 + )   2 r r 0.5  1 1  Y2 = 2.0778RgTc[ (r + 1) + X ]   0.5 4  RTc RTc  − Tc ·  Vm2 + 0.0778(1 √2) − Vm2 + 0.0778(1 + √2)  − pc pc 5. Veriﬁcation of the Accuracy of the Theoretical Model with the Ejection Test

5.1. Verification of the Accuracy of the Theoretical Model Based on Real Gas Effects Comparing the analytical results in the real gas state with the data from ejection test can help one understand how reliable the model is. Based on the theoretical model developed, an analysis program is worked out. Then, the analytical results of the ejection based on the real gas effects are obtained, whichEntropy 2020 are, compared 22, x FOR PEER with REVIEW the test data of the ejection obtained in Section 3.2, as shown in Figure14 12of 17.

25 15 Ejection test Ejection test Real gas state Real gas state 20 12

15 9

6 10

Piston stroke (m) stroke Piston 3 Pressure in cylinder (MPa) cylinder in Pressure

0 0 0.00 0.15 0.30 0.45 0.60 0.75 0.90 0.00 0.15 0.30 0.45 0.60 0.75 0.90 Time (s) Time (s) (a) The piston stroke (b) The pressure in the cylinder

Figure 12. TheThe analytical analytical results results and and test data of the pressure in the cylinder.

5.2. ComparisonAs indicated of inTheoretical Figure 12 Models, it can of be High-Pressure seen that the Pneumatic analytical Catapult results are in good agreement with the test data.The comparison When the initial of analytical pressure ofresu workinglts for the medium ejection in the in gasthe sourceideal gas is set state to 30and MPa, the thereal maximum gas state aredeviation shown of in ejection Figures stroke 13–15. between Figure 13 the shows analytical the anal resultsytical and results the test of data the pressure is 3.7%, and in the the gas maximum source pressureand cylinder. deviation It can of be working seen that medium the analytical in the cylinder results in is 4%.the ideal It shows gas thatstate the are theoretical clearly larger model than of thathigh-pressure in the real pneumatic gas state, catapultwith the developed maximum in deviation the real gas of 22%, state haswhile ahigh the initial precision. parameters of the gas5.2. source Comparison are the of Theoreticalsame. Figure Models 14 shows of High-Pressure the temper Pneumaticature of the Catapult working medium in the gas source and cylinder. Whether to consider dynamic leakage and convective heat transfer or not, the temperatureThe comparison deviation of analyticalof working results medium for the re ejectionached 12% in the in idealthe gascylinder, state andand the the real maximum gas state temperatureare shown in difference Figures 13 is– 1555.4. Figure K, which 13 shows indicates the the analytical effects of results dynamic of the leakage pressure and in convective the gas source heat transferand cylinder. cannot It canbe ignored. be seen thatMoreover, the analytical Figure results15 shows in the the ideal deviation gas state of arethe clearlyejection larger overload than coefficientthat in the realand gasthe piston state, with velocity the maximumreaches 20 deviationand 12%, ofrespectively. 22%, while the initial parameters of the gas source are the same. Figure 14 shows the temperature of the working medium in the gas source and cylinder. Whether to consider dynamic35 leakage and convective heat transfer or not, the temperature 30

20 (MPa)

In gas source, real gas state

Pressure 10 In gas source, ideal gas state 5 In cylinder, real gas state In cylinder, ideal gas state 0 0.00 0.15 0.30 0.45 0.60 0.75 0.90 Time (s) Figure 13. The analytical results of pressure.

420

In gas source, real gas state 390 In gas source, ideal gas state

In cylinder, real gas state 360 In cylinder, ideal gas state

330

300 Temperature (K) 270

240 0.00 0.15 0.30 0.45 0.60 0.75 0.90 Time (s) Figure 14. The analytical results of temperature.

EntropyEntropy 20202020,, 2222,, xx FORFOR PEERPEER REVIEWREVIEW 1414 ofof 1717

2525 1515 Ejection test Ejection test Ejection test Ejection test Real gas state Real gas state Real gas state Real gas state 2020 1212

1515 99

10 66 10

Piston stroke (m) stroke Piston 3

Piston stroke (m) stroke Piston 3 Pressure in cylinder (MPa) cylinder in Pressure Pressure in cylinder (MPa) cylinder in Pressure

0 00 0 0.000.00 0.15 0.15 0.30 0.30 0.45 0.45 0.60 0.60 0.75 0.75 0.90 0.90 0.000.00 0.15 0.15 0.30 0.30 0.45 0.45 0.60 0.60 0.75 0.75 0.90 0.90 Time (s) TimeTime (s)(s) Time (s) ((aa)) TheThe pistonpiston strokestroke ( (bb)) TheThe pressurepressure inin thethe cylindercylinder

FigureFigure 12.12. TheThe analyticalanalytical resultsresults andand testtest datadata ofof thethe pressurepressure inin thethe cylinder.cylinder.

