One-Dimensional Optimization Design of Supercritical Carbon Dioxide Radial Inflow Turbine and Performance Analysis under Off-Design Conditions

Abstract

In the present study, the one-dimensional optimization design and the performance analysis based on a three-dimensional model were performed for a 2.1 MW supercritical carbon dioxide radial inflow turbine. Firstly, an in-house code was developed with MATLAB language for one-dimensional optimization design to maximize the total-to-static efficiency. Then, the three-dimensional radial inflow turbine model was constructed based on the one-dimensional optimization design result. Finally, the flow field and the aerodynamic performance were studied using the commercial software NUMECA. It is shown that the total-to-static efficiency obtained from three-dimensional simulation under the nominal design condition is 85.77%, with a relative deviation of 0.55%, as compared with that from the one-dimensional optimization design. Furthermore, the static temperature and pressure from the turbine inlet to the outlet drop uniformly, and there is no obvious flow separation under the nominal design condition. The values of total-to-static efficiency are always higher than 75% at the 80–110% relative rotating speed and the expansion ratio of 1.75–4.48, which also demonstrates good performance under the off-design working conditions.

1. Introduction

The Brayton cycle with supercritical carbon dioxide (s-CO₂) as a working fluid has raised universal attention in the last few decades [1]. The s-CO₂ Brayton cycle exhibits a few advantages compared with the Brayton cycle with the working fluid. A higher cycle thermal efficiency can be achieved with a relatively lower turbine inlet temperature, which could enhance the operation’s dependability, reduce the system’s complexity and save on costs. Furthermore, the lower turbine inlet temperature of the s-CO₂ Brayton cycle makes the applications possible in solar energy and geothermal power generation, waste heat recovery, and nuclear power generation [2,3,4,5,6]. Moreover, because it is an environmentally friendly working fluid, no pollution is generated in the power generation process of the s-CO₂ Brayton cycle. However, more challenges bring to the key power generation component, i.e., the s-CO₂ radial inflow turbine design, resulting from unique thermal-physical properties. The geometry scaling effect is a salient issue that needs to be resolved due to the smaller geometry of the s-CO₂ turbine [7]. Hence, the radial inflow design with high efficiency is a fundamental issue for applying the advanced s-CO₂ Brayton cycle.

Sandia National Laboratory [8] performed the pioneering experimental work to test the s-CO₂ Brayton cycle and key component performance. In their work, the No. 2 radial inflow turbine was designed with a power of 213 kW and a total-to-static efficiency of 87%. Bechtel Marine Propulsion Corporation [9] designed the s-CO₂ radial inflow turbine with a total inlet temperature of 299 °C. However, due to several confines in their experimental test, the turbine inlet’s total temperature was limited to 232 °C, and the rotating speed could not achieve the nominal value. In the experimental test carried out by the Tokyo Institute of Technology [10], the rotating speed of the s-CO₂ radial inflow was only 55,000 rpm, which was much lower than the design value of 100,000 rpm; this was attributed to the technical limitations of the shaft bearing and sealing. Hence, the output power and the efficiency measured were lower than the designed values. The experimental tests performed by the Korea Institute of Energy Research [11] found the same astriction. The designed rotating speed of their s-CO₂ radial inflow turbine was 70,000 rpm; however, the actual experimental value was limited to 30,000 rpm.

