Centrifugal turbine which has less land occupation, simple structure, and high aerodynamic efficiency is suitable to be used as small to medium size steam turbines or waste heat recovery plant. In this paper, onedimensional design of a multistage centrifugal steam turbine was performed by using inhouse onedimensional aerodynamic design program. In addition, threedimensional numerical simulation was also performed in order to analyze design and offdesign aerodynamic performance of the proposed centrifugal steam turbine. The results exhibit reasonable flow field and smooth streamline; the aerodynamic performance of the designed turbine meets our initial expectations. These results indicate that the onedimensional aerodynamic design program is reliable and effective. The offdesign aerodynamic performance of centrifugal steam turbine was analyzed, and the results show that the mass flow increases with the decrease of the pressure ratio at a constant speed, until the critical mass flow is reached. The efficiency curve with the pressure ratio has an optimum efficiency point. And the pressure ratio of the optimum efficiency agrees well with that of the onedimensional design. The shaft power decreases as the pressure ratio increases at a constant speed. Overall, the centrifugal turbine has a wide range and good offdesign aerodynamic performance.
With the continuous reduction of fossil energy and the enhancement of people’s environment awareness, there has been increasing attention to the high efficiency utilization of energy, where the turbine is energy conversion component; if the turbine is improved effectively, it can help to improve the efficiency utilization of energy.
Centrifugal turbine is a new type of turbine engine which has many advantages. In the centrifugal turbine, the gas flows outward the center and the channel area of the flow path increase naturally as the fluid volume flow increases, during the expansion process. That meets the principle of aerodynamic and geometric matching. In addition, centrifugal turbine can be used to achieve multistage design [
For turbine as the key component, many scholars in this area have done some relevant researches, but there is not much research on centrifugal turbine. Domestically, mainly Jing and Peng had studied the pneumatic analysis of a rocket centrifugal turbine prototype starter and designed the modification turbine [
In this paper, onedimensional design method of centrifugal steam turbine is proposed by drawing on the conventional turbine [
A onedimensional design of centrifugal turbine has been developed, which is based on the onedimensional design method of conventional turbine. The onedimensional thermodynamic calculation program was developed by FORTRAN. The main design principles of centrifugal turbine onedimensional design program are as follows:
The expansion flow is assumed to be adiabatic, steady, and onedimensional in the cascade channel.
The properties of working fluids are obtained by calling Refpro 9.0, which is applicable to a variety of working fluid.
In order to simplify the design of the blade, the blade is designed to be of straight and constant height.
Stator and rotor velocity coefficients
Each stage of the rotor absolute flow angle is 90 degrees.
The program is mainly composed of continuity and energy equations to achieve onedimensional design.
Figure
Onedimensional design process of centrifugal turbine.
Figure
In the stator, the steam expands from state 0 to state 1. Lines 01 represent the actual expansion and lines 01
Then, in the rotor, the steam continues to expand from state 1 to state 2. Lines 12 represent the actual expansion and lines 12
The rotor actual outlet density
Similar to the axial flow turbine, the relevant adiabatic efficiency is given by (
Turbine expansion process.
The wheel efficiency is calculated at different speed ratio to select the optimum speed ratio and other corresponding parameters at maximum efficiency. Geometry and thermodynamic parameters are identified at the optimum speed ratio. The abovementioned steps are also adopted to design other stages.
The onedimensional design program of the centrifugal turbine is used to design the multistage centrifugal turbine. The initial design thermal parameters are derived from a small axial turbine and summarized in Table
The initial design parameters of the centrifugal turbine.
Total pressure 
1275000 
Total temperature 
613 
Back pressure 
294000 
Rotational speed 
6500 
Mass flow 
16.7 
Rated power 
750 
Because there is no stage number information about the original axial turbine from the product information, the different stage number was tested to design the centrifugal turbine by using the onedimensional design program. The results show that if the stage number is 1 and 2, there is supersonic in the centrifugal turbine. As the stage number increased, there is no supersonic in the centrifugal turbine, but the size of the centrifugal turbine becomes more and more large. When the stage number is 3, and the diameter ratio is 1.12, the centrifugal turbine is subsonic, and it can satisfy the enthalpy drop as well as high efficiency. So the centrifugal turbine is designed with three stages.
Then, the same blade height and the absolute airflow angle of each stage are 90 degrees as the criteria, impeller diameter ratio is estimated to be 1.12 for each stage, each stage outlet flow angle of stator is assumed to be 12 degrees, and radial clearance between stator and rotor is set to be 2 mm. The optimal efficiency and reasonable structure are the objectives. The design parameters are calculated by using the abovementioned onedimensional calculation program, with iterative and screening methods. In this paper, the designed threestage centrifugal turbine just meets the required enthalpy drop and has a very high wheel efficiency. The main aerodynamic parameters of each stage are shown in Table
The main aerodynamic parameters of each stage.
Stage number  1st  2nd  3rd 
Speed ratio 
0.69  0.86  0.87 
Reaction degree 
0.322  0.453  0.468 
Flow coefficient 
0.97  0.97  0.97 
Flow coefficient 
0.95  0.95  0.95 
Mach number of stator 
0.55  0.58  0.76 
Shaft power 
1205  1551  2436 
Wheel efficiency 
91.20%  90.97%  90.98% 
The main aerodynamic parameters of overall centrifugal turbine.
Overall wheel efficiency 
90.50% 
Overall shaft power 
5192 
Mass flow 
16.7 
Back pressure 
294000 
The main geometrical parameters of the centrifugal turbine.
Stage number  1st  2nd  3rd 
Stator inlet diameter 
0.597  0.758  0.959 
Rotor outlet diameter 
0.754  0.955  1.208 
Radial clearance 
0.002  0.002  0.002 
Blade height 
0.03  0.03  0.03 
Symmetrical cone angle 
0  0  0 
Diameter ratio of stator 
1.12  1.12  1.12 
Diameter ratio of rotor 
1.12  1.12  1.12 
The speed triangle data of the centrifugal turbine.
Stage number  1st  2nd  3rd 

