Parameter Optimization and Performance Research: Radial Inﬂow Turbine in Ocean Thermal Energy Conversion

: Combining one-dimensional parameter optimization and three-dimensional modeling optimization, a 30 kW radial inﬂow turbine for ocean thermal energy conversion was designed. In this paper, the effects of blade tip clearance, blade number, twist angle, and wheel–diameter ratio on the radial inﬂow turbine were analyzed. The results show that the model prediction method based on 3D numerical simulation data can effectively complete secondary optimization of the radial turbine rotor. The prediction model can be used to directly obtain the optimal modeling parameter of the rotor. The tip clearance, blade number, twist angle, wheel–diameter ratio, and shaft efﬁciency were found to be 0.273 mm, 16, 43.378 ◦ , 0.241, and 88.467%, respectively. The optimized shaft efﬁciency of the turbine was found to be 2.239% higher than the one-dimensional design result, which is of great signiﬁcance in reducing the system’s power generation costs and promoting the application of this approach in engineering power generation using ocean thermal energy.


Introduction
Ocean thermal energy conversion (OTEC) is the main method of using ocean thermal energy to generate electricity.The principle of this approach's function lies in using the warm water on the surface of the ocean to heat low-boiling-point working mediums to gasify them; alternatively, the pressure is reduced, allowing the sea water to vaporize, driving a turbine and producing electricity.At the same time, the cold sea water extracted from the seabed is used to condense the steam into liquid again so that a circulation system is formed [1].
The radial inflow turbine is a power component that is widely used in the field of OTEC.It has many advantages, such as compact structure, a large single-stage enthalpy drop, a high rotational speed, strong adaptability to variable working conditions with variable guide stator vanes, and a high operating efficiency under small-flow conditions [2][3][4][5].At present, research achievements made by domestic and foreign scholars on OTEC system radial inflow turbines are gradually increasing.These researchers have been focusing on the aerodynamic design, working medium selection, parameter optimization algorithm, profile design method, loss model, and performance prediction of variable working conditions.For example, Nithesh et al. [6] designed a 2 kW OTEC turbine with an R134a working medium.The three-dimensional flow field of this turbine was numerically simulated using CFX and differences in the flow characteristics of small-sized turbines with different blade thicknesses, blade tip clearances, and blade edge shapes were given.Yue et al. [7] designed a toluene medium radial inflow turbine for a medium-high-temperature solarpower-generation system based on the organic Rankine cycle (ORC).In addition, CFD Zhai et al. [21] adopted a constrained genetic algorithm (GA) to optimize the radial inflow turbine.Based on four working mediums-pentane, R245fa, R365mfc, and R123-the influence of heat source outlet temperature on the design parameters of the radial inflow turbine and the performance of the ORC system was studied.Kiyarash et al. [22] proposed a new method combining mean line modeling and DIRECT optimization for small radial inflow turbines.This method can obtain dynamic efficiency based on turbine losses and then quickly screen turbine size parameters under different working conditions.
Although there has been substantial research on OTEC radial inflow turbines, most researchers focus on the optimization of initial parameters and loss coefficient and less on the modeling parameters influence of performance.Therefore, based on the one-dimensional design scheme, CFX simulation, and the support vector regression method, the present study carried out a secondary optimization of the rotor modeling parameters of the radial inflow turbine.In addition, this research compared the two schemes before and after optimization and described the influence of modeling parameters on turbine performance.

Circulation System Parameters Selection
The inlet and outlet working conditions of the radial turbine were determined through the circulation system.The annual average temperature of the surface water in the South China Sea is in the range of 24.6~30.1 • C and the temperature of the cold sea water below 800 m is generally stable at about 4 • C. The inlet and outlet temperature difference of seawater in the heat exchanger is 2~4 • C. In order to maintain the power consumption of the warm sea pump and the cold sea pump at a low level, the terminal temperature difference of medium and large OTEC evaporator is generally 2 • C [23].The pipe friction loss from the evaporator to the turbine and the fluctuation of the turbine inlet condition were ignored.According to the evaluation of the five different working substances in presented in [24], the net power generated using R134a in the highest performance cycle is higher than that generated using n-pentane and other working substances.In addition, R134a has the characteristics of low critical pressure, low critical temperature, low flammability, low toxicity, and low impact on the environment [25].It was selected as the circulating working medium and its physical parameters were obtained through invoking NIST REFPROP V9.1, the international authoritative database of working mediums and their physical properties.We chose organic Rankine cycle as the system circulation mode, and the circulation system diagram is shown in Figure 1.The system thermal parameters were determined using cyclic calculation.The calculation model we used is the model mentioned in [26] and the results are shown in Table 1.
the performance of the ORC system was studied.Kiyarash et al. [22] proposed method combining mean line modeling and DIRECT optimization for small radial turbines.This method can obtain dynamic efficiency based on turbine losses an quickly screen turbine size parameters under different working conditions.
Although there has been substantial research on OTEC radial inflow turbines researchers focus on the optimization of initial parameters and loss coefficient and l the modeling parameters influence of performance.Therefore, based on the one-d sional design scheme, CFX simulation, and the support vector regression method, th sent study carried out a secondary optimization of the rotor modeling parameters radial inflow turbine.In addition, this research compared the two schemes befo after optimization and described the influence of modeling parameters on turbine p mance.

