Analysis of the Influence of Thermophysical Properties of a Droplet Liquid on the Work Processes of a Two-Stage Piston Hybrid Power Machine

Abstract

The article considers the influence of the main thermophysical properties of the power liquid on the working processes and energy characteristics of a new highly efficient two-stage piston hybrid power machine designed to compress gas to medium and high pressures. Mathematical model of the working processes of the machine under study with a pro-filed working chamber of stage 2 has been developed. To verify the developed mathematical model of working processes, a prototype and a stand for its study were created. The visualization of the movement of the liquid piston, measurement of the instant pressure in the working chamber of the machine, energy and consumption characteristics verified the developed mathematical model. As a result of a numerical experiment, we determined that there is an optimal value of the dynamic viscosity of the power liquid, which depends on the angular velocity of the crankshaft and is in the range from 0.005 to 0.02 Pa∙s, with large values referring to large values of the angular velocity. With a decrease in the density of the power liquid, the work of friction forces decreases and the total indicator efficiency increases. The findings of the research can be used when choosing coolants in reciprocating compressors with a liquid piston.

1. Introduction

The ever-growing demand for new technology prototypes stimulates the development of research and development works, mostly in energy-intensive branches of technology. These branches include pump and compressor construction. More than 20% of the power used is spent on the drive of pumping units [1,2]. As a result, the task of increasing the efficiency of recently produced pumping and compressor units is important and urgent.

An increase in the total efficiency of a piston compressor is connected with an increase in indicator efficiency, mechanical efficiency, and transmission efficiency. Indicator efficiency can be increased in the following ways: by developing the cooling of the compressed gas [2], a decrease in the leakage of the compressed gas through the leaks of the cylinder-piston group and valves [3,4,5], a decrease in losses of work in suction and discharge by reducing the gas and valve closure oscillations [3,4,5]. An increase in the displacement ratio of a reciprocating compressor is due to a decrease in the leakage of the compressed gas, a decrease in dead space, and an increase in the density of the intake gas due to an increase in pressure (resonant or mechanical boost), as well as a decrease in the temperature of the intake gas [6]. It should be noted that the most in the indicator efficiency can be achieved by approaching compression to isothermal by intensive cooling of the compressed gas [7,8]. Currently, there are three main types of cooling used in compressors: air, water and coolant injection.

From a thermodynamic point of view, the coolant injection is the most efficient due to the developed heat exchange surface and absence of thermal resistance between the compressed gas and the coolant [9,10]. However, significant energy costs connected with liquid spraying and its separation, as well as operational difficulties, significantly slow down the widespread introduction of this type of cooling.

Air cooling is simple for practical implementation, however, due to its low efficiency, mostly it is used in general-purpose compressors with a discharge pressure of up to 1.0 MPa of low capacity.

The majority of medium and high pressure piston compressors of small, medium, and high capacity are water-cooled [11].

As a result, the idea of combining a piston compressor and a piston pump into a single unit called the “piston hybrid power machine” (PHPM) appeared. The schematic diagram of the cross-head PHPM is shown in Figure 1.

The PHPM upper part is the compressor section, and the PHPM lower part is the pumping section. One piston is used to compress and move gas and liquid; there is liquid in the gap between the piston and the cylinder, which eliminates the leakage of the compressed gas. In the first case, the pumping section can be independent and be considered a pump for pumping liquid. In this case, the compressor is cooled along the way without energy costs. In the second case, the pumping section can be considered a pump for pumping liquid through the cooling jacket. Then, the cost of driving the pump section must be added to the cost of energy to drive the compressor.

When combining the pump and compressor into a single unit, we achieve intensive cooling of the compressed gas, elimination of leaks in the cylinder–piston group, significant reduction or elimination of dead space, and reduction of friction forces in the cylinder–piston group due to the regulation of liquid friction.

All the aspects result in improved performance and overall compressor efficiency.

Combining with a compressor is useful for a pump. When combined, the suction cavitation margin increased due to the flow of liquid from the gap in the cylinder–piston group; the work supplied in the cycle decreases due to the utilization of the compressed gas heat [12].

Multistage reciprocating compressors are used to compress gas to medium and high pressures. The number of stages varies from 3 to 4, depending on the type of cooling. When using air cooling, the degree of pressure increase in the stage is within 5–6, and when using water cooling, it can be increased to 10–12. When using coolant injection, its value can be raised up to 20, even for gases with a high adiabatic index, such as helium.

One of the promising schemes for multistage gas compression to high pressures is its preliminary compression in a reciprocating compressor and final compression in the cylinder using a liquid piston [13,14].

The use of a liquid piston regulates its intensive cooling, eliminates leaks in the cylinder-piston group, and the size of the dead space. These factors lead to an increase in the indicator efficiency and the compressor delivery ratio.

Thus, this gas compression scheme needs a compressor for preliminary gas compression and a pump for its final compression. This condition is satisfied by the two-stage PHPM shown in Figure 2. The PHPM stage 1 is a single-stage crosshead power machine with piston 1 located in cylinder 2. Piston 1 divides the working volume of cylinder 2 into compressor chamber 5 and pump chamber 13. The pumping chamber is connected by pipeline 14 with stage 2 of the machine, which has liquid volume 16 (liquid piston) and gas volume 17.

The compressor section of stage 1 has suction chamber 6 and discharge chamber 7 connected through suction valve 3 and discharge valve 4 to working chamber 5. The discharge chamber 7 of stage 1 is connected by pipeline 8 to suction chamber 9 of stage 2. Suction chamber 9 is connected by suction valve 11 of stage 2 of the compressor section to working chamber 17, and the latter is connected to the pressure chamber of stage 2 by pressure valve 12. Compressed gas is supplied from discharge chamber 10. There are heat exchangers 15 and 18 to remove heat from the preliminary compressed gas and the power liquid.

The two-stage PHPM works as follows: During the upward stroke of piston 1, gas is compressed and pumped from working chamber 5 through discharge valve 4 into pumping chamber 7. At the same time, pumping chamber 13 increases and the liquid from chamber 16 of stage 2 enters chamber 13 through connecting pipeline 14. The preliminary compressed gas enters suction chamber 9 through connecting pipeline 8, and through suction valve 11 enters gas chamber 17 of stage 2, where the gas pressure decreases.

