Quantitative Expression of Outlet Deviation Angle of Turbomachine Stator Based on Equivalent Moment of Momentum Principle

In order to express outlet deviation angle of turbomachine stator, outlet deviation angle dominantly expressed by fluid velocity distribution was derived based on equivalent moment of momentum under the condition of certain CFD simulation design parameters. On this basis, response surface function of outlet deviation angle was constructed by CFD simulation data of orthogonal experiment. The validity of the response surface function was proved by CFD simulation data of confirmatory models. An effective method is provided for calculating outlet deviation angle of turbomachine stator.

the parameters on outlet deviation angle and CFD simulation data of orthogonal experiment. The average deviation of the response surface function was proved 0.404 degree by CFD simulation data of confirmatory models.

Equivalent Moment of Momentum Principle
The classical turbomachine design method is based on one-dimension flow theory. According to one-dimension flow theory, a value is chosen as the outlet flow angle instead of the flow angles of all particles. But, the outlet flow angle and the flow angles of all particles must be equivalent. (The particle maybe distributes along stream wise direction or circumferential direction) Among all of the physical quantities in turbomachine, the moment of every impeller is the macroscopic output physical quantity. The moment is determined by the moment of momentum [1]. So the moment of momentum is chosen as the equivalent principle of the outlet flow angle and the flow angles of all particles. It's called the equivalent moment of momentum principle in the paper.
The moment of momentum can be expressed by all the fluid particles as: r v (1) where L identifies moment of momentum; r identifies radius; m identifies mass; v identifies velocity vector; The direction of shaft is defined as direction z, then, the direction of moment of momentum is in direction z. The coordinate of fluid particle i is defined as , , , and the velocity component in each direction is , .
respectively. Then, it can be derived that: The moment of momentum can also be expressed by radius in the middle flow line, flow velocity, and total mass as: where R identifies radius vector in the middle flow line.
According to the definition of multiplication cross: (4) where the D is the angle between R and v , as showed in figure 1.  (5) where E identifies flow angle based on one-dimension flow theory; m v identifies normal velocity component.
It can be derived in Fig.1 that the relationship between D and E can be expressed as 2 S D E (6) According to one-dimension flow theory, normal velocity component can be expressed by flow rate and normal area as m m Q F v (7) where Q identifies flow rate; m F identifies normal area.
For fluid, the mass in micro time t can be expressed by flow rate, density and time as where U identifies density.
The mass of all particles is equal to the total mass. So, it can be derived that The relationship among the flow angle, blade angle and deviation angle is where y E identifies blade angle; E ' identifies deviation angle.
According to the equivalent moment of momentum principle, formula (1) is equal to formula (3). According to the formulas above, the deviation angle can be expressed as

Simulation Models for CFD
The Besides the high precision, another characteristic of CFD is that almost all of the physical quantities can be observed.
If CFD is used to calculate the flow field in turbomachine, the physical quantities in formula 11, such as normal area, flow rate, density, velocity and particle coordinate can be observed. So CFD is chosen as the research method.
When the turbomachine is under design, the blade angles of the impellers are the design variables. Circulating flow rate is a very important physical quantity in turbomachine and it has effect on the turbomachine design. Therefore, the simulation models with different inlet blade angles, different outlet blade angles and different flow rates need to be built for CFD.
The inlet impact angle of simulation model is zero, as the turbomachine is under design. Revolving spiral flow passage model was built to control the inlet impact angle.
The stator of some type hydraulic torque converter is taken as a turbomachine stator. The whole flow passage simulation models of the stator with different flow rates, inlet and outlet blade angles and zero inlet impact angles were built for CFD simulation. One of the models is shown in figure 3.  It can be seen that, the relationship between inlet blade angle and outlet deviation angle is almost quadric. Quadratic fitting is taken to fit the points. The Residual Sum of Squares (RSS) is 1.79×10-6. It can be concluded that the relationship between inlet blade angle and outlet deviation angle is almost quadric.

Effect of Outlet Blade Angle on Outlet Blade Angle
The method to analyse the effect of outlet blade angle on outlet deviation angle is similar to that of inlet blade angle. The design range of outlet blade angle is also 20 degree [1]. So, another five models need to be built to analyse the effect of outlet blade angle. According to table 1, the inlet blade angle and the flow rates of the models are 90 degree and 0.097m3/s. The outlet blade angles of the five models are 25 degree, 30 degree, 35 degree, 40 degree and 45 degree respectively.
According the parameters, five models can be built and simulation parameters for CFD can be set. The outlet deviation angles of stator models with different outlet blade angles can be got in the same way above. The corresponding relationship between outlet blade angle and outlet deviation angle is shown in figure 5.
It can be seen that, the relationship between outlet blade angle and outlet deviation angle is almost linear. Linear fitting is taken to fit the points. The RSS is 7.83×10-2. It can be concluded that the relationship between outlet blade angle and outlet deviation angle is almost linear.

Effect of Flow Rate on Outlet Blade Angle
The method to analyse the effect of flow rate on outlet deviation angle is similar to that of inlet blade angle. When the turbomachine works normally, the flow rate range is displayed in table 1. So, another five models need to be built to analyse the effect of flow rate. According to table 1, the inlet blade angle and the outlet blade angle of the models are 90 degree and 35 degree. The flow rates of the five models are 0.04 m3/s, 0.069m3/s, 0.097m3/s, 0.126m3/s and 0.154m3/s, respectively.
According the parameters, five models can be built and simulation parameters for CFD can be set. The outlet deviation angles of stator models with different flow rates can be got in the same way above. The relationship between flow rate and outlet deviation angle is shown in Fig. 6. It can be seen that, the relationship between flow rate and outlet deviation angle is almost quadric. Quadratic fitting is taken to fit the points. The RSS is 3.01×10-5. It can be concluded that the relationship between flow rate and outlet deviation angle is almost quadric.

Orthogonal Experiment
According to the analyses above, the relationships between inlet blade angle, outlet blade angle, flow rate and outlet deviation angle are got. That means the response surface function form of outlet deviation angle is determined, but the coefficients of the function aren't obtained. It can be expressed as:

Verification
The accuracy of the response surface function needs to be verified to prove the function can reflect the real relationship among outlet deviation angle and design parameters.
In order to verify the accuracy of the response surface function, another seven models with different groups of design parameter are built. The parameters are random and in the design range.
CFD simulation of the seven models can be done, and outlet deviation angles can be calculated by formula 11 and fluid velocity distribution obtained from CFD simulation. Outlet deviation angles can also be calculated by the response surface function. The results are shown in figure 7.
As introduced above, design range of outlet blade angle is between 25 degrees to 45 degrees [1]. It can be seen from That means the accuracy of formula quantitatively calculating outlet deviation angle in design range can satisfy the engineering demand.

Conclusion
In this paper, the equivalent moment of momentum principle was proposed according to the macroscopic output physical quantity of turbomachine. Outlet deviation angle dominantly expressed by fluid velocity distribution was derived based on the principle under the condition of certain CFD simulation parameter.
The stator of some type hydraulic torque converter was taken as a turbomachine stator. The effects of CFD simulation parameters on outlet deviation angle were analysed respectively by CFD simulation and dominantly expression of outlet deviation angle. Response surface function of outlet deviation angle was constructed by the effects of the parameters on outlet deviation angle and CFD simulation data of orthogonal experiment. The average deviation of the response surface function was proved 0.404 degree by CFD simulation data of confirmatory models. The average error of moment of momentum in design range was 1.43%~1.71%. The accuracy of the response surface function could satisfy the engineering demand in design range.