Zhou et al. [12] designed the s-CO₂ radial inflow turbine with the power of 1.5 MW and performed the performance prediction based on the Computational Fluid Dynamics (CFD) method using the software package ANSYS CFX. They found that the predicted total-to-static efficiency is 83.49%, with a deviation less than 5% from the calculated value from their one-dimensional design code. Lv et al. [13] presented an optimization work for the one-dimensional design of the s-CO₂ radial turbine based on the Sequential Quadratic Programming (SQP) optimization algorithm. They revealed the superiority of the optimization work with an improvement of 2.27% on the total-to-static efficiency. Saeed et al. [14] designed s-CO₂ turbines using deep neural networks. Results suggest that the employed multifaceted approach reduces computational time and resources significantly and is required to completely understand the effects of various turbine design parameters on its performance and sizing. Lee et al. [15] developed a code to predict the aerodynamic performance of an s-CO₂ radial inflow turbine under the off-design working condition. The variations were less than 10% between the one-dimensional calculations and the three-dimensional CFD predictions. The energy loss empirical or semi-empirical correlations determine the predicted accuracy. However, limited correlations are available, especially for the s-CO₂ radial inflow turbine. Persky and Sauret [16] evaluated the analytical energy loss correlations used in the preliminary design for the air or gas radial inflow turbine in designing the s-CO₂ turbine. They found that the published loss model configurations may not suit off-design analysis for the s-CO₂ radial inflow turbine. They developed a new loss model configuration with less than a 2% difference in efficiency prediction over the CFD result. With the aid of the CFD method, Keep and Jahn [17] studied the loss contribution for a 300 kW low specific speed s-CO₂ radial inflow turbine. Their results concluded that the end-wall viscous losses in the stator are dominated, which plays a more significant role than that for the gas radial inflow turbine. Yang et al. [18] focused on the leakage loss in the impeller backface cavity for the s-CO₂ radial inflow turbine and revealed the relevance of the leakage loss to the labyrinth seal geometrical parameter.

The present paper provides an optimization design platform based on the one-dimensional design method in Shu et al. [19] to carry forward the efficient s-CO₂ radial inflow turbine design. An in-house code is developed in MATLAB language. The design variables are selected with nozzle velocity coefficient, rotor velocity coefficient, reaction degree, diameter ratio, absolute flow angle rotor inlet, and relative flow angle at the rotor outlet. The optimization function maximizes the total-to-static efficiency with aerodynamic and structural constraints. The build-in function, fmincon, is treated as the optimization algorithm. The optimization code is then applied to design the s-CO₂ radial inflow turbine with a power of 2.1 MW. The comparison of the total-to-static efficiency is performed between the CFD simulation and the one-dimensional calculation under the design working condition to validate the current code. The flow field and the performance are also analyzed in detail under the off-design working conditions.

2. One-Dimensional Optimization Design

The general flow chart of the current one-dimensional optimization design for the s-CO₂ radial inflow turbine is shown in Figure 1. The optimization target, the design variables and the relevant lower and upper limits, the initial values of the design variables, and the restricted conditions are required to be specified to run the optimization work. The built-in function in MATLAB, fmincon, is selected as the optimization algorithm. Because the fmincon function is optimized locally, the predicted optimization result strongly relies on the given starting value. Hence, the final optimization result is obtained by adjusting the initial values and the trial. When the optimization value in the current step is equal to or larger than that in the former step, the optimization process is completed. In the one-dimensional design method for the s-CO₂ radial inflow turbine, the calculated value of the total-to-static efficiency also depends on the thermo-physical properties of the s-CO₂, the selection of the energy loss models, and the specific constrained conditions.

2.1. Thermo-Physical Property Calculation

Unlike conventional air or gas, working fluid s-CO₂ should be treated as the real working fluid with varying thermo-physical properties. In the current code, the thermo-physical properties are acquired from a featured software package NIST REFPROP [20]. Hence, the thermodynamic parameter at the nozzle inlet, rotor inlet, and rotor exit and the total-to-static efficiency of the s-CO₂ radial inflow turbine can be accurately calculated.

2.2. Optimization Algorithm

The optimization algorithm is the fundamental part of an optimization issue, determining the local or global way and the optimization time. The optimization function, fmincon, is commonly applied in constrained nonlinear multivariable nonlinear system optimization problems [21]. The option SQP is chosen for the algorithm option. The Hessian matrix is updated using the BFGS algorithm in the quadratic programming problem. Its major advantages include succinct code, simple usage, and fewer iterations. The comparison test for the current one-dimensional optimization design showed that the iteration steps are 36, much lower than the iteration step of 120 for the Simulated Annealing Approach. The confinement factor is that only the local optimum result can be acquired, which may differ from the optimum global result. In the present study, the initialized variables are carefully given using the screening method [22] for the radial inflow turbine design. Furthermore, the additional iterative process shown in Figure 1 is applied to pursue the global optimum solution.

