Development and Research of Crosshead-Free Piston Hybrid Power Machine

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Article Development and Research of Crosshead-Free Piston Hybrid Power Machine

Viktor Shcherba 1,* , Viktor Shalay 2, Evgeniy Nosov 1, Evgeniy Pavlyuchenko 1 and Ablai-Khan Tegzhanov 1

1 Hydromechanics and Machines Department, Omsk State Technical University, Prospect Mira st., 11, 644050 Omsk, Russia; [email protected] (E.N.); [email protected] (E.P.); [email protected] (A.-K.T.) 2 Transport, Oil and Gas Storage, Standardization and Certiﬁcation Department, Omsk State Technical University, Prospect Mira st., 11, 644050 Omsk, Russia; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +7-3812-65-31-77

Abstract: This article considers the development and research of a new design of crosshead-free piston hybrid power machine. After veriﬁcation of a system of simplifying assumptions based on the fundamental laws of energy, mass, and motion conservation, as well as using the equation of state, mathematical models of the work processes of the compressor section, pump section, and liquid ﬂow in a groove seal have been developed. In accordance with the patent for the invention, a prototype of a crosshead-free piston hybrid power machine (PHPM) was developed; it was equipped with the necessary measuring equipment and a stand for studying the prototype. Using the developed mathematical model, the physical picture of the ongoing work processes in the compressor and pump sections is considered, taking into account their interaction through a groove seal. Using the developed plan, a set of experimental studies was carried out with the main operational parameters

of the crosshead-free PHPM: operating processes, temperature of the cylinder–piston group and integral parameters (supply coefﬁcient of the compressor section, volumetric efﬁciency of the pump

Citation: Shcherba, V.; Shalay, V.; section, etc.). As a result of numerical and experimental studies, it was determined that this PHPM Nosov, E.; Pavlyuchenko, E.; design has better cooling of the compressor section (decrease in temperature of the valve plate is Tegzhanov, A.-K. Development and from 10 to 15 K; decrease in temperature of intake air is from 6 to 8 K, as well as there is increase in Research of Crosshead-Free Piston compressor and pump section efﬁciency up to 5%). Hybrid Power Machine. Machines 2021, 9, 32. https://doi.org/10.3390/ Keywords: reciprocating compressor; piston pump; piston hybrid power machine; work processes; machines9020032 indicator efﬁciency of the compressor

Academic Editor: Hamid Reza Karimi Received: 16 December 2020 Accepted: 31 January 2021 1. Introduction Published: 5 February 2021 Pumps and compressors designed for the compression and displacement of low- compressible liquids and gases are widely used as in traditional industries: power, oil and Publisher’s Note: MDPI stays neutral gas, transport [1] and aviation, as in new ones: measuring equipment [2], medical industry, with regard to jurisdictional claims in published maps and institutional affil- etc. In general, from the point of liquid and gas mechanics, “liquid” as a term includes iations. low-compressible (droplet) liquid and compressible liquid (gaseous). They simply use “liquid” instead of “low-compressible liquid” as a term in the theory of pumps, and “gas” is used instead of “compressible liquid” in the theory of compressors. Thus, we also use “liquid” and “gas” in the article. Due to the fact that 20% of the energy used in industry is spent on pump units’ Copyright: © 2021 by the authors. drive [3], the task to increase their efficiency is important and relevant. Licensee MDPI, Basel, Switzerland. At present, one of the main ways to increase the work efficiency and effectivity This article is an open access article distributed under the terms and of compressors and displacement pumps, as well as to improve their weight and size conditions of the Creative Commons indicators, is to unite them into a single unit called a piston hybrid power machine. Piston Attribution (CC BY) license (https:// hybrid power machines have the leading role among variety of hybrid power machines. creativecommons.org/licenses/by/ When a pump and a compressor are combined into a single power unit, the compressor 4.0/). cooling is improved, leaks and overflows of the compressed gas are eliminated in the

Machines 2021, 9, 32. https://doi.org/10.3390/machines9020032 https://www.mdpi.com/journal/machines Machines 2021, 9, x FOR PEER REVIEW 2 of 39

cooling is improved, leaks and overflows of the compressed gas are eliminated in the cylinder–piston group, friction forces are reduced in the cylinder–piston group, positive suction head is increased, and gas compression heat is to be utilized in the pump section [4]. There are two main line diagrams among hybrid power machines. Machines 2021, 9, x FOR PEER REVIEW The first diagram is a crosshead hybrid power machine (see Figure 1), in which a2 disk of 39 Machines 2021, 9, 32 2 of 38 piston divides the working space of the cylinder into two working chambers: the compressor section is located above the piston and the pump section is located under the piston.cooling is improved, leaks and overflows of the compressed gas are eliminated in the cyl- cylinder–pistoninder–piston group, group, friction friction forces forces are are reduc reduceded in the in thecylinder–piston cylinder–piston group, group, positive positive suc- suctiontion head head is increased, is increased, and and gas gas compression compression heat heat is is to to be be utilized utilized in in the pump sectionsection [[4].4]. ThereThere are are two two main main line line diagrams diagrams among among hybrid hybrid power power machines. machines. TheThe ﬁrstfirst diagram is is a a crosshead crosshead hybrid hybrid powe powerr machine machine (see (see Figure Figure 1),1 in), which in which a disk a diskpiston piston divides divides the working the working space space of the of cy thelinder cylinder into intotwo twoworking working chambers: chambers: the com- the compressor section is located above the piston and the pump section is located under pressor section is located above the piston and the pump section is located under the pis- the piston. ton.

Figure 1. Construction diagram of piston hybrid power machine (PHPM). (1—cylinder; 2—piston; 3, 4—compressor and pump chamber; 5, 6—discharge and suction liquid nozzles; 7—plunger).

The second diagram has a differential piston and two working chambers. The inner working chamber is a compressor section, and the external working chamber is a pump section (see Figure 2). Design, development, and research of the crosshead PHPM operation are given in [4]. Experimental and numerical studies of crosshead PHPM with various types of groove FigureFigure 1. 1.Construction Construction diagram diagram of of piston piston hybrid hybrid power power machine machine (PHPM). (PHPM). (1—cylinder; (1—cylinder; 2—piston; 2—piston; 3, seals were carried out there. Findings presented that the crosshead PHPM with a step-like 4—compressor3, 4—compressor and and pump pump chamber; chamber; 5, 6—discharge 5, 6—discharge and and suction suction liquid liquid nozzles; nozzles; 7—plunger). 7—plunger). groove seal is the most effective. The crosshead PHPM defects are: − ThePoorThe second secondcooling diagramdiagram of the cylinder–pist hashas aa differentialdifferentialon group piston piston and and and the two twocompressed working working chambers.gas; chambers. The The inner inner working−working Many chamberchamber moving isis parts aa compressorcompressor with reciprocating section,section, andan movement;d thethe external external workingworking chamberchamber isis aa pump pump section−section Specific (see(see FigureFigure weight2 ).2). and size indicators have low values. Design, development, and research of the crosshead PHPM operation are given in [4]. Experimental and numerical studies of crosshead PHPM with various types of groove seals were carried out there. Findings presented that the crosshead PHPM with a step-like groove seal is the most effective. The crosshead PHPM defects are: − Poor cooling of the cylinder–piston group and the compressed gas; − Many moving parts with reciprocating movement; − Specific weight and size indicators have low values.

FigureFigure 2.2. Construction diagram diagram of of crosshead-free crosshead-free PHPM PHPM (1—crankcase; (1—crankcase; 2—piston; 2—piston; 3—pumping 3—pumping workingworking chamber;chamber; 4—gas4—gas workingworking chamber).chamber).

Design, development, and research of the crosshead PHPM operation are given in [4]. Experimental and numerical studies of crosshead PHPM with various types of groove seals were carried out there. Findings presented that the crosshead PHPM with a step-like groove seal is the most effective. The crosshead PHPM defects are:

- Poor cooling of the cylinder–piston group and the compressed gas; Figure 2. Construction diagram of crosshead-free PHPM (1—crankcase; 2—piston; 3—pumping - Many moving parts with reciprocating movement; working chamber; 4—gas working chamber). - Speciﬁc weight and size indicators have low values.

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TheseThese defectsdefects areare partiallypartially eliminated eliminated in in the the crosshead-free crosshead-free PHPM PHPM [ 5[5].]. WeWe developed developed a a new new powerful powerful design design of of crosshead-free crosshead-free PHPM PHPM based based on on the the analysis analysis ofof thethe above-mentionedabove-mentioned defectsdefects typicaltypical forfor crossheadcrosshead andand crosshead-freecrosshead-free PHPM;PHPM; itsits line line diagramdiagram isis shownshown inin FigureFigure3 .3.

FigureFigure 3.3. SchemeScheme of of longitudinal longitudinal section section of crosshea of crosshead-freed-free PHPM PHPM test model. test model. 1—crankcase; 1—crankcase; 2— 2—crankshaft;crankshaft; 3—connecting 3—connecting rod; rod; 4—piston 4—piston pin; 5— pin;piston; 5—piston; 6—cylindrical 6—cylindrical surface surface of the ofworking the work- chamber of the compressor section; 7—outer cylinder; 8—suction valve of the compressor section; ing chamber of the compressor section; 7—outer cylinder; 8—suction valve of the compressor 9—discharge valve of the compressor section; 10, 13—channels for supplying and removing com- section; 9—discharge valve of the compressor section; 10, 13—channels for supplying and removing pressible gas; 11—suction valve of the pump section; 12—pressure valve of the pump section; 14— compressibleinner cylinder gas; attaching 11—suction to the valveouter ofcylinder; the pump 15—working section; 12—pressure chamber of valvethe compressor of the pump section; section; 14—inner16—annular cylinder П-shaped attaching protrusion to the of outer the piston; cylinder; 17—i 15—workingnner cylinder; chamber 18—working of the clearance; compressor 19— section;O-rings; 16—annular 20—screw;Π-shaped 21—working protrusion chamber of theof the piston; pump 17—inner section. cylinder; 18—working clearance; 19—O-rings; 20—screw; 21—working chamber of the pump section. In the developed PHPM, there is annular П-shaped protrusion 16 installed on piston 5 formingIn the developedtwo working PHPM, chambers there iswith annular the innerΠ-shaped surface protrusion of outer 16 cylinder installed 7 onand piston outer 5surface forming of twoinner working cylinder chambers 17: working with chamber the inner 15 surface is for gas of outer compressing cylinder and 7 and moving, outer surfaceworking of chamber inner cylinder 21 is for 17: liquid working compressing chamber and 15 moving. is for gas Inner compressing cylinder 17 and is mounted moving, workingconcentrically chamber to outer 21 is forcylinder liquid 7. compressing Pipe 10 is for and gas moving. supply Innerand pipe cylinder 13 is 17for is its mounted removal concentricallyin inner cylinder to outer 17. The cylinder pipes 7.have Pipe suction 10 is for valve gas supply8 and discharge and pipe valve 13 is for9, which its removal allow ingas inner to be cylinder sucked 17.into The working pipes chamber have suction 15 of valve the compressor 8 and discharge section valve and 9,its which supply allow from gasthe toworking be sucked chamber into working to the consumer. chamber There 15 of theare compressorpipes in inner section cylinder and 17, its where supply suction from thevalve working 11 and chamber discharge to valve the consumer. 12 are installed; There are they pipes supply in inner liquid cylinder to the 17,working where cavity suction of valve 11 and discharge valve 12 are installed; they supply liquid to the working cavity of pump section 21 and then to the discharge. pump section 21 and then to the discharge. The developed assembly of the power machine, including inner cylinder 17 centered The developed assembly of the power machine, including inner cylinder 17 centered relative to working surface of cylinder 7, produces parallel cylindrical surfaces; piston 5 relative to working surface of cylinder 7, produces parallel cylindrical surfaces; piston 5 is centered with П-shaped annular protrusion 16 along them. Both external and inner is centered with Π-shaped annular protrusion 16 along them. Both external and inner working surfaces of П-shaped protrusion 16 are cylindrical surfaces of one part located working surfaces of Π-shaped protrusion 16 are cylindrical surfaces of one part located on on the same axis and can be made concentrically in one assembling on processing equip- the same axis and can be made concentrically in one assembling on processing equipment. ment. Reed valves were used as gas distribution bodies in the compressor section; their Reed valves were used as gas distribution bodies in the compressor section; their photos are shown in Figure4. photos are shown in Figure 4.

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Figure 4. Reed valves installed in the valve body of the compressor section.

The inner cylinder of the compressor section (Figure 5, item 2) is T-shaped and is shown in Figure 6. It has pipes for connecting suction and discharge lines to valve plate 8 of the compressor section; the valve plate is installed in the lower part of inner cylinder 2. Holes are made in the pump cavity in the upper part of inner cylinder 2; they are united by crescent-shaped windows for connection with the suction and discharge lines of the pump. The crosshead-free piston hybrid power machine operates as follows (see Figure 3): whenFigureFigure piston 4. 4.Reed Reed 5 valveswithvalves annular installed installed П in in-shaped the the valve valv protrusione body body of of the the 6 compressormounted compressor on section. section. it is up (from bottom dead center to top dead center), gas is compressed in working cavity of compressor section 15, and liquidTheThe inner inneris compressed cylindercylinder of ofin the thethe compressor compressorworking cavi sectionsectionty of pump (Figure(Figure section5 ,5, item item 21. 2) 2) isWhen is T-shaped T-shaped the nominal and and is is dischargeshownshown in in pressure Figure Figure6 .6. is It It in has has the pipes pipes pump for for section, connecting connecting valve suction suction12 opens, and and and discharge discharge the liquid lines lines flows to to valve fromvalve plate cavity plate 8 8 21ofof to the the the compressorcompressor consumer. section; section;With further the the valve valve upward plateplate isstisroke installedinstalled and inreachingin thethe lowerlower the partnominalpart ofof innerinner gas cylindercompres-cylinder 2.2. sionHolesHoles pressure are are made made in the in in the workingthe pump pump cavity cavity cavity of in incomp the the upper ressorupper part section part of innerof 15, inner discharge cylinder cylinder 2; valve they 2; they are9 opens, unitedare united and by gascrescent-shapedby flowscrescent-shaped to the consumer windows windows forthrough connection for connectionpipe 13. with thewith suction the suction and discharge and discharge lines of lines the pump. of the pump. The crosshead-free piston hybrid power machine operates as follows (see Figure 3): when piston 5 with annular П-shaped protrusion 6 mounted on it is up (from bottom dead center to top dead center), gas is compressed in working cavity of compressor section 15, and liquid is compressed in the working cavity of pump section 21. When the nominal discharge pressure is in the pump section, valve 12 opens, and the liquid flows from cavity 21 to the consumer. With further upward stroke and reaching the nominal gas compression pressure in the working cavity of compressor section 15, discharge valve 9 opens, and gas flows to the consumer through pipe 13.