Circulation System Parameters Selection
The inlet and outlet working conditions of the radial turbine were deter through the circulation system.The annual average temperature of the surface w the South China Sea is in the range of 24.6~30.1 °C and the temperature of the co water below 800 m is generally stable at about 4 °C.The inlet and outlet temperatu ference of seawater in the heat exchanger is 2~4 °C.In order to maintain the powe sumption of the warm sea pump and the cold sea pump at a low level, the termina perature difference of medium and large OTEC evaporator is generally 2 °C [23].Th friction loss from the evaporator to the turbine and the fluctuation of the turbin condition were ignored.According to the evaluation of the five different workin stances in presented in [24], the net power generated using R134a in the highest p mance cycle is higher than that generated using n-pentane and other working subst In addition, R134a has the characteristics of low critical pressure, low critical tempe low flammability, low toxicity, and low impact on the environment [25].It was se as the circulating working medium and its physical parameters were obtained th invoking NIST REFPROP V9.1, the international authoritative database of working ums and their physical properties.We chose organic Rankine cycle as the system c tion mode, and the circulation system diagram is shown in Figure 1.The system th parameters were determined using cyclic calculation.The calculation model we u the model mentioned in [26] and the results are shown in Table 1.According to the system parameters presented in Table 1, the one-dimensional design model of the radial inflow turbine was established using a screening method and the isentropic efficiency of the radial inflow turbine was calculated using the loss model [27].In the above process, the flow losses of the working medium in the volute were ignored.Otherwise, we used the ISIGHT 2020 software to carry out optimal Latin hypercube sampling of seven initial parameters.The sampling results were input into the one-dimensional design program written by MATLAB 2020b.With the goal of attaining an isentropic efficiency greater than 85%, the thermodynamic parameters optimization results of the radial inflow turbine were obtained through iteration.On this basis, a data regression prediction method was used to optimize the modeling parameters of the radial inflow turbine rotor.The process is shown in Figure 2.  According to the system parameters presented in Table 1, the one-dimensional d model of the radial inflow turbine was established using a screening method and entropic efficiency of the radial inflow turbine was calculated using the loss model [ the above process, the flow losses of the working medium in the volute were ig Otherwise, we used the ISIGHT 2020 software to carry out optimal Latin hypercub pling of seven initial parameters.The sampling results were input into the one-d sional design program written by MATLAB 2020b.With the goal of attaining an isen efficiency greater than 85%, the thermodynamic parameters optimization results radial inflow turbine were obtained through iteration.On this basis, a data regressio diction method was used to optimize the modeling parameters of the radial inflow tu rotor.The process is shown in Figure 2.  Multiple one-dimensional design schemes of the radial inflow turbine were obtained through iterative calculations [28].After checking, the schemes which were found to now meet the structural strength requirements, or ones in which the outlet relative speed was less than inlet relative speed, were eliminated.The one-dimensional design scheme with the highest isentropic efficiency of the remaining schemes was selected as the initial scheme of modeling optimization.The parameters of the initial scheme are shown in Table 2.

Numerical Simulation of Radial Turbines
Although the one-dimensional design of the radial inflow turbine can obtain the optimal parameters of the rotor, the calculation of its isentropic efficiency is based on the loss model.The calculation results cannot fully reflect the viscous and unsteady characteristics of the flow of organic working fluid in the radial inflow turbine.Therefore, based on the three-dimensional numerical simulation results, 40 groups of training samples were established in this research and the regression prediction method was used to optimize the turbine rotor parameters, so as to further improve the performance of the initial scheme.

Three-Dimensional Modeling
Based on the one-dimensional design results in Table 2, in this study, we designed the volute using the equal circulation method; we selected the TC-2P subsonic radial blade type for the blade.The three-dimensional modeling of the volute and stator was completed using SolidWorks.The rotor modeling was completed with reference to Ansys-Bladegen [29].The three-dimensional model of the initial scheme is shown in Figure 3.

