Research on Performance Prediction Model of Impeller-Type Breather

Featured Application: The equations established in this paper can be used to evaluate the separation and resistance performance of an impeller-type breather in an aeroengine lubrication system. The dimensional analysis method used in this paper can be extended to the performance evaluation of multi-parameter complex systems. Abstract: To investigate the characteristics of separation and resistance of an impeller-type breather in an aeroengine lubrication system, orthogonal test design is used in calculation of the operating condition. Also, phase coupling of the RNG(Renormalization Group) k − ε model and the DPM model (Discrete Phase Model) is used in calculating the selected operating condition. Through analysis of the results, combined with dimensional analysis, it shows the signiﬁcance of various inﬂuencing factors and the optimal level. Based on this, a general formed dimensionless group equation is established for comprehensive separation e ﬃ ciency, breather separation e ﬃ ciency, and ventilation resistance. Also, through the least squares method, the performance prediction model of the breather is obtained considering ﬁve operating conditions and six structural parameters. The theoretical calculation of separation e ﬃ ciency and ventilation resistance of an impeller-type breather can be performed. The results show that: the main factors a ﬀ ecting the separation e ﬃ ciency are the rotating speed and the number of impeller blades; the main factors a ﬀ ecting the ventilation resistance are the ventilation rate and the diameter of the vent hole; the variation trends of the calculated values of the performance prediction model and the experimental values are consistent. The mean error of the comprehensive separation e ﬃ ciency is 0.97% and the mean error of the ventilation resistance is 11.73%. The calculated values and the experimental values remain consistent, which proves that this performance prediction model can provide references to the assessment and the design of an impeller breather.


Introduction
When an aircraft engine lubrication system starts operating, part of the sealed air will enter the bearing chamber and mix with the lubricating oil forming an oil-gas mixture. Direct discharge of the oil-gas mixture from the bearing chamber will cause loss of lubricating oil. Since the amount of the lubricating oil is certain, the loss of the lubricating oil will affect the mechanical properties of the engine [1]. Therefore, installation of a breather in the air passage of the bearing chamber will separate the oil-gas mixture effectively.
Breathers come in various forms. Typical breathers include centrifugal breathers and impeller-type breathers; both types share the same operating principle. When the engine is operating, the oil carried by the gas enters the breather. The spinning rotors create a centrifugal force to the oil-gas mixture, thus forcing the oil droplets to the inner wall of the shell and flowing back to the transmission chamber 3 of 13 was 5%. In 2016, Han et al. [15] used the Euler-Lagrange equation to simulate the two-phase flow field of an ultra-high speed centrifugal breather. The results showed that the separation efficiency can be improved by optimizing the structure of the rotating hallow shaft at ultra-high speed. In 2016, Zhong et al. [16] used VOF (Volume of Fluid) and a liquid-solid wetting model to study the impact of droplets with different sizes on the dry wall. It was found that the properties of droplets at the micron level would have a stronger impact on the diffusion process of droplets after impact than droplets at the millimeter level, and the critical parameter of droplet splash at the micron level was K = 122. In 2017, Tianyu et al. [17] proposed a new model based on the experiment of droplet impact on the dry wall and droplet morphology dynamics. The model considered the wall conditions, summarized and analyzed various splash modes, and was verified under different conditions. At present, the research methods of breather performance are mostly experiments and numerical simulation. Experiments are often limited by the amount of specimen processing, flow field control, and measurements. Numerical simulation is mainly to explore the influence of the working condition and main structural parameters on the performance of the breather. Many factors can affect the performance of the breather. When evaluating the significance of factors and the optimal level, a performance prediction model for a breather is established in consideration of multiple factors, which has practical significance for the engineering design and optimization of the breather. This paper verifies the accuracy of the numerical simulation method by comparison with the experimental data. The impeller breather is numerically studied by means of orthogonal experimental design. The significance and optimal level of different influencing factors are determined by range analysis. Based on the dimensional analysis, a dimensionless group equation in general form is established for comprehensive separation efficiency, breather separation efficiency, and ventilation resistance. Also, through the least squares method, the performance prediction model of the breather is obtained, which provides a reference for the design, evaluation, and optimization of the impeller breather.