2.3. One-Dimensional Design Method

The total-to-static efficiency η_s can be calculated by considering several energy losses, which is given by:

$${\eta }_{\mathrm{s}}=\left[\left({\eta }_{\mathrm{u}}-{\zeta }_{\mathrm{f}}\right){\eta }_{\Delta }-{\zeta }_{\mathrm{i}}\right]{\eta }_{\mathrm{e}}$$

(1)

where η_u is the peripheral efficiency, ζ_f is the disk friction energy loss coefficient, η_Δ is the clearance efficiency, ζ_i is the incidence loss coefficient, η_e is the partial admission efficiency.

From the energy standpoint, the peripheral efficiency η_u is calculated by:

$${\eta }_{\mathrm{u}}=1-{\zeta }_{\mathrm{n}}-{\zeta }_{\mathrm{r}}-{\zeta }_{\mathrm{c}2}$$

(2)

where ζ_n, ζ_r, and ζ_c2 are the energy loss coefficients in the nozzle, rotor blade, and the leaving velocity at rotor exit.

The peripheral efficiency η_u can also be expressed by the design variables as follow:

$${\eta }_{\mathrm{u}}=2x_{\mathrm{a}}(\phi \cos {\alpha }_1\sqrt{1-\Omega }-{\overline{D}}_2^2{x}_{\mathrm{a}}+{\overline{D}}_2\psi \cos {\beta }_2\sqrt{\Omega +{\phi }^2\left(1-\Omega \right)+{\overline{D}}_2^2{x}_{\mathrm{a}}^2-2x_{\mathrm{a}}\phi \cos {\alpha }_1\sqrt{1-\Omega }})$$

(3)

where x_a is the velocity ratio, φ and ψ are the nozzle velocity coefficient and rotor blade velocity ratio, α₁ is the absolute flow angle at rotor inlet, β₂ is the relative flow angle at rotor exit, Ω is the reaction degree, ${\overline{D}}_2$ is wheel diameter ratio. The definitions of x_a, φ, ψ, and ${\overline{D}}_2$ can be referred to in [19].

The disk friction loss coefficient ζ_f is given by:

$${\zeta }_{\mathrm{f}}=\frac{1000}{G\Delta h_{\mathrm{s}}}\cdot \frac{f{\rho }_1{D}_1^2}{1.36}\cdot {\left(\frac{u_1}{100}\right)}^3$$

(4)

where G is the mass flow rate, Δh_s is the turbine isentropic enthalpy drop, f is the coefficient with a constant value of 4, ρ₁, D₁, and u₁ are the density, diameter, and peripheral velocity at the rotor inlet.

The expression of clearance efficiency η_Δ is as follows:

$${\eta }_{\Delta }=1-1.3(\Delta r/l_{\mathrm{m}})$$

(5)

where Δr is the radial clearance between the shroud and the casing, l_m is the arithmetic mean blade height.

The current radial inflow design assumes the full admission pattern and the partial admission efficiency η_e equals 1.

The incidence loss ζ_i is additionally considered in the current study, which is given by:

$${\zeta }_{\mathrm{i}}=\frac{{w_1}^2{\sin }^2{i}_1}{2\Delta h_{\mathrm{s}}}$$

(6)

where w₁ is the relative velocity and relative flow angle at the rotor inlet, i₁ is the incidence angle at the rotor inlet. The definition of i₁ is as follows:

$$i_1={\beta }_{1\mathrm{b}}-{\beta }_1$$

(7)

where β_1b is the rotor inlet blade angle of 90°, β₁ is the relative flow angle at the rotor inlet.

2.4. Design Constraints and Optimization Function

As seen in the one-dimensional design method, seven design parameters influence the final total-to-static efficiency of the radial inflow turbine. The upper and lower limits of each design parameter for the current design are listed in

Table 1. Design variable and the relevant value region.

Design Variable	Value Region
Velocity ratio xa	0.63~0.72
Reaction degree Ω	0.36~0.5
Nozzle velocity coefficient φ	0.92~0.97
Rotor inlet absolute flow angle α1 (°)	14~25
Rotor exit relative flow angle β2 (°)	20~45
Wheel diameter ratio ${\overline{D}}_2$	0.35~0.55
Rotor blade velocity coefficient ψ	0.75~0.90

, following the suggested values in [19]. Note that the design parameters vary within a wide region, providing sufficient wide design space for the following optimization.