During the downward stroke of piston 1, gas is sucked from suction chamber 6 through suction valve 3 into working chamber 5. The volume of pumping chamber 13 decreases, thus the liquid enters chamber 16 through pipeline 14, its volume increases, and volume of gas chamber 17 decreases.

A decrease in chamber 17 leads to an increase in its pressure. When the discharge pressure is reached, discharge valve 12 opens and gas enters discharge chamber 10.

When liquid in the pumping section of stage 1 exceeds the volume of the working chamber of stage 2, the excess liquid enters the gas cap installed on the connecting pipeline. In the considered machine, there is no gas cap and the volume of stage 1 pumping section is equal to the working volume in stage 2 compressor section.

Thus, the two-stage PHPM provides intensive cooling in stages 1 and 2, elimination of leaks in their cylinder-piston groups, elimination of dead space in the stages of the machine, which provides a high indicator efficiency and compressor delivery coefficient at high weight and dimensional characteristics.

Completed works on the study of reciprocating compressors with a liquid piston [15,16,17] were aimed at studying the processes of heat transfer and its effect on the compression efficiency. In [18], we analyzed the effect of inserts from a porous medium on the operation of a liquid reciprocating compressor. The influence of the speed of the liquid piston on the working processes of the compressor was studied in [19]. The authors of [20] considered the influence of the main size and operational parameters on the dynamics of liquid movement and work processes in a reciprocating compressor with a liquid piston.

The references review shows that mostly the influence of size, operational, and thermal parameters on the operation of a liquid piston was studied. However, studies on the influence of the main thermophysical properties of a liquid on the dynamics of the movement of a liquid piston could not be found.

A significant influence on the dynamics of the motion of a droplet liquid, which ensures gas compression in stage 2 of the compressor, is exerted by its main thermophysical properties: density, viscosity, coefficient of volumetric compression, and coefficient of thermal expansion. Taking into account that the effective pressure drop on the liquid is not more than 10 MPa, and the modulus of elasticity of the droplet liquids is about 2000 MPa, the effect of the compressibility of the liquid on the working processes in the machine can be neglected. The performed theoretical studies established that, in the range of the investigated pressures (up to 15 MPa), the compressibility of a liquid in a two-stage PHPM under the action of gas forces, inertial forces, and resistance forces does not exceed 1%, and the decrease in the instant velocity of the liquid piston is within 5%. Thus, we assume that the liquid is incompressible.

It should be noted that the temperature coefficient β_T for dripping liquids is higher than the volumetric compression ratio β_p, but its absolute value remains small. So, for water, it is 0.63$\times$10⁻⁴ K⁻¹. It should be noted that low absolute β_p and β_T even if they are changed several times will not have a significant effect on the working processes of the machine.

The density ρ_w and the dynamic viscosity of the liquid μ_w are different. Values ρ_w and μ_w change significantly both when changing thermodynamic parameters, primarily temperature, and when using various liquids. So, for example, the dynamic viscosity of water at T = 293K is μ_w = 1004$\times$10⁻³ Pa·s, but the dynamic viscosity of industrial oil M-20A at T = 313K is μ_w = 26.7$\times$10⁻³ Pa·s, i.e., 26 times greater. We observe similar trends with the density of the liquid.

This research is devoted to the analysis of the influence of the density and viscosity of the working fluid on the working processes and integral characteristics (indicative power and pressure loss in connecting communications) of a two-stage PHPM. The issue of designing and manufacturing a prototype of a piston hybrid power machine with a liquid piston, designed to compress gas to medium and high pressures, is considered; the physical picture of the working processes in the machine under study is studied. With the developed mathematical model of the working processes of the prototype, the influence of the main thermophysical properties of the working fluid: dynamic viscosity and density on the working processes and energy characteristics of the new two-stage PHPM will be considered.

2. Theoretical Research

2.1. Determination of Basic Geometric Parameters

For a given capacity and number of revolutions, using well-known techniques [11], the diameter of stage 1 piston, stage 1 piston stroke, the main parameters of the self-acting valves, and the diameters of the interstage pipelines are determined. In our case, we coordinate the operation of stage 1 of the pump section and stage 2 of the compressor section. In this case, we determine the volume of the working chamber of stage 2 of the compressor section and the diameter of the stem in the pump section. In addition, to improve the operation of stage 2 of the PHPM, profiling the working chamber has been completed.

2.1.1. Determination of Working Chamber of Stage 2

We determine the amount of intake gas into stage 2 of the compressor, neglecting leaks and overflows of gas in stage 1 of the compressor, from the equality of the masses of gas entering stages 1 and 2. Assuming that the compressible gas follows the laws of an ideal gas, we obtain:

$$V_{sc2}=V_{sc1}\frac{T_{sc2}}{T_{sc1}}\frac{P_{sc1}}{P_{sc2}}$$

(1)

Considering that $V_{sc1}=V_{h1}/{\lambda }_1$, but $V_{sc2}=V_{h2}/{\lambda }_2$ Equation (1) is transformed to:

$$V_{h2}=V_{h1}\frac{{\lambda }_2}{{\lambda }_1}\frac{T_{sc2}}{T_{sc1}}\frac{P_{sc1}}{P_{sc2}}$$

(2)

Values $T_{sc2}$ and $P_{sc2}$ can be determined as $T_{sc2}=T_{sc1}+\Delta T_{H0}$, $P_{sc2}=P_{D1}-\Delta P_c$.

In view of the above, Equation (2) is transformed to:

$$V_{h2}=V_{h1}\frac{{\lambda }_2}{{\lambda }_1}\left(1+\frac{\Delta T_{HO}}{T_{sc1}}\right)\left(\frac{1}{{\epsilon }_1-\frac{\Delta P_c}{P_{sc1}}}\right)$$

(3)

Values ${\lambda }_1$,${\lambda }_2$ and $\Delta T_{H0}$, $\Delta P_c$ are determined on well-known recommendations [11].

The working volume of the pumping section of stage 1 $V_{cp1}$ equals to the working volume of stage 2. Thus, the rod diameter is determined as:

$$d_r=\sqrt{d_1^2-\frac{4V_{h2}}{\pi S_{h1}}}$$

(4)

2.1.2. Profiling of Stage 2 Working Chamber

When designing and operating compressor units with a liquid piston, a number of measures are used to improve the kinematics and integrity of the liquid piston by reducing foaming, crushing, and carrying part of the liquid through the discharge valve out.