Three-Dimensional Modeling
Based on the one-dimensional design results in Table 2, in this study, we designed the volute using the equal circulation method; we selected the TC-2P subsonic radial blade type for the blade.The three-dimensional modeling of the volute and stator was completed using SolidWorks.The rotor modeling was completed with reference to Ansys-Bladegen [29].The three-dimensional model of the initial scheme is shown in Figure 3.The rotor is the core moving part of a radial inflow turbine, and its efficiency is much lower than those of the volute and the guide vane.Optimizing the design of the rotor is the focus of the radial inflow turbine optimization design.This paper first used Isight to perform Latin hypercube sampling on seven shape parameters of the rotor: wheel-diameter ratio  , blade number  , blade twist angle , radial clearance ∆ , blade tip clearance ∆ , blade outlet thickness  , and blade outlet fillet radius  [30].The sampling range of the seven parameters was set according to [8], as shown in Table 3.The definition of  is shown in Equation ( 1), and the schematic diagram of other parameters is shown in Figure 4.The rotor is the core moving part of a radial inflow turbine, and its efficiency is much lower than those of the volute and the guide vane.Optimizing the design of the rotor is the focus of the radial inflow turbine optimization design.This paper first used Isight to perform Latin hypercube sampling on seven shape parameters of the rotor: wheel-diameter ratio D 2 , blade number Z 2 , blade twist angle θ, radial clearance ∆ 1 , blade tip clearance ∆ 2 , blade outlet thickness T 2 , and blade outlet fillet radius r [30].The sampling range of the seven parameters was set according to [8], as shown in Table 3.The definition of D 2 is shown in Equation ( 1), and the schematic diagram of other parameters is shown in Figure 4.
Table 3. Sampling range of modeling parameters.The wheel-diameter ratio  is the ratio of the rotor outlet average diameter to the rotor inlet diameter, which reflects the relative size of the rotor inlet and outlet diameters.The authors of [21] pointed out that the wheel-diameter ratio has an important influence on the flow channel shape and the blockage coefficient of the inlet and outlet.The blade The wheel-diameter ratio D 2 is the ratio of the rotor outlet average diameter to the rotor inlet diameter, which reflects the relative size of the rotor inlet and outlet diameters.The authors of [21] pointed out that the wheel-diameter ratio has an important influence on the flow channel shape and the blockage coefficient of the inlet and outlet.The blade number Z 2 and blade twist angle θ have an impact on the off-design performance of the radial inflow turbine.While radial clearance ∆ 1 and blade tip clearance ∆ 2 are too large, they will both cause internal leakage loss to increase.When ∆ 1 is too small, it will cause uneven mixing of the working fluid when it flows out of the stator, resulting in secondary flow loss [31].When ∆ 2 is set by equal clearance, if it is too small, it will cause the blade to contact with the rotor shell during the transient process of rotation and heat transfer, resulting in large friction loss.The blade outlet thickness T 2 and blade outlet fillet radius r mainly affect the turbine shaft efficiency by affecting the wake and residual velocity of the airflow.

Numerical Simulation Settings
We used the professional rotating machinery simulation software Ansys-CFX 2020 R2 to perform a three-dimensional viscous flow steady-state simulation of the radial inflow turbine.For robustness, we used a first-order upwind scheme to discretize the Reynolds-averaged Navier-Stokes (RANS) equations.The k-ε model was selected as turbulence modeling for its scalable wall function, fine stability, and fast convergence, which has been widely used in the fields of aviation and turbomachinery [32,33].Details of the governing equations used in this study can be found in the ANSYS CFX-Solver Theory Guide [34].The boundary conditions were set according to Table 1: inlet total pressure-0.64MPa; inlet total temperature-297.15K; outlet static pressure-0.393MPa; mass flow rate-3.82kg/s.The connection between the volute and the stator was static, and the options Frame Change/Mixing and Pitch Change were set to Stage and Automatic, respectively; the connection between the stator and the rotor was dynamic and was defined as a frozen rotor model [35].In addition, all walls were set as no-slip boundaries, the wall roughness was set as Ra3.2, and the numerical calculation convergence residual was set as 10 −6 .Timescale control option was set to auto timescale, the length scale option was set to conservative, and the timescale factor was set to the value of one.

Mesh Generation
The volute was meshed with tetrahedral elements using Ansys-ICEM 2020 R2, and the grid was partly refined at the tongue.The stator model was imported into Ansys-Turbogrid using a configuration file to generate a single-channel structured grid, and then a full-channel grid was generated using Ansys-ICEM.The initial scheme and the other 40 rotor models were directly parameterized using Ansys-Bladegen and then imported into Turbogrid and ICEM to generate single-channel and full-channel structured grids [18].The value of y+ associated with the near-wall element size specification was set to 30.The single-channel grids of the stator and rotor of the initial scheme are shown in Figure 5.
In order to ensure sufficient accuracy of the numerical simulation results, this paper used the same grid-generation method for all schemes, and used ICEM to check the grid quality.Total grid number of the first scheme of the 40 schemes was set to 10.5 million based on the grid quality, and then grid independence verification was performed on this basis.The verification result of Scheme 1 is shown in Figure 6, and the verification process of other schemes was similar to it.The results showed that, after the grid number of 9 million, increasing the grid number had little effect on the results, indicating that the grid number set could meet the accuracy and independence requirements.The grid numbers of the volute and stator set were 700,229 and 4,309,184, respectively, and the rotor grid number range was 4,553,272~5,658,086.
rotor models were directly parameterized using Ansys-Bladegen and then imported into Turbogrid and ICEM to generate single-channel and full-channel structured grids [18].The value of y+ associated with the near-wall element size specification was set to 30.The single-channel grids of the stator and rotor of the initial scheme are shown in Figure 5.In order to ensure sufficient accuracy of the numerical simulation results, this paper used the same grid-generation method for all schemes, and used ICEM to check the grid quality.Total grid number of the first scheme of the 40 schemes was set to 10.5 million based on the grid quality, and then grid independence verification was performed on this basis.The verification result of Scheme 1 is shown in Figure 6, and the verification process of other schemes was similar to it.The results showed that, after the grid number of 9 million, increasing the grid number had little effect on the results, indicating that the grid number set could meet the accuracy and independence requirements.The grid numbers of the volute and stator set were 700,229 and 4,309,184, respectively, and the rotor grid number range was 4,553,272~5,658,086.

