Research on the Volumetric Efficiency of 2D Piston Pumps with a Balanced Force

Axial piston pumps with high rotational speeds are required in many fields to increase the power-to-weight ratio. However, three main sliding friction pairs in the pump restrict the increase in rotational speed. To solve this problem, we propose a 2D piston pump with a balanced force that contains a sliding friction pair. Firstly, the mechanical structure and working principle of the pump are described. Then, the pump volumetric efficiency is studied by mathematical modeling, and volumetric losses containing backflow and leakage are analyzed and discussed from the perspectives of load pressure and rotational speed. A test bench that verifies the mathematical model is built to measure the volumetric efficiency of the tested pump. We have found that the increase in rotational speed can help to increase the pump volumetric efficiency, and the mathematical model is consistent with the tested data for 1000 rpm but demonstrates a remarkable difference from the tested data for 3000 rpm. Thus, the temperature field of the pump and the viscosity-temperature characteristics of the oil must be taken into account to increase volumetric efficiency further.


Introduction
Hydraulic systems are widely used in many fields because of their high power density, quick dynamic response, and strong load capacity [1][2][3]. In recent years, due to their high power-to-weight ratio, electro-hydrostatic actuator (EHA)-installed axial piston pumps as the oil source are used to replace the centrifugal hydraulic system, which uses centrifugal pumps as the oil source in the aerospace industry [4].
A high power density is the most important requirement for axial piston pumps used in the aerospace industry, and the power-to-weight ratio of energy components needs to be further improved [5]. This is often achieved by increasing rotational speed [6]. The displacements of axial piston pumps of various manufacturers at various rotational speeds are shown in Figure 1. In aerospace applications, the rotational speed of the axial piston pump is required to be greater than 10,000 rpm [7]. When the rotational speed exceeds 10,000 rpm, as shown in Figure 1, the pump displacement decreases to less than 5 mL/r. Increasing the rotational speed and reducing the displacement of the pump can increase the power density, known as the "reduction effect". This reduction effect is significant for low rotational speed pumps, but it becomes less noticeable when the rotational speed exceeds 10,000 rpm [6]. On the other hand, the rotational speed of axial piston pumps is limited by two factors in practical applications [9]. The first one is the presence of three main sliding friction pairs in axial piston pumps [10][11][12]. The friction pairs subjected to overturning moments generate mechanical losses during rotation [13]. Especially at high speeds, the rise of the overturning torque causes direct metal-to-metal contact between the sliding surfaces in the three main lubrication surfaces [14]. This problem can be improved to a certain extent by coating on the surface of these friction pairs [15]. The second factor that limits the rotational speed is the thermal problem. The combination of small displacement and high speed generates heat and reduces the heat dissipation capacity of the axial piston pump [16,17]. In addition, the churning losses are generated by the rotating parts and increase the oil temperature at high rotational speeds [18][19][20].
Ruan group has proposed a new type of axial piston pump based on a "two-dimensional" concept, named 2D piston pump [21][22][23]. As shown in Figure 2, through the cooperation of the rollers and guiding rails, the rotation of the 2D piston is converted into the reciprocation, thereby achieving sucking and discharging oil. The 2D piston pump has only a sliding friction pair between the 2D piston and cylinder block, and the radial force of the 2D piston is balanced and is not affected by the overturning torque [24]. Thus, the 2D piston pump has the potential to achieve high rotational speed.  On the other hand, the rotational speed of axial piston pumps is limited by two factors in practical applications [9]. The first one is the presence of three main sliding friction pairs in axial piston pumps [10][11][12]. The friction pairs subjected to overturning moments generate mechanical losses during rotation [13]. Especially at high speeds, the rise of the overturning torque causes direct metal-to-metal contact between the sliding surfaces in the three main lubrication surfaces [14]. This problem can be improved to a certain extent by coating on the surface of these friction pairs [15]. The second factor that limits the rotational speed is the thermal problem. The combination of small displacement and high speed generates heat and reduces the heat dissipation capacity of the axial piston pump [16,17]. In addition, the churning losses are generated by the rotating parts and increase the oil temperature at high rotational speeds [18][19][20].
Ruan group has proposed a new type of axial piston pump based on a "two-dimensional" concept, named 2D piston pump [21][22][23]. As shown in Figure 2, through the cooperation of the rollers and guiding rails, the rotation of the 2D piston is converted into the reciprocation, thereby achieving sucking and discharging oil. The 2D piston pump has only a sliding friction pair between the 2D piston and cylinder block, and the radial force of the 2D piston is balanced and is not affected by the overturning torque [24]. Thus, the 2D piston pump has the potential to achieve high rotational speed. On the other hand, the rotational speed of axial piston pumps is limited by two factors in practical applications [9]. The first one is the presence of three main sliding friction pairs in axial piston pumps [10][11][12]. The friction pairs subjected to overturning moments generate mechanical losses during rotation [13]. Especially at high speeds, the rise of the overturning torque causes direct metal-to-metal contact between the sliding surfaces in the three main lubrication surfaces [14]. This problem can be improved to a certain extent by coating on the surface of these friction pairs [15]. The second factor that limits the rotational speed is the thermal problem. The combination of small displacement and high speed generates heat and reduces the heat dissipation capacity of the axial piston pump [16,17]. In addition, the churning losses are generated by the rotating parts and increase the oil temperature at high rotational speeds [18][19][20].
Ruan group has proposed a new type of axial piston pump based on a "two-dimensional" concept, named 2D piston pump [21][22][23]. As shown in Figure 2, through the cooperation of the rollers and guiding rails, the rotation of the 2D piston is converted into the reciprocation, thereby achieving sucking and discharging oil. The 2D piston pump has only a sliding friction pair between the 2D piston and cylinder block, and the radial force of the 2D piston is balanced and is not affected by the overturning torque [24]. Thus, the 2D piston pump has the potential to achieve high rotational speed. However, vibration is generated by the pump at high rotational speeds. When the 2D piston rotates, the cylinder block is moved by a force F1, which is generated by the rollers. The force acts on the left side of the cylinder block at the left displacement chamber discharging oil, and the force acts on the right side of the cylinder block at the right displacement chamber discharging oil. Due to the  However, vibration is generated by the pump at high rotational speeds. When the 2D piston rotates, the cylinder block is moved by a force F 1 , which is generated by the rollers. The force acts on the left side of the cylinder block at the left displacement chamber discharging oil, and the force acts on the right side of the cylinder block at the right displacement chamber discharging oil. Due to the traditional 2D pump designed to discharge oil alternately between the left and right displacement chambers, the cylinder block is moved by an alternate force and generates vibration [25].
To solve this problem, we propose a 2D piston pump with a balanced force using a well-balanced force. By adding a balancing set, the cylinder block is subjected to a pair of balanced forces, thereby eliminating the vibration [26]. Firstly, the mechanical structure and working principle of the proposed piston pump are described in Section 2. Then, the volumetric efficiency of the pump is researched through mathematical modeling in Section 3. The pump volumetric losses are analyzed and discussed from the perspectives of load pressure and rotational speed in Section 4. Finally, a test bench is built to measure the volumetric efficiency of the pump and verify the mathematical model.

