Preliminary Design and Blade Optimization of a Two-Stage Radial Outflow Turbine for a CO₂ Power Cycle

Abstract

Recently, the CO₂ power cycle has attracted attention because of tightening environmental regulations. The turbine is a factor that greatly affects the efficiency of the cycle. The radial outflow turbine is a turbomachine with the various advantages of an axial flow turbine and a radial inflow turbine, but the design theory for the turbine is uncertain. In this study, a preliminary design algorithm for a radial outflow turbine with a multi-stage configuration is presented. To verify the preliminary design algorithm, a preliminary design for a two-stage radial outflow turbine for a CO₂ power cycle was carried out, and a computational fluid dynamic analysis was performed. Consequently, values close to the target performance were obtained, but blade optimization was performed to obtain more satisfactory results. The final geometry of the radial outflow turbine was obtained through optimization considering the blade exit angle related to the deviation angle, blade maximum thickness-true chord ratio, and incidence angle. In the final geometry, the error rates of power ($\dot{W}$), efficiency (${\eta }_{ts}$), and pressure ratio ($P{R}_{ts}$) between target performance and computational fluid dynamic results were improved to 5.0%, 4.8%, and 1.8%, respectively. The performance and flow characteristics of the initial and final geometries were analyzed.

1. Introduction

Regulations to overcome environmental pollution and energy depletion in various fields are continuously being developed [1]. In response, studies on new and renewable energy sources are actively being conducted. Among these studies, a field deals with systems that produce energy using a low-temperature heat source [2]. As a method for generating electricity using a low-temperature heat source, the CO₂ power cycle is receiving attention [3].

CO₂ is an economical, stable, safe, and low-polluting natural working fluid [4]. In particular, this working fluid is very effective in the supercritical state. CO₂ in the supercritical state has a higher density than the gas; therefore, it is possible to miniaturize the equipment constituting the cycle. In addition, they have the advantages of high heat capacity and excellent fluidity [5]. The CO₂ power cycle is shown in Figure 1. It can comprise a supercritical cycle (Brayton cycle) in which the entire process is performed only in a supercritical state and a transcritical cycle (Rankine cycle) in which the state of the working fluid changes [3]. Each cycle is applicable to low-temperature heat sources, such as waste heat, solar heat, and geothermal heat [6,7]. The advantages of the CO₂ power cycle are being highlighted daily, and numerous studies are being conducted. In particular, there are numerous studies on thermodynamic analysis based on various heat sources and cycles, and the importance of system efficiency has been emphasized.

The turbine is a factor that significantly affects the efficiency of the power generation system [8]. Axial flow turbines and radial inflow turbines can be considered for the CO₂ power cycle. Because the axial flow turbine is mainly composed of multiple stages, it is used in large systems that handle high-pressure ratios, high flow rates, and high efficiencies [9]. On the other hand, the radial inflow turbine is suitable for small flow rates and small-scale systems, has the advantages of being easy to fabricate, inexpensive, and has excellent performance even at the off-design point [10]. However, an axial flow turbine is difficult to manufacture and expensive because the blade must be twisted owing to the velocity triangle difference between the hub and shroud [11]. Meanwhile, because a radial inflow turbine is difficult to construct in multiple stages, there is a limit to the applicable pressure ratio [12].

The radial outflow turbine, which has recently attracted attention, is a turbomachinery that has the advantages of both an axial flow turbine and a radial inflow turbine. Figure 2 shows the typical structure of a radial outflow turbine. In the turbine, the working fluid flows in the axial direction and then expands in the radial direction [13]. Because the radius of the blade increases with the expansion of the working fluid, the turbine can be designed such that the blade height remains the same or the difference is not large. In addition, because there is no change in the peripheral velocity over the span of the blade, the velocity triangles of the hub and shroud are identical. Thus, the blades of a radial outflow turbine do not need to be twisted, unlike those in an axial flow turbine. Because it is easy to configure in multiple stages, it can be used even under operating conditions that require a high-pressure ratio. In other words, this turbine has the characteristics of an axial flow turbine that can be used at high-pressure ratios because it is composed of multiple stages and a radial inflow turbine that is easy to manufacture and has excellent performance even under off-design conditions [3,8,12].