The aerodynamic limitations and structural constraints must be considered for the one-dimensional optimization design. In the current study, the followings constraints are covered and listed in

Table 2. Constraint conditions.

Constrained Variable	Value
Incidence angle at rotor inlet i1 (°)	−20~+10
Mach number at rotor inlet	≤1.4
Absolute flow angle at rotor exit (°)	85~95
Relative shroud diameter at rotor exit	≤0.85
Relative hub diameter at rotor exit	≥0.12
Relative blade height at rotor inlet	0.02~0.17

. The incidence angle limitation avoids severe reverse flow and the excessive incidence loss at the rotor inlet. The point of the Mach number constraint is to prevent the energy loss and the secondary flow loss from the shock wave. The absolute flow angle at the rotor exit is bounded from 85° to 95°, which guarantees the close axial flow direction for the working fluid at the rotor exit to avoid excessive leaving velocity energy loss. The meridional passage geometry depends on the specific values of the relative shroud diameter and the hub diameter. Hence, the current constraints are assumed to avoid excessive expansion in the meridional passage. When the relative blade height is smaller than its lower limit, the blade height may be too small, especially for the s-CO₂ radial inflow turbine with a heavy working fluid. However, the relative blade height with a large value would influence the shape of the meridional passage.

Under the consideration of the constraints mentioned above, the optimization function is given as follows:

$$fmincon\left(1/{\eta }_{\mathrm{s}}\right)=f\left(x_{\mathrm{a}},\Omega ,{\alpha }_1,{\beta }_2,\phi ,\psi ,{\overline{D}}_2\right)$$

(8)

The fmincon function is to find the minimum. When the minimum of 1/η_s is achieved, the maximum of η_s is obtained equivalently. Hence, the reciprocal of η_s is supposed to be the optimization function.

2.5. Optimization Result

A design case is performed to validate the current in-house one-dimensional optimization design code. The design specifications of the s-CO₂ radial inflow turbine are listed in

Table 3. Design specifications.

Parameter	Specification
Mass flow rate (kg/s)	16.0
Rotating speed (r/min)	42,000
Inlet total pressure (MPa)	24.6
Inlet total temperature (K)	823
Outlet static pressure (MPa)	8.5
Output power (MW)	2.1

.

In the optimization process, the numbers of nozzle and rotor blades are fixed as constant values of 33 and 19, respectively. The one-dimensional design results originated from the initial design parameters, and the optimization results are listed in

Table 4. Initial and optimum design results.

Parameter	Initial Value	Optimum Value
Velocity ratio xa	0.650	0.662
Reaction degree Ω	0.470	0.483
Nozzle velocity coefficient φ	0.95	0.97
Rotor inlet absolute flow angle α1 (°)	16.0	16.0
Rotor exit relative flow angle β2 (°)	21.4	25.0
$\mathrm{Wheel} \mathrm{diameter} \mathrm{ratio} {\overline{D}}_2$	0.430	0.417
Rotor blade velocity coefficient ψ	0.85	0.90
Rotor inlet relative flow angle β1 (°)	85.55	85.00
Rotor exit absolute flow angle α2 (°)	78.0	75.0
Nozzle exit diameter (mm)	170.3	170.4
Rotor inlet diameter (mm)	163.8	166.8
Rotor inlet blade height (mm)	3.0	3.3
Rotor exit hub diameter (mm)	43.1	48.6
Rotor exit shroud diameter (mm)	89.8	85.6
Specific speed	13.76	13.84
Output power (MW)	2.056	2.118
Total-to-static efficiency ηs (%)	83.76	85.30

. It is seen that the optimal set of the seven design variables results in a large rotor geometry and a slightly higher specific speed than the initial design. The optimal design provides an output power of 2.118 MW and a total-to-static efficiency of 85.30%.

3. Numerical Methodology and Setup

The steady-state, three-dimensional Navier-Stokes equations are solved using the commercial software NUMECA. The turbulence model comparison work for the radial inflow turbine in [23] concluded that the Spalart-Allmaras turbulence model was a suitable choice with good convergence and adaptability. Hence, the Spalart-Allmaras turbulence model is selected for the turbulent closure in the current study.

