Optimization of the structure of water axial piston pump and cavitation of plunger cavity based on the Kriging model

2.1. Research object

The research object of this paper is the water axial piston pump. Its structure is shown in Fig. 1. The angle of the swash plate is 15°. The cross angle $\alpha$ is 4°. The number of the piston $i$ is 9. The piston diameter is 21 mm. Piston distribution circle diameter is 81 mm. Rotation rate $n$ is 1500 r/min.

Fig. 1Structural representation of axial piston pump

Fig. 2Internal fluid in axial piston pump

2.2. Internal flow channel geometry model of axial piston pump

On the premise of no neglect of the leakage of the axial piston pump, the flow channel geometry model of axial piston pump is shown in Fig. 2. In this model, the position of valve plate, suction port and drainage port are fixed. The kidney shape port and plunger cavity rotate with the cylinder, and the volume of the plunger cavity changes with the piston movement. The structure of plunger cavity of axial piston pump, valve plate, and kidney shape port is shown in Fig. 3.

2.3. Mesh division

The axial piston pump is divided by the structure and un-structure grid due to its complex internal flow channel and its changeable size. The mesh is encrypted in small throttling groove, as shown in partial enlargement of Fig. 4. The grid consists of 168813 elements.

2.4. Boundary conditions

As it is shown in Fig. 4, nine plunger cavity of the cylinder barrel is rotated around with the $Z$ axis, and the movement speed of the piston is expressed as:

The pump inlet and outlet are set as pressure inlet and outlet respectively. The paper determined the values of inlet pressure and outlet pressure as 3.3E5 Pa and 5 MPa respectively. The saturated vapor pressure of water is 2367.8 Pa. The value of the gas volume fraction of pump inlet and outlet is zero. The compressibility of the liquid and gaseous fluid and the piston velocity are set through UDF by using the commercial fluent software and the mixture model, RNG $k$-$\epsilon$ model, dynamic mesh and sliding mesh. The control equations were space dispersed by finite volume method and non-staggered grid. The two order upwind scheme is adopted in turbulent kinetic energy term. The others adopt the first order upwind scheme. The time dispersion adopts two order full implicit schemes. The SIMPLE algorithm is used in velocity component in momentum equation and pressure coupling problem. Every iterative calculation accuracy is set as 10^-8, and time step is set as 10^-4.

Fig. 3Valve plate and the plunger cavity structure

Fig. 4Internal flow channel mesh division of axial piston pump

Fig. 5Distribution contours of gas volume fraction of No. 1 plunger cavity

2.5. Cavitation process analysis of plunger cavity

According to the simulation model, the inlet pressure of axial piston pump is 1.01×10⁵ Pa and the outlet pressure is 3.0×10⁶ Pa. The simulation results are shown in Fig. 5. and Fig. 6.

Fig. 5 and Fig. 6 shows that the plunger cavity of water axial piston pump appears the cavitation when the plunger cavity sucks water. Due to the high rotation speed of the plunger cavity, the bubbles in the plunger cavity operated by means of centrifugal force are concentrated in the inner side. When piston cavity drains water, its pressure is high with no cavitation now. When the drainage is over, the plunger cavity passes through the bottom dead centre (BDC) and enters suction port. The pressure of plunger cavity fluid is greater than outlet pressure when the plunger cavity just contacts V shape throttling groove. The fluid in plunger cavity flows backward to outlet in an instant. At this time, the cavitation is not obvious. With the rotation of pump cylinder barrel, the piston moves with variable velocity along the axial, and the plunger cavity pressure drops. The gas in the fluid will precipitate when the fluid pressure in the plunger cavity is lower than the saturated vapor pressure of fluid or air separation pressure of fluid. At this time, the plunger cavity cavitation has been occurrence. The degree of plunger cavity cavitation is little at the beginning and end of drainage period. The degree of plunger cavity cavitation is serious at 0.028 second to 0.03 second this time. The plunger cavity cavitation is most serious especial at 0.0295 second.

Fig. 6Curve chart of mean value of volume fraction of No. 1 plunger cavity

Fig. 7Comparison of the mean value of volume fraction of plunger cavity with different rotating rate

4.1. V shaped and L shaped throttling groove flow area and hydraulic radius

Fig. 8 shows geometric sketch map of V and L shape throttling groove.

Calculation formula of the flow area and hydraulic radius of V shape throttling groove is expressed as follows:

Fig. 8Geometric sketch map of V and L shape throttling groove

Calculation formula of the flow area and hydraulic radius of L shape throttling groove is express as follows:

The flow area tone-up of V shape throttling groove is very small when the $x$ is small, but the flow area tone-up of L shape throttling groove is large when the $x$ is smaller. Therefore, high speed axial piston pump with shape throttling groove is more appropriate.