Numerical Method
In the breather, there are strong swirling two-phase flows, in which the two phases are separated and there are physical phenomena, such as oil droplets hitting the wall surface. In this paper, the Euler-Lagrange equation was adopted to establish a two-way coupling model to show the two-phase flows. A liquid film model was used to describe oil droplet wall interaction. The splash, spreading, stick, rebound, and other phenomena of droplets were also considered.

Two-Phase Flow Calculation Method
In an aeroengine impeller-type breather, the internal flow is gas-liquid two-phase flow. The oil/gas ratio is approximately 1% [18]. The volume fraction of the oil droplets is much less than 10%, which is considered as dilute two-phase flow. Therefore, the Euler-Lagrange equation was adopted to simulate its internal flow field.

Governing Equations for Gas Phase
The gas phase governing equation was established based on the two-way coupling method, with consideration of the effect of droplet motion on airflow. The governing equation of gas phase per unit volume is as follows: where ρ is the density of the gas; u is the velocity of the gas; S m is the source term. In the oil and gas two-phase flow in the breather, the evaporation of oil droplets will cause the source term to change if operating in a high heat environment.
where p is static pressure;τ is stress tensor; S u is the source term. The influence of oil droplets on airflow momentum includes the momentum carried by oil droplet evaporation and the momentum exchange caused by oil droplet movement.

Oil Droplet Motion Equation
The Euler-Lagrange method was used to calculate the trajectory of the oil droplets. According to Newton's law, the motion equation is as follows: where u p is the velocity of droplets; F D is the frictional force known as Stokes' drag; ρ p is the density of the droplets; F add is additional force, including additional force generated by oil droplets in the rotating coordinate system, Saffman lift force, Magnus force, virtual mass force, pressure gradient force, etc.

Wall Liquid Film Model
When droplets hit a solid surface, the droplets may splash, spread, stick, or rebound, depending on the impact energy and surface temperature. Scholars from all over the world have done a lot of experimental research and proposed some evaluation criterion. In this paper, the interaction model between oil droplets and the wall surface was based on the research of Farrall [18], as shown in Figure 1.
where p is static pressure; is stress tensor; Su is the source term. The influence of oil droplets on airflow momentum includes the momentum carried by oil droplet evaporation and the momentum exchange caused by oil droplet movement.

Oil Droplet Motion Equation
The Euler-Lagrange method was used to calculate the trajectory of the oil droplets. According to Newton's law, the motion equation is as follows: where up is the velocity of droplets; FD is the frictional force known as Stokes' drag; p is the density of the droplets; Fadd is additional force, including additional force generated by oil droplets in the rotating coordinate system, Saffman lift force, Magnus force, virtual mass force, pressure gradient force, etc.

Wall Liquid Film Model
When droplets hit a solid surface, the droplets may splash, spread, stick, or rebound, depending on the impact energy and surface temperature. Scholars from all over the world have done a lot of experimental research and proposed some evaluation criterion. In this paper, the interaction model between oil droplets and the wall surface was based on the research of Farrall [18], as shown in Figure 1. To determine the result of droplet-wall collision, splashing parameter K and the Weber number Wed have been used and the result is presented as follows: where ur is the velocity of the oil droplets relative to the wall; dp is the diameter of the droplet; p is the surface tension of the droplet; h0 is the initial thickness of the oil film; δbl is the thickness of the boundary layer; up,n is the normal component of the oil drop velocity. To determine the result of droplet-wall collision, splashing parameter K and the Weber number We d have been used and the result is presented as follows: where u r is the velocity of the oil droplets relative to the wall; d p is the diameter of the droplet; σ p is the surface tension of the droplet; h 0 is the initial thickness of the oil film; δ bl is the thickness of the boundary layer; u p,n is the normal component of the oil drop velocity. When the splashing parameter K is greater than the critical value 57.7, the oil droplets will splash and produce new oil droplets in a smaller size. The parameters such as the diameter of the oil droplet will be determined by the cumulative probability distribution function fitted by the experiment. When the splashing parameter K is less than the critical value 57.7, oil droplets do not break and remain their original size. After collision with the wall, there are three main kinds of phenomenon, which include spread, stick, and rebound. When the Weber number is less than 5 and greater than 10, stick and spread of the oil droplets will happen, respectively. These two phenomena are considered as completely collected by the wall surface. However, when the Weber number is between 5 and 10, the oil droplets will rebound, that is, the oil droplets return to the flow inside the breather and are not collected by the wall surface.