FigureFigure 5. 5. Three-dimensionalThree-dimensional (3D) (3D) image image of of the the layout layout of of the the test test model model of of aa piston piston hybrid hybrid machine. machine. 1—piston1—piston unitunit of of the the base base compressor; 2—inner cylinder;cylinder; 3—annular3—annularΠ П-shaped-shaped protrusion protrusion of of the the pis- piston;ton; 4—outer 4—outer cylinder; cylinder; 5—pump 5—pump cavity cavity suction suctio lines;n lines; 6—pump 6—pump cavity cavity discharge discharge lines; lines; 7—compressor 7—com- pressorsection section discharge discharge lines; 8—valve lines; 8—valve body of body compressor of compressor section. section. The crosshead-free piston hybrid power machine operates as follows (see Figure3): when piston 5 with annular Π-shaped protrusion 6 mounted on it is up (from bottom dead center to top dead center), gas is compressed in working cavity of compressor section 15, and liquid is compressed in the working cavity of pump section 21. When the nominal discharge pressure is in the pump section, valve 12 opens, and the liquid ﬂows from cavity 21Figure to the 5. consumer.Three-dimensional With further (3D) image upward of the stroke layout and of reachingthe test model the nominal of a piston gas hybrid compression machine. pressure1—piston in unit the of working the base cavitycompressor; of compressor 2—inner cylinder; section 3—annular 15, discharge П-shaped valve 9protrusion opens, and of the gas ﬂowspiston; to 4—outer the consumer cylinder; through 5—pump pipe cavity 13. suction lines; 6—pump cavity discharge lines; 7—compressor section discharge lines; 8—valve body of compressor section.

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Figure 6. Inner cylinder: ( (aa)) assembled assembled with with the the valve valve body, body, ( (bb)) top top view, view, ( c) bottom view. 1—holes for for positioning positioning pins; pins; 2—fixation2—fixation holes; 3—holes into the pump section; 4—channel to the discharge discharge valve of the the compressor compressor section; section; 5—channel 5—channel to the suction valvevalve ofof thethe compressorcompressor section; section; 6—hole 6—hole for for assembling assembling the the instant instant pressure pressure sensor sensor in thein the compressor compressor section. section. Thus, when the piston moves upward, liquid and gas are compressed and move to theThus, discharge; when the it should piston bemoves noted upward, that the liquid discharge and gas valve are of compressed the pump section and move opens to theearlier discharge; than the it dischargeshould be valve noted of that the the compressor discharge section valve andof the the pump liquid section is supplied opens to ear- the lierdischarge than the from discharge the pump valve section of the at compressor a larger crank section angle. and When thethe liquid piston is supplied reaches the to topthe dischargedead center, from the the discharge pump section valve from at a thelarger pump crank section angle. and When compressor the piston section reaches closes, the andtop deadthe flow center, of liquid the discharge and gas tovalve the dischargefrom the pump stopped. section When and the compressor piston moves section down closes, (from andtop deadthe flow center of toliquid bottom and dead gas to center), the disc thereharge is an stopped. expansion When process the piston and a pressuremoves down drop (fromto the top suction dead pressure center to inbottom the pump dead and center), compressor there is an section. expansion Then, process with a and further a pressure piston dropstroke, to thethe suctionsuction valvespressure open in (11—inthe pump the and pump compressor section, 8—in section. the Then, compressor with a section), further pistonand working stroke, chambersthe suction 21 andvalves 15 areopen filled (11— within the liquid pump and section, gas. 8—in the compressor section),The and process working of back chambers expansion 21 and is 15 much are filled shorter with in liquid the pump and gas. section than in the compressorThe process section, of back and suctionexpansion valve is much 11 opens shorter in the in pump the pump section section before than suction in the valve com- 8 pressorof the compressor section, and section suction opens. valve Then, 11 opens the cyclein the repeats. pump section before suction valve 8 of the compressorThe certain section advantages opens. of Then, this diagram the cycle are repeats. enhanced cooling of the cylinder–piston groupThe and certain compressed advantages gas, asof thethis pump diagram section are enhanced is located aroundcooling theof the compressor cylinder–piston section groupand there and arecompressed pipes operating gas, as the as heatpump exchangers section is located in the valve around plate. the Thiscompressor design section should andreduce there the are temperature pipes operating of suction as heat gas, improveexchangers the in compressed the valve gasplate. cooling, This design and have should high reduceeconomic the efficiency temperature as well of suction as specific gas, weight improve and the size compressed indicators. gas cooling, and have high economicCurrently, efficiency when simulating as well as the specific working weight processes and size of compressors indicators. and displacement pumps,Currently, mathematical when simulating models with the working lumped andprocesses distributed of compressors parameters and are displacement used [6–8]. pumps,Unlike vanemathematical machines models [9,10], modelswith lumped with lumped and distributed parameters parameters are most widelyare used used [6–8]. in displacement machines. This is due to following reasons: Unlike vane machines [9,10], models with lumped parameters are most widely used in displacement- Working machines. chamber volumeThis is due and to boundaries following reasons: change, the computational grid is to be rebuilt constantly, and it complicates the implementation of existing software for − Working chamber volume and boundaries change, the computational grid is to be non-steady flow of viscous compressible liquid as for laminar and turbulent flows; rebuilt constantly, and it complicates the implementation of existing software for - Linear dimensions of positive displacement machines are rather small, so there is non-steady flow of viscous compressible liquid as for laminar and turbulent flows; no need to use models with distributed parameters, as when there are small linear − Linear dimensions of positive displacement machines are rather small, so there is no dimensions, uneven distribution of thermodynamic parameters is very small in the need to use models with distributed parameters, as when there are small linear di- working chamber. This is true for displacement pumps, as the speed of sound (speed mensions, uneven distribution of thermodynamic parameters is very small in the of propagation of simple waves of compression and expansion) is greater than in gas; working chamber. This is true for displacement pumps, as the speed of sound (speed - The existing patterns of distribution of thermodynamic parameters in the working ofchambers propagation of compressors of simple waves and displacement of compression pumps and accordingexpansion) to is the greater angle than of rotation in gas; − Theare well-known,existing patterns and researchersof distribution are oftenof thermodynamic interested in averaged parameters parameters in the working at each chamberspoint of the of compressors process under and study. displacement Therefore, pumps when according using models to the withangle distributed of rotation areparameters, well-known, averaging and researchers at each angle are ofoften rotation interested is to bein carriedaveraged out, parameters and the results at each of pointaveraged of the thermodynamic process under parametersstudy. Therefore, will be when close using to the models thermodynamic with distributed parameters parameters,obtained whenaveraging using at models each angle with of lumped rotation parameters; is to be carried out, and the results of averaged thermodynamic parameters will be close to the thermodynamic parameters obtained when using models with lumped parameters;

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− Costs of material and physical resources necessary for the development and implementation of models with distributed parameters are higher than for models with lumped parameters; Machines 2021, 9, 32 − The time to implement models with distributed parameters is higher than the 6time of 38 to implement models with lumped parameters; the difference in the results is nominal [6,7]. - Thus,Costs the of materialaim of this and article physical is a comprehe resourcesnsive necessary numerical for the and development experimental and study imple- of a crosshead-freementation ofpiston models hybrid with power distributed machine parameters to identify are the higher main physical than for laws models in work with processeslumped and parameters;to establish the influence of the main operational parameters on the work processes- The timeand tointegral implement characteristics models with of distributedthe machine parameters for their further is higher use than in thethe timedevelop- to im- ment plementand operation models of with the machine. lumped parameters; the difference in the results is nominal [6,7]. TheThus, results the aim obtained of this on article the cooling is a comprehensive of the cylinder–piston numerical group and experimental and the work study pro- cessesof a crosshead-free of the machine piston under hybrid study are power to be machine compared to with identify the results the main of previous physical studies laws in forwork crosshead processes PHPM. and to The establish article thewill inﬂuence make a si ofgnificant the main contribution operational to parameters the study onof the the workwork processes processes of and PHPM integral and characteristicswill be useful for of thereaders machine engaged for theirin the further research use and in de- the signdevelopment of reciprocating and operation compressors, of the pumps, machine. and hybrid power machines. The results obtained on the cooling of the cylinder–piston group and the work pro- 2.cesses Mathematical of the machine Simulating under studyof Work are Processes to be compared of the withMachine the results of previous studies for crossheadConsidering PHPM. the above The articleaspects, will it seems make appropriate signiﬁcant to contribution use a model to with the studylumped of pa- the rameterswork processes when developing of PHPM and mathematical will be useful mode for readersls of a crosshead-free engaged in the PHPM. research The and scheme design ofof the reciprocating crosshead-free compressors, PHPM is pumps,shown in and Figure hybrid 7. power machines. When developing a mathematical model of work processes, we distinguish three 2. Mathematical Simulating of Work Processes of the Machine main stages: the mathematical model of the compressor section (see Figure 7, item 3); the mathematicalConsidering model the of above the pump aspects, section it seems (see appropriateFigure 7, item to use1); and a model the mathematical with lumped modelparameters of a piston when seal developing (see Figure mathematical 7, item 2) connecting models of athe crosshead-free pump and compressor PHPM. The sections. scheme Theof the allocated crosshead-free working PHPM volumes is shown in figures in Figure are shown7. with dashed lines.

FigureFigure 7. 7. ConstructionConstruction diagram diagram of of a a crosshead-free crosshead-free PH PHPMPM for for developing developing a a mathematical mathematical model model of ofwork work processes. processes. (1—displacement (1—displacement of of the the pump pump section; section; 2—gap2—gap clearanceclearance betweenbetween the pump and andcompressor compressor sections; sections; 3—displacement 3—displacement of the of compressorthe compressor section). section).

When developing a mathematical model of work processes, we distinguish three main stages: the mathematical model of the compressor section (see Figure7, item 3); the mathematical model of the pump section (see Figure7, item 1); and the mathematical model of a piston seal (see Figure7, item 2) connecting the pump and compressor sections. The allocated working volumes in ﬁgures are shown with dashed lines. Machines 2021, 9, 32 7 of 38

2.1. Mathematical Simulation of Work Processes of Compressor Section Based on the analysis of existing assumptions accepted when constructing the mathematical model of displacement compressors with lumped parameters, we accept the following system of assumptions without additional justification [11]: - Simulated processes are reversible and equilibrium. - Working liquid is continuous. - Liquid entering the working chamber is localized as liquid layer on the piston surface. - Compressible gas is clean air. - Changing in gas potential and kinetic energy is negligible. - Working liquid is perfect gas. - Heat exchange of the working liquid with the walls of the working chambers of the compressor section is carried out only by convection and is described by the Newton–Richmann hypothesis. When simulating mass flows through valves and leaks, there are assumptions typical for mathematical models with lumped parameters [12,13]: gas flow is one-dimensional, isotropic; dependencies for steady flow are used to describe the flow; flow coefficients obtained with stationary purges are used, etc. Operation of a piston hybrid power machine can be carried out in the following main modes: - PHPM operating mode with priority liquid flow from the pump section to the compressor (discharge pressure in the pump section is equal or higher than the discharge pressure in the compressor section); - PHPM operating mode with priority gas flow from the compressor section to the pump section (discharge pressure in the compressor section exceeds the discharge pressure in the pump section); - PHPM operating mode with equality of liquid mass per cycle from the pump section to the compressor section and back (the discharge pressure in the compressor section and the pump section are comparable). The first and third modes of PHPM operation are the most effective. Implementation of the third mode is very difficult, so the first two modes of operation are the most practical. We consider a mathematical model of work processes in the compressor section with priority liquid flow from the pump section to the compressor. The following basic fundamental laws are the basis of the mathematical model of the compressor section work processes: - Energy conservation equation as the first law of thermodynamics of a variable mass body; - Mass conservation equation; - Equation of motion; - Equation of state. The system of differential equations in full derivatives describing changes in thermodynamic parameters of gas in the compressor section is written as:

 N1 N2  dU = dQ − p dV − dM16−dM17 + i dM − i dM  kin ρw ∑ ni ni ∑ 0 0i  i=1 i=1   N1 N2  dM = ∑ dMni − ∑ dM0i = dM3 − dM4 + dM5 + dM10 − dM9 i=1 i=1 (1)  p =(k − 1)U/V   T =pV/MR  2  m d hc = F = F + F + F ± m g  spr dτ2 ∑ i g spr f r spr i=1 Machines 2021, 9, 32 8 of 38

where dU = d(Cv MT); dVkin = υpFpdτ is elementary change in the control volume 2 Sh λt πd1 due to the kinematics of the drive mechanism; υp = 2 ω sin φ + 2 sin 2φ ; Fp = 4 ; ini = CpiTni. There is heat exchange between the compressed gas and the surface of the walls of the working chamber in the working chamber of the compressor section, as it is between the gas and liquid layer above the piston. Then, an elementary amount dQ, diverted from the compressed gas is determined as dQ = αT(Fcl + Fcov) Tavt − T dτ + αT Fp Tw − T dτ (2) h i Sh λt where Fcl = πd(S − δ); S = Sd + 2 (1 − cos φ) + 4 (1 − cos 2φ) . The distribution of the temperature field over the surface of the cylinder liner and the surface of the valve plate cover is complex and not stationary over time. The performed studies prove that temperature fluctuations of the cylinder bore and surface of the valve plate are less than 1 K for single-acting compressors, so we assume that the temperature distribution field is stationary. In most cases, the temperature distribution fields are determined experimentally [14]. The averaged integral temperature Tavt is determined as