Since a radial outflow turbine does not require a high blade height at all stages, it has the obvious mechanical merit of withstanding high pressure at the inlet [11]. Meanwhile, in the CO₂ power cycle, the working fluid enters the turbine at very high pressure. For this reason, a radial outflow turbine can be expected to provide significant benefits to the CO₂ power cycle.

Unlike axial flow turbines or radial inflow turbines, there are only a few studies on radial outflow turbines. A representative research institute studying radial outflow turbines is the Laboratory of Fluid-Dynamics of Turbomachinery at the Polytechnic University of Milan. Here, the aerodynamic design, optimization, and flow pattern of a radial outflow turbine for organic Rankine cycles were studied. Various types of radial outflow turbines for organic Rankine cycles from kW to MW were studied [14,15,16,17,18,19]. The University of Shanghai for Science and Technology in China has also actively studied the radial outflow turbine [5,9,20,21,22,23]. The design of radial outflow turbines for ideal gases and organic working fluids was studied, and the design of volutes and transonic turbines was addressed. In particular, Luo et al. [5] performed a preliminary design for a radial outflow turbine for a supercritical CO₂ power cycle and optimized the turbine until the target output and efficiency were satisfied using computational fluid dynamics (CFD).

In addition to the above research institutes, Al Jubori et al. [24,25] designed an axial flow turbine, radial inflow turbine, and radial outflow turbine for an organic Rankine cycle under identical design conditions and then comparatively analyzed the strengths and weaknesses of each turbine using CFD. Maksiuta et al. [26] conducted a study on a radial outflow turbine for an organic Rankine cycle for waste-heat recovery. In this study, the efficiency of the radial outflow turbine was excellent at a velocity ratio of 0.7. This implies that the enthalpy drop of the downstream stage with a large peripheral velocity must be greater than that of the upstream stage with a small peripheral velocity. Dogu et al. [13] designed a 150 kW-class multi-stage radial outflow turbine for an organic Rankine cycle, optimized it using CFD and analyzed the turbine performance according to the blade tip spacing.

Although some studies have been conducted on radial outflow turbines, the design method for the turbine has not been established. The preliminary design methods proposed so far require different input parameters and limit them to a constant or a specific range without clear evidence [5,21,22,23,24,25]. Recently, the velocity ratio and loading coefficient were found to be parameters that significantly influenced the performance of the radial outflow turbine [12]. Therefore, developing a preliminary design algorithm for a multi-stage radial outflow turbine is required using these dimensionless variables.

Meanwhile, the existing related studies are mainly concerned with the maximum output and efficiency of the radial outflow turbine. However, it is also important to clarify the target performance of the turbine and develop a preliminary design algorithm that can meet it. This is because the cycle must be redesigned to account for the new turbine efficiency when it is different from that considered in the thermodynamic cycle [12]. In addition, for the optimal design of a turbine, it is necessary to identify the blade shape factors that determine the performance and to clearly indicate the blade optimization process accordingly. However, existing studies do not clearly demonstrate the blade optimization procedure using CFD. Also, although radial outflow turbines are not the only turbomachine-type of organic Rankine cycle, research on the turbines is overly focused on the cycle. Therefore, it is necessary to evaluate the applicability of the radial outflow turbine other than the organic Rankine cycle, such as the CO₂ power cycle.

In this study, a preliminary design algorithm is presented to determine the basic geometry of the radial outflow turbine according to the target performance. It was developed to enable multi-stage configuration regardless of the working fluid. In addition, parameters that have a significant impact on the performance of the turbine were reflected in the algorithm. The proposed algorithm has been applied to a CO₂ power cycle that has recently received attention but has not been well applied in a radial outflow turbine for no apparent reason. Using this algorithm, a two-stage radial outflow turbine was designed and revealed geometrical parameters significantly affecting turbine performance. In addition, through the performance analysis of the turbine according to the change of the parameters, the attempts applicable in the optimization process for the blade geometry were shown in detail.

2. Preliminary Design

2.1. Basic Concept

The basic concepts required for the preliminary design algorithm of the radial outflow turbine can be summarized as follows: The velocity triangles and the blading terminology for each stage are shown in Figure 3 and Figure 4, respectively. Here, the incidence angle is the angle difference between the inlet blade and inlet flow, and the deviation angle is the angle difference between the exit blade and the exit flow. Figure 5 shows the H–S diagram of the two-stage radial outflow turbine.