The three-dimensional model is generated based on the one-dimensional optimization design geometry, as shown in Figure 2. In the current simulation, one nozzle passage and the rotor blade passage are selected as the computational domain to save the computational cost. The structured meshes are generated using IGG-AutoGrid5, shown in Figure 3. The O4H and HI topology structures are applied for the mesh generation in the nozzle passage and the rotor blade passage. The meshes around the endwall, nozzle, and rotor blade surfaces are refined with an expansion ratio of 1.2 to describe the velocity boundary layers. The computational meshes stratify the numerical simulation standards with an orthogonality angle of less than 20°, an aspect ratio of less than 2500, and an extension ratio of less than 2.

A grid independence test is performed to verify the accuracy of the numerical results. Figure 4 shows the total-to-static efficiency variation at five computation mesh resolutions. It is observed that the value of total-to-static efficiency is nearly unchanged since the mesh resolution is higher than 1.72 million. Hence, the mesh resolution of 1.72 million is adopted for the following simulations to balance the prediction accuracy and computation time. The mesh resolutions for the nozzle passage and rotor blade passages are 0.89 million and 0.83 million, respectively.

Alireza et al. [24] pointed out that the thermal-physical properties of the s-CO₂ significantly impact the prediction accuracy for the performance and flow field of the radial inflow turbine. Hence, the real gas thermal-physical properties are generated using a built-in code, TabGen, in the software NUMECA. Because the table generation technique is adopted in the code TabGen, table resolution must be determined in advance. Based on the design specifications for the radial inflow turbine listed in Table 3, the pressure ranges from 5 MPa to 30 MPa, and the temperature ranges from 400 K to 1000 K for the table generation. Note that the thermal-physical properties follow the perfect gas law with explicit functions, which change successively depending on the temperature and pressure at a specific thermodynamic state of CO₂. It is seen in

Table 5. Table resolution test of the thermal-physical properties.

Gas Type	Resolution	Total-to-Static Efficiency
Perfect gas	/	82.82%
Real gas	101 × 101	85.78%
	201 × 201	85.77%
	301 × 301	85.77%

that the perfect gas thermal-physical properties lead to a deviation of nearly 3% less in the total-to-static efficiency than the real gas ones. Combatively, only a deviation of 0.01% occurs among different table resolutions. Therefore, the table resolution of 201 × 201 is adopted in the following simulations.

However, the thermal-physical properties with a high resolution degrade the computation stability; hence, a low value of the CFL number is required to improve the stability. The CFL number is set as 0.5 in the trade-off between the computation time and stability for the current simulations.

The boundary conditions in simulations are set up as follows. The total pressure and temperature are imposed at the nozzle inlet, and static pressure is set at the rotor exit. The domain of the nozzle passage is stationary, and the domain of the rotor blade passage is rotational. The boundary conditions can be seen in Table 3 for the simulation of the design working conditions. For the simulations under off-design working conditions, the static pressure at the rotor exit varies to achieve the given total-to-static expansion ratio, and the rotational speed differs. The wall surfaces at the hub, blade, and shroud are assumed to be adiabatic, smooth, and in no-slip condition. When the root means square residuals of all variables are lower than 10–6, and the variation of mass flow rate entering into the nozzle passage is less than 0.5%, the convergence of the CFD simulation is achieved.

4. Numerical Result Analysis and Discussion

4.1. Design Working Condition

The comparison between the predicted performance obtained from the three-dimensional simulation and the designed data is listed in

Table 6. Evolution of the one-dimensional design.

Parameter	Design	Prediction	Error (%)
Total-to-static expansion ratio	2.896	2.896	0.00
Mass flow rate (kg/s)	16	16.213	1.33
Output power (MW)	2.118	2.135	0.80
Total-to-static efficiency (%)	85.30	85.77	0.47

. It is seen that the prediction of mass flow rate, output power, and total-to-static pressure agree well with the designed data in the one-dimensional optimization design. The predictions are slightly higher than the designed data, with a maximum difference of 1.33% for the mass flow rate. The deviation of the total-to-static efficiency is only 0.47%.