F F Rcov Rcl Tcov(F)dF + Tcl(F)dF 0 0 Tavt = . (3) Fcl + Fcov The value of the liquid temperature above the piston, as well as the surface of the walls of the working chamber, has an uneven distribution over the surface and in time. However, this unevenness is very small and can be neglected. The value of the liquid temperature can be considered constant and equal to the temperature of the liquid in the pump section. The heat transfer coefficient is a variable at each point on the surface of the working chamber. While calculating the averaged over the surface heat-transfer coefficient that has been adopted, its value is determined experimentally based on [6,15,16]. While determining the mass flows through the valves, the Bernoulli equation is used with the introduction of correction factors to consider the compressibility of gases. Movement of the blocking element is considered as unsteady movement of a material point with effective mass. Movement of the blocking element is plane-parallel, perpendicular to the seating under the influence of differential pressure and spring [17]. When gas is completely displaced from the working chamber of the compressor section, liquid is displaced from the working chamber, and the compressor section works in the pumping mode. Liquid from the working chamber of the compressor section enters the suction line through leaks in the suction valve dM4w, through the discharge valve to the discharge line of the compressor section dM5w, through leaks of the piston seal and into the working chamber of the pump section dM17. For calculations, we assume that the liquid temperature in the working chamber of the compressor section remains constant (Tw = const). Figure7 shows the mass flows of gas and liquid for compressor and pump sections through the main elements of the machine (pressure and suction valves, piston seal) in forward and reverse directions. If forward flow is determined, e.g., mass flow dM9, then dM10 is assumed to be zero and vice versa, if dM10 is determined, then dM9 is assumed to be zero. The direction of gas and liquid flow was determined by the pressure gradient. In this case, only deformation and mass transfer interaction will be observed in the working chamber and in accordance with [18] pw, pressure is determined as Vw0 Mw pw = pwd + Ew `n + `n . (4) Vw Mw0 Machines 2021, 9, 32 9 of 38

The system of equations describing change in the thermodynamic parameters of liquid is written as h i  Vh λt  Vw = 2 (1 − cos φ) + 4 (1 − cos 2φ) + Vdc   Vw0 Mw  pw =pwd + Ew `n V + `n M  w w0  N4 N5 dMw = ∑ dMwni − ∑ dMw0i = dM6w − dM4w − dM16. (5)  i=1 i=1  T =const  w  2 N6  m d hc = F  spr dτ2 ∑ wi i=1

N6 It should be noted that acting forces on the blocking element ∑ Fwi when injecting i=1 liquid are the same as when injecting gas. However, their deﬁnition differs and will be considered later when there is simulating of work processes in the pump section. Elementary masses of liquid passing through the valves can be determined as q dMw0 = αgrw fgr ρw(pw − pi)dτ. (6)

2.2. Mathematical Model of the Work Processes of the Pump Section The main assumptions while developing a mathematical model of the pump section work processes are written as follows: - The working liquid of the pump section is viscous compressible dropping liquid that follows the laws of Newton and Hooke. - Mass transfer processes at the gas–liquid interface can be neglected. - The interface between liquid and gas is a plane parallel to the earth. - Coefficients of local resistances and the coefficient of friction along the length obtained in stationary modes of liquid flow can be used in unsteady flow. Displacement pump cycle consists of four processes: compression, discharge, reverse expansion, and suction. According to physical phenomena, four processes can be divided into two groups. The first group includes compression and expansion. In these processes, an increase (decrease) in pressure occurs due to deformation interaction. The second group includes processes of discharge and suction. Mass transfer has a decisive effect. It should be noted that all four processes are connected with thermal interaction, which is insignificant and neglected. We made a sequential review of the calculations of the above work processes. Compression and reverse expansion processes In the processes of compression and reverse expansion, the processes of deformation (reduction in the working chamber volume due to piston movement), mass transfer (leaks and leakage of liquid), and heat transfer (convective heat transfer with surface of the walls of the working chamber) interactions take place when there is no liquid movement, i.e., kinetic energy. The main fundamental work on the calculation of compression and reverse expansion processes is [19]. Recently published work [18] continues [19] and determines the change in pressure in the working chamber as if there is separately deformation, mass transfer, and thermal interactions, as in their combination. Machines 2021, 9, 32 10 of 38

Due to the fact that change in liquid temperature is very small in the processes of compression and reverse expansion, the system of equations written in [19] can be converted as:  h i V =V + Vh (1 − cos φ) + λt (1 − cos 2φ)  w dw 2 4   N3 N4  dMw = ∑ dM − ∑ dM niw 0iw (7) i=1 i=1  Vscw Mw  pw =pscw + Ew `n + `n  Vw Mscw  Tw =const

2− 2 π 2 2 Shπ(d2 d1) where Vscw = Vdw + Vhp; Vhp = 4 d2 − d1 S; Mscw = Vscw · ρw, Vh = 4 . The current liquid mass change is determined as

dMw = −dM12 + dM14 − dM17. (8)

Mathematical model of discharge and suction Deformation, mass transfer, and thermal interactions are followed with conversion of the flow kinetic energy into pressure potential energy and back in the processes of discharge and suction. A method was developed in [20] for calculating the processes of discharge and suction based on the first law of thermodynamics for an open moving thermodynamic system. The calculation is carried out in two stages. At first, using the Bernoulli equation, we determine change in liquid pressure in the working chamber of the pump as a result of conversion of potential energy into kinetic and pressure loss (pressure process). At the second stage using the first law of thermodynamics, we determine change in temperature and pressure of the working liquid due to deformation, heat, and mass transfer processes; then, we determine the change in pressure, temperature, and mass of the working liquid. Considering that the external heat transfer is insignificant, the compressibility of the droplet liquid is insignificant, and leakages and overflows of the working liquid are small compared to the main flow; thus, the Bernoulli equation supplemented by the equation of dynamics of a self-acting valve in a single-mass formulation [21] should be based on the calculation

 2 2 w2−w1  pw =pwd + ρwg(SDw + Sw) cos α + ρw 2 + ∆hT1−2 + ∆h1−2 2 N6 . (9) m d hw = F  drw dτ2 ∑ iw i=1

The pressure loss due to inner friction is the sum of pressure loss along the length and the pressure loss to overcome local resistance.

∆h1−2 = ∆h` + ∆hζ (10)

Values ∆h` and ∆hζ are determined according to Darcy–Weisbach law

2 (SDw + Sw) w1 ∆h` = λ f r (11) d1 2g

w2 ∆h = 1 (12) ζ 2g

where λ f r is length friction coefficient determined using [22,23]. As first approximation, we can assume that the main pressure losses occur during a sudden flow expansion in the valve, which can be determined as

2 ζ = 1 − fgr/ f2 (13) Machines 2021, 9, 32 11 of 38

2 πdp where fgr = πdvl hw; f2 = 4 .

2.3. Mathematical Model of Liquid Flow in a Groove Seal The compressor and pump sections of the PHPM are interconnected by a groove seal, where there is always liquid under the PHPM investigated operation mode, and gas breakthrough from the pump section into the compressor section is impossible. In general, the flow of viscous liquid in a groove seal is unsteady and not axisymmetric, because pressure in pump and compressor sections varies in angle of rotation, and the piston has an eccentricity. The dynamics of liquid movement in a groove seal is as follows: (1) When the piston moves at an angle of rotation from π ≤ φ ≤ 2π, compression processes are in the pump and compressor cavities. Initially, as the pressure in the pump cavity increases much faster than in the compressor, liquid from the pump section flows through the groove seal into the compressor and forms a liquid layer above the piston. (2) If the pressure is equal to pdp = pc, liquid movement into the compressor cavity ceases and then, when the pressure in the compressor cavity pc exceeds discharge pressure pdp in the pump, liquid flows from the compressor cavity to the pump. (3) When the piston moves from 0 < ϕ < π, there is suction in the compressor and pump cavities. If suction pressure in the compressor and pump section is equal, then there is no liquid movement in the groove seal. There is liquid movement from the compressor cavity to the pump as the pressure loss in the pump section during suction is more than that in the compressor. In general, the liquid flow in a groove seal is described with a system of equations that includes the continuity equation, the equation of motion of viscous liquid, the energy conservation equation, and the equation of state. Considering the above approach to simulate PHPM work processes, we consider liquid flow in a groove seal as laminar and quasistationary with constant liquid properties and groove geometry. At present, various types of groove seals are used in PHPM: smooth, step-like, and profiled. The analysis of the PHPM groove seals operation proves that a smooth groove seal will be the most effective for a crosshead-free PHPM. We carry out the calculation for two principal cases: the concentric arrangement of the piston with and without friction movement and the eccentric arrangement of the piston without friction [24,25]. The task of calculating the piston seal is to determine the flow coefficient through the groove seal. With concentric arrangement of the piston, the flow coefficient is determined as

3 πd1δ1(pc − pw) 1 Q = ± υpπd1δ1. (14) 12µ`u 2

With an eccentric arrangement of the piston the ﬂow coefﬁcient is determined as

3 πd1δ1 2 Q = (pc − pw) 1 + 3/2ε . (15) 12µ`u

It should be noted that the length of the piston seal in the compression and discharge processes increases, and the instantaneous flow coefficient decreases. The current piston seal length ù is determined as

S λ ` = ` + S − h (1 − cos φ) + t (1 − cos 2φ) . (16) u u0 h 2 4 Machines 2021, 9, 32 12 of 38

3. Experimental Study The main aim of the experimental research is to obtain new knowledge about the object under study and to verify the developed mathematical model. That is why an experimental model of a crosshead-free PHPM was designed.

3.1. Object, Stand for Its Research, Measuring Equipment The developed experimental model of crosshead-free PHPM has the following main design and operational parameters: - Inner diameter of the piston is 55 mm; - The outer diameter of the piston is 65 mm; - Connecting rod length is 81 mm; - Piston stroke is 47 mm; - Total piston length is 119 mm; - Discharge pressure in the compressor section is up to 10 bar; - Discharge pressure in the pump section is up to 20 bar; - Crank angular velocity is from 250 to 750 min−1; - Suction pressure in the pump section is 1 bar; - Suction pressure in the compressor section is 1 bar. When designing a crosshead-free PHPM, the Solid Works software was used. Air was used as working liquid for the compressor section, and MGE-46V hydrolic oil (speciﬁcations are shown in Table1) was used for the pump section.

Table 1. Rosneft MGE-46V hydraulic oil speciﬁcations.

Indicator GOST Standart (TU) Kinematic viscosity, mm2/s: at 100 ◦C, not less than 6.0 at 50 ◦C- at 40 ◦C 41.4–50.6 at 0 ◦C, not more than 1000 Viscosity index, not less 90 Temperature, ◦C: ﬂashes in an open crucible, not lower 190 solidiﬁcation, not more −32 Acid number, mg KOH/g 0.7–1.5 Mass fraction: mechanical impurities, no more absence water absence Metal Corrosion Test pass density at 20 ◦C, kg/m3, no more 890 Oxidation stability: sediment, %, no more 0.05 change in acid number, mg KOH/g oil, not more than 0.15 Tribological characteristics at FBM: wear indicator at axial load 196 N, mm, no more 0.45

The prototype was tested at the developed stand to perform the following functions: - To change smoothly the speed of the drive shaft while ﬁxing the frequency of the piston reciprocating movement; - Measurement and maintenance of stationary parameters on the lines of liquid and gas suction and discharge; - Measurement of instantaneous values of pressures in working chambers with their indicator diagrams in digital and graphic forms; - Measuring the productivity of gas and liquid working chambers; - Measurement of leaks of working liquid; - High ergonomics to inﬂuence the main initial parameters; Machines 2021, 9, x FOR PEER REVIEW 13 of 39

− To change smoothly the speed of the drive shaft while fixing the frequency of the piston reciprocating movement; − Measurement and maintenance of stationary parameters on the lines of liquid and gas suction and discharge; − Measurement of instantaneous values of pressures in working chambers with their Machines 2021, 9, 32 indicator diagrams in digital and graphic forms; 13 of 38 − Measuring the productivity of gas and liquid working chambers; − Measurement of leaks of working liquid; − High ergonomics to influence the main initial parameters; -− MeasuringMeasuring the the temperature temperature of of gas gas and and liquid liquid at suctionat suction and and discharge, discharge, the temperaturethe tempera- ofture the of inner the inner surface surface of the of valve the valv platee plate of the of compressor the compressor section. section. A hydropneumatic diagram of the stand is shown in Figure8. A hydropneumatic diagram of the stand is shown in Figure 8.