4.2. Effect of V shape and L shape throttling grooves on the plunger cavity cavitation

The simulation analysis of the plunger cavity cavitation is carried out on the axial piston pump with V and L shape throttling groove of valve plate under the rotating speed of 1500 r/min and 3000 r/min. The simulation result is shown in Fig. 9. The high flow tone-up performance of L shape throttling groove is not obvious when the rotation rate of the axial piston pump is low but it is very obvious when the rotation rate of the axial piston pump is high. Because L shape throttling groove has high flow tone-up and the fluid which flow into the plunger cavity is increased, cavitation degree of the plunger cavity decreases significantly.

5.1. Kidney shape port of cylinder barrel structure

Fig. 10 shows kidney shape port of cylinder barrel structure. The kidney shape port is divided into three types including straight kidney shape, one-way inclined type kidney shape and two-way inclined type kidney shape. Table 1 is the structural parameters of the kidney shape port of cylinder barrel.

5.2. Influence of the inclined type kidney shape port structure of cylinder barrel on self-sucking property

5.2.1. Analysis on the self-sucking property of plunger cavity with ${\mathit{\delta }}_{\mathit{x}\mathit{y}}$

Fig. 11 (A-A view in Fig. 10) shows the one-way inclined kidney-shape port. Because the kidney-shape port inclines certain angle relative to the $XY$ plane, the fluid in the kidney-shape port is influenced by centrifugal force when the cylinder rotates at high speed.

Fig. 9Comparison of the mean value of gas volume fraction of No. 1 plunger cavity under different rotation rate and different throttling groove

Fig. 10Kidney-shape port structure of cylinder barrel

Fluid press along the path is expressed as:

8

$\mathrm{\Delta }p=\frac{32\mu lv}{d^2},$

where $\mu$ is dynamic viscosity, $l$ is pipe length, $v$ is average flow velocity, $d$ is pipe diameter, for non-circular pipe with hydraulic diameter $d_h$ instead of $d$.

Table 1Structural parameters of the cylinder barrel kidney shape port

Kidney shape port of cylinder barrel structure	${\delta }_{xy}$(°)	${\delta }_{xz}$ (°)	$H$ (mm)
Straight kidney shape	90	0	7-15
One-way inclined kidney shape	65-85	0	7-15
Two-way inclined kidney shape	65-85	10-14	7-15

9

$dm=\rho sdz,\mathrm{}\mathrm{}\mathrm{}\mathrm{}d{F}_c=dm{r}_m{\omega }^2,$
$s=\left(\pi r_2^2-\pi r_1^2\right)\frac{\phi }{360}+\pi r_3^2,r_m=\left(\frac{r_1+r_2}{2}\right)+\frac{z}{\mathrm{t}\mathrm{a}\mathrm{n}{\delta }_{xy}},$
$d{F}_c=\rho s{r}_m{\omega }^{2}dz,d{F}^{\mathrm{\text{'}}}=d{F}_c\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{xy},$

where $dm$ is the quality of $dz$ segment fluid of cylinder kidney-shape port, $d{F}_c$ is centrifugal force of $dm$, $d{F}^{\text{'}}$ is the component force of $d{F}_c$ along axis of the cylinder kidney-shape port.

When the cylinder rotates at high speed, the component of centrifugal force of fluid along the axis of cylinder kidney-shape port is expressed as:

10

$F={\int }_0^H\rho s{r}_m\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{xy}{\omega }^{2}dz.$

where $s$ is sectional area of cylinder barrel kidney-shape port.

When ${\delta }_{xy}$ is smaller, the centrifugal force of cylinder kidney-shape port fluid and the component force $F$ along the axis of cylinder kidney-shape port is higher. Therefore, self-sucking capacity of the plunger cavity is improved. When the height $H$ of cylinder kidney-shape port is higher and the ${\delta }_{xy}$ is smaller, the length $L$ of cylinder kidney-shape port is increased. Meanwhile, loss pressure of fluid along the way is increased and pressure of fluid in the plunger cavity is reduced, which promote the generation of cavitation. So the cavitation degree of plunger cavity is related to the parameter ${\delta }_{xy}$ and $H$ of cylinder barrel kidney-shape port.