The output of each stage can be defined as Equation (1), according to Euler’s equation and enthalpy drop [27]. The radial outflow turbine differs from the axial flow turbine in that the peripheral velocities of the rotor inlet and exit are not the same [11]. Because the absolute tangential velocity of the rotor exit ($C_{3\theta }$) is a factor that reduces the turbine output, it is desirable to design it to a minimum [12]. In Equation (1), the mass flow rate was the same as that in Equation (2). The continuity equation determines the density ($\rho$), meridian absolute velocity ($C_m$), flow radius ($r$), and height ($he$) of each point:

$$\dot{W}=\dot{m}\left(h_{01}-h_{03}\right)=\dot{m}\left(U_2{C}_{2\theta }-U_3{C}_{3\theta }\right)$$

(1)

$$\dot{m}={\rho }_1{C}_{1m}2\pi r_{1}h{e}_1={\rho }_2{C}_{2m}2\pi r_{2}h{e}_2={\rho }_3{C}_{3m}2\pi r_{3}h{e}_3$$

(2)

In Figure 5, the total-total efficiency (${\eta }_{tt}\left(i\right)$) and total-static efficiency (${\eta }_{ts}\left(i\right)$) of the $i$th stage can be expressed by Equations (3) and (4), respectively [27]. In the two-stage case, the total-total efficiency (${\eta }_{tt}$) and total-static efficiency (${\eta }_{ts}$) of the overall stage are as shown in Equations (5) and (6), respectively [28]. In addition, the total enthalpy ($h_0$) and static enthalpy ($h$) at a specific point have the same relationship as in Equation (7):

$${\eta }_{tt}\left(i\right)=\left(h_{01}\left(i\right)-h_{03}\left(i\right)\right)/\left(h_{01}\left(i\right)-h_{03s}\left(i\right)\right)$$

(3)

$${\eta }_{ts}\left(i\right)=\left(h_{01}\left(i\right)-h_{03}\left(i\right)\right)/\left(h_{01}\left(i\right)-h_{3ss}\left(i\right)\right)$$

(4)

$${\eta }_{tt}=\frac{h_{01}\left(1\right)-h_{03}\left(1\right)+h_{01}\left(2\right)-h_{03}\left(2\right)}{h_{01}\left(1\right)-h_{03s}\left(1\right)+h_{01}\left(2\right)-h_{03s}\left(2\right)}$$

(5)

$${\eta }_{ts}=\frac{h_{01}\left(1\right)-h_{03}\left(1\right)+h_{01}\left(2\right)-h_{03}\left(2\right)}{h_{01}\left(1\right)-h_{03s}\left(1\right)+h_{01}\left(2\right)-h_{3ss}\left(2\right)}$$

(6)

$$h_0=h+0.5C^2$$

(7)

The enthalpy loss coefficients ($\xi$) of the nozzle and rotor at each stage are given by Equations (8) and (9), respectively [29]:

$${\xi }_N=\frac{h_2-h_{2s}}{0.5C_2^2}$$

(8)

$${\xi }_R=\frac{h_3-h_{3s}}{0.5W_3^2}$$

(9)

The velocity ratio ($\nu$) is defined by Equation (10). The rotor inlet peripheral velocity ($U_2$) and spouting velocity ($C_0$) of each stage are given by Equations (11) and (12), respectively [29]:

$$\nu =U_2/C_0$$

(10)

$$U_2=r_2\omega$$

(11)

$$C_0=\sqrt{2\left(h_{01}-h_{3ss}\right)}$$

(12)

The loading coefficient ($\psi$) of each stage is given by Equation (13) [11]:

$$\psi =\frac{U_2{C}_{2\theta }-U_3{C}_{3\theta }}{U_2^2}$$

(13)