FigureFigure 8.8. HydroHydro pneumatic pneumatic diagram diagram of ofthe the test test bench bench to study to study a crosshead-free a crosshead-free PHPM. PHPM. Legend: Leg- end:1—fluid1—fluid controlcontrol valve; valve; 2, 46—tank; 2, 46—tank; 3—electromotor 3—electromotor;; 4, 54—pump; 4, 54—pump; 5, 34, 45, 47—oil 5, 34, 45,filter; 47—oil 6, 20, fil- 29—safety valve; 7, 15, 26, 35, 41, 55—pressure sensor; 8—air filter; 9, 24, 40, 51—ratemeter; 10, 14, ter; 6, 20, 29—safety valve; 7, 15, 26, 35, 41, 55—pressure sensor; 8—air filter; 9, 24, 40, 16, 31, 37, 44, 52, 56—temperature sensor; 11, 23, 28—manometer; 12—rotation sensor; 13—com- 51—ratemeter; 10, 14, 16, 31, 37, 44, 52, 56—temperature sensor; 11, 23, 28—manometer; 12—rotation pressor; 17—oil-and-moisture separator; 18, 21, 25—globe valve; 19—receiver; 22—pressure sensor;switch gauge;13—compressor 27—fluid-power; 17—oil-and-moisture motor; 30—heat separator;exchanger; 18,32—throttle; 21, 25—globe 33, 39, valve; 42, 43, 19—receiver; 50—ball 22—pressurevalve; 36—hydropneumaticswitch gauge; 27—fluid-poweraccumulator; 38—mea motor;suring 30—heat container; exchanger; 48—fluid 32—throttle; level gauge; 33, 49— 39, 42, 43,filler 50—ball neck strainer; valve; 36—hydropneumaticSV1-SV3—safety valve; accumulator; GV1-GV3—globe 38—measuring valve; PSG1—pressure container; 48—fluid switch gauge; level gauge;RC1—receiver;49—filler MN1-MN3—manometer;neck strainer; SV1-SV3—safety RM1-RM4—ra valve; GV1-GV3—globetemeter; OS1—oil-and-moisture valve; PSG1—pressure separator; switchTS1-TS8—temperature gauge; RC1—receiver sensor;; PS1-PS7—pressure MN1-MN3—manometer; sensor; CM1—compressor; RM1-RM4—ratemeter; AF1—airOS1—oil filter;-and- F1- moistureF4—oil filter; separator; FN1—filler TS1-TS8—temperature neck strainer; T1, T2 sensor;—tank; PS1-PS7—pressure P1, P2—pump; RS1—rotation sensor; CM1—compressor sensor; FM1—; AF1—airfluid-power filter; motor;F1-F4—oil FV1—fluidfilter; control FN1—filler valve; TT1—throttle; neck strainer; HE1—heat T1, T2—tank; exchanger; P1, BV1-BV5—ball P2—pump; valve; AC1—hydropneumatic accumulator; FL1—fluid level gauge; MC1—measuring container. RS1—rotation sensor; FM1—fluid-power motor; FV1—fluid control valve; TT1—throttle; HE1—heat exchanger; BV1-BV5—ball valve; AC1—hydropneumatic accumulator; FL1—fluid level gauge; To change the speed of rotation of the drive shaft, a displacement hydraulic drive is MC1—measuring container. used, which includes a displacement axial piston pump model 313.3.56.804 with a dis- chargeTo changepressure the 6.3 speed MPa and of rotation an axial of piston the drive hydraulic shaft, amotor displacement 310.3.56.01.03.V.U. hydraulic model. drive isStatic used, pressures which includes are measured a displacement with pressure axial gauges; piston instant pump pressures model 313.3.56.804 are measured with with a dischargetensometric pressure sensors. 6.3 MPa and an axial piston hydraulic motor 310.3.56.01.03.V.U. model. StaticTo pressures reduce the are unevenness measured with of the pressure pump gauges;cavity supply, instant a pressures pneumatic are accumulator measured with with tensometrica Danfoss MBS sensors. 3000 (Danfoss, Chelyabinsk, Russia) pressure sensor is installed on the dischargeTo reduce line. the The unevenness same sensor of theis used pump to cavity measure supply, the liquid a pneumatic suction accumulator pressure. Liquid with aflow Danfoss rates MBSare measured 3000 (Danfoss, with TPR Chelyabinsk, 20-8 (“Arzamas Russia) instrument-making pressure sensor is plant installed named on after the dischargeP.I. Plandin”, line. Chelyabinsk, The same sensor Russia) is used flow tome measureters, suction theliquid and discharge suction pressure. temperatures Liquid are flow rates are measured with TPR 20-8 (“Arzamas instrument-making plant named after P.I. Plandin”, Chelyabinsk, Russia) flow meters, suction and discharge temperatures are measured with TW N PT100 (Autonics, Chelyabinsk, Russia) sensors, and the liquid temperature in tanks is controlled with DTS 035 50M.V3 (OVEN, Chelyabinsk, Russia) sensors. Leakage is measured with volumetric method. Air flow is measured with SMCPF2A751-F04-67N-M (SMC Corporation, Lukhovitsy, Russia), Vector-04 (NPP SKYMETER, Chelyabinsk, Russia), and SGV-15 Betar model flowmeters (OOO PKF “BETAR”, Chelyabinsk, Russia) at suction and discharge of the Machines 2021, 9, x FOR PEER REVIEW 14 of 39

measured with TW N PT100 (Autonics, Chelyabinsk, Russia) sensors, and the liquid temperature in tanks is controlled with DTS 035 50M.V3 (OVEN, Chelyabinsk, Russia) sen- Machines 2021, 9, 32 sors. Leakage is measured with volumetric method. 14 of 38 Air flow is measured with SMCPF2A751-F04-67N-M (SMC Corporation, Lukhovitsy, Russia), Vector-04 (NPP SKYMETER, Chelyabinsk, Russia), and SGV-15 Betar model flowmeters (OOO PKF "BETAR", Chelyabinsk, Russia) at suction and discharge of the compressor cavity; we used PSE530 M5 l l sensors sensors (SMC (SMC Corporation, Corporation, Lukhovitsy, Lukhovitsy, Russia) forfor measuring measuring suction suction and and discharge discharge pressures. pressures. An An AD22100 AD22100 STZ STZ sensor sensor (Analog (Analog Devices, Devices, Chelyabinsk, Russia) is used to measure the stationary temperature of the valve plate and the upper part of the cylinder. the upper part of the cylinder. The hydraulic and pneumatic lines of the stand are equipped with liquid and gas The hydraulic and pneumatic lines of the stand are equipped with liquid and gas puriﬁcation devices, safety devices, and controllers to organize the work of the tested purification devices, safety devices, and controllers to organize the work of the tested ma- machine in different modes. A general view of the tested machine installed on the stand is chine in different modes. A general view of the tested machine installed on the stand is shown in Figure9. shown in Figure 9.

Figure 9. PHPM of crosshead-free and test bench in the installation area of crosshead-free PHPM. Figure 9. PHPM of crosshead-free and test bench in the installation area of crosshead-free PHPM.

3.2. Verification Verification of the Mathematical Model of Work Processes of Crosshead-Free PHPMPHPM VerificationVerification of the developed mathematical model is based on the qualitative and quantitative matchingmatching ofof the the results results of of numerical numerical simulating simulating and and experimental experimental studies; studies; i.e., i.e.,it is it to is verify to verify that that the mathematicalthe mathematical model model responds responds to changes to changes in operational in operational parameters param- etersand howand ithow describes it describes the work the work processes processes and integraland integral parameters parameters of the of compressor the compressor and andpump pump sections. sections. VerificationVerification of the developed mathematical model was confirmedconfirmed over the range of dp dc rev changes in the independent parameters: pdp, ,ppdc, ,and and nnrev. . We compared compared the the indicator indicator diagrams diagrams in the in thecompressor compressor and andpump pump cavities, cavities, the main the integralmain integral characteristics: characteristics: compressor compressor sectionsection feed coefficient, feed coefficient, volumetric volumetric efficiency efficiency for the pumpfor the section, pump and section, the amount and the of amount liquid carried of liquid into carried the discharge into the line discharge of the compressor line of the section.compressor section. Figures 10–1310–13 compare indicator diagrams for the compressor and pump sections at different nominal discharge pressures. A comparative analysis of the indicator diagrams in the compressor section proves: - Discrepancy in determining instant pressure is insignificant in compression, expansion, and suction, and it is less than 5%. - Maximum discrepancy from 10% to 15% is observed during discharge, due to significant inertial forces acting on the liquid when the valve is opened, leading to a significant pressure jump in the experimental data. We compared indicator diagrams in the pump section and made conclusions: - Compression and suction are described with an error 5% max. The maximum discrepancy in determining instant pressure is observed during the

discharge, at the beginning and end of the process when the discharge valve is opened Machines 2021, 9, 32 15 of 38

and closed. The maximum discrepancy from 10% to 15% is because inertial forces of liquid displacement are not considered while calculating the discharge process. It should be noted that the test bench has long discharge and suction pipelines; investigated PHPM has a significant irregularity in the liquid supply—π. According to [24], MachinesMachines 2021 2021, ,9 9, ,x x FOR FOR PEER PEER REVIEW REVIEW 1515 of of 39 39 it leads to significant pressure fluctuation in pipelines (primarily on the discharge line), despite the installed gas cap (see Figures8 and9).

FigureFigure 10. 10. Dependence Dependence of of the the instantaneous instantaneousinstantaneous pressure pressure in in the the compressor compressor sectionsection section inin in thethe the crosshead-freecrosshead- crosshead- freefreePHPM PHPM PHPM determined determined determined experimentally experimentally experimentally and and byand calculationby by calculation calculation (1—Calculation; (1—Calculation; (1—Calculation; 2—Experiment). 2—Experiment). 2—Experiment).

FigureFigure 11. 11.11. Dependence Dependence of of the the instantaneous instantaneous pressure pressure in in the the pump pump section section in in a a crosshead-free crosshead-free PHPM PHPM determined determined experi- experi- mentallymentally and and by byby calculation calculationcalculation (1—Calculation; (1—Calculation;(1—Calculation; 2—Experiment). 2—Experiment).2—Experiment).

Machines 20212021,, 99,, 32x FOR PEER REVIEW 16 of 3839

Figure 12.12. The dependence of the instantaneous pressure in the compressorcompressor sectionsection in crosshead-freecrosshead-free PHPMPHPM determineddetermined experimentally andand byby calculationcalculation (1—Calculation;(1—Calculation; 2—Experiment).2—Experiment).

Figure 13.13. Dependence of the instantaneous pressure in the pump section in a crosshead-free PHPM determined experimentally and by calculation (1—Calculation;(1—Calculation; 2—Experiment).2—Experiment).

ATable comparative2 presents analysis the results of ofthe numerical indicator and diagrams experimental in the compressor studies. The section ﬂow coefﬁcient proves: of the compressor section was determined as − Discrepancy in determining instant pressure is insignificant in compression, expansion, and suction, and it is less thanH dM5%. − H dM = 6 5 − Maximum discrepancy fromλ 10% to 15% is observed. during discharge, due to signif- (17) ρscVhc icant inertial forces acting on the liquid when the valve is opened, leading to a significant pressure jump in the experimental data. We compared indicator diagrams in the pump section and made conclusions:

Machines 2021, 9, 32 17 of 38

Table 2. The results of numerical and experimental studies of crosshead-free PHPM.

∆V ∆V p , p , n , λ λ η η w w № dc dp rev vol vol ¯ mL/min mL/min bar bar rpm num. exp. num. exp. Tavt,K num. exp. 1 6 3 300 0.737 0.687 0.7475 0.86 320 0 0.7 2 6 5 300 0.738 0.7467 0.742 0.82 320 0 11 3 6 7 300 0.739 0.7168 0.737 0.774 320 0 14 4 6 9 300 0.739 0.7168 0.733 0.7 320 0 20 5 4 3 300 0.84 0.836 0.747 0.674 310 0 8 6 5 3 300 0.739 0.8064 0.747 0.686 310 0 5 7 6 3 300 0.737 0.7467 0.747 0.704 320 0 3 8 7 3 300 0.6934 0.7168 0.747 0.716 320 0 1 9 8 3 300 0.644 0.6868 0.747 0.728 325 0 0.5

The volumetric efﬁciency of the pump section was determined as H H dM13 − dM14 ηvol = . (18) ρwVhw

Table2 presents ∆Vw as the amount of liquid carried out from the compressor section to the discharge line. The distribution bodies in the compressor and pump sections in the experimental prototype had the following parameters: maximum lift height of the shut-off element of the suction valve in the pump section hscpmax = 0.0016 m; minimum lift −6 height of the shut-off element of the suction valve in the pump section hscpmin = 5×10 m; spring stiffness of the suction valve in the pump section csprscp = 716 N/m; mass of the shut-off element of the suction valve in the pump section mshscp = 9 gr; passage width in the seating of the suction valve of the pump section dvlscp = 0.03 m; maximum lift height of the shut-off element of the suction valve in the compressor section hsccmax = 0.00175 m; minimum lift height (conditional clearance) of the shut-off element of the suction valve in −6 the compressor section hsccmin = 3.5×10 m; passage width in the seating of the suction valve of the compressor section dvlscc = 0.01775 m; spring stiffness of the suction valve in the compressor section csprscc = 2052 N/m; mass of the shut-off element of the suction valve of the compressor section mshscc = 0.624 gr. The parameters for the discharge valve of the compressor section are similar: −6 hdcmax = 0.00175 m; hdcmin = 3.5×10 m; dvldc = 0.01775 m; csprdc = 2052 N/m; mshdvl = 0.624 gr. The parameters for discharge valve of the pump section are similar as well: hdpmax = 0.0016 m; −6 hdpmin = 5×10 m; dvldp = 0.03 m; csprdc = 338 N/m; mshdp = 23 gr. Figures 14 and 15 show changes in the compressor feed coefficient obtained experimentally and numerically with change in the discharge pressure in the compressor section with a fixed pdc and vice versa. According to the theory of the work processes of reciprocating compressors, an increase in the discharge pressure in the compressor section decreases the productivity of the reciprocating compressor due to a decrease in the volumetric ratio and leakage ratio (see Figure 14). Increasing the discharge pressure in the pump section increases the amount of liquid coming to the compressor section. If not all the dead space in the compressor section is filled with liquid (see curve 2 in Figure 15), then there is an increase in the delivery coefficient. If the dead space is completely filled with liquid, the delivery coefficient does not change (see curve 1 in Figure 15). Incomplete filling of the dead space with liquid in a real machine is due to partial removal of the liquid in the form of drops by compressed gas, which was not considered when developing a mathematical model. Machines 2021, 9, x FOR PEER REVIEW 18 of 39

with a fixed pdc and vice versa. According to the theory of the work processes of reciprocating compressors, an increase in the discharge pressure in the compressor section decreases the productivity of the reciprocating compressor due to a decrease in the volumetric ratio and leakage ratio (see Figure 14). Increasing the discharge pressure in the pump section increases the amount of liquid coming to the compressor section. If not all the dead space in the compressor section is filled with liquid (see curve 2 in Figure 15), then there is an increase in the delivery coefficient. If the dead space is completely filled with liquid, the delivery coefficient does not change (see curve 1 in Figure 15). Incomplete filling of the dead space with liquid in a real machine is due to partial removal of the liquid in the form of drops by compressed gas, which was not considered when developing a mathematical model. It should be noted that increasing the discharge pressure in the pump section, we improve cooling of the suction gas and decrease its leaks and overflows. This leads to an increase in the flow ratio and the capacity of the compressor section. The presented results prove that we observe a good qualitative and quantitative Machines 2021, 9, 32 18 of 38 matching. The discrepancy between the results of numerical and experimental studies is from 3% to 5%.