Fig. 11Centrifual force of one-way inclined kidney-shape port of cylinder

Fig. 12Analysis on the influence of the fluid drainage of the cylinder kidney-shape port on the δxz

5.3. Simulation analysis on the influence of the structure of cylinder kidney-shape port on the cavitation of the plunger cavity

Comprehensive the previous analysis, the structure parameters (such as $H$, ${\delta }_{xy}$, ${\delta }_{xz}$) of cylinder barrel have an effect on the self suction of the plunger cavity. The simulation analysis on the cavitation of the plunger cavity is carried out by using FLUENT software.

5.3.1. Influence of $\mathit{H}$ on cavitation of plunger cavity

The simulation analysis on the structure parameters of the cylinder kidney-shape port is described as follows

1) The values of $H$ range from 7 to 15 mm.

2) The value of ${\delta }_{xy}$ is 85°.

3) The value of ${\delta }_{xz}$ is 0°.

4) The value of rotation rate is 3000 r/min.

Fig. 13 shows that the maximum value of the gas volume fraction of the plunger cavity is larger as the value of $H$ increases, when the value of $H$ ranges from 7 to 15 mm. However, the maximum value of the gas volume fraction of the plunger cavity decreases as the value of $H$ increases when the value of $H$ exceeds 11 mm. It is because only the value of $H$ increases to a certain value as the value of ${\delta }_{xy}$ is large that the component of centrifugal force of fluid along the axis of cylinder barrel kidney-shape port can overcome the pressure loss along the side of cylinder barrel kidney-shape port.

Fig. 13Comparison of the mean value of gas volume fraction of No. 1 plunger cavity under different H

Fig. 14Comparison of the mean value of gas volume fraction of No. 1 plunger cavity under different δxy

5.3.3. Influence of ${\mathit{\delta }}_{\mathit{x}\mathit{z}}$ on cavitation of plunger cavity

The simulation analysis on the structure parameters of the cylinder kidney-shape port is described as follows:

1) The value of $H$ is 13 mm.

2) The value of ${\delta }_{xy}$ is 65°.

3) The values of ${\delta }_{xz}$ is 0°, 10°, 12°, 14°.

4) The value of rotation rate is 3000 r/min.

From the curve variation law of Fig. 15, it is found that the value of ${\delta }_{xz}$ has obvious effect on cavitation of plunger cavity. The time quantum that ranges from 0.01 second to 0.014 second is the period when the plunger cavity passes through the suction throttling groove. In this period when the value of ${\delta }_{xz}$ becomes greater, the gas volume fraction of plunger cavity is greater. Due to the increase of value of ${\delta }_{xz}$, the self suction of the plunger cavity becomes larger. At present the fluid pressure of the valve plate throttle groove drop sharply, and the cavitation is serious which is mainly caused by the small gain of valve plate throttling groove. In order to decrease the cavitation degree at this time, according to the results of the analysis in Section 4.2, the V shape throttling groove can be transformed into L shape throttling groove at high rotation rate.

Fig. 15Comparison of the mean value of gas volume fraction of No. 1 plunger cavity under different δxz

Fig. 16Kriging agent model approximate surface

The time quantum that ranges from 0.01 second to 0.014 second is the period when the plunger cavity has passed through the suction throttling groove. The gas volume fraction of the plunger cavity at this time first decreases rapidly and then increases, which is caused by the rotation rate from being slow to fast. In this period, when ${\delta }_{xz}$ is not equal to 0, cavitation of the plunger cavity is more likely to be inhibited than when ${\delta }_{xz}$ is equals 0. When the value of ${\delta }_{xz}$ is equal to 10°, the increased gradient of the value of gas volume fraction is the smallest after 0.018 second. It is mainly because when the value of ${\delta }_{xz}$ is equal to 10°, the ${\theta }_1$ and 90° – ${\delta }_{xz}$ are best matched. Strong vortex phenomenon is reduced by the fluid in the cylinder kidney hole and the flow resistance is reduced as well.

6.1. Basic principle of Kriging model

Kriging approximation model is a statistical foresting method which is widely used in mathematical geology based on stochastic processes. It can be used to optimization of regional variable, estimate of linear and unbiased interpolation. It has the statistical characteristics of smoothing effect and estimate of minimum variance. It has played an important role in linear geological statistics [18-20].

The Kriging model assumes that the true relationship between the system response values and the independent variables can be expressed as follows:

where $y\left(x\right)$ is Kriging unknown model, $f\left(x\right)$ is Known function (return part) of $x\text{,}$$\beta$ is Corresponding undetermined parameter, $f{\left(x\right)}^T\beta$ is a deterministic part and called the deterministic drift, $z\left(x\right)$ is the error of random distribution and is a statistical process. Its mean value is 0 and the variance is not 0.