Model Building and Model Validation
Impeller-type breathers are usually centrifugal impellers placed in an accessory casing or a bearing cavity and are usually mounted directly on a drive shaft. Compared with the centrifugal breather, it has no outer shell, so it needs to be placed in the experimental section for research. The structure of the experimental section is shown in Figure 2. The oil-gas mixture enters the inner chamber from the inlet and is separated by a rotating impeller breather. The separated lubricating oil is deposited at the bottom and collected. The separated gas is discharged from the hollow shaft.
splash and produce new oil droplets in a smaller size. The parameters such as the diameter of the oil droplet will be determined by the cumulative probability distribution function fitted by the experiment. When the splashing parameter K is less than the critical value 57.7, oil droplets do not break and remain their original size. After collision with the wall, there are three main kinds of phenomenon, which include spread, stick, and rebound. When the Weber number is less than 5 and greater than 10, stick and spread of the oil droplets will happen, respectively. These two phenomena are considered as completely collected by the wall surface. However, when the Weber number is between 5 and 10, the oil droplets will rebound, that is, the oil droplets return to the flow inside the breather and are not collected by the wall surface.

Model Building and Model Validation
Impeller-type breathers are usually centrifugal impellers placed in an accessory casing or a bearing cavity and are usually mounted directly on a drive shaft. Compared with the centrifugal breather, it has no outer shell, so it needs to be placed in the experimental section for research. The structure of the experimental section is shown in Figure 2. The oil-gas mixture enters the inner chamber from the inlet and is separated by a rotating impeller breather. The separated lubricating oil is deposited at the bottom and collected. The separated gas is discharged from the hollow shaft.

Physical Model
In order to use the experimental results to verify the numerical simulation, the computational physical model was established on the basis of the experimental section of the impeller-type breather, as shown in Figure 2, through simplification, which is shown in Figure 3. In Figure 3, it shows the inlet, shell, impeller breather, and the hollow shaft on which three vents were distributed.

Physical Model
In order to use the experimental results to verify the numerical simulation, the computational physical model was established on the basis of the experimental section of the impeller-type breather, as shown in Figure 2, through simplification, which is shown in Figure 3. In Figure 3, it shows the inlet, shell, impeller breather, and the hollow shaft on which three vents were distributed. splash and produce new oil droplets in a smaller size. The parameters such as the diameter of the oil droplet will be determined by the cumulative probability distribution function fitted by the experiment. When the splashing parameter K is less than the critical value 57.7, oil droplets do not break and remain their original size. After collision with the wall, there are three main kinds of phenomenon, which include spread, stick, and rebound. When the Weber number is less than 5 and greater than 10, stick and spread of the oil droplets will happen, respectively. These two phenomena are considered as completely collected by the wall surface. However, when the Weber number is between 5 and 10, the oil droplets will rebound, that is, the oil droplets return to the flow inside the breather and are not collected by the wall surface.

Model Building and Model Validation
Impeller-type breathers are usually centrifugal impellers placed in an accessory casing or a bearing cavity and are usually mounted directly on a drive shaft. Compared with the centrifugal breather, it has no outer shell, so it needs to be placed in the experimental section for research. The structure of the experimental section is shown in Figure 2. The oil-gas mixture enters the inner chamber from the inlet and is separated by a rotating impeller breather. The separated lubricating oil is deposited at the bottom and collected. The separated gas is discharged from the hollow shaft.

Physical Model
In order to use the experimental results to verify the numerical simulation, the computational physical model was established on the basis of the experimental section of the impeller-type breather, as shown in Figure 2, through simplification, which is shown in Figure 3. In Figure 3, it shows the inlet, shell, impeller breather, and the hollow shaft on which three vents were distributed.

Grid Generation
Due to the large size difference of each part of the physical model, block processing of the whole computational domain was performed. A structural grid was applied to the shell and the hollow shaft. The blade passages are geometrically irregular; therefore, unstructured grids were used. Local encryption was carried out at the flow passage and vents, and the total number of grids was about 700,000. The grid generation is shown in Figure 4.
Due to the large size difference of each part of the physical model, block processing of the whole computational domain was performed. A structural grid was applied to the shell and the hollow shaft. The blade passages are geometrically irregular; therefore, unstructured grids were used. Local encryption was carried out at the flow passage and vents, and the total number of grids was about 700,000. The grid generation is shown in Figure 4.