Machines 2021, 9, x FOR PEER REVIEW 19 of 39 Figure 14. Dependence of the supply coefﬁcientcoefficient ofof thethe compressorcompressor sectionsection on on the the discharge discharge pressure pres- sureof the of compressor the compressor section section (1—Theory; (1—Theory; 2—Experiment). 2—Experiment).

FigureFigure 15. 15. DependenceDependence of of the the supply supply coefficient coefﬁcient of the compressor section on thethe dischargedischarge pressurepres- sureof the of pumpthe pump section section (1—Theory; (1—Theory; 2—Experiment). 2—Experiment).

FiguresIt should 16 beand noted 17 show that increasingthe changes the in dischargevolumetric pressure efficiency in theof the pump pump section, sections we improve cooling of the suction gas and decrease its leaks and overflows. This leads to an obtained experimentally and numerically by changing pdc at fixed pdp and vice versa. increaseBy increasing in the flow the ratio discharge and the capacitypressure of in the the compressor compressor section. section, we increase the The presented results prove that we observe a good qualitative and quantitative amount of liquid flowing from the compressor section to the pump section. This increase matching. The discrepancy between the results of numerical and experimental studies is is during pumping in the pump section and increases the volumetric efficiency. It is ob- from 3% to 5%. served due to the underfilling of the working chamber in the real pump; there is no such Figures 16 and 17 show the changes in volumetric efficiency of the pump sections underfilling in the developed mathematical model (see Figure 16). obtained experimentally and numerically by changing pdc at fixed pdp and vice versa.

Figure 16. Volumetric efficiency the pump section from the discharge pressure of the compressor section (1—Theory; 2—Experiment).

Machines 2021, 9, x FOR PEER REVIEW 19 of 39

Figure 15. Dependence of the supply coefficient of the compressor section on the discharge pressure of the pump section (1—Theory; 2—Experiment).

Figures 16 and 17 show the changes in volumetric efficiency of the pump sections obtained experimentally and numerically by changing pdc at fixed pdp and vice versa. By increasing the discharge pressure in the compressor section, we increase the amount of liquid flowing from the compressor section to the pump section. This increase is during pumping in the pump section and increases the volumetric efficiency. It is ob- Machines 2021, 9, 32 served due to the underfilling of the working chamber in the real pump; there is no19 such of 38 underfilling in the developed mathematical model (see Figure 16).

Machines 2021, 9, x FOR PEER REVIEW 20 of 39 Figure 16. Volumetric efficiency efﬁciency the the pump pump section section from from the the discharge discharge pressure pressure of of the the compressor compressor section (1—Theory; 2—Experiment).

FigureFigure 17. 17. VolumetricVolumetric efficiency efficiency of the pumppump sectionsection fromfrom the the discharge discharge pressure pressure of of the the pump pump section section(1—Theory; (1—Theory; 2—Experiment). 2—Experiment). By increasing the discharge pressure in the compressor section, we increase the amount As the discharge pressure in the pump section increases, leaks and liquid spills in- of liquid flowing from the compressor section to the pump section. This increase is during crease and lead to a decrease in volumetric efficiency of the pump section both in the real pumping in the pump section and increases the volumetric efficiency. It is observed due to machine and in the mathematical model (see Figure 17). the underfilling of the working chamber in the real pump; there is no such underfilling in The presented results prove correct qualitative and satisfactory quantitative match- the developed mathematical model (see Figure 16). ing. The maximum discrepancy in definition ηvol is within 10%. As the discharge pressure in the pump section increases, leaks and liquid spills increase Thus, taking into account the complete qualitative matching between the numerical and lead to a decrease in volumetric efficiency of the pump section both in the real machine and experimental methods, and the allowable quantitative discrepancy (in determining and in the mathematical model (see Figure 17). instant pressures, it is from 10 to 15% and in integral parameters, it is from 5 to 10%), we The presented results prove correct qualitative and satisfactory quantitative matching. can conclude that the developed mathematical model of the work processes is true to life. The maximum discrepancy in definition ηvol is within 10%. 4. The Results of Numerical and Experimental Studies When conducting experimental and numerical studies, a classical plan with fractional replicas was used [26]. These operational parameters with following ranges were selected as independent parameters in experimental studies:

− Pressure in the compressor discharge line (receiver) pdc (from 3 to 7 bar); − Pump discharge pressure pdp (from 2 up to 10 bar); − Crankshaft speed nrev (from 250 up to 600 rpm). The experimental studies were carried out as follows:

− At pdp = 3 bar and nrev = 250 rpm, discharge pressure of the compressor section varied in the range 3 bar ≤ pdc ≤ 7 bar. − At pdc = 5 bar and nrev = 250 rpm, the pump section discharge pressure varied in the range 2 bar ≤ pdp ≤ 10 bar. − At constant discharge pressures in the compressor and pump sections: pdp = 2 bar; pdc = 4 bar angular speed of the crankshaft varied from 250 to 450 rpm. Water was used as liquid, and air was also used as gas in numerical simulating. In addition, when conducting numerical studies, the following independent parameters were added to the independent operational parameters listed above:

− Relative dead space value (ad). − Radial clearance in the piston seal (δ1). − Initial groove seal length (lu0).

Machines 2021, 9, 32 20 of 38

Thus, taking into account the complete qualitative matching between the numerical and experimental methods, and the allowable quantitative discrepancy (in determining instant pressures, it is from 10 to 15% and in integral parameters, it is from 5 to 10%), we can conclude that the developed mathematical model of the work processes is true to life.

4. The Results of Numerical and Experimental Studies When conducting experimental and numerical studies, a classical plan with fractional replicas was used [26]. These operational parameters with following ranges were selected as independent parameters in experimental studies:

- Pressure in the compressor discharge line (receiver) pdc (from 3 to 7 bar); - Pump discharge pressure pdp (from 2 up to 10 bar); - Crankshaft speed nrev (from 250 up to 600 rpm). The experimental studies were carried out as follows:

- At pdp = 3 bar and nrev = 250 rpm, discharge pressure of the compressor section varied in the range 3 bar ≤ pdc ≤ 7 bar. - At pdc = 5 bar and nrev = 250 rpm, the pump section discharge pressure varied in the range 2 bar ≤ pdp ≤ 10 bar. - At constant discharge pressures in the compressor and pump sections: pdp = 2 bar; pdc = 4 bar angular speed of the crankshaft varied from 250 to 450 rpm. Water was used as liquid, and air was also used as gas in numerical simulating. In addition, when conducting numerical studies, the following independent parameters were added to the independent operational parameters listed above:

- Relative dead space value (ad). - Radial clearance in the piston seal (δ1). - Initial groove seal length (lu0). The developed mathematical model considers a physical picture of the work processes in the compressor and pump sections of the crosshead-free PHPM.

4.1. Physical Picture of Ongoing Work Processes (Based on the Results of Numerical Analysis) We consider bottom dead center as a starting point. There is a liquid layer δ ≈ 0.001 m (Figure 18) above the piston that corresponds to the amount of dead space in the compressor section. When the piston moves up to the top dead center, there is a decrease in volume in the compressor and pump sections, which causes an increase in pressure. Consider- ing a high modulus of elasticity of liquid, pressure in the pump cavity increases almost immediately, but it increases rather slowly in the compressor section (Figures 19 and 20). At ϕ ≈ 3.4 rad, the pumping process begins in the pump section. Due to the significant pressure difference between the pump and compressor sections, the liquid from the pump section flows into the compressor section and the height of the liquid layer increases. It should be noted that with an increase in the angle of rotation, the length of the groove seal increases (Figure 21) and the pressure drop between the cavities decreases (Figures 19 and 20), leading to a decrease in the instant liquid flow rate from the pump section to the compressor section and the growth rate of the total liquid flow from the pump section to the compressor section (Figure 22). Machines 2021, 9, x FOR PEER REVIEW 21 of 39

The developed mathematical model considers a physical picture of the work processes in the compressor and pump sections of the crosshead-free PHPM.

4.1. Physical Picture of Ongoing Work Processes (Based on the Results of Numerical Analysis) We consider bottom dead center as a starting point. There is a liquid layer ≈ 0.001 m (Figure 18) above the piston that corresponds to the amount of dead space in the compressor section. When the piston moves up to the top dead center, there is a decrease in volume in the compressor and pump sections, which causes an increase in pressure. Con- sidering a high modulus of elasticity of liquid, pressure in the pump cavity increases al- Machines 2021most, 9, 32 immediately, but it increases rather slowly in the compressor section (Figures 19 21and of 38 20).

Machines 2021Figure, 9, x FOR 18. Dependence PEER REVIEW of the current height of the liquid above the piston in the compressor section on the angle of rotation 22 of 39 Figure 18. Dependence of the current height of the liquid above the piston in the compressor sec- of the crankshaft at different speeds. (1—n = 400 rpm; 2—n = 500 rpm; 3—n = 600 rpm). tion on the angle of rotation ofrev the crankshaft atrev different speeds.rev (1— = 400 rpm; 2— = 500 rpm; 3— = 600 rpm).

At φ ≈ 3.4 rad, the pumping process begins in the pump section. Due to the significant pressure difference between the pump and compressor sections, the liquid from the pump section flows into the compressor section and the height of the liquid layer increases. It should be noted that with an increase in the angle of rotation, the length of the groove seal increases (Figure 21) and the pressure drop between the cavities decreases (Figures 19 and 20), leading to a decrease in the instant liquid flow rate from the pump section to the compressor section and the growth rate of the total liquid flow from the pump section to the compressor section (Figure 22).

Figure 19. The indicator diagram of the compressor section at different speeds of the crankshaft (1—n = 400 rpm; Figure 19. The indicator diagram of the compressor section at different speeds of the crankshaft (1—rev = 400 rpm; 2— 2—n = 500 rpm; 3—n = 600 rpm). = 500rev rpm; 3— = 600rev rpm).

Figure 20. The indicator diagram of the pump section at different speeds of the crankshaft (1— = 400 rpm; 2— = 500 rpm; 3— = 600 rpm).

Machines 2021, 9, x FOR PEER REVIEW 22 of 39

Machines 2021, 9, 32 22 of 38 Figure 19. The indicator diagram of the compressor section at different speeds of the crankshaft (1— = 400 rpm; 2— = 500 rpm; 3— = 600 rpm).

MachinesFigure 2021, 20. 9, x FORThe PEER indicator REVIEW diagram of the pump section at different speeds of the crankshaft (1—=n 400rev rpm= 400 rpm 23; of= 39 Figure 20. The indicator diagram of the pump section at different speeds of the crankshaft (1— ; 2— 500 rpm2—; 3—nrev= 500= 600 rpm; rpm 3—n).rev = 600 rpm).

FigureFigure 21.21. DependenceDependence ofof thethe lengthlength ofof thethe gapgap sealseal onon thethe angleangle ofof rotationrotation ofof thethe crankshaft.crankshaft.

Figure 22. The dependence of the total liquid flow from the pump section to the compressor section on the angle of rotation of the crankshaft at different speeds (1— = 400 rpm; 2— = 500 rpm; 3— = 600 rpm).

At the angle of rotation φ ≈ 5.3 rad, the pressure in the compressor section exceeds the pressure in the pump section (Figures 19 and 20), liquid moves into the opposite direction, and the total amount of liquid flowing from the compressor section to the pump section increases (Figure 23). It should be noted that there is a discharge process in the compressor and pump sections this time. When the piston reaches the top dead center, the height of the liquid layer above the piston exceeds the dead volume and all gas from the compressor section is pushed into the discharge line (Figure 24); then, the “pump”

Machines 2021, 9, x FOR PEER REVIEW 23 of 39

Machines 2021, 9, 32 23 of 38

Figure 21. Dependence of the length of the gap seal on the angle of rotation of the crankshaft.

FigureFigure 22. 22.The The dependence dependence of of the the total total liquid liquid ﬂow flow from from the thepump pump sectionsection to the compressor compressor section section on on the the angle angle of ofrotation rotation of theof the crankshaft crankshaft at differentat different speeds speeds (1— (1—nrev==400 400 rpm; rpm 2—; 2—nrev==500 500 rpm; rpm;3— 3—nrev==600 600 rpm).rpm).

AtAt the the angle angle of of rotation rotationϕ φ≈ ≈ 5.3 rad, the pressure in in the the compressor compressor section section exceeds exceeds thethe pressure pressure in in thethe pumppump section section (Figures (Figures 19 19 and and 20), 20 liquid), liquid moves moves into intothe opposite the opposite di- direction,rection, and and the the total total amount amount of of liquid liquid flowing ﬂowing from from the the compressor compressor section section to tothe the pump pump sectionsection increases increases (Figure (Figure 2323).). ItIt should be be no notedted that that there there is is a adischarge discharge process process in inthe the Machines 2021, 9, x FOR PEER REVIEWcompressor and pump sections this time. When the piston reaches the top dead center,24 of 39 compressor and pump sections this time. When the piston reaches the top dead center, the heightthe height of the of liquid the liquid layer layer above above the pistonthe pist exceedson exceeds the the dead dead volume volume and and all all gas gas from from the compressorthe compressor section section is pushed is pushed into theinto discharge the discharge line (Figureline (Figure 24); then,24); then, the “pump” the “pump” stroke starts,stroke and starts, part and of the part liquid of the from liquid the from compressor the compressor cavity enterscavity theenters discharge the discharge line through line thethrough discharge the discharge valve of the valve compressor of the compressor section (Figure section 18(Figure). 18).