Optimized Method of Radial Turbines-Data Regression Prediction Based on Support Vector Machine
Support vector regression (SVR) is one of the most commonly used machine learning regression methods; it has a rigorous statistical theory foundation and has high robustness in dealing with complex problems such as small samples, nonlinear regression, and  In order to ensure sufficient accuracy of the numerical simulation results, th used the same grid-generation method for all schemes, and used ICEM to check quality.Total grid number of the first scheme of the 40 schemes was set to 10.5 based on the grid quality, and then grid independence verification was performed basis.The verification result of Scheme 1 is shown in Figure 6, and the verification of other schemes was similar to it.The results showed that, after the grid num million, increasing the grid number had little effect on the results, indicating that number set could meet the accuracy and independence requirements.The grid n of the volute and stator set were 700,229 and 4,309,184, respectively, and the ro number range was 4,553,272~5,658,086.

Optimized Method of Radial Turbines-Data Regression Prediction Based on Suppo Machine
Support vector regression (SVR) is one of the most commonly used machine regression methods; it has a rigorous statistical theory foundation and has high rob in dealing with complex problems such as small samples, nonlinear regressi

Optimized Method of Radial Turbines-Data Regression Prediction Based on Support Vector Machine
Support vector regression (SVR) is one of the most commonly used machine learning regression methods; it has a rigorous statistical theory foundation and has high robustness in dealing with complex problems such as small samples, nonlinear regression, and multiple features; additionally, it has excellent generalization ability [36].When dealing with high-dimensional problems, SVR can be used to effectively reduce the computational consumption by choosing a suitable kernel function.Compared with other methods such as neural network regression and random forest regression, SVR has the advantages of easy implementation, computational complexity independent of the input space dimension, and high prediction accuracy.It has been successfully applied in many fields.
In order to achieve stable prediction of radial turbine efficiency using different modeling parameters, this study used the above numerical simulation data to establish a training sample library; further, we obtained an SVR model through training.The SVR model adopts radial basis function as the kernel function [37], and the mathematical model is as follows: The sample set is {(x 1 , y 1 ), (x 2 , y 2 ), • • • , (x 40 , y 40 )}.The independent variable is x i ∈ R 7 and it contains 7 rotor modeling parameters.y i ∈ R and it is the dependent variable of the sample set.The data {x 1 , x 2 , • • • , x 40 } can be mapped to a higher dimensional space through a nonlinear mapping method and the regression function can be defined as follows: In addition, ω, Φ(x), b are the weight vector, the mapping transformation to a high dimensional space, and the offset.The above problem can be transformed into a quadratic programming problem through introducing Lagrange multipliers.The objective function and constraints of this problem are constructed as Equations ( 3) and ( 4) [38].
In the equations above, c, ε are penalty factor, insensitive loss coefficient and ξ i , ξ * i are the relaxation factors.Through introducing Lagrange multipliers α i and α * i , the objective function can be reduced as Equation (5).
K(x i , x) is RBF kernel function, calculated by Equation (6).

Numerical Simulation Results
Figure 7 shows the simulation results of the 3D streamline distribution and pressure distribution of the radial inflow turbine initial scheme.
As shown in Figure 7a, the working fluid flow in the turbine is stable, and the streamlines are evenly distributed in each channel.There is no large-scale swirling vortex or flow disturbance.The working fluid flows into the stator evenly with a certain circumferential angle, and gradually accelerates in the stator channel, reaching the maximum value at the outlet.The average outlet velocity is about 104.4 m/s, and the velocity is not completely evenly distributed along the circumferential direction, which is caused by the small radial clearance, but there is no transonic flow phenomenon at the stator outlet.The working fluid flows into the rotor with a relative flow angle close to 90 • , and gradually decelerates until it flows out of the outlet, producing small shock loss and residual velocity loss.As shown in Figure 7b, the pressure inside the turbine decreases gradually along the flow direction, and the pressure gradient is basically consistent with the flow direction.The working fluid pressure drops in the stator, and the flow velocity increases.In the rotor, it further expands to drive the rotor to rotate and do work.
Considering various internal losses of radial inflow turbines, shaft power P T and shaft efficiency η P were used as the criteria to evaluate the power performance of radial inflow turbines when optimizing rotor modeling.P T and η P were calculated by the following two equations.