Mechanical Structure and Working Principle of the 2D Piston Pump with a Balanced Force
As shown in Figure 3a, the 2D piston pump with a balanced force consists of successively, from left to the right, a fork shaft, left roller set, left guiding rail, cylinder block, 2D piston, pump shell, right guiding rail, right roller set, and right fork. The roller sets are divided into driving rollers and balancing rollers. As shown in Figure 3b,c, the orange rollers are the driving rollers, and the blue rollers are the balancing rollers. The driving rollers are fixed to the 2D piston, and the balancing rollers are fixed to the piston ring. The two sets are combined and have a phase difference of 90 deg between them in the circumferential direction through the two forks, whereas they have relatively independent axial motions. The 2D piston is hollow to install the transmission shaft, and it is precisely contained with the cylinder block.
Energies 2020, 13, x FOR PEER REVIEW 3 of 16 traditional 2D pump designed to discharge oil alternately between the left and right displacement chambers, the cylinder block is moved by an alternate force and generates vibration [25].
To solve this problem, we propose a 2D piston pump with a balanced force using a well-balanced force. By adding a balancing set, the cylinder block is subjected to a pair of balanced forces, thereby eliminating the vibration [26]. Firstly, the mechanical structure and working principle of the proposed piston pump are described in Section 2. Then, the volumetric efficiency of the pump is researched through mathematical modeling in Section 3. The pump volumetric losses are analyzed and discussed from the perspectives of load pressure and rotational speed in Section 4. Finally, a test bench is built to measure the volumetric efficiency of the pump and verify the mathematical model.