In Equation (10), the velocity ratio $\nu$ of the stage is defined as the ratio of $U_2$ to $C_0$. Maksiuta et al. [26] stated that a velocity ratio ($\nu$) of 0.7 was appropriate for a radial outflow turbine. In the turbine, $U_2$ increased in the downstream stage compared with that in the upstream stage. When the velocity ratio ($\nu$) of the stage is determined within a certain range where high performance can be expected, to design all stages with high efficiency, the spouting velocity ($C_0$) of the downstream stage must be different from that of the upstream stage, as in Maksiuta et al. [26]. In this study, the stage load ratio ($\gamma$) was applied to determine the difference in the spouting velocity ($C_0$) between the stages. In a turbine composed of n stages, the enthalpy drop of the $i$th stage can be expressed using Equation (14) using the stage load ratio ($\gamma$): Taking a two-stage turbine as an example, the total enthalpy drop of the turbine ($h_{01}\left(1\right)-h_{03}\left(2\right)$) is divided by the ratio of 1/(1 + $\gamma$) and $\gamma$/(1 + $\gamma$) in the first and second stages, respectively, so that the enthalpy drop and the spouting velocity between stages are different. The enthalpy drop in Equation (14) can be expressed using the loading coefficient ($\psi$) and peripheral velocity of the rotor inlet ($U_2$) with reference to Equations (1) and (13), respectively:

$$\mathsf{\Delta }\mathrm{h}\left(i\right)=h_{01}\left(i\right)-h_{03}\left(i\right)=\frac{{\gamma }^{i-1}}{{{\displaystyle \sum }}_{j=1}^n{\gamma }^{j-1}}\left(h_{01}\left(1\right)-h_{03}\left(n\right)\right)=\psi \left(i\right)U_2^2\left(i\right)$$

(14)

2.2. Preliminary Design Algorithm

Figure 6 shows a flowchart of the preliminary design algorithm for the radial outflow turbine developed in this study. This algorithm was developed to enable multi-stage turbine design. This was performed using MathWorks MATLAB R2016a [30], and NIST REFPROP V9.1 [31] was used to determine the properties of the working fluid. This algorithm assumes the following and performs the preliminary design of the turbine. There are no studies on the radial outflow turbine related to the blade maximum thickness-true chord ratio ($t_{max}/c$). Therefore $t_{max}/c$ was set to 0.20, which is generally used in axial flow turbines [32].

• Standard stage is applied.
• The meridian absolute velocity ($C_m$) is constant.
• The height of the nozzle blade is constant.
• The blade maximum thickness-true chord ratio ($t_{max}/c$) is 0.20.

The developed preliminary design algorithm uses a genetic algorithm (GA). The preliminary design provides information on the basic geometry of the turbine that can satisfy the target design conditions through one-dimensional analysis. This information includes the number of blades and the radius, height, thickness, and angle of the blade.

Currently, there is no definitive methodology for stage partition for multi-stage radial outflow turbines. In this study, the stage was divided so that the efficiency of each stage did not differ significantly while satisfying the target efficiency. Therefore, the objective function of the GA, which is the core of the preliminary design algorithm, is expressed by Equation (15):

$$\mathrm{y}=\left|\max \left({\eta }_{ts}\right)-\min \left({\eta }_{ts}\right)\right|=f\left(h{e}_1\left(1\right),RPM,\gamma ,\psi ,s_3\right)$$

(15)

In Equation (15), the boundary conditions of the independent variables are as follows in Equations (16)–(20). In the case of a two-stage turbine, the second-stage load is greater than the first-stage load by $\gamma$ times. In other words, as $\gamma$ increases, the load difference between the upstream and downstream stages increases. In this study, a range was assigned, as shown in Equation (18), to obtain the appropriate load difference at each stage:

$$8\mathrm{mm}\le h{e}_1\left(1\right)\le 12\mathrm{mm}$$

(16)

$$5000\le RPM\le \text{10,000}$$

(17)

$$1.1\le \gamma \le 1.6$$

(18)

$$0.85\le \psi \le 1.30$$

(19)

$$s_1\left(1\right)\le s_3\le s_3\left(\mathrm{n}\right)$$

(20)

The constraint conditions of the GA are given by Equations (21) and (22), respectively. Through Equation (21), the difference in total-total efficiency (${\eta }_{tt}$) of each stage was minimized, and a preliminary design was performed such that the performance between the stages was similar.

$$\left|\max \left({\eta }_{tt}\right)-\min \left({\eta }_{tt}\right)\right|\le 0.001$$

(21)

$$0.57\le \nu \le 0.70$$

(22)

Kim and Kim [12] revealed that the loading coefficient ($\psi$) and velocity ratio ($\upsilon$) have a significant influence on the performance of the radial outflow turbine and suggested their appropriate ranges. In this study, the ranges of Equations (19) and (22) were determined with reference to Kim and Kim [12].