5.2.5.2. ComparisonComparison ofof TheoreticalTheoretical ModelsModels ofof High-PressureHigh-Pressure PneumaticPneumatic CatapultCatapult TheThe comparisoncomparison ofof analyticalanalytical resuresultslts forfor thethe ejectionejection inin thethe idealideal gasgas statestate andand thethe realreal gasgas statestate areare shownshown inin FiguresFigures 13–15.13–15. FigureFigure 1313 showsshows thethe analanalyticalytical resultsresults ofof thethe pressurepressure inin thethe gasgas sourcesource andand cylinder.cylinder. ItIt cancan bebe seenseen thatthat thethe analyticalanalytical resultsresults inin thethe idealideal gasgas statestate areare clearlyclearly largerlarger thanthan that in the real gas state, with the maximum deviation of 22%, while the initial parameters of the thatEntropy in2020 the, 22real, 1010 gas state, with the maximum deviation of 22%, while the initial parameters 14of ofthe 16 gasgas sourcesource areare thethe same.same. FigureFigure 1414 showsshows thethe tempertemperatureature ofof thethe workingworking mediummedium inin thethe gasgas sourcesource andand cylinder.cylinder. WhetherWhether toto considerconsider dynamicdynamic leakagleakagee andand convectiveconvective heatheat transfertransfer oror not,not, thethe temperaturedeviationtemperature of working deviationdeviation medium ofof workingworking reached mediummedium 12% in the rere cylinder,achedached 12%12% and theinin maximumthethe cylinder,cylinder, temperature andand thethe di maximummaximumfference is temperature55.4temperature K, which differencedifference indicates theisis 55.455.4 effects K,K, whichwhich of dynamic indicatesindicates leakage thethe effectseffects and convective ofof dynamicdynamic heat leakageleakage transfer andand cannot convectiveconvective be ignored. heatheat transferMoreover,transfer cannotcannot Figure bebe 15 ignored. ignored.shows the Moreover,Moreover, deviation FigureFigure of the ejection1515 showsshows overload thethe deviationdeviation coefficient ofof and thethe theejectionejection piston overloadoverload velocity coefficientreachescoefficient 20 and and the 12%,the pistonpiston respectively. velocityvelocity reachesreaches 2020 andand 12%,12%, respectively.respectively.

3535

3030

2525

2020 (MPa) (MPa)

1515

In gas source, real gas state

Pressure 10 In gas source, real gas state

Pressure 10 InIn gasgas source,source, idealideal gasgas statestate 55 InIn cylinder,cylinder, realreal gasgas statestate InIn cylinder,cylinder, idealideal gasgas statestate 00 0.000.00 0.15 0.15 0.30 0.30 0.45 0.45 0.60 0.60 0.75 0.75 0.90 0.90 TimeTime (s)(s) FigureFigure 13. 13. TheThe analyticalanalytical results results of of pressure. pressure.

420420

InIn gasgas source,source, realreal gasgas statestate 390 390 InIn gasgas source,source, idealideal gasgas statestate

InIn cylinder,cylinder, realreal gasgas statestate 360360 InIn cylinder,cylinder, idealideal gasgas statestate

330330

300300 Temperature (K) Temperature (K) 270270

240240 0.000.00 0.15 0.15 0.30 0.30 0.45 0.45 0.60 0.60 0.75 0.75 0.90 0.90 TimeTime (s)(s) Entropy 2020, 22, x FOR PEER REVIEW 15 of 17 FigureFigure 14. 14. TheThe analyticalanalytical results results of of temperature. temperature.

8 35 Real gas state Ideal gas state Real gas state 28 Ideal gas state 6 ) s / ) m 2 ( 21 ty 4 i oc l 14

(g=9.8m/s 2 ston ve ston 7 Pi

0 Overload coefficient of ejection of coefficient Overload 0 0.00 0.15 0.30 0.45 0.60 0.75 0.90 0.00 0.15 0.30 0.45 0.60 0.75 0.90 Time (s) Time (s)

(a) The overload coefficient of ejection (b) The piston velocity

Figure 15. The analytical results of ejection performance parameters.

All thethe valuesvalues related related to to the the ideal ideal gas gas state state are are higher higher than than those those to the to real the gasreal state. gas state. In the In initial the stage,initial thestage, deviation the deviation is small is and small however, and however, it becomes it becomes increasingly increasingly large. This large. means This that means the ability that the of theability high-pressure of the high-pressure pneumatic pneumatic catapult in thecatapult ideal gasin the state ideal is exaggerated. gas state is Therefore,exaggerated. a high-precision Therefore, a assessmenthigh-precision of pneumatic assessment catapult of pneumatic requires catapult taking re thequires eﬀects taking of dynamic the effects leakage of dynamic and convective leakage heatand transferconvective into heat account transfer in the into theoretical account model, in the in th ordereoretical to gain model, the necessary in order accuracy to gain ofthe the necessary model. accuracy of the model.