P T =
T r n 9549 ( 7) In addition, T r , n, m, h s , and i are the torque, speed, mass flow, isentropic enthalpy, and enthalpy value, respectively.T r , n, h s , and i are obtained through the CFD results.The units of T r , n, and P T are N•m, r/min, and kW, respectively.
Based on CFD-Post 2020 R2 software, simulation results of 40 schemes were obtained through the batch post-processing method.The simulation results of the shaft power are distributed between 31.5 kW and 35.0 kW, and the degree of dispersion is not high.As shown in Figure 7a, the working fluid flow in the turbine is stable, and the streamlines are evenly distributed in each channel.There is no large-scale swirling vortex or flow disturbance.The working fluid flows into the stator evenly with a certain circumferential angle, and gradually accelerates in the stator channel, reaching the maximum value at the outlet.The average outlet velocity is about 104.4 m/s, and the velocity is not completely evenly distributed along the circumferential direction, which is caused by the small radial clearance, but there is no transonic flow phenomenon at the stator outlet.The working fluid flows into the rotor with a relative flow angle close to 90°, and gradually decelerates until it flows out of the outlet, producing small shock loss and residual velocity loss.As shown in Figure 7b, the pressure inside the turbine decreases gradually along the flow direction, and the pressure gradient is basically consistent with the flow direction.The working fluid pressure drops in the stator, and the flow velocity increases.In the rotor, it further expands to drive the rotor to rotate and do work.
Considering various internal losses of radial inflow turbines, shaft power  and shaft efficiency  were used as the criteria to evaluate the power performance of radial inflow turbines when optimizing rotor modeling. and  were calculated by the following two equations.

Model Training and Optimization Results
Through the SVR model, we can establish a nonlinear mapping relationship between the seven rotor shape parameters and the radial inflow turbine shaft efficiency and we can obtain the shaft efficiency prediction results.Different from the one-dimensional model, this model is a non-parametric model based entirely on the numerical simulation results; it does not need to give the specific calculation process from the rotor shape parameters to the shaft efficiency results.Ten sets of radial inflow turbine rotor shape parameters were arbitrarily given, and the actual values of shaft efficiency were obtained through the same modeling and simulation process; these were used as a test set to verify the accuracy of the SVR model.The comparison of the shaft efficiency prediction results for the training set and the test set is shown in Figure 8.
As can be seen from the figure, there is some error between the predicted value and the simulation result.The test set error is slightly larger than the training set error, but it is also below 0.5%, which indicates that 40 training samples can meet the prediction accuracy requirements.
In order to find the rotor shape parameters that can maximize the radial inflow turbine shaft efficiency, we performed a full-factor sampling in the value range of the rotor shape parameters using the MATLAB program; next, we input all the parameter combinations into the above SVR model, and then used a genetic algorithm to screen the prediction results.The optimal rotor shape scheme under the rated condition was obtained, and the optimization results are shown in Table 4.
it does not need to give the specific calculation process from the rotor shape parameters to the shaft efficiency results.Ten sets of radial inflow turbine rotor shape parameters were arbitrarily given, and the actual values of shaft efficiency were obtained through the same modeling and simulation process; these were used as a test set to verify the accuracy of the SVR model.The comparison of the shaft efficiency prediction results for the training set and the test set is shown in Figure 8.As can be seen from the figure, there is some error between the predicted value and the simulation result.The test set error is slightly larger than the training set error, but it is also below 0.5%, which indicates that 40 training samples can meet the prediction accuracy requirements.
In order to find the rotor shape parameters that can maximize the radial inflow turbine shaft efficiency, we performed a full-factor sampling in the value range of the rotor shape parameters using the MATLAB program; next, we input all the parameter combinations into the above SVR model, and then used a genetic algorithm to screen the prediction results.The optimal rotor shape scheme under the rated condition was obtained, and the optimization results are shown in Table 4.