Figure 5 represents the streamlines in the nozzle and rotor blade passages. It is observed that the velocity varies gradually along the flow passage, and the streamlines distribute uniformly. There is no obvious swirl flow and separation, which demonstrates a good flow property for the current radial inflow turbine at the design working condition.

As presented in Figure 6, the Mach number at 50% blade span gradually increases along the nozzle passage, and the maximum value of 1.26 occurs at the nozzle exit. Although the transonic flow exists in the nozzle passage, the maximum is still within the limits of the Mach number in Table 2.

The static pressure and temperature distribution at 50% blade span are shown in Figure 7. The static pressure and temperature drop gradually inside the turbine. The distributions of the static pressured drop and the temperature drop are reasonable with no abrupt changes.

The variations of the enthalpy and entropy in the radial inflow turbine are represented in Figure 8. It is seen that the enthalpy drops uniformly along the flow passage in Figure 8a. As shown in Figure 8b, the entropy reduction changes smoothly along most flow passages. The entropy varies from 2.667 to 2.856 kJ/(kg·K). There is an abrupt increase of about 0.1 kJ/(kg·K) at the interface between the stationary and rotational domains due to the irreversible shock wave energy at the nozzle exit.

4.2. Off-Design Working Conditions

The performance curves of the mass flow rate to the expansion ratio at 70–110% nominal rotational speeds are presented in Figure 9. The benchmark value is set as the nominal rotational speed of 42,000 r/min. As seen in Figure 9, the mass flow rate increases first at the nominal rotational speed and then chokes with the expansion ratio. The mass flow rate approaches the choked mass flow rate when the expansion ratio is 2.896 at the design working condition. At the off-design working conditions, the rotational speed significantly impacts the mass flow rate when the expansion ratio is lower than 3.0. The mass flow rate increases at the same expansion ratio as the rotational speed increases. However, if the expansion ratio is higher than 3.0, the rotational speed variation has little impact on the mass flow rate. The radial inflow turbine is always choked with an expansion ratio higher than 3.788 regardless of the rotational speed. The maximum allowable expansion ratio is 5.47, and the corresponding choked mass flow rate is 16.358 kg/s.

Figure 10 shows the variation curves of the total-to-static efficiency to the expansion ratio at different rotational speeds. At the nominal rotational speed, the value of the total-to-static efficiency increases and decreases with the expansion ratio. The total-to-static efficiency at the design working condition is 85.77%, close to the maximum value at the performance curve. This means that the current one-dimensional optimization design is appropriate. Moreover, the total-to-static efficiency always yields a value higher than 70%, with the expansion ratio region of 1.76–5.47. The performance curve and the local maximum total-to-static efficiency point shift towards the left with the decrease in rotational speed. The total-to-static efficiencies are higher than 75%, with an expansion ratio of 1.75–4.48 at 80–110% nominal rotational speed. However, the total-to-static efficiency is lower than 70%, with most expansion ratios at the 70% nominal rotational speed. Hence, it is suggested that the current radial inflow turbine avoid operating under these working conditions.

5. Conclusions

In this paper, the one-dimensional optimization design code is developed for the s-CO₂ radial inflow turbine. The key optimization function is the built-in function fmincon in MATLAB. A case study is performed using the developed code for the radial inflow turbine design with an output power of 2.1 MW. The predicted performance and detailed flow field are obtained from the simulation results for the three-dimensional model using the software package NUMECA. Several major conclusions are reached as follows:

(1)

The developed one-dimensional optimization design method is validated by comparing the initial and optimum designs. The total-to-static efficiency for the optimum design is 85.30%, with an absolute improvement of 1.54%.

(2)

The performance deviations between the three-dimensional prediction result and the one-dimensional design data are lower than 2%. Under the design working condition, the total-to-static efficiency is close to the maximum value at the nominal rotational speed. The static pressure, temperature, enthalpy, and entropy distribute uniformly in the nozzle and rotor blade flow passages, demonstrating good performance for the current radial inflow turbine under the designed working conditions.

(3)

The performance analysis revealed that the total-to-static efficiency is higher than 75% at 80–110% nominal rotational speed.