FigureFigure 23. 23.Dependence Dependence of theof the total total liquid liquid ﬂow flow from from the the compressor compressor section section to to the the pump pump section section onon thethe angleangle ofof rotationrotation of theof crankshaft the crankshaft at different at different speeds speeds (1— n(1—rev =400= rpm;400 rpm 2—; n2—rev=500= 500 rpm; rpm 3—; 3—nrev= 600= 600 rpm). rpm).

Figure 24. Dependence of the height of the gas volume of the compressor section on the angle of rotation of the crankshaft at different speeds.

By the end of the discharge process, the height of the liquid layer will be equal to the linear dead volume—0.001 m. It should be noted that the amount of liquid flowing from the compressor section to the pump section (Figure 23) sharply increases at the end of the discharge process. This is due to a sharp increase in the discharge pressure of the com-

Machines 2021, 9, x FOR PEER REVIEW 24 of 39

stroke starts, and part of the liquid from the compressor cavity enters the discharge line through the discharge valve of the compressor section (Figure 18).

Machines 2021, 9, 32 24 of 38 Figure 23. Dependence of the total liquid flow from the compressor section to the pump section on the angle of rotation of the crankshaft at different speeds (1— = 400 rpm; 2— = 500 rpm; 3— = 600 rpm).

FigureFigure 24.24. DependenceDependence ofof thethe heightheight ofof thethe gasgas volumevolume ofof thethe compressorcompressor sectionsection onon thethe angleangle ofof rotationrotation ofof thethe crankshaftcrankshaft atat differentdifferent speeds.speeds.

ByBy the end of of the the discharge discharge process, process, the the height height of of the the liquid liquid layer layer will will be beequal equal to the to thelinear linear dead dead volume—0.001 volume—0.001 m. It m. should It should be no beted noted that the that amount the amount of liquid of liquid flowing flowing from fromthe compressor the compressor section section to the to pump the pump section section (Figure (Figure 23) sharply 23) sharply increases increases at the atend the of end the ofdischarge the discharge process. process. This is This due isto due a sharp to a sharpincrease increase in the indischarge the discharge pressure pressure of the of com- the compressor section (Figure 19) due to the implementation of the “pump” stroke in the compressor section. When implementing the pump stroke, the increase in pressure is due to an increase in the density of the working liquid tens of times, and dropping liquid is discharged instead of gas. When the piston moves down, there are reverse expansion processes in the compressor and pump sections of the hybrid power machine. If all the dead space is filled with liquid in the compressor section, the reverse expansion processes in the pump and compressor sections are identical and proceed almost the same, especially if the dead volumes are the same. If there is compressed gas in the dead space of the compressor section, the reverse expansion will take place at a larger angle of rotation of the crankshaft, and the suction will be in the pump cavity. This will lead to liquid flow into the pump section from the compressor section and thereby will prevent the flow of “fresh” portions of liquid through the suction valve into the pump section reducing volumetric efficiency. During the suction process, when the nominal suction pressures and pressure losses in the pump and compressor sections are equal, there will be no liquid flow through the groove seal into forward and reverse directions (see Figures 22 and 23). If the pressure loss in the pump section exceeds the pressure loss in the compressor section, the liquid will flow from the compressor section to the pump section (Figure 23, end of the suction process). If the pressure loss in the compressor section is more than in the pump section, liquid from the pump section through the groove seal enters the compressor section (Figure 22). Thus, in the suction process, the liquid fills not only the entire space of the working chamber of the pump section but also part of the space of the compressor section. Machines 2021, 9, 32 25 of 38

4.2. Analysis of the Inﬂuence of the Main Operational Parameters on the Work Processes of Crosshead-Free PHPM (Based on the Results of Experimental Studies)

Discharge pressure in the pump section pdp (pdc = 0.6 MPa, nrev = 300 rpm) For PHPM operation with liquid supply from pump section to the compressor section, the liquid removal into the discharge line of the compressor section is typical. If there is increase in the discharge pressure of the pump section, then the amount of discharged liquid increases. Figure 25 shows that at pdp with 0.3 MPa, there is an intensive removal of liquid ∆Vw to the discharge line of the compressor section. This dependence (curve 3) has a parabolic nature (inverted parabola). At pdp = 0.7 MPa, ∆Vw is 15 mL/min. This value is less than Machines 2021, 9, x FOR PEER REVIEW∆V w for PHPM with a stepped groove seal (curve 1). Thus, the crosshead-free PHPM 26 of 39 in terms of the amount of liquid taken out is an intermediate between the PHPM with a smooth groove seal and the PHPM with a stepped groove seal, and it is closer to the last.

Figure 25. Dependence ∆Vw on the discharge pressure of the pump section (1—PHPM with step-type Figure 25. Dependence ΔVw on the discharge pressure of the pump section (1—PHPM with step- gap seals; 2—PHPM with smooth gap seals; 3—Crosshead-free PHPM). type gap seals; 2—PHPM with smooth gap seals; 3—Crosshead-free PHPM). Due to enhanced cooling of the cylinder in the developed new design, as well as the suction and discharge pipes, there is intensive cooling of the entire cylinder–piston group and, in particular, the valve plate as one of the most heated parts. Experimental results presented in Figure 26 lead to conclusions:

- Valve plate temperature has a maximum at pdp = 5 bar–314 K. If pdp increases, Tvl decreases. - Decreasing Tvl causes an increase in the discharge pressure of the pump section, which is also typical for earlier studies (curves 1 and 2). The most significant decrease Tvl is observed for a PHPM with a stepped groove seal, which is due to a more intensive liquid flow from the pump section to the compressor section. The temperature of the valve plate of the crosshead-free PHPM is from 10 to 12 K lower than the temperature of the cover of the PHPM with a stepped groove seal, which is a very significant result and fully corresponds to the stated aim

Figure 26. Dependence of the temperature of the valve body on the discharge pressure of the pump section (1—PHPM with step-type gap seals; 2—PHPM with smooth gap seals; 3—Cross- head-free PHPM).

Figure 27 shows the dependence of the temperature of the suction gas on pdp. It should be noted that in this case, as for Tvl, we observe the above trends. For a crosshead-free PHPM, the maximum value of the intake air temperature is observed at pdp = 0.5 MPa. Change in intake gas temperature when changing pdp from 0.3 to 0.9 MPa is about 1 K. The presented results prove that a decrease in temperature of intake air throughout the change pdp amounts to about 8 K, which is a very significant result and leads to an increase in the flow coefficient of the compressor section (Figure 28) and to the following conclusions:

− In the range of pressure change 0.3 MPa ≤ pdp ≤ 0.6 MPa, the feed coefficient of the compressor section for the crosshead-free PHPM exceeds the values of the feed coefficients for the PHPM with smooth and stepped groove seals. So, at pdp = 0.5 MPa, the excess feed rate is 0.2 and 0.1 compared with smooth and stepped groove seals, which is, respectively, 27% and 13.5%. Such increase is due to , which is a high-temperature coefficient.

Machines 2021, 9, x FOR PEER REVIEW 26 of 39

Machines 2021, 9, 32 26 of 38 Figure 25. Dependence ΔVw on the discharge pressure of the pump section (1—PHPM with step- type gap seals; 2—PHPM with smooth gap seals; 3—Crosshead-free PHPM).

Figure 26. Dependence of the temperature of the valve body on the discharge pressure of the Figure 26. Dependence of the temperature of the valve body on the discharge pressure of the pump section (1—PHPM with step-type gap seals; 2—PHPM with smooth gap seals; 3—Crosshead- freepump PHPM). section (1—PHPM with step-type gap seals; 2—PHPM with smooth gap seals; 3—Cross- head-free PHPM). Figure 27 shows the dependence of the temperature of the suction gas on pdp. It should T be notedFigure that 27 in thisshows case, the as fordependencevl, we observe of the the temperature above trends. of For the a crosshead-freesuction gas on pdp. It should PHPM, the maximum value of the intake air temperature is observed at pdp = 0.5 MPa. be noted that in this case, as for Tvl, we observe the above trends. For a crosshead-free Change in intake gas temperature when changing pdp from 0.3 to 0.9 MPa is about 1 K. The presentedPHPM, the results maximum prove that value a decrease of inthe temperature intake air of temperature intake air throughout is observed the change at pdp = 0.5 MPa. pChangedp amounts in tointake about gas 8 K, temperature which is a very when significant changing result and pdp leadsfrom to 0.3 an to increase 0.9 MPa in the is about 1 K. The flowpresented coefficient results of the prove compressor that a section decrease (Figure in 28temp) anderature to the following of intake conclusions: air throughout the change p- dp amountsIn the range to about of pressure 8 K, changewhich 0.3is a MPa very≤ significantpdp ≤ 0.6 MPa, result the feedand coefficientleads to an of increase in the flowthe coefficient compressor of section the compressor for the crosshead-free section (Figure PHPM exceeds 28) and the to values the following of the feed conclusions: coefficients for the PHPM with smooth and stepped groove seals. So, at pdp = 0.5 MPa, − theIn excessthe range feed rateof pressure is 0.2 and 0.1change compared 0.3 withMPa smooth ≤ pdp ≤ and 0.6 stepped MPa, the groove feed seals, coefficient of the whichcompressor is, respectively, section 27% for the and crosshead-free 13.5%. Such λ increase PHPM is exceeds due to λ theT, which values is aof the feed coef- high-temperature coefficient. ficients for the PHPM with smooth and stepped groove seals. So, at pdp = 0.5 MPa, the - In the range 0.6 MPa ≤ pdp ≤ 0.9 MPa, the λ feed coefficient for the crosshead-free PHPMexcess exceeds feed rate the valuesis 0.2 and of the 0.1 feed compared coefficients wi forth the smooth PHPM and with stepped smooth groove groove seals, which seals,is, respectively, but λ is lower 27% for the and PHPM 13.5%. with steppedSuch groove increase seals. is This due is to due to, thewhich fact is a high-tem- thatperature in the PHPM coefficient. with a stepped groove seal, the amount of incoming liquid from the pump section to the compressor is the maximum among the options considered. This leads to a decrease in dead space and an increase in λ0 volumetric coefficient and λ.

Increasing pdp, the volumetric efficiency of the pump section decreases (Figure 29). This decrease is typical for all types of positive displacement pumps. Decreasing ηvol is significant and amounts to ∆ηvol = 0.15 when increasing pdp from 0.3 to 0.9 MPa. It should be noted that this dependence is close to linear, and the angle of inclination for all three of the presented dependences is approximately the same. Crosshead-free PHPM ηvol exceeds the volumetric efficiency of PHPM with stepped and smooth groove seals for 5–10%, respectively. Machines 2021, 9, x FOR PEER REVIEW 27 of 39 Machines 2021, 9, x FOR PEER REVIEW 27 of 39

− In the range 0.6 MPa ≤ pdp ≤ 0.9 MPa, the feed coefficient for the crosshead-free − PHPMIn the exceeds range 0.6the MPavalues ≤ ofpdp the ≤ 0.9 feed MPa, coef theficients feed for thecoefficient PHPM withfor the smooth crosshead-free groove seals,PHPM but exceeds is lower the forvalues the PHPMof the feed with coef steppedficients groove for the seals. PHPM This with is due smooth to the groove fact thatseals, in the but PHPM is lower with for a stepped the PHPM groove with seal, stepped the amountgroove seals.of incoming This is dueliquid to thefrom fact thethat pump in the section PHPM to thewith compressor a stepped isgroove the maximum seal, the amongamount the of optionsincoming considered. liquid from Thisthe leads pump to section a decrease to the in compressor dead space is and the anmaximum increase amongin volumetric the options coefficient considered. andThis . leads to a decrease in dead space and an increase in volumetric coefficient and . Increasing pdp, the volumetric efficiency of the pump section decreases (Figure 29). dp This decreaseIncreasing is typical p , the for volumetric all types ofefficiency positive ofdisplacement the pump section pumps. decreases Decreasing (Figure is29). significantThis decrease and amounts is typical to for Δ all types = 0.15 of when positive increasing displacement pdp from pumps. 0.3 to 0.9 Decreasing MPa. It should is besignificant noted that andthis amountsdependence to Δ is close = 0.15 to linear, when and increasing the angle pdp of from inclination 0.3 to 0.9 for MPa. all three It should of thebe presented noted that dependences this dependence is approximately is close to linear, the and same. the Crosshead-freeangle of inclination PHPM for all three ex- of ceedsthe presentedthe volumetric dependences efficiency is of approximately PHPM with steppedthe same. and Crosshead-free smooth groove PHPM seals for 5– ex- Machines 2021, 9, 32 10%,ceeds respectively. the volumetric efficiency of PHPM with stepped and smooth groove27 ofseals 38 for 5– 10%, respectively.

FigureFigure 27. 27. DependenceDependence of of the the temperature temperature of of the the suct suctionion gas on the discharge pressurepressure of of the the pumppumpFigure section section 27. Dependence (1—PHPM (1—PHPM with withof the step-type step-type temperature gap gap seals; seals;of 2—PHPMthe 2—PHPMsuction with gas with smooth on thesmoothgap discharge seals; gap 3—Crosshead-seals; pressure 3—Cross- of the head-freefreepump PHPM). section PHPM). (1—PHPM with step-type gap seals; 2—PHPM with smooth gap seals; 3—Cross- head-free PHPM).

Figure 28. Dependence of the ﬂow coefﬁcient of the compressor section on the discharge pressure of the pump section (1—PHPM with step-type gap seals, 2—PHPM with smooth gap seals, 3—Crosshead-free PHPM).

Machines 2021, 9, x FOR PEER REVIEW 28 of 39 Machines 2021, 9, x FOR PEER REVIEW 28 of 39

Figure 28. Dependence of the flow coefficient of the compressor section on the discharge pressure Figure 28. Dependence of the flow coefficient of the compressor section on the discharge pressure Machines 2021, 9, 32 of the pump section (1—PHPM with step-type gap seals, 2—PHPM with smooth gap seals, 3—28 of 38 of the pump section (1—PHPM with step-type gap seals, 2—PHPM with smooth gap seals, 3— Crosshead-free PHPM). Crosshead-free PHPM).