12  12  12 

37.93  64.47  68.44 

322.1  328.7  408.1 

108.9  75.74  91.22 

229.1  290.3  367.0 

90  90  90 

14.18  12.32  13.84 

64.81  71.02  101.3 

264.6  332.8  423.3 

256.6  325.1  411.0 
The speed triangle of the centrifugal turbine.
The working fluid flows are outward the center in the centrifugal turbine. The passage area along the flow path increases naturally with the expansion of the working fluid, which makes the variety of specific volume matches the change of flow passage area. It can be seen that the centrifugal turbine is much superior to the conventional turbine from the structure. In the centrifugal turbine, the blade plane flow channel is expanding outward with the increase of the radius. It is obviously inappropriate to use the airfoil of conventional turbine at this time. So designing a suitable airfoil for centrifugal turbine is needed.
Angle and thickness design method is adopted to design airfoil, by using the inlet and outlet geometry angle, blade height, and leading and trailing edge diameters, which are based on onedimensional design results in Tables
Initial profile of the firststage stator
Initial profile of the firststage rotor
In the optimization process of blade, the blade number, the blade inlet and outlet geometry angles, and the leading and trailing edge thicknesses are the fixed parameters, and two control points of the tangent angle and two control points of the blade thickness are taken as the optimization parameters. Workbench is as the optimization software, and NLPQL algorithm is used as optimization method.
In the stator optimization process, the minimum loss coefficient is the object and stator back pressure is the constraint condition. Stators of threestage centrifugal turbine are optimized, respectively. The optimization mathematical model of stator is expressed as
Tangential angular distribution of medial camber line and control points selection of rotor blade
The thickness distribution and control points selection of rotor blade
Then the rotor is added behind the optimized stator. The maximum shaft power is the object and rotor back pressure is the constraint condition. The optimization mathematical model of rotor is expressed as
Tangential angular distribution of medial camber line and control points selection of stator blade
The thickness distribution and control points selection of stator blade
The optimization strategy and process are shown in Figures
Optimization strategy.
Optimization process.
Figure
Since the SST model is used in turbomachinery, in most cases, it requires
The contrast of SST and
Variable 