The deviation angle of the blade was based on the model of the axial flow turbine presented in the literature of Aungier [32], and it can be expressed as Equations (23)–(26).

$$\delta =\left\{\begin{array}{c}{\delta }_0 , M_{exit}\le 0.5\hfill \\ {\delta }_0\left[1-10X^3+15X^4-6X^5\right], M_{exit}>0.5\hfill \end{array}\right.$$

(23)

where

$${\delta }_0={\sin }^{-1}\left\{\left(o/s\right)\left[1+(1-o/s){\left({\beta }_g/90\right)}^2\right]\right\}-{\beta }_g$$

(24)

$$sin{\beta }_g=\mathrm{o}/s$$

(25)

$$X=2M_{exit}-1$$

(26)

The Zweifel criterion is applied to the number of blades, and the pitch of the blade is defined by Equation (27) using this basis [33].

$$s=\frac{Z·b}{2co{s}^2{\alpha }_2\left|tan{\alpha }_1-tan{\alpha }_2\right|}=2\pi r/N$$

(27)

3. Validation of the Preliminary Design Algorithm

3.1. Design Condition

A radial outflow turbine was designed to evaluate the preliminary design algorithm. The design conditions were referred to in a study by Wu et al. [34] dealing with the CO₂ transcritical power cycle using geothermal heat.

Table 1. Design conditions for a radial outflow turbine.

Variables	Unit	Values
Working fluid	-	CO2
$\dot{W}$	MW	3.80
$P_{01}$	MPa	14.0
$T_{01}$	K	403.15
$P_3$	MPa	6.43
${\eta }_{tt}$	%	90.0
${\eta }_{ts}$	%	85.0

lists the design conditions of a radial outflow turbine for the CO₂ power cycle.

$\dot{W}$$P_{01}$$T_{01}$$P_3$${\eta }_{tt}$${\eta }_{ts}$

3.2. Result of Preliminary Design

When the design conditions in Table 1 were input into the preliminary design algorithm, calculations were performed according to the flowchart in Figure 6. Here, the pressure ratio reaches approximately 2.18 under this design condition. According to the radial outflow turbine of Kim and Kim [12], when the pressure ratio was approximately 1.45, the maximum Mach number in the flow field reached 0.79. When the radial outflow turbine is designed as a single stage under the condition of a pressure ratio of 2.18, the maximum Mach number in the flow field will increase more than this. The developed program aims to design a multi-stage radial outflow turbine to handle the subsonic flow. In this study, a two-stage was selected to avoid choking at a given pressure ratio and to secure the reliability of the design algorithm for multi-stage radial outflow turbines.

Table 2. Main results of the two-stage radial outflow turbine according to the preliminary design.

Variables	Overall Stage
$\dot{W}$ [MW]	3.80
${\eta }_{tt}$ [%]	89.6
${\eta }_{ts}$ [%]	84.6
$\dot{m}$ [kg/s]	109.62
RPM	8700
$\gamma$	1.16
Variables	First-Stage	Second-Stage
$\dot{W}$ [MW]	1.76	2.04
${\eta }_{tt}$ [%]	89.5	89.6
${\eta }_{ts}$ [%]	79.4	80.7
${\xi }_N$	0.09	0.05
${\xi }_R$	0.08	0.10
$\nu$	0.57	0.69
$\psi$	1.20	0.85

and

Table 3. Specifications of the two-stage radial outflow turbine according to the preliminary design.

Specifications	First-Nozzle	First-Rotor	Second-Nozzle	Second-Rotor
$N$	61	85	71	58
$r_{in}$ [mm]	112.2	128.9	145.5	164.4
$r_{out}$ [mm]	124.9	141.5	160.4	179.2
$h{e}_{in}$ [mm]	9.96	9.96	9.76	9.76
$h{e}_{out}$ [mm]	9.96	9.76	9.76	10.67
$t_{max}$ [mm]	3.17	2.85	3.66	4.36
${\alpha }_{1b}$ [°]	0.0	-	0.0	-
${\alpha }_{2b}$ [°]	65.1	-	63.0	-
${\beta }_{2b}$ [°]	-	19.1	-	−18.1
${\beta }_{3b}$ [°]	-	−63.8	-	−68.2

present the main results and turbine specifications of the preliminary design algorithm developed in this study.