Influence of Modeling Parameters on Turbine Performance
To avoid the possibility that the optimal shape parameters obtained are "isolated" points in the parameter space, it is necessary to analyze the influence trend of the adjacent shape parameters on the performance.The sensitivity of the input parameters to the shaft efficiency in the regression prediction model was calculated using the experimental design module in the Isight 2020 software, and the results are shown in Figure 9.As can be seen from the figure, the first four parameters have the greatest impact on the shaft efficiency.
The blade tip clearance ∆ 2 , blade number Z 2 , wheel-diameter ratio D 2 , and twist angle θ were taken as variables, and the other parameters were kept consistent with those in Table 4.The distribution maps of the shaft efficiency with the wheel-diameter ratio and the twist angle at different blade numbers and blade tip clearances were drawn, as shown in Figures 10-12.These figures show that, for different rotor blade numbers and blade tip clearances, the surface shape of shaft efficiency distribution is similar, and there will always be a maximum value of shaft efficiency, which is not an isolated parameter point.Near this value, the shaft efficiency changes continuously with the wheel-diameter ratio and twist angle.
shape parameters on the performance.The sensitivity of the input par efficiency in the regression prediction model was calculated using th sign module in the Isight 2020 software, and the results are shown in seen from the figure, the first four parameters have the greatest impa ciency.The blade tip clearance ∆ , blade number  , wheel-diameter angle  were taken as variables, and the other parameters were k those in Table 4.The distribution maps of the shaft efficiency with t ratio and the twist angle at different blade numbers and blade tip clea as shown in Figures 10-12.These figures show that, for different rotor blade tip clearances, the surface shape of shaft efficiency distribution i will always be a maximum value of shaft efficiency, which is not an point.Near this value, the shaft efficiency changes continuously with t ratio and twist angle.The blade tip clearance ∆ , blade number  , wheel-diameter angle  were taken as variables, and the other parameters were k those in Table 4.The distribution maps of the shaft efficiency with t ratio and the twist angle at different blade numbers and blade tip clea as shown in Figures 10-12.These figures show that, for different rotor blade tip clearances, the surface shape of shaft efficiency distribution i will always be a maximum value of shaft efficiency, which is not an point.Near this value, the shaft efficiency changes continuously with t ratio and twist angle.Figure 10 shows the change in the radial inflow turbine shaft effic angle and the wheel-diameter ratio when the blade tip clearance is blade number is 16.As can be seen from the figure, at the upper and allowable range of the wheel-diameter ratio and the twist angle, the creases to a certain extent; when both the wheel-diameter ratio and to the upper limit of the allowable range, the radial inflow turbine Figure 10 shows the change in the radial inflow turbine shaft efficiency with the twist angle and the wheel-diameter ratio when the blade tip clearance is 0.273 mm and the blade number is 16.As can be seen from the figure, at the upper and lower limits of the allowable range of the wheel-diameter ratio and the twist angle, the shaft efficiency decreases to a certain extent; when both the wheel-diameter ratio and the twist angle tend to the upper limit of the allowable range, the radial inflow turbine shaft efficiency will drop rapidly.This should be avoided as much as possible in the design process.As can be seen from Figure 11, when the blade number is 16, the peak value of shaft efficiency corresponds to a twist angle range of 42 • ~46 • and a wheel-diameter ratio range of 0.385~0.420.When the blade tip clearance value is less than 0.273 mm, the change in the shaft efficiency with a clearance value is very small.When the blade tip clearance value is greater than 0.273 mm, the influence of blade tip clearance on shaft efficiency increases, and the peak value of shaft efficiency gradually moves to a smaller wheel-diameter ratio and a larger twist angle area.Considering the actual processing difficulty and off-design operating conditions, a blade tip clearance greater than 0.273 mm is preferred.Figure 12 shows the influence of different blade numbers (14~19) on radial inflow turbine performance when rotor blade tip clearance is 0.273 mm.It can be seen that, as the number of blades increases, the shaft efficiency first increases and then decreases, and when there are 16 blades, it can achieve optimal performance.In addition, the blade number has a relatively independent effect on radial inflow turbine shaft efficiency.When the blade number changes, the wheeldiameter ratio corresponding to the peak value of the shaft efficiency hardly changes, and the corresponding twist angle change range is less than 2 • .16 blades, it can achieve optimal performance.In addition, the blade number has a relatively independent effect on radial inflow turbine shaft efficiency.When the blade number changes, the wheel-diameter ratio corresponding to the peak value of the shaft efficiency hardly changes, and the corresponding twist angle change range is less than 2°.

Comparative Analysis of Optimization Scheme Performance
Figure 13a,b show the streamline distribution and static entropy distribution at 50% blade height of the rotor before optimization.It can be seen that the suction surface of the rotor obtained from the one-dimensional design always has recirculation vortices of different sizes, which cause the flow in the rotor passage to be uneven, and subject the blades to unsteady cyclic loads, reducing the efficiency and strength of the turbine.The suction surface vortices are mainly caused by the mismatch between the operating speed, blade geometry parameters, and rotor shape parameters.
Figure 13c,d show the flow field and static entropy contour of the optimized rotor model.From the streamline plot, it can be seen that the recirculation vortices on the suction surface of the blade almost disappear, and the overall streamline distribution is more uniform.In the entropy contour plot, the entropy increase amplitude in the passage is significantly reduced, and the entropy increase from the stator inlet to the rotor outlet is reduced by 0.28 J/(kg•K).This indicates that using the regression prediction method to optimize the rotor shape with the highest shaft efficiency as the objective can effectively avoid the vortex structure on the suction surface of the blade, achieve the adaptation of shape parameters and speed, and reduce losses.