6. Conclusions Based on the experimental and theoretical work of the high-pressure pneumatic catapult presented, the following conclusions can be drawn. (1) It is found that the analytical results based on the ideal gas model give an overestimated performance of the catapult, in comparison to the test data. The maximum deviation of the piston stroke and the cylinder gas pressure is 16.7% and 24.5%. Consequently, the precision of the ideal gas model is unacceptable for the engineering applications. (2) The relationship of the leakage rate, pressure and stroke is fitted. It is found that the maximum leakage rate of the pneumatic catapult does not exceed 5% within the whole piston stroke, showing a good sealing performance. The leakage rate model is a key factor that affects the accuracy of the theoretical model. Taking leakage into account can improve the accuracy of the theoretical model and make the theoretical calculation more consistent with the actual situation. Regardless of the leakage, the theoretical model is not a true high precision model, and cannot be used to evaluate actual pneumatic catapults, and its practicality will be limited. (3) A corresponding convective heat transfer model between the working medium and the metal wall has been developed. In the heat transfer process, the choice of laminar flow model or turbulent heat transfer model is based on the Reynolds number. (4) Based on the Peng–Robinson equation, a theoretical model of the high-pressure pneumatic catapult has been developed, in which the effects of dynamic leakage and the forced convective heat transfer between the gas and the metal wall are taken into account. The results from the theoretical model are consistent with the data form ejection tests, with a maximum deviation of 4%, indicating much higher precision than the ideal gas model. (5) The theoretical model established can be applicable for dry gases, such as an air, N2 and CO2. It cannot be applied to catapults that use water vapor as it involves phase change of working medium in ejection. The equation of state is not applicable to the two-phase regions.

Entropy 2020, 22, 1010 15 of 16

6. Conclusions Based on the experimental and theoretical work of the high-pressure pneumatic catapult presented, the following conclusions can be drawn.

(1) It is found that the analytical results based on the ideal gas model give an overestimated performance of the catapult, in comparison to the test data. The maximum deviation of the piston stroke and the cylinder gas pressure is 16.7% and 24.5%. Consequently, the precision of the ideal gas model is unacceptable for the engineering applications. (2) The relationship of the leakage rate, pressure and stroke is fitted. It is found that the maximum leakage rate of the pneumatic catapult does not exceed 5% within the whole piston stroke, showing a good sealing performance. The leakage rate model is a key factor that affects the accuracy of the theoretical model. Taking leakage into account can improve the accuracy of the theoretical model and make the theoretical calculation more consistent with the actual situation. Regardless of the leakage, the theoretical model is not a true high precision model, and cannot be used to evaluate actual pneumatic catapults, and its practicality will be limited. (3) A corresponding convective heat transfer model between the working medium and the metal wall has been developed. In the heat transfer process, the choice of laminar flow model or turbulent heat transfer model is based on the Reynolds number. (4) Based on the Peng–Robinson equation, a theoretical model of the high-pressure pneumatic catapult has been developed, in which the effects of dynamic leakage and the forced convective heat transfer between the gas and the metal wall are taken into account. The results from the theoretical model are consistent with the data form ejection tests, with a maximum deviation of 4%, indicating much higher precision than the ideal gas model.

(5) The theoretical model established can be applicable for dry gases, such as an air, N2 and CO2. It cannot be applied to catapults that use water vapor as it involves phase change of working medium in ejection. The equation of state is not applicable to the two-phase regions.

Author Contributions: J.R. conceived the idea of the study and established the structure scheme of the high-pressure pneumatic catapult and derived the theoretical model with dynamic leakage and convection. He analyzed the data and wrote the paper. The remaining authors contributed to refining the idea, carrying out additional analyses and finalizing this paper. J.Z. was responsible for the comparative analysis of the pneumatic drive device performance based on real gas state equation and ideal gas state equation, including the analysis of dynamic thermodynamic processes. He is also responsible for data logging and data accuracy checks in the experimental section. L.Y. developed a research program to study the real gas effects, by using a five-step four-order Runge-Kutta method to solve the closed-form pneumatic equations and calculated the compression factor based on the P-R equation. Z.G. gave guidance on research and participated in constructive discussions. He proposed many revisions and contributed a lot to the realization of the research work and the article. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Natural Science Foundation of Jiangsu Province (Grant number BK20170837) and the China Scholarship Council (Grant number 201906845017). Conflicts of Interest: The authors declare that there is no conflict of interest regarding the publication of this paper.

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