Optimization of swash plate multistage compressor based on dynamics

The swash plate compressor can provide high-pressure gas through multistage compression using its different cylinders, which is an ideal structure for high-pressure compressor miniaturization. The swash plate multistage compressor (SPMC) with variable sections cylinder and inter-stage cooler is proposed and its complex structure results in complex multi-body dynamics and thermodynamics. To research the dynamic characteristics of SPMC, the dynamic model of SPMC is established, which includes the thermodynamics of compression and inter-stage cooler and the dynamics of all the check valves and pistons. The optimization objective function considering the dynamic characteristics and efficiency is proposed. The constraint conditions and sensitivity of the optimized parameters are studied and the final parameters are achieved through an optimization algorithm. Based on the above work, it can be found that the optimization way has high convergence and accuracy, and the comprehensive performance of SPMC is improved significantly.


Introduction
High-pressure gas, as a green energy, is widely applied in aerospace [1][2].How to miniaturize highpressure compressor becomes an important research field [3][4].High-pressure gas can be achieved by multistage compression and multistage compressors are often large in volume and weight in industry so far.Swashplate compressor can drive multiple pistons at the same time, which makes multistage compression possible in the same device [5][6].So, SPMC has become a research focus in the field of miniature high-pressure compressors [7][8].
When the swash plate compressor works, each cylinder is in a different compression state because of the phase differences among them and their piston forces are also different, which increases the vibration and noise significantly.How to improve the dynamic characteristics becomes very important.Tojo et al. [9] investigated the dynamic of variable stroke swash plate compressors and established a mathematical model to improve the inertia balance.Estupinan and Santos [10] studied the dynamic characteristics of multi-body dynamics and proposed the parameter matching method based on vibration reduction.Zhang et al. [11] studied and improved the dynamic balance of the compressor.He et al. [12] studied the dynamic characteristics of oscillating compressors under different exhaust pressures.As a multi-parameter coupled dynamic system, the performance improvement of the compressor is heavily influenced by multiple structural parameters, and the multi-parameter optimization method becomes an important approach for improving compression performance.Sultan and Kalim [13] optimized the parameters of reciprocating compressors based on a hybrid two-stage optimization method of kinematic and thermodynamic performance for the driving mechanism.Shin et al. [14] established and solved the dynamic equation of variable stroke swash plate compressor and the influence of system parameters (speed, piston mass, etc.) on piston stroke is analyzed.Deng et al. [15] studied the variation of temperature, mass flow, and pressure of multi-stage swash-plate compressors in different ambient temperatures.Baek and Kim [16] proposed a structural design optimization method based on the Chebyshev orthogonal polynomials and optimization techniques.These research achievements have promoted the rapid development of the swash plate compressor.SPMC, as a new kind of compressor, has not yet been investigated from the view of its dynamic characteristics.
The SPMC with variable sections is proposed to improve the dynamic characteristics.The optimization method of SPMC based on dynamics and efficiency is studied on the basis of establishing the SPMC dynamics model and the design principles for the key parameters are achieved, which are of great significance for the designing and performance improvement of SPMC.

Working principle of SPMC
The SPMC (Figure 1), as a four-stage compression, mainly consists of six units: swash plate, piston, cylinder, check valve, inter-stage cooler, and high-pressure gas cylinder.The swash plate converts the rotation motion into the reciprocating straight motion of the piston (Figure 1(b)).The four pistons are distributed in the same circle.As the swash plate rotates a round, the four pistons all move back and forth once and the gas is compressed four times.The check valve is used to keep the gas flowing in one direction and is mainly installed between the cylinder and the inter-stage cooler.The diameters of the four cylinders are different (Figure 1(a)) and they are used as the first, second, third, and fourth stage compression from large to small in diameter.The gas in the larger cylinder is compressed into the smaller one through check valves and an inter-stage cooler, which can reduce the gas force difference between different cylinders and improve efficiency and dynamic characteristics.The interstage cooler is to cool the compressed gas between the front stage compression and rear stage compression.