Mechanical Structure and Working Principle of the 2D Piston Pump with a Balanced Force
As shown in Figure 3a, the 2D piston pump with a balanced force consists of successively, from left to the right, a fork shaft, left roller set, left guiding rail, cylinder block, 2D piston, pump shell, right guiding rail, right roller set, and right fork. The roller sets are divided into driving rollers and balancing rollers. As shown in Figure 3b,c, the orange rollers are the driving rollers, and the blue rollers are the balancing rollers. The driving rollers are fixed to the 2D piston, and the balancing rollers are fixed to the piston ring. The two sets are combined and have a phase difference of 90 deg between them in the circumferential direction through the two forks, whereas they have relatively independent axial motions. The 2D piston is hollow to install the transmission shaft, and it is precisely contained with the cylinder block.  The left and right displacement chambers, both having the same working principle, are composed of the 2D piston, piston rings, and cylinder block, but have a phase difference of 90 deg.  The left and right displacement chambers, both having the same working principle, are composed of the 2D piston, piston rings, and cylinder block, but have a phase difference of 90 deg. Taking the left displacement chamber as an example, the working principle of oil suction and discharge is introduced.
The pump's state described in Figure 4a is assumed to be zero at the initial state, where the left displacement chamber is at the maximum volume and prepares to discharge oil. From Figure 4a-c, the 2D piston rotates from 0 deg to 90 deg. The left displacement chamber is compressed from the maximum, causing the oil to be discharged into the discharging port out of the left displacement chamber. From Figure 4a-c, the 2D piston rotates from 90 deg to 180 deg. The left displacement chamber expands from the minimum, causing the oil to be sucked from the intaking port into the left displacement chamber. Taking the left displacement chamber as an example, the working principle of oil suction and discharge is introduced. The pump's state described in Figure 4a is assumed to be zero at the initial state, where the left displacement chamber is at the maximum volume and prepares to discharge oil. From Figure 4a-c, the 2D piston rotates from 0 deg to 90 deg. The left displacement chamber is compressed from the maximum, causing the oil to be discharged into the discharging port out of the left displacement chamber. From Figure 4a-c, the 2D piston rotates from 90 deg to 180 deg. The left displacement chamber expands from the minimum, causing the oil to be sucked from the intaking port into the left displacement chamber. As soon as the left shaft-fork is rotated by a motor, both the driving and balancing sets are rotated at the same pace. When the rollers rotate under the constraint of the guiding rails, the rotation of the sets is transformed into the relative reciprocating motion. As shown in Figure 5, when the rollers rotate from 0 deg to 45 deg, the driving set moves left with uniform acceleration, and the balancing set moves right with a uniform acceleration, decreasing the volume of the left displacement chamber and increasing that of the right displacement chamber. Therefore, the force acting on the cylinder block by the driving set can be balanced by force from the balancing set. As soon as the left shaft-fork is rotated by a motor, both the driving and balancing sets are rotated at the same pace. When the rollers rotate under the constraint of the guiding rails, the rotation of the sets is transformed into the relative reciprocating motion. As shown in Figure 5, when the rollers rotate from 0 deg to 45 deg, the driving set moves left with uniform acceleration, and the balancing set moves right with a uniform acceleration, decreasing the volume of the left displacement chamber and increasing that of the right displacement chamber. Therefore, the force acting on the cylinder block by the driving set can be balanced by force from the balancing set.

Mathematical Modeling
Since the working principle of the left displacement chamber is consistent with that of the right displacement chamber, the left displacement chamber is taken as an example to model its volumetric efficiency. As shown in Figures 4a and 6, the rotational degree is assumed to be zero at the initial moment, where the left displacement chamber is at the maximum volume and starts to discharge oil. Assuming the oil in the left displacement chamber is compressible, the instantaneous left displacement chamber pressure is governed by the following pressure build-up Equation (1) where pL is the instantaneous left displacement chamber pressure, t is the rotating time from 0 deg, βe is the bulk modulus of the oil, VL is the instantaneous left displacement chamber volume, Qo is the outlet flow of the left displacement chamber, Qi is the inlet flow of the left displacement chamber, and QL is the leakage flow of the left displacement chamber. The instantaneous left displacement chamber volume is determined by the positions of the left piston ring and 2D piston, where their motions are accorded to uniform acceleration and deceleration [24]. The left piston ring velocity vpr and the left 2D piston velocity vp are described by Equation (2).
where a is the acceleration and decided by the rotational speed n and the 2D piston stroke Ls that is consistent with the left piston ring stroke, and ω is the rotational angular velocity. When the pump rotates from 0 deg to 45 deg, the 2D piston moves left with uniform acceleration, and the axial displacement is half of the 2D piston stroke. The acceleration is calculated by Equation (3).
where t45deg is the rotating time from 0 deg to 45 deg. Integrating the left piston ring velocity and 2D piston velocity, the instantaneous left displacement chamber volume is calculated by Equation (4).