Model Validation
The inside of the breather is a three-dimensional vortex flow with strong turbulence. The purpose of single-phase model validation is to find the most suitable turbulence model. In this paper, a single-phase numerical simulation was performed using the RSM model, the realizable k   model, and the RNG k   model. The turbulence model was finalized by comparing the calculated value with the experimental data. As shown in Figure 5, the calculated values of ventilation resistance of the three turbulence models were consistent with the experimental data. The mean error of the RSM model, realizable k   model, and RNG k   model was 6.80%, 6.25%, and 4.01%, respectively. Since the mean error of RNG k   model was the smallest, it will be used as the turbulence model in this paper.

Two-Phase Model
In this paper, the Euler-Lagrange method was used to describe the two-phase flow, the RNG k   model was used to solve the continuous phase, and the DPM model was used to solve the oil droplet phase. In order to verify the accuracy of the coupling calculation of the RNG k   model and the DPM model, the calculated value was compared with the experimental value. The results are shown in Figure 6.

Single-Phase Model
The inside of the breather is a three-dimensional vortex flow with strong turbulence. The purpose of single-phase model validation is to find the most suitable turbulence model. In this paper, a single-phase numerical simulation was performed using the RSM model, the realizable k − ε model, and the RNG k − ε model. The turbulence model was finalized by comparing the calculated value with the experimental data.
As shown in Figure 5, the calculated values of ventilation resistance of the three turbulence models were consistent with the experimental data. The mean error of the RSM model, realizable k − ε model, and RNG k − ε model was 6.80%, 6.25%, and 4.01%, respectively. Since the mean error of RNG k − ε model was the smallest, it will be used as the turbulence model in this paper.
Due to the large size difference of each part of the physical model, block processing of the whole computational domain was performed. A structural grid was applied to the shell and the hollow shaft. The blade passages are geometrically irregular; therefore, unstructured grids were used. Local encryption was carried out at the flow passage and vents, and the total number of grids was about 700,000. The grid generation is shown in Figure 4.

Model Validation
The inside of the breather is a three-dimensional vortex flow with strong turbulence. The purpose of single-phase model validation is to find the most suitable turbulence model. In this paper, a single-phase numerical simulation was performed using the RSM model, the realizable k   model, and the RNG k   model. The turbulence model was finalized by comparing the calculated value with the experimental data. As shown in Figure 5, the calculated values of ventilation resistance of the three turbulence models were consistent with the experimental data. The mean error of the RSM model, realizable k   model, and RNG k   model was 6.80%, 6.25%, and 4.01%, respectively. Since the mean error of RNG k   model was the smallest, it will be used as the turbulence model in this paper.

Two-Phase Model
In this paper, the Euler-Lagrange method was used to describe the two-phase flow, the RNG k   model was used to solve the continuous phase, and the DPM model was used to solve the oil droplet phase. In order to verify the accuracy of the coupling calculation of the RNG k   model and the DPM model, the calculated value was compared with the experimental value. The results are shown in Figure 6.

Two-Phase Model
In this paper, the Euler-Lagrange method was used to describe the two-phase flow, the RNG k − ε model was used to solve the continuous phase, and the DPM model was used to solve the oil droplet phase. In order to verify the accuracy of the coupling calculation of the RNG k − ε model and the DPM model, the calculated value was compared with the experimental value. The results are shown in As it shows in Figure 6, the calculated values and the experiment values of the ventilation resistance and separation efficiency are identical. The mean error of ventilation resistance was 4.05%, and the maximum error was 8.60%. The mean error of separation efficiency was 0.82%, and the maximum error was 2.08%. The results indicate that the coupling of the RNG k   model and the DPM model can be used for the calculation of breather performance.

Orthogonal Calculation Method Design
In order to study the influence of operating condition and structure on the performance of the breather, five operating condition (A~E) parameters and six structural parameters (F~K) were selected for the research. For each parameter, five levels were selected, as shown in Table 1. If the influence of all 11 parameters on the performance of the separator was studied in detail, the workload will be huge since there are 5 11 types of operating conditions to be considered. Therefore, with reference to the experimental design method, L50(5 11 ) orthogonal table was selected to optimize the 11 parameters on 5 levels, and 50 groups of calculation conditions were finally determined.