Thermodynamics of compression
When the swash plate rotates and the piston moves, the gas in the cylinder satisfies the first law of thermodynamics (Equation (1)), gas state equation (Equation ( 2)), and conservation equation (Equation (3)).
where U is the internal energy of the gas in the cylinder, Qgc is the energy of the heat exchange between the cylinder and the gas in it, Vc is the volume of gas in the cylinder, and mg and hg are the mass and enthalpy of the gas, respectively.P, T, R0, mi, and mo are the gas pressure, the gas temperature, the gas constant, and the mass of gas through the inlet and outlet of the check valve, respectively.
When the angular velocity of the SPMC is w, the piston stroke and working volume of compression are shown in Equation (4) and Equation (5).

tan (1 cos(
) where l is the stroke of the piston, α is the inclination angle of the swash plate, R is the piston distribution circle radius, i α is the phase angle between the first stage cylinder and the ith stage one, d is the diameter of the piston, and V0 is the clearance volume.
The energy of the heat exchange between the cylinder and the gas in it is shown in Equation (6).
where Hgc and S are the coefficients of convection heat transfer and heat convection area, respectively.Tg and Tc are the temperatures of the gas and the cylinder, respectively.v c and p c are the specific heat at constant volume and constant pressure.
When the cylinder absorbs heat Qgc, its temperature is changed.
where c C , c m , c T and c0 T are the specific heat capacity, the mass, the current temperature, and the initial temperature of the cylinder, respectively.

Check valve
In the compressor, the check valve is used to keep gas flowing in one direction (The valve parameters are shown in Figure 2).The valve core moves as its inlet gas pressure is higher than the outlet pressure.Therefore, the gas satisfies the mass flow equation (Equation ( 11)) and the valve can be described as a single freedom spring-mass-damper system (Equation ( 12)).
where Ae is the effective orifice area of the valve, Cm is the flow parameter, Cq is the flow coefficient, Pi is the inlet gas pressure, and Ti is the inlet gas temperature.me, ce and ke respectively are the equivalent mass, damping coefficient and stiffness.xe is the opening of the check valve, Fi and Fo respectively are the inlet and outlet gas force, and F0 is the pre-pressure of the valve.
Assuming the gas is the perfect gas, Cm can be expressed as Equation ( 13) [17].Figure 3. Force on the swash plate.

Inter-stage cooler
The inter-stage cooler is a thermodynamic system with heat dissipation and variable mass gas.The gas in the inter-stage cooler also satisfies the first law of thermodynamics (Equation ( 14)), gas state equation (Equation ( 15)) and conservation equation (Equation ( 3)).The relationships among parameters satisfy Equation ( 7) -Equation (10).
where Uic is the internal energy of the gas in the inter-stage cooler, and Qic is the heat exchange between the gas and the inter-stage cooler.
When the cooling coefficient of the inter-stage cooler is Hic, its cooling energy is shown in Equation (16).
where Hic is the heat extraction coefficient of the inter-stage cooler, Tic is the gas temperature in the inter-stage cooler, and Tic0 is the temperature of the inter-stage cooler.

Dynamics analysis
The piston force Fp acting on the swash plate can be decomposed into resistance force Ft and normal force Fn (Figure 3).The compressor resistance moment is generated by the resistance force (Equation ( 17)).The friction moment between the piston and the swash plate is generated under the action of normal force (Equation ( 18)).The total resistance moment of the compressor is the sum of all the resistance and friction moments (Equation ( 19)).where Mti and Mfi are respectively the resistance moment and friction moment generated by the action of the ith piston.Mr is the total resistance moment.βi is the phase angle of the ith stage, and Fsf is the friction force between the piston and swash plate.
The piston force Fp mainly includes the inertia force Fa, gas force Fg, and friction force between the piston and the cylinder wall Ff.

Simulation
The SPMC is a four-stage compression system and the four cylinders are distributed in the same diameter circle.The phase differences between the first stage and the second one, third one and fourth one are 180°, 90°and 270°, As the gas pressure in the front stage of the cylinder is lower than that in the rear stage, the diameter of the front stage cylinder should be larger than the rear stage cylinder.It can not only reduce the maximum gas force but also improve the compression efficiency.
The same parameters are used on the Eight check valves.In the simulation, the compressed gas is air.The parameters in Table 1 are the initial parameters of SPMC.When the maximum output pressure is 256 bar, and the working time is 120 s, the simulation results of the last two cycles are shown in Figure 4.The gas pressure in each cylinder changes periodically and the maximum gas pressure of the former stage is always much lower than that of the latter (Figure 4(a)).The maximum gas pressure in the former stage is always larger than the minimum pressure in the latter stage because of the check valve (Figure 4(a)).The piston forces of each stage also vary periodically (Figure 4(b)).As the phase differences exist among different cylinders, the maximum piston force occurs at different times (Figure 4

Optimization
As is shown in Figure 4(c), both the resistance moment and overturning moment fluctuate widely in one cycle.The overturning moment with larger fluctuation will intensify vibration and noise, which aggravates fatigue damage and reduces the service life of SPMC.The resistance moment with larger fluctuation will put the motor in an unstable state all the time, which will reduce the life of the motor and increase its noise.Therefore, the key parameters should be studied and optimized to improve SPMC dynamic characteristics.