For this purpose, the compressor chamber is profiled (the working chamber of stage 2 of the PHPM compressor section) and devices for stabilizing the movement of liquid of various designs are used, for example, made in the form of movable and stationary permeable solid inserts and elements.

The main reason for the liquid piston destruction is the effect on the liquid of inertial forces caused by the acceleration of the pump piston supplying liquid to the working chamber. In order to reduce inertial forces, it is necessary to bring the liquid velocity in the working chamber closer to a certain constant value, for example, the average piston speed in the pump.

To achieve this goal, it is necessary to profile the working chamber, i.e., change the flow area (F) along the length of the working chamber (S).

Currently, various profiling techniques have been developed based on different optimization criteria or their combination [21]. In practice, it is advisable to use working chambers obtained using simple bodies of revolution: a cylinder and a cone (see Figure 3).

In [21], after analyzing the working processes and operating features of stage 2 of the high-pressure compressor, it was proposed to replace the cylindrical chamber (position “a” Figure 3) with a combined one (position “b” Figure 3) having two conical parts and one cylindrical. Conducted theoretical studies [21] established that the working chamber, consisting of two truncated cones, provides a high heat exchange surface, good stabilization of the liquid piston speed with high ergonomic parameters.

Considering the above, we use the working chamber of stage 2 consisting of two truncated cones (see Figure 3, position “c”), its working volume can be determined as:

$$V_{h2}=2S_1\frac{1}{3}\pi \left[R_1^2+R_1\cdot R_2+R_2^2\right]$$

(5)

The ratio of the height of the cone to the diameter of its base, in accordance with [15] is to be chosen in the range from 10 to 20.

2.2. Review of Mathematical Models and System of Accepted Assumptions

The two-stage piston hybrid power machine has stages 1 and 2 of compression and inter-stage connections for gas and liquid. Stage 1 of the investigated machine is a single-stage piston hybrid power machine. In [22], the mathematical model of working processes of a cross-head PHPM with a smooth seal is considered for the first time. The works [13,23] are devoted to the development of a mathematical model of the working processes of a cross-head PHPM with a gas chamber in the pump section. In these works, the simultaneous compression of liquid and gas in the working processes of a piston pump was considered for the first time.

The developed mathematical models are based on the following basic equations:

• for the chambers of the compressor section: the first law of thermodynamics of a body of variable mass, the equation of conservation of mass, the equation of volume change, the equation of state, the equation of the dynamics of motion of the closure body of the self-acting valve;
• for the working chamber of the pump section: the energy conservation equation written in the form of the Bernoulli equation; Hooke’s equation, volume conservation equation, mass conservation equation. In [24,25], a general approach was developed for calculating work processes in the working chamber of a compressor and a positive displacement pump.

It should be noted that the developed models simulate the working processes in the compressor section, pump section, piston seal, as well as the flow of gas and liquid in the connecting pipelines.

When calculating the gas flow in the connecting pipelines of the compressors, we use a system of equations including the continuity equation, the equation of motion, and the equation for the conservation of internal and kinetic energy. The equation of state for an ideal gas is used as the equation of state. One of the effective methods for the numerical solution of such a system of partial differential equations as for single-phase flow or two-phase flow is the “large particles” method [7].

Currently, when modeling work processes in machine chambers, both distributed-parameter and lumped-parameter models are used. Taking into account the small linear dimensions of the working chambers of the displacement machines [11], as well as significant mathematical difficulties and implementation time, the researchers use models with lumped parameters. As a result, the use of mathematical models with lumped parameters in the above works is verified and useful. It seems reasonable to use a mathematical model with lumped parameters when developing a mathematical model of a two-stage PHPM.

When considering the flow of gas and liquid in the connecting pipes of the machine, a one-dimensional flow is considered either in an unsteady or in a quasi-stationary setting [26]. If the pipeline is simple, without local resistance, and a mathematical model with distributed parameters is used to describe the working processes, we use a non-stationary one-dimensional flow model. If there are complex pipelines with closure and control valves, as well as when using a mathematical model with lumped parameters to describe thermogasdynamic processes in machine chambers, we use a quasi-stationary model for calculating the flow of gas and liquid in connecting pipes.

System of accepted basic assumptions:

• taking into account the above, we accept a mathematical model with lumped parameters to simulate the working processes of a two-stage PHPM;
• taking into account that the mathematical models of the working processes of single-stage cross-head and crosshead-free PHPMs are well developed, we consider the working processes in stage 2 of the PHPM and the liquid movement from the pump section of stage 1 through the connecting pipeline to stage 2;
• the thermodynamic gas parameters in stage 2 suction chamber are constant and equal to their rated values;
• the motion of the liquid at each moment of time is steady and stationary, i.e., the motion of the liquid is quasi-stationary;
• the working liquid has constant temperature and follows Newton’s laws;
• the thermophysical properties of the liquid are constant;
• coefficients of friction along the length and local coefficients of resistance obtained for stationary conditions are also valid for unsteady flow;
• the bottom of the liquid piston is a plane parallel to the ground;
• there is no droplet liquid in the compressed gas in stage 2 of the PHPM;
• compressed gas follows the laws that are true for an ideal gas;
• the simulated processes in a compressible gas are equilibrium and reversible;
• the change in kinetic and potential energy for a compressible gas can be neglected.
• the movement of the closure body of a self-acting valve is considered as the movement of a material point;
• the walls of the pipeline and working chambers are considered absolutely rigid.

2.3. Mathematical Model of the Working Processes of a Two-Stage PHPM

The working chamber of stage 2 consists of two truncated cones (see Figure 4), having the same geometric dimensions and located symmetrically about the axis passing through the center of the working chamber and perpendicular to the vector of the liquid velocity. Thus, we have a working chamber consisting of a diffuser and confuser parts, when the liquid moves in both forward and reverse directions.