Significance of Factors and Optimal Method
The two-phase flow in the breather is complex and the performance can be affected by many factors. An orthogonal test can reduce the calculation workload effectively. At the same time, the significance of factors and the optimal levels can be obtained through the range analysis of the calculation results. By giving an estimation of fluctuant range and achievable index value, it provides a reference for the determination of the operating condition and structure optimization. As it shows in Figure 6, the calculated values and the experiment values of the ventilation resistance and separation efficiency are identical. The mean error of ventilation resistance was 4.05%, and the maximum error was 8.60%. The mean error of separation efficiency was 0.82%, and the maximum error was 2.08%. The results indicate that the coupling of the RNG k − ε model and the DPM model can be used for the calculation of breather performance.

Orthogonal Calculation Method Design
In order to study the influence of operating condition and structure on the performance of the breather, five operating condition (A~E) parameters and six structural parameters (F~K) were selected for the research. For each parameter, five levels were selected, as shown in Table 1. If the influence of all 11 parameters on the performance of the separator was studied in detail, the workload will be huge since there are 5 11 types of operating conditions to be considered. Therefore, with reference to the experimental design method, L 50 (5 11 ) orthogonal table was selected to optimize the 11 parameters on 5 levels, and 50 groups of calculation conditions were finally determined.

Significance of Factors and Optimal Method
The two-phase flow in the breather is complex and the performance can be affected by many factors. An orthogonal test can reduce the calculation workload effectively. At the same time, the significance of factors and the optimal levels can be obtained through the range analysis of the calculation results.
By giving an estimation of fluctuant range and achievable index value, it provides a reference for the determination of the operating condition and structure optimization.

Characteristics of Separation
Separation efficiency of the breather is defined as the percentage of the mass of oil droplets separated from the breather in the mass of oil droplets entering the breather, which is an important index to evaluate the performance of the breather. Range analysis was made on the separation efficiency in the calculated results to determine the significance and optimal level of the various factors, as shown in Table 2.
As shown in Table 2, the rotating speed has the largest influence on separation efficiency among all operating conditions, followed by ventilation rate and average inlet particle size. The temperature and oil/gas ratio have little influence. Number of blades has the largest influence among all structural factors, followed by external diameter of the blades and thickness of the breather. Blade angle and diameter of vent hole have little influence.

Characteristics of Resistance
Ventilation resistance is defined as the pressure difference between the inlet and outlet of the breather; it indicates flow loss, which is another important index to evaluate the performance of the breather. The range analysis of the ventilation resistance in the calculation results was carried out to determine the significance and optimal level of the various influential factors. The results are shown in Table 3. Table 3. Range analysis of each influential factor on ventilation resistance.

Operating Condition Factors Structural Factors
Among all operating condition factors, the ventilation rate has the largest influence on ventilation resistance, while other factors have a small influence. Among all structural factors, the diameter of the vents has the greatest influence, while the other factors have a small influence.

Performance Prediction Model
In this paper, the dimensional analysis method was used to establish the empirical formula of the ventilation performance calculation for an engineering application. The factors related to the breather performance were sorted into dimensionless numbers using the Buckingham π theorem, forming a dimensionless group equation in general form. By using the least squares method, the engineering formulas for the comprehensive separation efficiency, the ventilation efficiency, and the ventilation resistance were obtained.
As shown in Table 1, there are 11 main factors affecting the performance of the breather. Their units and dimensions are listed in Table 4. In order to show the state of the two-phase mixture, v, ρ, and µ have been added to Table 4. Considering the relationship between mixture velocity (v) and ventilation rate, v = q π(d 1 /2) 2 , where d 1 is the inlet diameter, mixture velocity (v) can represent the effect of ventilation rate on the performance of breather. Mixture density (ρ) and dynamic viscosity (µ) is related to temperature, which can characterize the influence of temperature. Thus, a total of 12 influencing factors are shown in Table 4. The comprehensive separation efficiency η, breather separation η 0 , and ventilation resistance ∆P are expressed in the functions of various influencing factors, as shown in Equations (6)- (8). The difference between the comprehensive separation efficiency and the breather separation efficiency are given in later section of this paper.
In Table 4, the 12 factors contained in Equations (6)-(8) can be expressed by 3 individual physical dimensions, such as M (mass), L (length), and T (time). ρ, v, and r 1 are representing the basic dimensions. Dimensionless transformation is carried out using Buckingham's π theorem. Dimensionless Equations (9)-(11) can be obtained. ∆P Equations (9)-(11) can be rewritten into general form of dimensionless group equation as follows:

Goodness of Fit Test
According to the 50 groups of calculation conditions determined by the orthogonal experiment design, the corresponding numerical calculation was completed. The calculation process was simplified, and the least squares method was used for fitting. Finally, each index in the dimensionless group equation was determined, and each prediction model is shown in Equations (15)- (17).
Comprehensive separation efficiency prediction model: Breather separation efficiency prediction model: Ventilation resistance prediction model: The prediction model validation mainly includes two aspects-one is to analyze the errors in the process of prediction model fitting, and secondly, the reliability of the equation and model parameter estimation is tested by mathematical statistics, including goodness of fit test, significance test of equation, etc. The mean absolute percentage error (MAPS) of the prediction model fitting for the comprehensive separation efficiency, breather separation efficiency, and ventilation resistance were 3.46%, 5.49%, and 9.99%, which indicate that the accuracy of the model was relatively high. The F values of the models were 149.83, 141.43, and 1566.96, which were larger than the corresponding value of F 0.001 (9, 40), 4.02. The coefficients of determination (R 2 ) were 0.6196, 0.6059, and 0.9445, indicating that the equation has a high significance and a good fitting degree.

Prediction Model Verification
The research purpose of this paper was to offer a simple calculation model for engineering applications. The applicability of the model is subject to verification through experiment. In the experiment, the separator was placed in a shell, as shown in Figure 2. Therefore, the gas-liquid separation should be the joint action of the shell and the separator, and neither experimental nor engineering applications can separate the two. The comprehensive separation efficiency η reflects this function, so it is reasonable to evaluate the efficiency in engineering applications. Of course, the cavity where the breather is located in the engine must be different from the experimental cavity, which will cause errors. The numerical calculation can be carried out by extracting the breather separately. The design is more concerned about the influence of the breather's structure on separation. Thus, the separation efficiency of the breather (η 0 ) is given solely.
Equations (15) and (16) were used to calculate the separation efficiency under various operating conditions, and the results were compared with the experimental data, as shown in Figure 7. According to Figure 7, the calculated values and the experimental values of the comprehensive separation efficiency show a same variation trend. As the ventilation rate, rotating speed, and temperature changed, the mean error of the calculation was 0.48%, 1.72%, and 0.70%, respectively. Thus, the prediction model of comprehensive separation efficiency has a high accuracy. The separation efficiency η 0 is obviously lower than the experimental value, and η 0 decreases significantly with the increase of ventilation rate, which is obviously different from the experiment and η. The effect of rotating speed on η 0 is significantly stronger than that of η.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 13 experiment, the separator was placed in a shell, as shown in Figure 2. Therefore, the gas-liquid separation should be the joint action of the shell and the separator, and neither experimental nor engineering applications can separate the two. The comprehensive separation efficiency η reflects this function, so it is reasonable to evaluate the efficiency in engineering applications. Of course, the cavity where the breather is located in the engine must be different from the experimental cavity, which will cause errors. The numerical calculation can be carried out by extracting the breather separately. The design is more concerned about the influence of the breather's structure on separation. Thus, the separation efficiency of the breather (η0) is given solely. Equations (15) and (16) were used to calculate the separation efficiency under various operating conditions, and the results were compared with the experimental data, as shown in Figure 7. According to Figure 7, the calculated values and the experimental values of the comprehensive separation efficiency show a same variation trend. As the ventilation rate, rotating speed, and temperature changed, the mean error of the calculation was 0.48%, 1.72%, and 0.70%, respectively. Thus, the prediction model of comprehensive separation efficiency has a high accuracy. The separation efficiency η0 is obviously lower than the experimental value, and η0 decreases significantly with the increase of ventilation rate, which is obviously different from the experiment and η. The effect of rotating speed on η0 is significantly stronger than that of η. Equation (17) was used to calculate the ventilation resistance under various operating conditions. The results were compared with the experimental values, as shown in Figure 8. The results show that the calculated value of ventilation resistance has the same variation trend as the experimental value under different operating conditions. When there were changes in ventilation rate, rotation speed, and temperature, the mean error was 13.93%, 9.37%, and 11.89%, respectively. Thus, the prediction model of ventilation resistance has high reliability, which can be implemented in theoretical calculations of ventilation resistance and to provide good reference for engineering design.  Equation (17) was used to calculate the ventilation resistance under various operating conditions. The results were compared with the experimental values, as shown in Figure 8. The results show that the calculated value of ventilation resistance has the same variation trend as the experimental value under different operating conditions. When there were changes in ventilation rate, rotation speed, and temperature, the mean error was 13.93%, 9.37%, and 11.89%, respectively. Thus, the prediction model of ventilation resistance has high reliability, which can be implemented in theoretical calculations of ventilation resistance and to provide good reference for engineering design.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 13 experiment, the separator was placed in a shell, as shown in Figure 2. Therefore, the gas-liquid separation should be the joint action of the shell and the separator, and neither experimental nor engineering applications can separate the two. The comprehensive separation efficiency η reflects this function, so it is reasonable to evaluate the efficiency in engineering applications. Of course, the cavity where the breather is located in the engine must be different from the experimental cavity, which will cause errors. The numerical calculation can be carried out by extracting the breather separately. The design is more concerned about the influence of the breather's structure on separation. Thus, the separation efficiency of the breather (η0) is given solely. Equations (15) and (16) were used to calculate the separation efficiency under various operating conditions, and the results were compared with the experimental data, as shown in Figure 7. According to Figure 7, the calculated values and the experimental values of the comprehensive separation efficiency show a same variation trend. As the ventilation rate, rotating speed, and temperature changed, the mean error of the calculation was 0.48%, 1.72%, and 0.70%, respectively. Thus, the prediction model of comprehensive separation efficiency has a high accuracy. The separation efficiency η0 is obviously lower than the experimental value, and η0 decreases significantly with the increase of ventilation rate, which is obviously different from the experiment and η. The effect of rotating speed on η0 is significantly stronger than that of η. Equation (17) was used to calculate the ventilation resistance under various operating conditions. The results were compared with the experimental values, as shown in Figure 8. The results show that the calculated value of ventilation resistance has the same variation trend as the experimental value under different operating conditions. When there were changes in ventilation rate, rotation speed, and temperature, the mean error was 13.93%, 9.37%, and 11.89%, respectively. Thus, the prediction model of ventilation resistance has high reliability, which can be implemented in theoretical calculations of ventilation resistance and to provide good reference for engineering design.