Mathematical Modeling
Since the working principle of the left displacement chamber is consistent with that of the right displacement chamber, the left displacement chamber is taken as an example to model its volumetric efficiency. As shown in Figures 4a and 6, the rotational degree is assumed to be zero at the initial moment, where the left displacement chamber is at the maximum volume and starts to discharge oil. Assuming the oil in the left displacement chamber is compressible, the instantaneous left displacement chamber pressure is governed by the following pressure build-up Equation (1) where p L is the instantaneous left displacement chamber pressure, t is the rotating time from 0 deg, β e is the bulk modulus of the oil, V L is the instantaneous left displacement chamber volume, Q o is the outlet flow of the left displacement chamber, Q i is the inlet flow of the left displacement chamber, and Q L is the leakage flow of the left displacement chamber. The instantaneous left displacement chamber volume is determined by the positions of the left piston ring and 2D piston, where their motions are accorded to uniform acceleration and deceleration [24]. The left piston ring velocity v pr and the left 2D piston velocity v p are described by Equation (2).
where a is the acceleration and decided by the rotational speed n and the 2D piston stroke L s that is consistent with the left piston ring stroke, and ω is the rotational angular velocity. When the pump rotates from 0 deg to 45 deg, the 2D piston moves left with uniform acceleration, and the axial displacement is half of the 2D piston stroke. The acceleration is calculated by Equation (3).
Energies 2020, 13, 4796 where t 45deg is the rotating time from 0 deg to 45 deg. Integrating the left piston ring velocity and 2D piston velocity, the instantaneous left displacement chamber volume is calculated by Equation (4).
where V M is the maximum volume of the left displacement chamber. Equation (5) is obtained by differentiating Equation (4).
The outlet flow and the inlet flow can be calculated by using the standard orifice Equations (6) and (7).
where C d is the orifice coefficient, A out and A in are the open areas of the discharging ports and intaking ports, respectively, p Load and p T are the pressures of the load and tank, respectively, and ρ is the oil density.
where VM is the maximum volume of the left displacement chamber. Equation (5) is obtained by differentiating Equation (4).
The outlet flow and the inlet flow can be calculated by using the standard orifice Equations (6) and (7).
where Cd is the orifice coefficient, Aout and Ain are the open areas of the discharging ports and intaking ports, respectively, pLoad and pT are the pressures of the load and tank, respectively, and ρ is the oil density.  By expanding the outer circle of the 2D piston into a plane, as shown in Figure 7, when the pump rotates from 0 deg to 90 deg, the state that the left displacement chamber connects with the discharging ports can be described in detail. Because the rotational speed and port length L g are constant, when the rotational angle is from 0 deg to 45 deg, the open area increases linearly; when the rotational angle is  (8) and (9) [23] where D is the 2D piston diameter.
Energies 2020, 13, x FOR PEER REVIEW 7 of 16 where D is the 2D piston diameter. Due to the clearances h between the piston rings, 2D piston, and cylinder block, as shown in Figure 8, the external leakage flow can be calculated using the standard leakage Equation (10).
where μ is the oil dynamic viscosity, Lpr is the contact length between the left piston ring and cylinder block, d is the small diameter of the 2D piston, LD is the minimum contact length between the left piston ring and cylinder block and the left piston ring and the 2D piston. Since the motions of the 2D Due to the clearances h between the piston rings, 2D piston, and cylinder block, as shown in Figure 8, the external leakage flow can be calculated using the standard leakage Equation (10).
where µ is the oil dynamic viscosity, L pr is the contact length between the left piston ring and cylinder block, d is the small diameter of the 2D piston, L D is the minimum contact length between the left piston ring and cylinder block and the left piston ring and the 2D piston. Since the motions of the 2D piston and left piston ring are relative, the external leakage between the left piston ring and 2D piston generated by the velocity is not considered. The length between the left piston ring and the cylinder block is calculated by integrating Equation (2) [22].
Energies 2020, 13, x FOR PEER REVIEW 8 of 16 piston and left piston ring are relative, the external leakage between the left piston ring and 2D piston generated by the velocity is not considered. The length between the left piston ring and the cylinder block is calculated by integrating Equation (2)  π ω π π π π ω ω ω ω π π π π ω ω ω ω π π π ω ω ω The internal leakage flow is also composed of two parts: the internal axial leakage flow QiLa and the internal circumferential leakage flow QiLr. As shown in Figure 8, there are two contact lengths between the 2D piston and cylinder block, Lp1 and Lp2, and each accounts for half of the circumference of the 2D piston. The internal axial leakage flow can be calculated using the standard leakage Equation (12) [23].
where pr is the instantaneous right displacement chamber pressure. The inter-circumferential leakage flow would occur during pump rotation, as shown in Figure  9. Since the inflow by the shear flow is equal to the outflow, the inter-circumferential leakage flow due to the shear flow is not considered. The internal circumferential leakage flow can be calculated using the standard leakage Equation (13).
where Lr is the circumferential contact length between the 2D piston and cylinder block and is obtained using Equation (14). The internal leakage flow is also composed of two parts: the internal axial leakage flow Q iLa and the internal circumferential leakage flow Q iLr . As shown in Figure 8, there are two contact lengths between the 2D piston and cylinder block, L p1 and L p2 , and each accounts for half of the circumference of the 2D piston. The internal axial leakage flow can be calculated using the standard leakage Equation (12) [23].
where p r is the instantaneous right displacement chamber pressure. The inter-circumferential leakage flow would occur during pump rotation, as shown in Figure 9. Since the inflow by the shear flow is equal to the outflow, the inter-circumferential leakage flow due to the shear flow is not considered. The internal circumferential leakage flow can be calculated using the standard leakage Equation (13).
where L r is the circumferential contact length between the 2D piston and cylinder block and is obtained using Equation (14).
Energies 2020, 13, 4796 9 of 16 Energies 2020, 13, x FOR PEER REVIEW 9 of 16 (a) (b) (c) When the rotational time nears 0 or , the inter-circumferential leakage flow approaches infinity, which is inconsistent with the real situation. Under the collective effect from the ports and clearance, the face sealing for the circumferential leakage is changed to a line sealing at a specific rotational angle, as shown in Figure 9a. This line sealing might be turned into a throttle, which leads to so-called transient leakage flow and can be considered as a unique phenomenon for 2D piston pumps. Thus, Equation (13) is modified to obtain Equation (15).
Finally, as shown in Equation (16), during the left displacement chamber discharging oil, all discharging flow can be calculated by integrating the outlet flow; then, volumetric efficiency η is obtained by comparing all discharging flow with the theoretical output flow.
where t90deg is the time that the pump rotates from 0 deg to 90 deg, and VD is the pump's displacement.