The system of differential equations describing the change in the thermodynamic parameters of gas in stage 2 of the PHPM and the motion of liquid includes: the first law of thermodynamics of a variable mass body for an open thermodynamic system; the equation of conservation of mass, the dynamics of motion of the closure body of the self-acting valve, the equation of state for a compressible gas, as well as the equation of conservation of energy and the equation of continuity for the liquid phase, can be written as:

$$d{U}_g=d{Q}_g-p_{g}d{V}_g+{\displaystyle \sum }_{i=1}^{N_1}{i}_{ni}d{M}_{ni}-{\displaystyle \sum }_{i=1}^{N_2}{i}_{0}d{M}_{0i}$$

(6)

$$d{M}_g={\displaystyle \sum }_{i=1}^{N_1}d{M}_{ni}-{\displaystyle \sum }_{i=1}^{N_2}d{M}_{0i}$$

(7)

$$m_{rdv}\frac{d^2{h}_c}{d{\tau }^2}={\displaystyle \sum }{F}_i$$

(8)

$$d{p}_g=\left(k-1\right)\left(\frac{d{U}_g}{V_g}-\frac{U_g}{V_g^2}d{V}_g\right)$$

(9)

$$d{T}_g=\frac{1}{k}\left(p_{g}d{V}_g+V_{g}d{p}_g\right)-T_{g}d{M}_g$$

(10)

$$d{z}_w+\frac{d{p}_w}{{\rho }_{w}g}+d\left(\frac{v_w^2}{2g}\right)+d\left(h_l+h_{\zeta }+h_{in}\right)=0$$

(11)

$$d{Q}_w=d\left(V_w{f}_w\right)=0$$

(12)

When determining the elementary amount of heat removed from the compressed gas, the Newton–Richmann hypothesis was used:

$$d{Q}_g={\alpha }_w{\displaystyle \sum }_{i=3}^{N_3=3}{F}_{ni}\left({\overline{T}}_w-T_g\right)d\tau$$

(13)

The total heat transfer surface during gas compression can be defined as:

$${\displaystyle \sum }_{i=1}^{N_3=3}{F}_{ni}=F_p+F_b+F_{vp}$$

(14)

Piston surface area F_p and valve plate F_vp can be defined as $F_p=\pi {\left(R_1+z_{21}tg\alpha \right)}^2$, $F_{vp}=\pi R_1^2$.

The side surface of the working chamber (see Figure 4) can be determined as

$$F_b=\pi \left(R_2+R_1\right)·\frac{S_1}{cos\alpha }+\frac{\pi \left(R_2+R_1+z_{21}·tg\alpha \right)\left(S_1-z_{21}\right)}{cos\alpha }$$

Heat transfer coefficient ${\alpha }_w$, in general for a reciprocating compressor is determined on the experimental studies. In most practical cases, an empirical ratio is used to determine the heat transfer coefficient in the cylinder of a single-acting compressor $N_u=f\left(A, B,R_e\right)$ [11]. The piston speed is used as the determining speed in defining the Reynolds number. In our case, we use the average mass velocity of the liquid in the working chamber of stage 2 as the determining speed.

The average integral value of the surface temperature of the working chamber can be determined as:

$${\overline{T}}_w=\frac{{\overline{T}}_{vp}{F}_{vp}+F_b{\overline{T}}_b+F_p·{\overline{T}}_w}{{{\displaystyle \sum }}_{i=1}^{N_3}{F}_{ni}}$$

(15)

When determining the mass flows of the attached (dM_ni) and detached (dM_0i) through the gas distribution bodies, the San Venant–Wanzel equation was used for a one-dimensional isentropic flow in a quasi-stationary setting. Changing working chamber dV_g was limited by a change in the volume in the pumping section of stage 1 in the absence of dropping liquid compressibility.

When integrating Bernoulli Equation (11), two sections were used: the first section (I-I) was aligned with the lower boundary of the mechanical piston of stage 1, and the second section (II-II) was aligned with the surface of the liquid piston.

The first coordinate can be determined as:

$$z_1=S_{ld}+\frac{S_h}{2}\left[\left(1-cos\phi \right)+\frac{{\lambda }_m}{4}\left(1-cos2\phi \right)\right]$$

(16)

The second coordinate can be determined as:

• the height of the surface of the liquid piston is less than the height of the truncated cone (S1), then $z_2=z_{20}+z_{21}$. Value z₂₁ is determined from the solution of the following nonlinear volume conservation equation

  $$\frac{1}{3}\pi z_{21}\left[R_1^2+R_1·\left(R_1+z_{21}tg\alpha \right)+{\left(R_1+z_{21}tg\alpha \right)}^2\right]=\left\{Sh-\frac{S_h}{2}\left[\left(1-cos\phi \right)+\frac{{\lambda }_m}{4}\left(1-cos2\phi \right)\right]\right\}·F_1$$

  (17)
• the surface of the liquid piston is greater than the height of the truncated cone (S₁), then $z_2=z_{20}+S_1+z_{22}$. Value z₂₂ is determined from the equation of conservation of the liquid volume

  $$\frac{1}{3}\pi S_1\left(R_1^2+R_1·R_2+R_2^2\right)+1/3\pi z_{22}·\left[R_2^2+R_2\left(R_2-z_{22}tg\alpha \right)+\left(R_2-z_{22}tg\alpha \right)\right]=\left\{S_h-S_h/2\left[\left(1-cos\phi \right)+{\lambda }_m/4\left(1-cos2\phi \right)\right]\right\}·F_1$$

  (18)

The liquid velocity in the first section V₁ is determined by the kinematics of stage 1 piston drive mechanism, and the liquid velocity in the second section V₂ is determined based on the continuity equation. Total head losses consist of head losses along the length $\Delta h_l$, head loss on local resistance $\Delta h_{\zeta }$ and inertial head losses $\Delta h_{in}$ ($\Delta h_{\mathsf{\Sigma }}$=$\Delta h_l$+$\Delta h_{\zeta }$+$\Delta h_{in}$).

We determine the values included in the total head losses $\Delta h_{\mathsf{\Sigma }}$.

Head losses along the length ($\Delta h_l$) are determined from the head losses along the length in the pumping chamber of stage 1, head losses along the length in pipeline 3 and head losses along the length in the profiled working chamber of stage 2 $\Delta h_l=\Delta h_{l1}+\Delta h_{l\mathrm{pp}}+\Delta h_{l2}$.