Conclusions
In this research, orthogonal test design was used in calculation of the operating condition. The RNG model and the DPM model were coupled to calculate the selected operating conditions. By analyzing the results, the significance of various influencing factors and the optimal levels were shown. Based on the dimensional analysis, a dimensionless group equation in general form was established by combining comprehensive separation efficiency, breather separation efficiency, and ventilation resistance. Also, through the least squares method, the performance prediction model of the breather was obtained considering five operating conditions and six structural parameters. Through the research in this paper, the following conclusions are drawn: 1.
The significance of the influential factors affecting the separation of impeller-type breathers and their optimal levels were determined. Among all the operating conditions, rotating speed has the largest influence on separation efficiency, followed by ventilation rate and average inlet particle size. Temperature and oil/gas ratio have relatively little influence. The optimal levels were C 5 B 1 A 5 D 1 E 2 . Among all the structural factors, number of blades has the largest influence, followed by external diameter of the blades and thickness of the breather. Blade angle and diameter of vent hole have little influence. The optimal level shows as F 5 H 1 J 1 G 4 I 4 K 2 . 2.
The significance of the factors affecting the resistance characteristics of the impeller-type breather and their optimal levels were determined. Among all operating condition factors, ventilation rate has the largest influence on ventilation resistance, while temperature, rotating speed, inlet particle average diameter, and gas/oil ratio have little influence. Their optimal levels were B 1 D 2 C 1 A 1 E 3. Among all the structural factors, diameter of the vents has a greater influence, while the other factors, such as number of blades, blades angle, thickness, and the external diameters of the blades, all have a small influence. The optimal levels were K 5 F 2 I 1 J 4 H 2 G 3 .

3.
The performance prediction model of an impeller-type breather was established. The comprehensive separation efficiency has a mean error of 0.97%, while the ventilation resistance has a mean error of 11.73%. The calculation values and the experimental values share the same trend change, which indicate that the prediction model can be implemented in theoretical calculation of comprehensive separation efficiency, ventilation separation efficiency, and ventilation resistance, providing good references for evaluation and design of an impeller-type breather.