Simulation Results
The volumetric efficiency of the 2D piston pump with a balanced force can be calculated by solving Equations (1)- (16). According to Table 1, parameters needed by simulation are consistent with the tested pump.  When the rotational time nears 0 or π 2ω , the inter-circumferential leakage flow approaches infinity, which is inconsistent with the real situation. Under the collective effect from the ports and clearance, the face sealing for the circumferential leakage is changed to a line sealing at a specific rotational angle, as shown in Figure 9a. This line sealing might be turned into a throttle, which leads to so-called transient leakage flow and can be considered as a unique phenomenon for 2D piston pumps. Thus, Equation (13) is modified to obtain Equation (15).
Finally, as shown in Equation (16), during the left displacement chamber discharging oil, all discharging flow can be calculated by integrating the outlet flow; then, volumetric efficiency η is obtained by comparing all discharging flow with the theoretical output flow.
where t 90deg is the time that the pump rotates from 0 deg to 90 deg, and V D is the pump's displacement.

Simulation Results
The volumetric efficiency of the 2D piston pump with a balanced force can be calculated by solving Equations (1)- (16). According to Table 1, parameters needed by simulation are consistent with the tested pump. The volumetric efficiency is affected by backflow and leakage. This paper describes their effects on volumetric efficiency in detail from the perspectives of load pressure and rotational speed.
The positive flow rate is assumed as the outflow from the left displacement chamber.

Backflow
When the left displacement chamber is connected to the discharging ports, backflow is generated due to the instantaneous left displacement chamber pressure, which is lower than the load pressure.
As shown in Figure 10, the flow peak and the duration of the backflow increase with the increase in load pressure. As the load pressure increases, the pressure difference between the instantaneous left displacement chamber pressure and load pressure increases at a rotational angle of 0 deg; thus, more flow is needed to help the left displacement chamber establish the pressure to discharge oil.
Energies 2020, 13, x FOR PEER REVIEW 10 of 16 The volumetric efficiency is affected by backflow and leakage. This paper describes their effects on volumetric efficiency in detail from the perspectives of load pressure and rotational speed.
The positive flow rate is assumed as the outflow from the left displacement chamber.

Backflow
When the left displacement chamber is connected to the discharging ports, backflow is generated due to the instantaneous left displacement chamber pressure, which is lower than the load pressure.
As shown in Figure 10, the flow peak and the duration of the backflow increase with the increase in load pressure. As the load pressure increases, the pressure difference between the instantaneous left displacement chamber pressure and load pressure increases at a rotational angle of 0 deg; thus, more flow is needed to help the left displacement chamber establish the pressure to discharge oil. As shown in Figure 11, the flow peak of the backflow increases with the rise in rotational speed. Due to the different rotational speeds, although the angle range of the backflow increases as the rotational speed increases, the duration of the backflow decreases. Since the load pressure is constant, the total amount of demanded backflow remains constant, and the flow peak of the backflow increases due to the decrease in the duration of the backflow.  As shown in Figure 11, the flow peak of the backflow increases with the rise in rotational speed. Due to the different rotational speeds, although the angle range of the backflow increases as the rotational speed increases, the duration of the backflow decreases. Since the load pressure is constant, the total amount of demanded backflow remains constant, and the flow peak of the backflow increases due to the decrease in the duration of the backflow. The volumetric efficiency is affected by backflow and leakage. This paper describes their effects on volumetric efficiency in detail from the perspectives of load pressure and rotational speed.
The positive flow rate is assumed as the outflow from the left displacement chamber.