Head losses along the length $\Delta h_{l1}$ and $\Delta h_{lpp}$ can be determined based on the Darcy–Weisbach law as follows $\Delta h_{l1}={\lambda }_{c1}\left(z_1/d_1\right)\left(v_1^2/2g\right)$, $\Delta h_{l\mathrm{pp}}={\lambda }_{cpp}\left(l_{\mathrm{pp}}/d_{\mathrm{pp}}\right)\left(v_{\mathrm{pp}}^2/2g\right)$.

Length friction coefficients ${\lambda }_{c1}$, ${\lambda }_{cpp}$, ${\lambda }_{c2}$ in general, are a function of the Reynolds number and relative roughness and is determined on the known recommendations.

The compression chamber of stage 2 consists of a diffuser and a confuser, and the head losses can be determined in them as follows:

• for the diffusor

  $$\Delta h_{l2}=\frac{{\lambda }_{c2}}{8\sin \alpha }\left(1-\frac{1}{n_g^2}\right)\frac{v_{21}^2}{2g}$$

  (19)

  where $n_g={\left(R_1+z_{21}\cdot tg\alpha \right)}^2/R_1^2$ is diffuser expansion ratio; $v_{21}=v_1{F}_1/\left(\pi R_1^2\right)$ is liquid velocity in the initial section of the diffuser.
• for the confusor

  $$\Delta h_{l2}=\frac{{\lambda }_{c2}}{8\sin \alpha }\left(1-\frac{1}{n_K^2}\right)\frac{v_{22}^2}{2g}$$

  (20)

  where $n_k=R_2^2/{\left(R_2-z_{22}tg\alpha \right)}^2$ is the degree of narrowing of the confuser; $v_{22}=v_1{F}_1/\left(\pi \left[R_2-z_{22}tg\alpha \right]\right)$ is liquid velocity in confuser.

The head losses on local resistances $\left(\Delta h_{\zeta }\right)$ depend on the direction of the liquid flow, as when moving forward the local resistance is a sudden narrowing, and with the reverse liquid movement—there is a sudden expansion.

As a consequence, when determining local resistances, it is necessary to take into account the direction of the piston movement.

All local resistances can be divided into four types:

• local resistance when connecting stage 1 pumping section to connecting pipeline 3—sudden narrowing (${\zeta }_1$);
• local resistance when flow turns (${\zeta }_2$);
• local resistance when connecting pipeline 3 to the working chamber of stage 2—sudden expansion (${\zeta }_3$);
• constant expansion in the diffuser part of the working chamber ${\zeta }_g$ and constant narrowing in the confuser part of the working chamber ${\zeta }_k$.

In sudden narrowing, ${\zeta }_1$ can be determined as:

$${\zeta }_1=0.5\left(1-n_{\zeta }\right)$$

(21)

where $n_{\zeta }={\omega }_2/{\omega }_1$ is the ratio of the areas of flow sections after narrowing and before narrowing. We assume that ${\zeta }_1=0.5$, as ${\omega }_1\to \infty$.

When turning the pipe, the value ${\zeta }_2$ is determined as [21]:

$${\zeta }_2=0.05+0.2\frac{d_{\mathrm{pp}}}{R_{\mathrm{pp}}}$$

(22)

where $R_{\mathrm{pp}}$ is bending radius of the pipeline (see Figure 4).

Local resistances during sudden expansion are determined by the Borda formula. Local resistances in the diffuser and confuser parts are determined according to [26].

Inertial head losses $\Delta h_{in}$ consist of inertial head losses in stage 1 of the pump section in the connecting pipeline and in stage 2 Δh_in=Δh_in1+Δh_inpp+Δh_in2.

In stage 1 of the pump section and in the connecting pipeline, the determination of the inertial head losses does not depend on the direction of liquid movement. Their value can be defined as $\Delta h_{\mathrm{in}1}=\frac{a_n}{g}{z}_1$; $\Delta h_{in\mathrm{pp}}=\frac{a_{\mathrm{pp}}}{g}{l}_{\mathrm{cpp}}$.

We consider the inertial head losses in stage 2 of the PHPM.

When the liquid piston moves up:

• At z_2i < S₁ the inertial head losses are determined as:

  $$\Delta h_{in2}=\frac{1}{g}{{\displaystyle \int }}_{R_1}^{R_i}\frac{a_i}{tg\alpha }dR$$

  (23)

  where $R_i=R_1+z_{21}\cdot tg\alpha$; $a_i=\frac{F_1}{f_i}{a}_n-\frac{F_1{v}_1}{f_i^2}\omega \frac{d{f}_i}{d\varphi }$; $f_i=\pi R_i^2$.
• At z_2i > S₁

  $$\Delta h_{in2}=\frac{1}{g}{{\displaystyle \int }}_{R_1}^{R_2}\frac{a_i}{tg\alpha }dR+\frac{1}{g}{{\displaystyle \int }}_{R_2}^{R_i}\frac{a_i}{tg\alpha }dR$$

  (24)

  where, $R_i=R_2-z_{22}\cdot tg\alpha$.

When the piston moves down, Δh_in2 is defined similarly.

3. Experimental Studies

The main purpose of the experiment is to check the performance of the two-stage PHPM, to gain new knowledge about its work, and to verify the developed mathematical model.

3.1. Subject of Research, Stand, and Measuring Equipment

A schematic diagram of a two-stage PHPM was previously shown in Figure 4. To exclude the influence of gas pulsations from the first stage, a 25 L receiver was additionally installed between the compressor stages.

The developed prototype of the two-stage PHPM had basic dimensions,

Table 1. Main dimensions PHPM.

Indicator Description	Values
The diameter of the first stage piston	dp = 0.050 m
Diameter of the rod of the first stage	dr = 0.036 m
Full stroke of the 1st stage piston	Sh = 0.050 m
Initial coordinate of the liquid in the second stage	Z20 = 0.100 m
Diameter of the supply pipeline	dsp = 0.016 m
The length of the supply pipeline	lp = 0.5 m
Linear dead space	Sd = 0.0025 m
Turning radius of the supply pipeline	Rp = 0.5 m

.

The basic geometrical dimensions of the second stage are shown in Figure 5a.

The profiling of the working chamber of the second stage was carried out by installing the insert in the cylinder. The insert consisted of two cones connected at their vertices. Some accesses are made on the upper cone to ensure the passage of gas to the suction and discharge ports, and the base of the lower cone is made conical to improve the flow of liquid around it. The 3D view of the redesigned second stage is shown in Figure 5b. A photograph of the finished layout of the modernized second stage is shown in Figure 5c. Lobe valves were used as gas distribution bodies in the compressor section. To visualize the dynamics of the movement of the liquid piston, the second stage of the PHPM was made of Plexiglas, and the value of the discharge pressure was limited.