Figure 29. Dependence of volumetric efficiency of the pump section from the discharge pressure of FigureFigure 29. 29.DependenceDependence of volumetric of volumetric efficiency efﬁciency of the of pump the pump section section from fromthe discharge the discharge pressure pressure of the pump section: (1—PHPM with step-type gap seals; 2—PHPM with a smooth gap seal; 3—Cross- theof pump the pumpsection: section: (1—PHPM (1—PHPM with step-type withstep-type gap seals; gap2—PHPM seals; with 2—PHPM a smooth with gap a seal; smooth 3—Cross- gap seal; head-free PHPM). head-free3—Crosshead-free PHPM). PHPM).

Discharge pressure in the compressor section pdcp (pdpp = 0.3 MPa, nrevn = 300 rpm) DischargeDischarge pressure pressure in the in the compressor compressor section section pdc (dcpdp( =dp 0.3= MPa, 0.3 MPa, nrev = rev300= rpm) 300 rpm) IncreasingIncreasing pdcp, the, the amount amount of ofliquid liquid carried carried into into the the discharge discharge line line of of the the compressor compressor Increasing pdc, thedc amount of liquid carried into the discharge line of the compressor section is reduced (Figure 30). sectionsection is reduced is reduced (Figure (Figure 30). 30 ).

Figure 30. Dependence ∆Vw from the discharge pressure of the compressor section (1—PHPM with Figurestep-type 30. Dependence gap seals; 2—PHPM ΔVw from with the discharge a smooth gappressure seal; 3—Crosshead-freeof the compressor section PHPM). (1—PHPM Figure 30. Dependence ΔVw from the discharge pressure of the compressor section (1—PHPM with step-type gap seals; 2—PHPM with a smooth gap seal; 3—Crosshead-free PHPM). with step-typeIn terms gap ofseals; the 2—PHPM discharged with liquid a smooth amount, gap seal; the 3—Crosshead-free crosshead-free PHPM PHPM). is close to the PHPM with a smooth groove seal and at pdc = 0.7 MPa, the amount of liquid they carry is the same and equal to 2 mL/min. It should be noted that the amount of liquid to be removed from the PHPM with a stepped groove seal is 5–10 times higher than the amount of liquid to be removed from the above-mentioned devices.

Decreasing the amount of liquid leads to increase in temperature of the valve plate (Figure 31) and temperature of the intake gas (Figure 32). Machines 2021, 9, x FOR PEER REVIEW 29 of 39

In terms of the discharged liquid amount, the crosshead-free PHPM is close to the PHPM with a smooth groove seal and at pdc = 0.7 MPa, the amount of liquid they carry is Machines 2021, 9, x FOR PEER REVIEWthe same and equal to 2 mL/min. It should be noted that the amount29 of 39of liquid to be re-

moved from the PHPM with a stepped groove seal is 5–10 times higher than the amount of liquid to be removed from the above-mentioned devices. Machines 2021, 9, 32 InDecreasing terms of the dischargedthe amount liquid of liquidamount, leads the crosshead-free to increase PHPM in temperature is close to the of the29 of valve 38 plate PHPM with a smooth groove seal and at pdc = 0.7 MPa, the amount of liquid they carry is the(Figure same and31) equaland temperature to 2 mL/min. It of should the intake be noted gas that (Figure the amount 32). of liquid to be removed from the PHPM with a stepped groove seal is 5–10 times higher than the amount of liquid to be removed from the above-mentioned devices. Decreasing the amount of liquid leads to increase in temperature of the valve plate (Figure 31) and temperature of the intake gas (Figure 32).

FigureFigure 31.31. DependenceDependence of the the temperature temperature of the the valve valve plate plate on on the the discharge discharge pressure pressure of of the com- Figure 31. Dependence of the temperature of the valve plate on the discharge pressure of the com- thepressor compressor section section(1—PHPM (1—PHPM with step-type with step-type gap seals; gap seals;2—PHPM 2—PHPM with withsmooth smooth gap gapseals; seals; 3—Cross- pressor section (1—PHPM with step-type gap seals; 2—PHPM with smooth gap seals; 3—Cross- head-free3—Crosshead-freehead-free PHPM). PHPM). PHPM).

Figure 32. 32. DependenceDependence of the oftemperature the temperature of the suctio ofn thegas on suction the discharge gas on pressure the discharge of the com- pressure of pressorthe compressor section (1—PHPM section with (1—PHPM step-type with gap seals; step-type 2—PHPM gap with seals; smooth 2—PHPM gap seals; with 3—Cross- smooth gap seals; head-free PHPM). 3—Crosshead-free PHPM).

Dependences Tvl and Tscc have a linear nature due to pdc. The presented results lead to Dependences T and T have a linear nature due to p . The presented results lead theFigure following 32. Dependence conclusions:vl of thescc temperature of the suction gasdc on the discharge pressure of the com- topressor the following section (1—PHPM conclusions: with step-type gap seals; 2—PHPM with smooth gap seals; 3—Cross- -head-freeThe temperature PHPM). of the valve plate for the crosshead-free PHPM is 10 K and 15 K, respectively; it is lower than that of the PHPM with a stepped and smooth groove seal. - IntakeDependences gas temperature Tvl and TTsccscc haveof the a crosshead-freelinear nature due PHPM to p isdc. lowerThe presented by 6 K and results 8 K, lead to the followingrespectively, conclusions: than PHPM with a stepped and smooth groove seal. Increasing the temperature of the intake gas and an increase in the dead space by reducing the liquid flow from the pump section to the compressor leads to a decrease in the temperature coefficient λT and volumetric coefficient λ0, and therefore the entire feed rate λ. Crosshead-free PHPM value λ is equal with λ for PHPM with a stepped groove seal (Figure 33) and exceeds λ for PHPM with a smooth groove seal from 10 to 20%.

Machines 2021, 9, x FOR PEER REVIEW 30 of 39

− The temperature of the valve plate for the crosshead-free PHPM is 10 K and 15 K, respectively; it is lower than that of the PHPM with a stepped and smooth groove seal. − Intake gas temperature Tscc of the crosshead-free PHPM is lower by 6 K and 8 K, respectively, than PHPM with a stepped and smooth groove seal. Increasing the temperature of the intake gas and an increase in the dead space by reducing the liquid flow from the pump section to the compressor leads to a decrease in the temperature coefficient and volumetric coefficient , and therefore the entire feed rate . Crosshead-free PHPM value is equal with for PHPM with a stepped Machines 2021, 9, 32groove seal (Figure 33) and exceeds for PHPM with a smooth groove seal from 10 to 30 of 38 20%.

Figure 33. DependenceFigure 33. Dependence of the supply of the coefﬁcient supply coefficient of the compressor of the compressor section on section the discharge on the discharge pressure ofpres- the compressor section (1—PHPMsure of withthe compressor step-type gap section seals; (1—PHPM 2—PHPM with with step-type smooth gap gap seals; seals; 3—Crosshead-free 2—PHPM with smooth PHPM). gap seals; 3—Crosshead-free PHPM). With increasing pdc, there is an increase in volumetric efﬁciency (Figure 34). It should Machines 2021, 9, x FOR PEER REVIEW 31 of 39 With increasingbe noted pdc, that thereηvol is anfor increase crosshead-free in volumetric PHPM efficiency exceeds η(Figurevol for PHPM34). It should with a stepped and smooth groove seal from 2 to 4%. be noted that for crosshead-free PHPM exceeds for PHPM with a stepped and smooth groove seal from 2 to 4%.

FigureFigure 34.34. VolumetricVolumetric efﬁciency efficiency of of the the pump pump section section from from the the discharge discharge pressure pressure of the of compressorthe compressor section section (1—PHPM (1—PHPM with step-typewith step-type gap seals; gap seals; 2—PHPM 2—PHPM with awith smooth a smooth gap seal; gap 3—Crosshead-freeseal; 3—Crosshead-free PHPM). PHPM).

Crank angular velocity ((ppdcdc = 0.6 0.6 MPa, MPa, ppdpdp == 0.3 0.3 MPa) MPa) With anan increaseincrease inin thethe crankcrank angularangular velocity,velocity, thethe amountamount ofof dischargeddischarged liquidliquid perper unit timetime decreasesdecreases for for a a crosshead-free crosshead-free PHPM PHPM (Figure (Figure 35), 35), and and it increases it increases for PHPM for PHPM with smoothwith smooth and stepped and stepped groove groove seals. seals. The amount of liquid to be discharged into the discharge line of the compressor section for a crosshead-free PHPM is practically equal to the quantity of liquid to be discharged to the PHPM with a smooth groove seal and is significantly less ΔVw for PHPM with stepped groove seals. If there is an increase in , then the valve plate temperature increases for all considered PHPM options.

Figure 35. Dependence ΔVw from the angular velocity of the crankshaft (1—PHPM with step-type gap seals; 2—PHPM with smooth gap seals; 3—Crosshead-free PHPM).

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Figure 34. Volumetric efficiency of the pump section from the discharge pressure of the compressor section (1—PHPM with step-type gap seals; 2—PHPM with a smooth gap seal; 3—Crosshead-free PHPM).

Crank angular velocity (pdc = 0.6 MPa, pdp = 0.3 MPa) With an increase in the crank angular velocity, the amount of discharged liquid per unit time decreases for a crosshead-free PHPM (Figure 35), and it increases for PHPM with smooth and stepped groove seals. The amount of liquid to be discharged into the discharge line of the compressor section for a crosshead-free PHPM is practically equal to the quantity of liquid to be discharged to the PHPM with a smooth groove seal and is significantly less ΔVw for PHPM Machines 2021, 9, 32 31 of 38 with stepped groove seals. If there is an increase in , then the valve plate temperature increases for all considered PHPM options.

Figure 35. V Figure 35. DependenceDependence Δ∆Vw wfromfrom the the angular angular velocity velocity of of the the crankshaft crankshaft (1—PHPM (1—PHPM with with step-type step-type gapgap seals; seals; 2—PHPM 2—PHPM with with smooth smooth gap gap seals; seals; 3—Crosshead-free 3—Crosshead-free PHPM). PHPM). The amount of liquid to be discharged into the discharge line of the compressor section for a crosshead-free PHPM is practically equal to the quantity of liquid to be discharged to Machines 2021, 9, x FOR PEER REVIEW 32 of 39 the PHPM with a smooth groove seal and is signiﬁcantly less ∆Vw for PHPM with stepped groove seals. If there is an increase in ηvol, then the valve plate temperature increases for all considered PHPM options. ItIt shouldshould bebe notednoted thatthat itsits valuesvalues fromfrom 314314 toto 318318 KK andand itsits valuesvalues fromfrom 12.512.5 toto 2222 KK are lower than thethe temperature ofof thethe valvevalve plateplate forfor thethe PHPMPHPM withwith smoothsmooth andand steppedstepped groove seals,seals, respectivelyrespectively (Figure(Figure 3636).).

Figure 36. TheThe dependence dependence of of the the temperature temperature of ofthe the valve valve plate plate on onthe theangular angular velocity velocity of the of the crankshaftcrankshaft (1—PHPM with step-type gap gap seals; seals; 2—PHPM 2—PHPM with with a a smooth smooth gap gap seal; seal; 3—Crosshead- 3—Crosshead- freefree PHPM).PHPM).

If there is an increase in nrev, the temperature of the intake gas decreases both for the If there is an increase in nrev, the temperature of the intake gas decreases both for the crosshead-free PHPM and for the PHPM with a smooth groove seal. For the PHPM with crosshead-free PHPM and for the PHPM with a smooth groove seal. For the PHPM with stepped groove seals, an inverse tendency is typical (Figure 37). It should be noted that for stepped groove seals, an inverse tendency is typical (Figure 37). It should be noted that a crosshead-free PHPM, the temperature of the intake air is from 8 to 10 K lower than all for a crosshead-free PHPM, the temperature of the intake air is from 8 to 10 K lower than all the options considered, which is very important and leads to an increase in productivity and indicator efficiency.

Figure 37. Dependence of the temperature of the suction gas on the angular veloci ty of the crankshaft (1—PHPM with step-type gap seals; 2—PHPM with smooth gap seals; 3—Crosshead-free PHPM).

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It should be noted that its values from 314 to 318 K and its values from 12.5 to 22 K are lower than the temperature of the valve plate for the PHPM with smooth and stepped groove seals, respectively (Figure 36).

Figure 36. The dependence of the temperature of the valve plate on the angular velocity of the crankshaft (1—PHPM with step-type gap seals; 2—PHPM with a smooth gap seal; 3—Crosshead- free PHPM).

If there is an increase in nrev, the temperature of the intake gas decreases both for the Machines 2021, 9, 32 32 of 38 crosshead-free PHPM and for the PHPM with a smooth groove seal. For the PHPM with stepped groove seals, an inverse tendency is typical (Figure 37). It should be noted that for a crosshead-free PHPM, the temperature of the intake air is from 8 to 10 K lower than theall the options options considered, considered, which which is very is very important important and and leads leads to anto an increase increase in in productivity productiv- andity and indicator indicator efﬁciency. efficiency.

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FigureFigure 37.37. DependenceDependence ofof the the temperature temperature of of the the suction suction gas gas on on the the angular angular veloci veloci ty ofty theof the crankshaft crank- (1—PHPMshaft (1—PHPM with step-type with step-type gap seals; gap 2—PHPMseals; 2—PH withPM smooth with smooth gap seals; gap 3—Crosshead-freeseals; 3—Crosshead-free PHPM). PHPM). If there is increase in crank angular velocity, and values decrease, as well as If there is increase in crank angular velocity, λT and λp values decrease, as well as λ0, due, due to decrease to decrease in the in the amount amount ofliquid of liquid coming coming from from the the pump pump section section to theto the compressor. compres- Thissor. This leads leads to a decreaseto a decrease in the in ﬂow the rate flow of rate the compressorof the compressor section sectionλ (Figure (Figure38). It should 38). It beshould noted be that noted in the that range in the of therange investigated of the investigated crank angular crank velocity, angular the velocity, feed coefﬁcient the feed forco- theefficient compressor for the section compressor of the crosshead-freesection of the PHPMcrosshead-free exceeds λPHPMfor PHPM exceeds with smooth for PHPM and steppedwith smooth groove and seals stepped from groove 5 to 10%. seals from 5 to 10%.