Backflow
When the left displacement chamber is connected to the discharging ports, backflow is generated due to the instantaneous left displacement chamber pressure, which is lower than the load pressure.
As shown in Figure 10, the flow peak and the duration of the backflow increase with the increase in load pressure. As the load pressure increases, the pressure difference between the instantaneous left displacement chamber pressure and load pressure increases at a rotational angle of 0 deg; thus, more flow is needed to help the left displacement chamber establish the pressure to discharge oil. As shown in Figure 11, the flow peak of the backflow increases with the rise in rotational speed. Due to the different rotational speeds, although the angle range of the backflow increases as the rotational speed increases, the duration of the backflow decreases. Since the load pressure is constant, the total amount of demanded backflow remains constant, and the flow peak of the backflow increases due to the decrease in the duration of the backflow.

Leakage
The leakage of the 2D piston pump with a balanced force is composed of external leakage, internal axial leakage, and internal circumferential leakage.
As shown in Figure 12, when the left displacement chamber connects with the discharging ports, and the rotational angle nears 0 deg, the external leakage approaches 0 L/min. As the backflow helps the instantaneous left displacement chamber pressure to increase rapidly, the external leakage reaches its maximum. Then, as the contacting length and the velocity of the left piston ring increase, the external leakage decreases. When the rotational angle exceeds 45 deg, due to the decrease in the velocity of the left piston ring, the external leakage increases.

Leakage
The leakage of the 2D piston pump with a balanced force is composed of external leakage, internal axial leakage, and internal circumferential leakage.
As shown in Figure 12, when the left displacement chamber connects with the discharging ports, and the rotational angle nears 0 deg, the external leakage approaches 0 L/min. As the backflow helps the instantaneous left displacement chamber pressure to increase rapidly, the external leakage reaches its maximum. Then, as the contacting length and the velocity of the left piston ring increase, the external leakage decreases. When the rotational angle exceeds 45 deg, due to the decrease in the velocity of the left piston ring, the external leakage increases. As the rotational speed decreases, the leakage from the shear flow decreases. As the directions of leakage due to the pressure difference and shear flow are opposite, the external leakage increases as the rotational speed decreases at the discharging oil.
When the load pressure is 1 MPa, the overall external leakage is negative at the discharging oil. The external leakage contributes to the volumetric efficiency of the pump at low load pressures.
As shown in Figure 13, the internal axial leakage is similar to external leakage. Because the two contacting lengths between the 2D piston and cylinder block are constant, as shown in Figure 13a, the internal axial leakage linearly declines and then rises for the rotational angle range 0-90 deg. When the load pressure is 1 MPa, the overall internal axial leakage also presents negative flow at the discharging oil.  As the rotational speed decreases, the leakage from the shear flow decreases. As the directions of leakage due to the pressure difference and shear flow are opposite, the external leakage increases as the rotational speed decreases at the discharging oil.
When the load pressure is 1 MPa, the overall external leakage is negative at the discharging oil. The external leakage contributes to the volumetric efficiency of the pump at low load pressures.
As shown in Figure 13, the internal axial leakage is similar to external leakage. Because the two contacting lengths between the 2D piston and cylinder block are constant, as shown in Figure 13a, the internal axial leakage linearly declines and then rises for the rotational angle range 0-90 deg. When the load pressure is 1 MPa, the overall internal axial leakage also presents negative flow at the discharging oil.