Performance Analysis of Variable Working Conditions
In actual sea conditions, the seawater temperature at the ocean surface will change in accordance with factors such as light intensity and season, which will cause the inlet condition of the radial inflow turbine to be inconsistent with the design condition.Therefore, it is necessary to analyze the performance of the radial inflow turbine under variable conditions.This paper simulates the shaft efficiency distribution of the optimized radial inflow turbine under different inlet conditions and speeds using CFX 2020 R2 software.The curves, fitted according to the simulation results under multiple conditions, are shown in Figure 14; here, the boundary condition assumes that the static pressure at the turbine outlet is constant.The numerical simulation speeds include three groups of values: design speed and design speed plus or minus 20%.The inlet conditions are all saturated conditions corresponding to each expansion ratio.It can be concluded from Figure 14 that the shaft efficiency of the radial inflow turbine increases first and then decreases with the increase in the expansion ratio.For different speeds, the corresponding expansion ratios that reach the peak shaft efficiency are not the same; the higher the speed, the larger the expansion ratio.When the operating speed is 80% of the design speed, 100% of the design speed, and 120% of the design speed, the turbine efficiency reaches its peak at expansion ratios of 1.53, 1.70, and 1.85, respectively, which are 86.337%,88.509%, and 88.736%, respectively.The optimized radial inflow turbine operates at design speed, and the shaft efficiency changes a little with the expansion ratio.Although its performance is weaker than that of the high-speed operation mode at a large expansion ratio, its performance degradation at low and high temperatures is not large, and its overall performance is excellent.Figure 13c,d show the flow field and static entropy contour of the optimized rotor model.From the streamline plot, it can be seen that the recirculation vortices on the suction surface of the blade almost disappear, and the overall streamline distribution is more uniform.In the entropy contour plot, the entropy increase amplitude in the passage is significantly reduced, and the entropy increase from the stator inlet to the rotor outlet is reduced by 0.28 J/(kg•K).This indicates that using the regression prediction method to optimize the rotor shape with the highest shaft efficiency as the objective can effectively avoid the vortex structure on the suction surface of the blade, achieve the adaptation of shape parameters and speed, and reduce losses.

Performance Analysis of Variable Working Conditions
In actual sea conditions, the seawater temperature at the ocean surface will change in accordance with factors such as light intensity and season, which will cause the inlet condition of the radial inflow turbine to be inconsistent with the design condition.Therefore, it is necessary to analyze the performance of the radial inflow turbine under variable conditions.This paper simulates the shaft efficiency distribution of the optimized radial inflow turbine under different inlet conditions and speeds using CFX 2020 R2 software.The curves, fitted according to the simulation results under multiple conditions, are shown in Figure 14; here, the boundary condition assumes that the static pressure at the turbine outlet is constant.The numerical simulation speeds include three groups of values: design speed and design speed plus or minus 20%.The inlet conditions are all saturated conditions corresponding to each expansion ratio.It can be concluded from Figure 14 that the shaft efficiency of the radial inflow turbine increases first and then decreases with the increase in the expansion ratio.For different speeds, the corresponding expansion ratios that reach the peak shaft efficiency are not the same; the higher the speed, the larger the expansion ratio.When the operating speed is 80% of the design speed, 100% of the design speed, and 120% of the design speed, the turbine efficiency reaches its peak at expansion ratios of 1.53, 1.70, and 1.85, respectively, which are 86.337%,88.509%, and 88.736%, respectively.The optimized radial inflow turbine operates at design speed, and the shaft efficiency changes a little with the expansion ratio.Although its performance is weaker than that of the high-speed operation mode at a large expansion ratio, its performance degradation at low and high temperatures is not large, and its overall performance is excellent.

Conclusions
Based on the one-dimensional design of a 30 kW ocean thermal energy radia turbine, this paper used a three-dimensional numerical simulation method to st effects of different rotor shape parameters on the working fluid flow and perform the rated condition, and used the machine learning method to optimize the roto parameters.The main conclusions are as follows: (1) A 30 kW ocean thermal energy radial inflow turbine was designed, and a one sional design optimization was performed using Isight 2020 and Matlab 20 taining the optimal scheme based on seven initial thermodynamic paramete a shaft efficiency of 86.228%.Based on the numerical simulation carried out u CFX 2020 R2 software, this paper obtained radial inflow turbine models wit ent rotor shapes using parameterization and batch processing methods and a their performance parameters.(2) Based on the one-dimensional optimal scheme, the rotor shape parameters w ther optimized based on the support vector regression method, achieving adaptation of the rotor shape parameters, stator geometry parameters, and condition.The regression prediction method used does not require a prior mo can directly obtain the nonlinear mapping relationship between the shape p ters and the shaft efficiency.The optimized rotor results are as follows: diam tio-0.421;blade number-16; twist angle-43.378°;radial clearance-2.5 blade tip clearance-0.273mm; blade thickness-1.582mm; outlet blade fi dius-3.152mm.(3) The radial inflow turbine scheme obtained through using the model pr method can effectively avoid the recirculation vortices on the suction surfac