Table 2. Size dimensions of self-acting valves of stage 2 compressor section.

Indicator Description	Values
Maximum lift height of the shut-off element of the suction valve in the compressor section	hscvmax = 0.0015 m
The width of the passage in the seat of the suction valve of the compressor section	dscvc = 0.023 m
Spring stiffness of the suction valve in the compressor section	csscv = 3200 N/m
Maximum lift height of the shut-off element of the discharge valve in the compressor section	hdvmax = 0.0015 m
The width of the passage in the seat of the discharge valve of the compressor section	ddvc = 0.023 m
Spring stiffness of the discharge valve in the compressor section	csdv = 3200 n/m

shows the main dimensions of self-acting valves in stage 2 compressor section.

Working bodies used during experimental research are air, mineral oil (MGE-46V).

The prototype was tested on the developed stand; its general view is shown in Figure 6. A general view of the prototype installed on the stand is shown in Figure 7a. An image of the prototype on the stand is shown in Figure 7b.

To change smoothly the rotational speed of the drive shaft of the machine, a volumetric hydraulic drive is used, which includes a variable displacement axial piston pump model 313.3.56.804 with a discharge pressure 6.3 MPa and an axial piston hydraulic motor model 310.3.56.01.03.V.U.

Measurement of static pressures is performed by manometers, instantaneous pressures are measured with strain gauges.

Suction and discharge temperatures are measured by TW-N PT100 sensors, the temperature of the liquid in the tanks is controlled by DTS-035-50M.B3 sensors.

The air flow rate at the suction of the first stage of the compressor is measured by the “Vector-04” model flow meters; we used the PSE530-M5-l sensors to measure the suction and discharge pressures.

The hydraulic and pneumatic lines of the stand are equipped with devices for liquid and gas purification, safety devices, and control equipment to operate the tested machine in different modes.

3.2. Verification of the Mathematical Model of Work Processes

One of the main tasks of the experimental research of the developed machine is to verify the mathematical model. The mathematical model verification was carried out at different values of the discharge pressure in stage 2, the suction pressure in stage 2 (discharge pressure in stage 1) and the crankshaft revolutions. An estimate of the experimental error and a comparison of the results of theoretical and experimental studies were published in [27].

It should be noted that the developed mathematical model describes the friction and inertia forces in moving liquid, as the data of indicator diagrams obtained experimentally and theoretically match; the maximum and minimum values of instantaneous pressures match as well.

Thus, the discrepancy in determining the instantaneous pressure in the machine cavities is within 10–15%.

4. Results of a Numerical Experiment and Their Discussion

To carry out a numerical experiment, we choose a two-stage piston hybrid power machine with a profiled working chamber shown in Figure 8. The main geometrical and operational parameters are shown in

Table 3. Main geometrical and operational parameters of two-stage PHPM.

Indicator Description	Values
The diameter of stage 1 piston	d1 = 0.050 m
Diameter of the stem of stage 1	dst = 0.036 m
Full stroke of stage 1 piston	Sh1 = 0.050 m
The number of revolutions of the crankshaft	nrev = 500 rpm
Suction pressure	Psc2 = 5 × 105 Pa
Discharge pressure	Pd2 = 25 × 105 Pa
The average temperature of the walls of the surface of the working chamber of stage 2	${\overline{T}}_w$ = 330 K
Intake air temperature	Tsc2 = 310 K
Initial coordinate of the liquid in stage 2	Z20 = 0.100 m
Diameter of the supply pipeline	dpp = 0.016 m
The length of the supply pipeline	lpp = 0.5 m
Linear dead space	Sds = 0.0025 m
Turning radius of the supply pipeline	Rpp = 0.5 m
The radius of the conical insert in the design scheme of stage 2 of the PHPM (see Figure 8)	R21 = 0.003 m, R11 = 0.015 m
The radii of the conical working chamber in the given design scheme of stage 2 of the PHPM:	R1 = 0.0132 m, R2 = 0.01977 m
The angle of inclination of the generatrix of the truncated cone	α = 0.235 rad
Truncated cone height	S1 = 0.0272 m
The maximum lifting height of the closure body of the discharge valve	hdmax = 0.001 m
The maximum lifting height of the closure body of the suction valve	hscmax = 0.001 m
The width of the passage in the saddle of the suction valve	dscv = 0.09 m
The width of the passage in the saddle of the discharge valve	ddv = 0.09 m
Spring rate of the discharge valve	Csprd = 16,280 N/m
Spring rate of the suction valve	Csprsc = 5980 N/m
Maximum clearance between the saddle and the closure body of the suction valve	δsc = 1 × 10−8 m
Minimum clearance between the saddle and the closure body of the discharge valve	δd = 1 × 10−8 m
Roughness of the inner surface of the connecting pipeline	K = 0.000005 m
Compressed gas	air

.

As independent variables, we take the above-stated coefficient of dynamic viscosity and density of the liquid.

The dynamic viscosity coefficient varied from 0.0005 Pa to 0.04 Pa, and the density of the liquid varied from 600 kg/m³ to 1400 kg/m³.

In carrying out the numerical experiment, we used the classical design with fractional replicas.

All response functions can be divided into two large groups. The first group is made up of functions that determine the flow of liquid in the machine: $\Delta {\overline{p}}_1=\Delta p_1/p_{\mathsf{\Sigma }}$; $\Delta {\overline{p}}_2=\Delta p_2/p_{\mathsf{\Sigma }}$; $\Delta {\overline{p}}_3=p_3/p_{\mathsf{\Sigma }}$, where $\Delta p_{\mathsf{\Sigma }}=\Delta p_1+\Delta p_2+\Delta p_3; \Delta p_1={{\displaystyle \int }}_0^{2\pi }\left|z_2-z_1\right|{\rho }_{w}gd\phi$; $\Delta p_2={{\displaystyle \int }}_0^{2\pi }{\rho }_w\left|\left({\alpha }_2{V}_2^2\right)/2-\left({\alpha }_1{V}_1^2\right)/2\right|d\phi$; $\Delta p_3={{\displaystyle \int }}_0^{2\pi }{\rho }_{w}g\left|\Delta h_l+\Delta h_{\zeta }+\Delta h_{in}\right|d\phi$.