Objective function
When the SPMC works, four cylinders are in different compression states and their piston forces change dynamically in one cycle.The characteristics of the resistance moment and overturning moment in one cycle are both closely related to all the piston forces and their distribution, which can comprehensively show the dynamics of SPMC.The maximum and average resistance moments determine the power and the maximum driving moment of the driving motor.The mean square error of the resistance moment is closely related to its fluctuation and the larger value of it will enhance the noise and reduce the life of the motor.The overturning moment is periodic and its maximum, average and mean square errors are all closely related to the fatigue and vibration of SPMC.Their larger values can greatly intensify the vibration and noise of SPMC.Considering the mass flow of exhaust is closely related to SPMC efficiency, a new objective function is proposed in Equation (24) after normalization.The weight coefficients of mass flow, resistance moment and overturning moment are the same.Their values are equal to the ratio between the values under the new and initial parameters (Table 1).When the parameters are the same in the Table 1, the Obj is 1.The Obj is larger, the efficiency is higher and the levels of vibration and noise are lower.
where o m ( % , rmax and v ρ ( are the ratio between the mean mass flow under the new and initial parameters in one period, the ratio between the maximum resistance moment under the new and initial parameter in one period, the ratio between the average resistance moment under the new and initial parameter in one period, the ratio between the mean square error of the resistance moment under the new and initial parameter in one period, the ratio between the maximum overturning moment under the new and initial parameter in one period, the ratio between the average overturning moment under the new and initial parameter in one period and the ratio between the mean square error of the overturning moment under the new and initial parameter in one period, respectively.

Optimized parameters and constraint conditions
The mass flow, resistance moment and overturning moment of SPMC are all closely related to all cylinder's diameters and the volume of the inter-stage cooler.The optimizing parameters should be the diameter of the second, third and fourth stages and the volume of all the inter-stage coolers.In order to determine the optimized parameters and their constraints, their sensitivity to the objective function is studied by quantitative method.When the optimized parameters are selected with different values, the objective function values are shown in Figure 5.If the objective function is larger than 1, the value of the studied parameter is better than its initial one.Both the cylinder diameter of the second stage and the inter-stage cooler's volume between the second and third have a great effect on the value of the objective function.The phase difference between the second and third stages is 90°.When the second stage has completed the compression, the third stage just starts to inhale.So, both the compression ratio and piston force of the second stage are larger (Figure 4), which increases the moment fluctuation.The cylinder diameter of the fourth stage and the inter-stage cooler's volume between the third and fourth have little effect on the value of the objective function, which is insensitive and need not be optimized.According to Figure 5, the final optimized parameters and their value ranges are shown in Table 2.The final optimized parameters are achieved by using this optimization method.The comparison before and after optimization is shown in Figure 6 and Table 4.The average and volatility of the resistance moment and overturning moment both decrease significantly.In particular, the mean square error of the overturning moment decreases by 64.9%.The dynamics characteristics of SPMC are improved significantly.There is little change in mass flow.When the sensitivity of the optimized parameters is not studied, their constraint conditions are shown in Table 5.The parameters of the optimization method are the same as in Table 3.The optimized structure parameters are almost the same as Table 4. But, its iterations are about 300 (Figure 7), which is about three times the constraint conditions in Table 2. Therefore, the optimization method after studying the sensitivity of optimized parameters has fast convergence and accuracy.

Conclusion
Based on the SPMC model, optimization method and the result of the above research, the following conclusions can be achieved.
1) The dynamic characteristic of the SPMC is improved significantly after the proposed optimization method.
2) Both the diameter of the second-stage cylinder and the volume of the inter-stage cooler between the second and third stages are sensitive to the dynamic characteristics of the SPMC.
3) This optimization method after studying the sensitivity of the constraint conditions has fast convergence and accuracy.
about the Oy and Oz axes (Figure (3)) respectively are My and Mz.The total overturning moment of compressor Mv is expressed as Equation (23).
(b)) and the moment fluctuates four times in one period (Figure 4(c)).The maximum resistance moment Mr and overturning moment Mv respectively are 11.4 N.m and 45.4 N.m.Their averages respectively are 7.1 N.m and 23 N.m and mean square errors respectively are 1.64 and 7.63 in one cycle.Both the resistance moment and overturning moment fluctuate significantly, which enhances the vibration and noise.The suction and exhaust mass flow of SPMC are shown in Figure 4(d), which is closely related to the movement of the valve core.

Figure 6 .
Figure 6.Comparison before and after optimization.Figure 7. Iterations at different conditions.

Figure 7 .
Figure 6.Comparison before and after optimization.Figure 7. Iterations at different conditions.

Table 3 .
The fitness function is Obj.

Table 4 .
Parameters of before and after optimization.