Leakage
The leakage of the 2D piston pump with a balanced force is composed of external leakage, internal axial leakage, and internal circumferential leakage.
As shown in Figure 12, when the left displacement chamber connects with the discharging ports, and the rotational angle nears 0 deg, the external leakage approaches 0 L/min. As the backflow helps the instantaneous left displacement chamber pressure to increase rapidly, the external leakage reaches its maximum. Then, as the contacting length and the velocity of the left piston ring increase, the external leakage decreases. When the rotational angle exceeds 45 deg, due to the decrease in the velocity of the left piston ring, the external leakage increases. As the rotational speed decreases, the leakage from the shear flow decreases. As the directions of leakage due to the pressure difference and shear flow are opposite, the external leakage increases as the rotational speed decreases at the discharging oil.
When the load pressure is 1 MPa, the overall external leakage is negative at the discharging oil. The external leakage contributes to the volumetric efficiency of the pump at low load pressures.
As shown in Figure 13, the internal axial leakage is similar to external leakage. Because the two contacting lengths between the 2D piston and cylinder block are constant, as shown in Figure 13a, the internal axial leakage linearly declines and then rises for the rotational angle range 0-90 deg. When the load pressure is 1 MPa, the overall internal axial leakage also presents negative flow at the discharging oil.  As shown in Figure 14, the transient leakage of the inter-circumferential leakage is consistent with theoretical analysis and can reach 1 L/min when the rotational angle nears 0 deg or 90 deg at a load pressure of 8 MPa. As the load pressure rises, since the pressure difference between the left and right displacement chamber rises, the transient leakage increases. As the rotational speed increases, the inter-circumferential leakage is almost unchanged, except for the reduction in the reflow region.
Energies 2020, 13, x FOR PEER REVIEW 12 of 16 As shown in Figure 14, the transient leakage of the inter-circumferential leakage is consistent with theoretical analysis and can reach 1 L/min when the rotational angle nears 0 deg or 90 deg at a load pressure of 8 MPa. As the load pressure rises, since the pressure difference between the left and right displacement chamber rises, the transient leakage increases. As the rotational speed increases, the inter-circumferential leakage is almost unchanged, except for the reduction in the reflow region.

Volumetric Efficiency
As shown in Figure 15a, the simulated volumetric efficiency distribution of the 2D piston pump with a balanced force is consistent with the above analysis. With the load pressure rising, since the backflow and the leakage increase, the volumetric efficiency decreases. Furthermore, as the rotational speed increases, the volumetric efficiency increases because the increased velocity of the backflow and leakage is lower than the increased velocity of the outlet flow.
In addition, due to wear and manufacturing accuracy, clearances between the piston rings, 2D piston, and cylinder block are easily changed. As shown in Figure 15b, the volumetric efficiency decreases significantly with the increase of the clearance.

Volumetric Efficiency
As shown in Figure 15a, the simulated volumetric efficiency distribution of the 2D piston pump with a balanced force is consistent with the above analysis. With the load pressure rising, since the backflow and the leakage increase, the volumetric efficiency decreases. Furthermore, as the rotational speed increases, the volumetric efficiency increases because the increased velocity of the backflow and leakage is lower than the increased velocity of the outlet flow.

Experimental Results
As shown in Figure 16a, the test rig consists of a tested pump, a torque sensor, a driving motor, a supply pump, a tank, two pressure sensors installed at the inlet and the outlet of the tested pump, a relief valve, a flow meter, and a data acquisition device. The tested pump and driving motor are connected by a flexible coupling, which increases the stability of the transmission chain at high rotational speeds. To prevent cavitation, the inlet pressure is added to 0.5 MPa by the supply pump. The accuracies of the related sensors are shown in Table 2.  In addition, due to wear and manufacturing accuracy, clearances between the piston rings, 2D piston, and cylinder block are easily changed. As shown in Figure 15b, the volumetric efficiency decreases significantly with the increase of the clearance.

Experimental Results
As shown in Figure 16a, the test rig consists of a tested pump, a torque sensor, a driving motor, a supply pump, a tank, two pressure sensors installed at the inlet and the outlet of the tested pump, a relief valve, a flow meter, and a data acquisition device. The tested pump and driving motor are connected by a flexible coupling, which increases the stability of the transmission chain at high rotational speeds. To prevent cavitation, the inlet pressure is added to 0.5 MPa by the supply pump. The accuracies of the related sensors are shown in Table 2.

Experimental Results
As shown in Figure 16a, the test rig consists of a tested pump, a torque sensor, a driving motor, a supply pump, a tank, two pressure sensors installed at the inlet and the outlet of the tested pump, a relief valve, a flow meter, and a data acquisition device. The tested pump and driving motor are connected by a flexible coupling, which increases the stability of the transmission chain at high rotational speeds. To prevent cavitation, the inlet pressure is added to 0.5 MPa by the supply pump. The accuracies of the related sensors are shown in Table 2.

Description Accuracy
Flow meter Range 0-100 L/min, accuracy ± 0.1% Torque/speed sensor Range 0-10 Nm, accuracy ± 0.1%, rotational speed range 0-18,000 rpm Pressure sensor Range 0-100 bar, accuracy ± 0.3% The load pressure-flow characteristic data of the tested pump are measured through fixing the speed of the driving motor and adjusting the relief valve installed at the outlet of the tested pump to control the load pressure. The load pressures, rotational speeds, and outlet flow are recorded, as shown in Table 3. On the other hand, the range and accuracy of the flowmeter are too large to accurately measure the output flow, and only one decimal place can be taken. The tested data of rotational speeds 1000 rpm and 3000 rpm are compared with the corresponding simulations, respectively, as shown in Figure 17.