$\Delta {\overline{p}}_1$ determines the ratio of the geometric head to the sum of the geometric head, velocity head, and total head losses per cycle. $\Delta {\overline{p}}_2$ determines the ratio of the velocity head to the sum of the geometric head, velocity head, and total head losses per cycle. $\Delta {\overline{p}}_3$ determines the ratio of total head losses to the sum of geometric head, velocity head, and total head losses per cycle. $\Delta {\overline{h}}_l=\left({{\displaystyle \int }}_0^{2\pi }\left|\Delta h_l\right|d\phi \right)/\Delta h_{\Sigma }$–the ratio of head losses to hydraulic resistance along the length to the total head losses per cycle; $\Delta {\overline{h}}_{\zeta }=\left({{\displaystyle \int }}_0^{2\pi }\left|\Delta h_{\zeta }\right|d\phi \right)/h_{\Sigma }$–the ratio of head losses to local resistances to total head losses per cycle; $\Delta {\overline{h}}_{i{n}_{\Sigma }}=\left({{\displaystyle \int }}_0^{2\pi }\left|\Delta h_{in\Sigma }\right|d\phi \right)/\Delta h_{\Sigma }$–ratio of inertial losses, taken in modulus, to total head losses per cycle, where $\Delta h_{\Sigma }={{\displaystyle \int }}_0^{2\pi }\left(\left|\Delta h_l\right|+\left|\Delta h_{\zeta }\right|+\left|\Delta h_{i{n}_{\Sigma }}\right|\right)d\phi$.

The second group is made of functions determining the efficiency of stage 2 of the PHPM.

Indicator work of the cycle in stage 2 is $L_{ind}={\displaystyle \oint }{V}_{2}dp$.

Technical work to overcome the frictional forces of the liquid in the machine $L_{fr}={\displaystyle \oint }{\rho }_{w}g\left(\left|\Delta h_l\right|+\left|\Delta h_{\zeta }\right|\right)dV$.

The ratio of the work of friction forces to indicator work ${\overline{L}}_{fr}=L_{fr}/L_{ind}$.

Indicator isothermal efficiency of stage 2 of PHPM is:

$${\eta }_{\mathit{ind}.\mathrm{is}}=\frac{G_{cp}R{T}_{sc}\ln \left(p_d/p_{sc}\right)}{L_{ind}}$$

(25)

Full indicator efficiency of stage 2 of PHPM:

$${\eta }_{\mathit{ind}.f}=\frac{G_{cp}R{T}_{sc}\ln \left(p_d/p_{sc}\right)}{L_{ind}+L_{fr}}$$

(26)

4.1. Analysis of the Influence of the Coefficient of Dynamic Viscosity

It should be noted that when analyzing the effect of liquid viscosity on the working processes of a two-stage PHPM, the total speed of liquid movement in the machine, including the pump section of the right-hand stage, the connecting pipeline and stage 2, as well as the liquid velocity only in the connecting pipeline, is of great importance, since it is the main source of hydraulic losses.

In the first case, the change in the total speed will be achieved by changing the angular speed of rotation, and in the second case—by changing the diameter of the connecting pipeline.

With a decrease in the dynamic viscosity coefficient μ_w at a constant density of the medium, the Reynolds number increases, the flow turbulization increases, leading to an increase in the hydraulic friction coefficient and an increase in the relative hydraulic losses along the length $\Delta {\overline{h}}_l$ and the value of the relative pressure losses per cycle $\Delta {\overline{p}}_3$ (see Figure 9 and Figure 10).

It should be noted that a sharp increase in $\Delta {\overline{h}}_l$ and $\Delta {\overline{p}}_3$ is observed from the value of the coefficient of dynamic viscosity equal to 0.0008 Pa∙s. Up to this value, we observe a linear drop in these values in the range of variation of μ_w from 0.02 Pa∙s to 0.0008 Pa∙s, which is fully consistent with the theory of laminar liquid motion. Against the sharp increase of $\Delta {\overline{h}}_l$ and $\Delta {\overline{p}}_3$, there is a sharp decrease in $\Delta {\overline{p}}_1$ and $\Delta {\overline{h}}_{in\mathsf{\Sigma }}$ with a smooth decrease in $\Delta {\overline{p}}_2$.

In the range of μ_w variation from 0.02 Pa∙s to 0.005 Pa∙s, a decrease in the work of friction forces $L_{fr}$ is observed and with a further decrease in μ_w the value $L_{fr}$ increases (with a small local maximum) to 6.17 J, which is 14.89% (see Figure 11) from the indicator work of compression: the maximum value of the total indicator efficiency is 73.2% at μ_w = 0.005 Pa∙s and its value drops to 64.2% at μ_w = 0.0005 Pa∙s, i.e., by almost 10%. Values $L_{\mathit{ind}}$ and ${\eta }_{\mathit{ind}.\mathrm{is}}$ remain constant when changing μ_w.

Thus, the minimum $L_{\mathit{ind}}$ and ${\eta }_{\mathit{ind}.f}$ does not coincide with the minimum $\Delta {\overline{h}}_l$ and $\Delta {\overline{p}}_3$ and is shifted towards large values of μ_w.

4.2. Influence of the Angular Velocity of the Crankshaft

With an increase in the angular velocity of the crankshaft, the speed of liquid movement in the stages of the machine and in the connecting pipeline increases. This leads to an increase in $\Delta {\overline{p}}_3$ and $\Delta {\overline{h}}_l$ while maintaining the previously described physical picture. The onset of flow turbulization and a sharp increase in the hydraulic friction coefficient along the length is shifted towards large values of μ_w.

Thus, if at n_rev = 300 rpm, minimum $\Delta {\overline{h}}_l$ is achieved at μ_w = 0.005 Pa∙s, then at n_rev = 500 rpm minimum $\Delta {\overline{h}}_l$ is achieved at μ_w = 0.010 Pa∙s, and at n_rev = 750 rpm minimum $\Delta {\overline{h}}_l$ is achieved at μ_w = 0.02 Pa∙s (see Figure 12).