Conclusions
Based on the one-dimensional design of a 30 kW ocean thermal energy radial inflow turbine, this paper used a three-dimensional numerical simulation method to study the effects of different rotor shape parameters on the working fluid flow and performance at the rated condition, and used the machine learning method to optimize the rotor shape parameters.The main conclusions are as follows: (1) A 30 kW ocean thermal energy radial inflow turbine was designed, and a one-dimensional design optimization was performed using Isight 2020 and Matlab 2020b, obtaining the optimal scheme based on seven initial thermodynamic parameters, with a shaft efficiency of 86.228%.Based on the numerical simulation carried out using the CFX 2020 R2 software, this paper obtained radial inflow turbine models with different rotor shapes using parameterization and batch processing methods and analyzed their performance parameters.(2) Based on the one-dimensional optimal scheme, the rotor shape parameters were further optimized based on the support vector regression method, achieving the best adaptation of the rotor shape parameters, stator geometry parameters, and design condition.The regression prediction method used does not require a prior model and can directly obtain the nonlinear mapping relationship between the shape parameters and the shaft efficiency.The optimized rotor results are as follows: diameter ratio-0.421;blade number-16; twist angle-43.378• ; radial clearance-2.553mm; blade tip clearance-0.273mm; blade thickness-1.582mm; outlet blade fillet radius-3.152mm.(3) The radial inflow turbine scheme obtained through using the model prediction method can effectively avoid the recirculation vortices on the suction surface of the moving rotor.The maximum shaft efficiency of the optimized scheme is 88.467%, which is 2.239% higher than that of the one-dimensional design optimal scheme, effectively

Figure 3 .
Figure 3. Three-dimensional model of initial scheme.

Figure 3 .
Figure 3. Three-dimensional model of initial scheme.

Figure 4 .
Figure 4. Impeller modeling parameters schematic diagram.(a) Rotor blade twist angle  at axial view; (b) radial clearance ∆ and tip clearance ∆ at meridian view; (c) rotor outlet blade thickness  and rotor outlet blade fillet radius  at 50% leaf height cross section.

Figure 4 .
Figure 4. Impeller modeling parameters schematic diagram.(a) Rotor blade twist angle θ at axial view; (b) radial clearance ∆ 1 and tip clearance ∆ 2 at meridian view; (c) rotor outlet blade thickness T 2 and rotor outlet blade fillet radius r at 50% leaf height cross section.

Figure 6 .
Figure 6.Verification of grid independence for numerical simulation of radial turbines.

Figure 6 .
Figure 6.Verification of grid independence for numerical simulation of radial turbines.

Figure 6 .
Figure 6.Verification of grid independence for numerical simulation of radial turbines.

Figure 8 .
Figure 8.The comparison of shaft efficiency simulative value and predictive value: (a) training set comparative result; (b) test set comparative result.

Figure 9 .
Figure 9. Input parameters effect on shaft efficiency.
e l d i a m e t e r r a t i o

Figure 9 .
Figure 9. Input parameters effect on shaft efficiency.

Figure 9 .
Figure 9. Input parameters effect on shaft efficiency.

Figure
Figure13a,b show the streamline distribution and static entropy distribution at 50% blade height of the rotor before optimization.It can be seen that the suction surface of the rotor obtained from the one-dimensional design always has recirculation vortices of different sizes, which cause the flow in the rotor passage to be uneven, and subject the blades to unsteady cyclic loads, reducing the efficiency and strength of the turbine.The suction surface vortices are mainly caused by the mismatch between the operating speed, blade geometry parameters, and rotor shape parameters.

Figure 13 .
Figure 13.Comparison of flow field at 50% blade height surface before and after modeling parameters optimization.(a) The streamlined distribution before optimization; (b) the static entropy distribution before optimization; (c) the streamlined distribution after optimization; (d) the static entropy distribution after optimization.

JFigure 14 .
Figure 14.Shaft efficiency distribution of radial inflow turbine at different speed and expans condition.

Figure 14 .
Figure 14.Shaft efficiency distribution of radial inflow turbine at different speed and expansion ratio condition.

Table 2 .
Initial scheme design results.

Table 3 .
Sampling range of modeling parameters.

Table 4 .
Optimal modeling parameters.The comparison of shaft efficiency simulative value and predictive value: (a) training set comparative result; (b) test set comparative result.