FigureFigure 38.38. DependenceDependence ofof thethe supplysupply coefﬁcientcoefficient of of the the compressor compressor section section on on the the angular angular velocity velocity of theof the crankshaft crankshaft (1—PHPM (1—PHPM with with step-type step-type gap gap seals; seals; 2—PHPM 2—PHPM with with smooth smooth gap gap seals; seals; 3—Crosshead- 3—Cross- freehead-free PHPM). PHPM).

IfIf therethere is increase in in the the crank crank angular angular ve velocity,locity, then then the the amount amount of of under under filling ﬁlling of ofthe the working working chamber chamber of the of thepump pump section section increases increases due dueto friction to friction forces forces and inertial and inertial pres- pressuresure losses, losses, which which leads leads to toa decrease a decrease in in the the volumetric volumetric efficiency efﬁciency of of the pump sectionsection (Figure(Figure 3939).). It should be noted that with nrev, the increase of for the crosshead-free PHPM decreases less—less than 5%, unlike for PHPM with smooth and stepped groove seals. In addition, it is essential that for the crosshead-free PHPM significantly exceeds nrev (more than 20%) for other options under consideration.

MachinesMachines2021 2021, 9, ,9 32, x FOR PEER REVIEW 3334 of of 38 39

FigureFigure 39. 39.Dependence Dependence of of volumetric volumetric efﬁciency efficiency the the pump pump section section from from the the angular angular velocity velocity of of the the crankshaft crankshaft (1—PHPM (1—PHPM withwith step-type step-type gap gap seals; seals; 2—PHPM 2—PHPM with with a a smooth smooth gap gap seal; seal; 3—Crosshead-free 3—Crosshead-free PHPM). PHPM).

5. ConclusionsIt should be noted that with nrev, the increase of λ for the crosshead-free PHPM decreases less—less than 5%, unlike λ for PHPM with smooth and stepped groove seals. In (1) Analysis of the advantages and disadvantages of existing PHPM constructions addition, it is essential that λ for the crosshead-free PHPM significantly exceeds nrev (more than 20%)helped for to other develop options a new under design consideration. of crosshead-free PHPM, which is protected by a Patent for Invention of the Russian Federation. 5.(2) Conclusions Based on the fundamental laws of energy, mass, and motion conservation, after ver- (1) Analysisification ofand the adoption advantages of a andsystem disadvantages of simplifying of existing assumptions, PHPM a constructions mathematical helped model toof developa crosshead-free a new design PHPM of with crosshead-free the primary PHPM, liquid whichflow from is protected the pump by section a Patent to for the Inventioncompressor of section the Russian has been Federation. developed, which includes a mathematical model of the (2) Basedcompressor on the section, fundamental operating laws both of energy, in compressor mass, and mode motion and conservation, in pump mode, after a verifi-math- cationematical and model adoption of the of pump a system section, of simplifying and a mathematical assumptions, model a mathematical of groove seals. model The ofmathematical a crosshead-free model PHPM determines with the the primary main inte liquidgral characteristics flow from the of pump the compressor section to theand compressor pump sections: section compressor has been developed, delivery ratio, which indicator includes efficiency, a mathematical volumetric model effi- of theciency, compressor mass of liquid section, removed operating from both the in compressor compressor section mode per and cycle, in pump etc. mode, a (3) mathematicalThe developed model mathematical of the pump model section, consider and as mathematicalthe physical picture model ofof groove the ongoing seals. Thework mathematical processes in model the compressor determines and the pump main integralsections, characteristics as well as in the of the groove compres- seals sorof the and crosshead-free pump sections: PHPM, compressor which include: delivery ratio, indicator efficiency, volumetric efficiency,− Changes mass in of thermodynamic liquid removed parameters from the compressor in the compressor section perand cycle,pump etc. sections in (3) The developedterms of crankshaft mathematical angle; model considers the physical picture of the ongoing work− Changes processes in in the the mass compressor of liquid and in the pump compressor sections, assection well asby in the the crankshaft groove seals angle; of the− crosshead-freeThe current height PHPM, of whichthe liquid include: layer above the piston in the compressor section - Changesaccording in to thermodynamic the crankshaft parametersangle; in the compressor and pump sections in − termsImplementation of crankshaft of angle;the “pumping stroke” in the compressor section determim- - Changesing the pressure in the mass and of flow liquid rate in of the the compressor liquid in the section compressor by the crankshaftdischarge line. angle; (4) -BasedThe on current the proposed height of new the liquiddesign layer of th abovee crosshead-free the piston in PH thePM, compressor a prototype section was developed,according and to an the experimental crankshaft angle; stand equipped with modern measuring equipment -was Implementationcreated. Experimental of the “pumpingstudies proved stroke” the in operability the compressor of the section created determiming prototype in the rangethe pressure of operational and flow parameters; rate of the they liquid confirmed in the compressor the operability discharge of the line. developed (4) Basedmathematical on the proposedmodel with new primary design liquid of the flow crosshead-free from the pump PHPM, section a prototype to the wascom- developed,pressor section. and anThe experimental verification standof the equippedmathematical with model modern was measuring carried out equipment at differ- wasent discharge created. Experimental pressures of the studies compressor proved and the operabilitypump sections. of the The created comparison prototype was

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in the range of operational parameters; they confirmed the operability of the developed mathematical model with primary liquid flow from the pump section to the compressor section. The verification of the mathematical model was carried out at different discharge pressures of the compressor and pump sections. The comparison was completed using indicator diagrams, feed rate, volumetric efficiency and the volume of liquid removed to the compressor discharge line per cycle. The maximum discrepancy in the instantaneous pressure was observed in the process of injection both in the pump section and in the compressor section, and it was from 10 to 15%. The discrepancy in integral characteristics was within 5–10%. The developed mathematical model qualitatively and quantitatively describes real physical processes and is true to life. (5) As a result of the experimental and theoretical studies, it was established: - Temperature of the valve plate of the crosshead-free PHPM is from 10 to 15 K less than that of the crosshead PHPM with smooth and step-like groove; - Due to more intensive cooling of the cylinder–piston group in the crosshead-free PHPM, a decrease in the temperature of the suction gas by 6–8 K is observed in the entire range of the studied parameters as compared to the crosshead PHPM with smooth and step-like groove; - Due to more intensive cooling of the suction gas, there is an increase in the feed coefficient and indicator efficiency of the compressor section by 3–5% on average, and in some cases, it is higher if compared with analogs; - Due to the design features of the crosshead-free PHPM, there is an increase in volumetric efficiency of the pump section by 5–10% compared to the studied analogs; - The liquid carryover in the crosshead-free PHPM from the compressor section into the discharge line is at the level of the liquid carryover in the crosshead PHPM with a smooth groove and significantly less than liquid carryover in the crosshead PHPM with a step-like groove. So, it is possible to reduce energy costs for separating liquid from the compressed gas.

Author Contributions: Conceptualization, V.S. (Viktor Shcherba), E.N., E.P. and A.-K.T.; method- ology, V.S. (Viktor Shcherba) and E.N.; software, E.P.; validation, V.S. (Viktor Shcherba) and E.P.; formal analysis, V.S. (Viktor Shcherba); investigation, V.S. (Viktor Shcherba); writing—original draft preparation, V.S. (Viktor Shcherba); writing—review and editing, V.S. (Viktor Shcherba), E.P. and E.N.; visualization, E.P., E.N. and A.-K.T.; supervision, V.S. (Viktor Shcherba); project administration, V.S. (Viktor Shcherba); funding acquisition, V.S. (Viktor Shcherba) and V.S. (Viktor Shalay). All authors have read and agreed to the published version of the manuscript. Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations The nomenclature of the paper is shown below: pc gas pressure in the compressor section; Tc gas temperature in the compressor section; Vc gas volume in the compressor section; Mc mass of gas in the compressor section; pw liquid pressure in the pump section; pscw liquid pressure in the beginning of suction into the pump section; Tw liquid temperature in the pump section; averaged integral temperature of the working chamber surface in the T avt compressor section; Vw liquid volume in the pump section; Vh working volume of the compressor section; actual thermodynamic parameters in the working chamber of the compressor p, V, T, M section-pressure, volume, temperature, and mass, respectively; Machines 2021, 9, 32 35 of 38

Mw liquid mass in the pump section; R gas constant; Ew liquid elastic modulus; k adiabatic index of the compressible gas; U internal energy of the compressible gas; Uc total internal energy of the compressible gas; dQc elementary amount of heat supplied to the gas; dMni elementary attachable gas mass; dMoi elementary detachable gas mass; υp instantaneous speed of piston; λt ratio of the piston stroke to twice the length of the rod; ω crank angular velocity; ϕ crank angle; τ process life; Cv specific isochoric heat capacity; Fp piston area in the compressor section; ini specific enthalpy of the attached gas mass; Tni temperature of the attached gas; Cpi specific isochoric heat capacity of the attached gas; N1, N2 number of attached and detachable gas mass; i0 specific enthalpy of the detached gas; ρw liquid density; N4, N5 number of attached and detachable liquid mass; mspr reduced valve closure mass in the compressor section; hc degree of valve closure in the compressor section; N =4 3 sum of effective forces to valve closure in the compressor section; ∑ FiK i=1 Fg gas force; Fsprc elastic force of the valve spring in the compressor section; Ffr friction force of the valve in the compressor section; Fcl current side surface of the cylinder in the compressor section; g gravitational acceleration; Sh full piston stroke; elementary change of compressor section volume during dτ time derived from dV kin kinematics of drive mechanism; Fcov heat-transfer area between gas and fluid liner; pcd discharge pressure in the compressor section; S current piston stroke; δ liquid layer thickness over the piston; Fvl valve body area; averaged heat-transfer coefficient in the working chamber of the α T compressor section; Tw averaged liquid temperature over the piston; T(F)cov, Tcl (F) thermal fields along valve body cover and fluid liner; Mw0, Vw0 mass and volume of liquid in the beginning of liquid discharge process; pwd pressure of liquid in the beginning of discharge process; Vw, Mw current volume and mass of liquid in the working chamber; dMwni, dMw0i elementary masses of attached and detached liquid; dMw elementary change of liquid mass; Vdc dead volume in the working chamber of the compressor section; αgrw, fgr flow coefficient during liquid flow and valve cross-sectional area; pi pressure in the control volume where liquid flows; Tw current temperature of the working liquid; Vdw dead volume in the pump section; d2, d1 inner and outer diameter of the piston; Vscw liquid volume in the beginning of compression process; Mscw actuation liquid mass in the beginning of compression process; dMniw i-th liquid leakage; dM0iw i-th liquid leak; Machines 2021, 9, 32 36 of 38

w1 liquid velocity in I–I cross-section matching with piston head; w2 liquid velocity in II–II cross-section located behind the discharge valve; friction losses derived by performing technical work between the ∆h T1−2 selected sections; ∆h1−2 friction losses (along the length and local losses); ∆h` friction losses; ∆hζ friction losses for local losses; α angle between the cylinder vertical and axis; mdrw derived mass of the valve closure of the self-acting valve of the pump section; hw current degree of the valve closure of the self-acting valve of the pump section;

N6 sum of forces acting on closures of self-acting valves during “pump stroke” ∑ Fwi into the compressor section; i=1 N 6 sum of forces acting on valve closure of the self-acting valve; ∑ Fiw i=1 λ f r friction coefficient along the length; λ actual volumetric efficiency; Sd linear dead space in the compressor section; SDw linear dead space in the pump section; Sw current piston stroke in the pump section; f2 discharge pipe area; fgr current discharge valve gap area; dp diameter of the discharge valve; dvl diameter of the valve closure; ρsc density of the suction gas; Vhc piston displacement volume in the compressor section; Vhp current pump section volume without considering dead volume; Vhw piston displacement volume in the pump section; ∆Vw volume of liquid from the compressor section to the discharge line; ε excentricity of the piston placement in the cylinder; pdp rated delivery pressure in the pump section; pdc rated delivery pressure in the compressor section; ad relative dead space; δ1 radial clearance in piston seal; µ coefficient of shear viscosity; lu0 initial length of the groove seal; lu current piston seal length nrev rpm per minute; elementary gas mass supplied from the suction line through suction valve into dM 3 working chamber 3 of the compressor section during dτ time; elementary gas mass supplied from the working chamber 3 into suction line of dM4 the compressor section during dτ time (gas leaks through suction valve leakages); elementary liquid capacity supplied through the discharge valve during dM 4w “pump stroke” into the compressor section; elementary gas mass supplied from the working chamber 3 of the compressor dM 6 section into the discharge line through the discharge valve during dτ time; elementary gas mass supplied from the discharge line of the compressor dM5 section into working chamber 3 during dτ time (gas overflows through discharge valve leakages); elementary liquid capacity supplied through suction valve gaps during “pump dM 6w stroke” into the compressor section per time; elementary gas mass supplied from the working chamber 3 into the groove dM 9 seal gap during dτ time; elementary gas mass supplied from the groove seal gap into working chamber dM 10 3 during dτ time; elementary liquid capacity supplied through the discharge pump gaps into the dM 14 pump section working chamber; elementary liquid mass supplied from the pump section into the groove seal dM 17 gap during dτ time; Machines 2021, 9, 32 37 of 38

elementary liquid mass supplied from the compressor section through groove dM 16 seal gap into pump section during dτ time; elementary liquid mass supplied from the suction line into working chamber 1 dM 11 of the pump section through the suction valve during dτ time; elementary liquid mass supplied from the working chamber 1 of the pump dM12 section in to suction line during dτ time (liquid overflows through suction valve leakages of the pump section); elementary mass of liquid supplied from working chamber 1 of the pump dM 15 section into discharge line during dτ time; temperature coefficient of actual volumetric efficiency of the λ T compressor section; λ0 clearance volumetric efficiency; λp throttling coefficient into compressor section; ηvol pump section volumetric efficiency; Tscc temperature of the suction gas in the compressor section; ∆Vw volume of liquid supplied into discharge line of the compressor section.

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