Description Accuracy Flow meter
Range 0-100 L/min, accuracy ± 0.1% Torque/speed sensor Range 0-10 Nm, accuracy ± 0.1%, rotational speed range 0-18,000 rpm Pressure sensor Range 0-100 bar, accuracy ± 0.3% The load pressure-flow characteristic data of the tested pump are measured through fixing the speed of the driving motor and adjusting the relief valve installed at the outlet of the tested pump to control the load pressure. The load pressures, rotational speeds, and outlet flow are recorded, as shown in Table 3. On the other hand, the range and accuracy of the flowmeter are too large to accurately measure the output flow, and only one decimal place can be taken. The tested data of rotational speeds 1000 rpm and 3000 rpm are compared with the corresponding simulations, respectively, as shown in Figure 17. Although high-precision machining is used to ensure the size of the clearance designed to be 2 × 10 −5 m during the machining process, errors still remain in the clearance because of the inevitable machining errors and wear. However, it is regrettable that this clearance cannot be accurately measured due to the lack of corresponding detection equipment. To improve the match between the simulation and experiment, the maximum clearance 2.5 × 10 −5 m and the minimum clearance 2 × 10 −5 m are considered. As shown in Figure 17a, when the clearance changes from 2 × 10 −5 m to 2.5 × 10 −5 m, the simulations are closer to the tested data, and the trends of the tested data and simulations linearly decline at a rotational speed of 1000 rpm. Especially when the size of the clearance is 2.4 × 10 −5 or 2.5 × 10 −5 m, the maximum numerical difference between simulation and experimental data does not exceed 3%. However, when the rotational speed is 3000 rpm, a numerical difference remains between the volumetric efficiencies of the tested data and simulations and increases as the load pressure rises. There are three explanations for this difference. Although high-precision machining is used to ensure the size of the clearance designed to be 2 × 10 −5 m during the machining process, errors still remain in the clearance because of the inevitable machining errors and wear. However, it is regrettable that this clearance cannot be accurately measured due to the lack of corresponding detection equipment. To improve the match between the simulation and experiment, the maximum clearance 2.5 × 10 −5 m and the minimum clearance 2 × 10 −5 m are considered. As shown in Figure 17a, when the clearance changes from 2 × 10 −5 m to 2.5 × 10 −5 m, the simulations are closer to the tested data, and the trends of the tested data and simulations linearly decline at a rotational speed of 1000 rpm. Especially when the size of the clearance is 2.4 × 10 −5 or 2.5 × 10 −5 m, the maximum numerical difference between simulation and experimental data does not exceed 3%. However, when the rotational speed is 3000 rpm, a numerical difference remains between the volumetric efficiencies of the tested data and simulations and increases as the load pressure rises. There are three explanations for this difference.
Firstly, during discharging the oil, the leakage due to the shear flow no longer contributes to volumetric efficiency and is not consistent with the standard leakage equation at a high rotational speed.
Secondly, as other studies have shown, the friction and churning losses in the pump exacerbate the increase in oil temperature and reduce the oil viscosity at high rotational speeds, resulting in leakage increase.
Thirdly, when the rotational speed increases, the 2D piston may be eccentric in the cylinder block, which causes an increase in leakage.
Fourthly, because of the lack of equipment for detecting the clearance and the inaccuracy of test methods and test equipment, the experimental data and simulation have deviated.

Conclusions
In this paper, a new 2D piston pump with a balanced force is proposed, and its volumetric efficiency is studied through mathematical modeling. The volumetric losses generated by the backflow and leakage are analyzed from the perspectives of load pressure and rotational speed. A test bench is built to measure the volumetric efficiency of the tested pump and compared to the mathematical model. According to the experimental results, the following conclusions can be drawn: Firstly, the mathematical model is consistent with the experimental data, and as the load pressure increases, the volumetric efficiency linearly decreases at various rotational speeds.
Secondly, because the rotational speed has little effect on the leakage and hardly affects the total amount of the backflow, increasing rotational speed is beneficial to increasing volumetric efficiency. However, the increase in the rotational speed affects the flow ripple, which requires further investigation.
Thirdly, the tested data and simulations differ when the rotational speed is 3000 rpm, and the difference increases as the load pressure rises. Leakage is the main reason for this difference. When the rotational speed increases, the standard leakage equation is no longer applicable, and the increase in the oil temperature causes the viscosity to decrease, and the eccentricity of the 2D piston in the cylinder block causes the leakage to increase.
Future research on volumetric efficiency will concern the pump temperature field and the viscosity-temperature equation. The gap flow and the effect of clearance on the volumetric efficiency in 2D pumps need to be studied in more detail. Finally, the rotational speed of the 2D piston pump with a balanced force needs to be further increased.

Conflicts of Interest:
The authors declare no conflict of interest.