With an increase in n_rev and an increase in $\Delta {\overline{h}}_l$ there is an increase in $L_{fr}$ and the total indicator efficiency decreases. Maximum ${\eta }_{\mathit{ind}.f}$ shifts towards larger values of μ_w while simultaneously decreasing it in absolute terms. The maximum ${\eta }_{\mathit{ind}.f}$ and minimum $L_{fr}$ points coincide with the minimum points $\Delta {\overline{h}}_l$. Absolute decrease in the maximum value of the indicator efficiency is 0.041 or 5.93%.

It should be noted that at the number of revolutions n_rev = 750 rpm and μ_w = 0.0005 Pa∙s, the value of the work of the friction forces $L_{fr}$ slightly exceeds the work for gas compression in the second stage of the machine $L_{\mathit{ind}.\mathrm{is}}$ and ${\eta }_{\mathit{ind}.f}$ drops to 0.362, i.e., decreases by almost half.

4.3. Influence of the Diameter of the Connecting Pipe

It should be noted that the connecting pipeline is of decisive importance in the balance of total hydraulic losses along the length. This led to its consideration.

A change in the diameter of the connecting pipeline leads to a change in the speed in it and, therefore, the physical picture of working processes and changes in energy characteristics will be similar to that considered. As a result, we consider the working processes of the machine under study and its energy characteristics at constant μ_w and n_rev and a variable value of the pipeline diameter (d_pp).

With an increase in the diameter of the pipeline (d_pp), velocity of the liquid in it decreases, which leads to a decrease in the hydraulic resistance and an increase in $\Delta {\overline{p}}_3$ with a simultaneous increase in $\Delta {\overline{p}}_1$ and $\Delta {\overline{p}}_2$.

It should be noted that when d_pp changes from 0.014 m to 0.022 m, there is a linear drop in $\Delta {\overline{p}}_3$ from 0.9924 to 0.981 (see Figure 13).

With a decrease in d_pp, there is a drop in $\Delta {\overline{h}}_l$ and $\Delta {\overline{h}}_{\zeta }$ with an increase in the relative inertial losses $\Delta {\overline{h}}_{in\mathsf{\Sigma }}$. The curve ${\overline{h}}_l=f\left(d_{pp}\right)$ has a parabolic character with a sharp increase with decreasing d_pp, but the curve ${\overline{h}}_{\zeta }=f\left(d_{pp}\right)$ is close to linear (see Figure 14).

With an increase in d_pp and a decrease in hydraulic losses along the length and on local resistances there is a decrease in the work of friction forces $L_{fr}$ and an increase in the total indicator efficiency. With an increase in d_pp, the value $L_{fr}/L_{compr}$ decreases from 8.8% to 1%, when increasing d_pp from 0.014 m to 0.022 m, but the value of the total indicator efficiency increases from 0.674 to 0.726, i.e., more than 0.05, which is almost 10%.

4.4. Analysis of the Effect of Liquid Density

To analyze the influence of the density of the coolant on the working processes and characteristics of the machine, we take: d_pp = 0.018 m; n_rev = 500 rpm; μ_w = 0.01 Pa∙s.

With a change in ρ_w values $\Delta {\overline{p}}_1$ and $\Delta {\overline{p}}_2$ have maximum at ρ_w = 1200 kg/m³, although these maxima are negligible and can be neglected, which allows us to conclude that $\Delta {\overline{p}}_1$, $\Delta {\overline{p}}_2$, and $\Delta {\overline{p}}_3$ remain constant when ρ_w change in the range from 600 kg/m³ to 1400 kg/m³.

In the range of density variation, we observe a minimum of ${\overline{h}}_l$ at ρ_w = 1200 kg/m³ and maximum ${\overline{h}}_l$. It should be noted that $\Delta {\overline{h}}_{\zeta }$ changes insignificantly, while ${\overline{h}}_l$ decreases almost twofold. The influence of the density of the working fluid is very complicated. With a change in density, the Re number changes at a constant μ_w, which leads to a change in the value of the coefficient of friction along the length, and the values of the hydrostatic and velocity heads change. The maximum relative inertial losses are achieved at ρ_w =1200 kg/m³ and make up the biggest share of the total inertial losses—92%.

With an increase in the density of the liquid the work of friction forces $L_{fr}$ increases almost linearly from ρ_w =600 kg/m³ to 1100 kg/m³, with a further increase in ρ_w the value $L_{fr}$ changes parabolically (see Figure 15). With a decrease in density due to the fact that $L_{fr}$ decreases, there is an increase in the total indicator efficiency. Its value increases from 0.7036 at ρ_w = 1400 kg/m³ to 0.722 at ρ_w = 600 kg/m³, i.e., by almost 2% (see Figure 15).

5. Conclusions

In the manuscript, a mathematical model of the working processes of a two-stage piston hybrid power machine designed to compress gas to medium and high pressures has been developed and validated. A numerical experiment was carried out to study the influence of the viscosity and density of the working fluid on the working processes in the machine under study.

As a result of the numerical experiment, we determined:

(1)

There is a value of the optimal value μ_w, which provides the maximum value of the total indicator efficiency and bypassing the work of the forces of liquid friction. The optimal μ_w is in the range from 0.005 Pa∙s to 0.02 Pa∙s, with a change in the number of revolutions of the crankshaft from 300 rpm to 750 rpm.

(2)

With an increase in the angular velocity of the crankshaft to obtain maximum efficiency, it is necessary to increase the dynamic viscosity coefficient. The use of water as a coolant is not effective, because as the temperature rises, it decreases and at the operating temperature T = 323 K; its value is μ_w = 0.0005494 Pa∙s, which corresponds to the minimum indicator efficiency and the maximum work of friction forces. An increase in the diameter of the connecting pipeline leads to a decrease in the work of friction forces and an increase in the indicator efficiency.

(3)

With a decrease in the density of the medium, the total indicator isothermal efficiency increases and the work of friction forces $L_{fr}$ decreases. The minimum relative hydraulic losses along the length and the maximum $\Delta {\overline{p}}_1$ and $\Delta {\overline{p}}_2$ correspond to ρ_w = 1200 kg/m³.

(4)

The analysis showed that industrial oil I-12A, which has ρ_w = 880 kg/m³ and μ_w/ρ_w = (13÷17)∙10⁻⁶ m²/s can be used as power liquid in the machines.