Take the Gauss function as correlation function ($R$), and that is:

The estimate of Kriging model in the experimental sample point estimate is:

where $m$ is sampling points, ${\theta }_k$ is the orthogonal correlation function of No. 13, $\beta$ is the Regression constant, $\widehat{\beta }$ is the estimated value of $\beta$, $r^T$ is the Correlation vector.

The likelihood function $L$ of the correlation function $R$ is defined as follows:

The least squares estimation of $\beta$ and $\sigma$ in the No. 15 formula is:

Thus, $\widehat{\beta }$ and $R$ can be determined for a certain sample space. The maximum likelihood method ${\theta }_K$ is:

The accuracy of predicted value $\widehat{y}\left(x\right)$ is determined by the distance between sample points and forecast point $X$. When the sample point is closer to the forecast point $X$, the error of the predicated value is smaller.

6.2. Establishment of Kriging agent model

Through simulation analysis above, it is found that the ${\theta }_1$ and 90° – ${\delta }_{xz}$ match best when the ${\delta }_{xz}$ equals 10°. At this moment, the cavitation degree of the plunger cavity is the smallest. At the same time, $H$ and ${\delta }_{xy}$ also affect the cavitation of the plunger cavity, but the $H$ and ${\delta }_{xy}$ have a certain correlation. Therefore, in this paper, the Kriging agent model is established using constant regression model by taking the parameters $H$ and ${\delta }_{xy}$ as independent variables and the gas volume fraction of plunger cavity as function value when the t equals 0.019 second. In this paper, the five level full factorial experimental design is used. Value of $H$ is 7, 9, 11, 13 and 15 mm. Value of ${\delta }_{xy}$ is 65°, 70°, 75°, 80° and 85°. The twenty-five structure models have been simulated and analyzed. The Kriging agent model is established by taking Gauss function as correlation function, and is shown in Fig. 16.

6.3. Optimization results of genetic algorithm (GA) and its simulation analysis

6.3.2. Genetic algorithm operators

Now the genetic algorithm is applied to optimize the agent model. First, genetic operators and other parameters must be defined in the genetic algorithm, namely: population number, crossover with rate, mutation with rate, maximum number of generations (MUG), generation gap and variable interval. The paper determined the values of $p_c$ and $p_m$ as 0.9 and 0.7 respectively. The values of other parameters are shown in Table 2. The optimal solution of the Kriging agent model is shown in Fig. 17.

Table 2The values of other parameters

Parameter	Population number	MUG	Generation gap	$H$ (mm)	${\delta }_{xy}$(°)
Value	60	400	0.9	7-15	65-85

When $H$ is 14.87 mm, and ${\delta }_{xy}$ is 70.61°, the minimum target value of the gas volume fraction of the plunger cavity is 26.24 %.

Fig. 17The change lines chart of the genetic iterative optimization and calculation

Fig. 18Curve charts of gas volume fraction in piston cavity of optimized structure of axial piston pump

6.4. Establishment of optimization structure of cylinder kidney-shape port of piston pump and its simulation analysis

According to the result of genetic optimization calculation, the main structural parameters of the one-way inclined cylinder kidney-shape port are determined. The value of $H$ is 14.87 mm and the value of ${\delta }_{xy}$ is 70.61°. Simulation analysis of gas fraction of plunger cavity is shown by curve 1 in Fig. 18.

To further reduce the cavitation degree of plunger cavity, the optimized structure of one-way inclined cylinder kidney-shape port can be transformed into the two-way inclined cylinder kidney-shape port. According to the result of the analysis in section 4.3.3, the value of ${\delta }_{xz}$ is 10°, which is determined. Simulation analysis of gas fraction of plunger cavity is shown by curve 2 in Fig. 18.

Comparing curve 2 with curve 1, the cavitation of the plunger cavity has some improvement from 0.016 second to 0.02 second. But in the range of 0.01 second to 0.014 second, the cavitation degree of the plunger cavity increases. The self-suction of the plunger cavity increases, which is mainly because of adopting two-way inclined cylinder kidney-shape port. However, a certain degree of air suction has occurred because the flow gain of the V shape throttling groove is small when the plunger cavity passes through the V shape throttling groove of valve plate. In order to improve the situation, with the analysis of section 4.2, the V shape throttling groove can be replaced by L shape throttling groove. Simulation analysis of gas fraction of plunger cavity is shown by curve 3 in Fig. 18. According to the change regulation of the three curves in Fig. 18, it is obvious that the axial piston pump structure corresponding to the third curve in Fig. 18 has obviously inhibition to the cavitation of the plunger cavity.