SST 


Wheel efficiency 
93.29%  93.01%  0.29% 
Shaft power 
5387  5256.8  2.42% 
Mass flow 
16.708  16.443  1.59% 
Back pressure 
291170  293999  0.97% 
The optimized centrifugal turbine is simulated by CFX, based on NavierStokes equations. Stators and rotors use structured grid generated in TurboGrid, and the total grid number is about 4010,000; the blade global size factors of mesh are 1.2, 1.2, 1.25, 1.25, 1.25, and 1.2 for each stator and rotor, respectively. The computational model mesh used for simulation is shown in Figure
Computational domain for the simulation of stages
Mesh distribution along the height and tangential direction of blade
The optimal parameters of stator and rotor are almost consistent with onedimensional aerodynamic design. The optimized stator and rotor of whole stages are both straight blades and the blade diagram is referred to in Figure
Optimized stators and rotors
3D model of stator and rotor blades
The numerical simulation results of the optimized centrifugal turbine are basically consistent with the onedimensional aerodynamic design, which shows that the onedimensional design program is reliable and effective. The overall performance data of centrifugal turbine is shown in Table
The contrast of design value and CFD value of the overall performance of the centrifugal turbine.
Variable  Design value  Simulation value 


Wheel efficiency 
90.50%  93.29% 

Shaft power 
5192  5387 

Mass flow 
16.7  16.708 

Back pressure 
294000  291170 

The contrast of design value and CFD value of each stage performance of the centrifugal turbine.
Variable  Stage number  

1st  2nd  3rd  
Design value  Simulation value 

Design value  Simulation value 

Design value  Simulation value 



0.55  0.55 








0.32  0.30 








0.69  0.63 








1205  1231 








91.20  93.34 








12.0  12.3 








90.0  78.9 







Figures
Static pressure distribution.
Mach number distribution.
Streamline distribution.
Figures
The variation of mass flow with pressure ratio at different speeds.
The variation of efficiency with pressure ratio at different speeds.
The variation of shaft power with pressure ratio at different speeds.
The threestage centrifugal turbine is simulated by CFX with Water Ideal Gas as working fluid, drawing on the conventional turbine aerodynamic design method and offdesign condition performance research method. Conclusions are as follows.
The numerical simulation results of the whole stages are basically consistent with the onedimensional design results, and the aerodynamic performance meets the expected requirements, which indicates the reliability and effectiveness of the centrifugal turbine design. Because each stage of centrifugal turbine is optimized, the simulated overall efficiency is 3.08% higher than the onedimensional design efficiency and the shaft power is 3.75% more than the onedimensional design shaft power. At the design condition, the streamline of threestage centrifugal steam turbine is smooth at cascade flow channel, the pressure distribution is uniform, and the flow field is reasonable.
At the offdesign condition, when the speed is constant, the pressure ratio
At a constant speed, the efficiency is highest at optimum pressure ratio
The shaft power curve trend with pressure ratio is similar at different rotation speeds. But as the rotation speed increases, the power curve drops faster when the pressure ratio increases.
Overall, it can be seen that the centrifugal turbine has a wide range and good offdesign performance, from the numerical simulation results of the threestage centrifugal turbine and the analysis of the offdesign conditions.
Temperature, K
Pressure, Pa
Enthalpy, kJ/kg
Entropy, kJ/(K·kg)
Rotational speed, r/min
Mass flow, kg/s
Diameter ratio
Diameter, m
Radius, m
Absolute velocity, m/s
Relative velocity, m/s
Circumferential velocity, m/s
Height, m
Stage number
Shaft power, kW
Mach number.
Density, kg/m^{3}
Radial symmetry cone angle, °
Radial clearance between stator and rotor, m
Absolute angle, °
Relative angle, °
Stator flow coefficient
Rotor flow coefficient
Speed ratio
Wheel efficiency
Reaction degree
Pressure ratio.
First stage
Inlet
Outlet
Stator
Stage number.
Stagnation condition.
The authors declare that they have no conflicts of interest.
This work was supported by National Natural science Foundation of China no. 51536006 and Shanghai Science and Technology Committee with Grant no. 17060502300 and funded by Opening Project of Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering no. 13DZ2260900.