The efficiencies specified in Table 1 were for a single-stage turbine. However, in the H–S diagram of Figure 5, when the entropy increases, the constant pressure lines diverge from each other so that the overall isentropic enthalpy drop of the multi-stage turbine increases compared to that of the single-stage turbine. Meanwhile, considering Equations (3)–(6) express the efficiency of a multi-stage turbine, an increase in the isentropic enthalpy drop indicates a decrease in turbine efficiency. That is, it can be seen that the efficiency of the multi-stage turbine is somewhat lower than that of the single-stage turbine under same pressure ratio. Because a two-stage turbine was selected in this study, the total-total efficiency (${\eta }_{tt}$) and total-static efficiency (${\eta }_{ts}$) target values of the overall stage were revised to 89.6% and 84.6%, respectively, as listed in Table 2.

3.3. Method of CFD Analysis

CFD was performed to verify the preliminary design of the algorithm. CFD simulations have gained considerable confidence in the field of turbomachinery when dealing with compressible flows. CFD analysis was performed as follows. Using the specification information of the turbine provided in Table 3, the geometry of the turbine was created using ANSYS BladeGen 2019 R2 [35]. This program uses a four-digit NACA thickness model to complete the blade geometry [12,35]. Figure 7 shows the full geometry of the two-stage radial outflow turbine. In this study, the one-passage geometry of the blade was analyzed for an efficient simulation. For the CFD analysis, the one-passage geometry of the blade was applied with a hexahedral mesh using ANSYS TurboGrid 2019 R2 [36].

Figure 8 shows the geometry and mesh of the corresponding control volume. In Figure 8, each passage’s lower and upper faces correspond to the hub and shroud faces, respectively. The no-slip condition was applied to the hub, shroud and blade surfaces, and the axisymmetric faces were set as a periodic boundary condition. The number of total elements in the control volume was set to about 8 million, where the maximum y+ was less than 30. ANSYS CFX 2019 R2 [37,38] was used for CFD analysis, and the conditions are listed in

Table 4. Main models and boundary conditions of the turbine for CFD analysis.

Variables	Values
Equation of state	Aungier–Redlich–Kwong
Turbulence model	Default Shear Stress Transport (SST)
$\mathrm{Inlet}\mathrm{boundary}(\dot{m}$)	1.80 kg/s (109.62 kg/s ÷ 61)
$\mathrm{Inlet}\mathrm{boundary}(T_{01}$)	403.15 K
$\mathrm{Exit}\mathrm{boundary}(P_3$)	6.43 MPa
RPM	8700 RPM
Interface between nozzle and rotor	Frozen Rotor
Physical time step	$0.1/\omega$

. In addition, the NASA format was used to calculate the specific heat capacity at constant pressure, specific static enthalpy, and specific static entropy, which are defined in Equations (28)–(30). The coefficients of each equation were determined using NIST REFPROP V9.1 [31].

$\mathrm{Inlet}$$\mathrm{Inlet}\mathrm{boundary}(T_{01}$$\mathrm{Exit}\mathrm{boundary}(P_3$$0.1/\omega$

3.4. Results of CFD Analysis

Table 5. Results of the preliminary design and CFD.

Variables	Unit	Preliminary Design	CFD
$\dot{W}$	MW	3.80	3.60
${\eta }_{ts}$	%	84.60	88.25
$P{R}_{ts}$	-	2.18	1.95

lists the preliminary design and CFD simulation results for the main performance of the turbine. It can be observed that the results are generally similar. More specifically, the error rates for power ($\dot{W}$), efficiency (${\eta }_{ts}$), and pressure ratio ($P{R}_{ts}$) between target performance and CFD results were 5.3%, 3.7%, and 10.6%, respectively. However, it can be found that the output ($\dot{W}$) and the pressure ratio ($P{R}_{ts}$) are slightly less than the target values, respectively. This seems to be because the preliminary design algorithm used some criteria and models of axial-flow turbines. Therefore, an optimization process is required to obtain a blade geometry closer to the design goal.

$$\frac{C_p^0}{R}=a_1+a_{2}T+a_3{T}^2+a_4{T}^3+a_5{T}^4$$

(28)

$$\frac{H^0}{R}=a_{1}T+\frac{a_2}{2}{T}^2+\frac{a_3}{3}{T}^3+\frac{a_4}{4}{T}^4+\frac{a_5}{5}{T}^5+a_6$$

(29)

$$\frac{S^0}{R}=a_{1}lnT+a_{2}T+\frac{a_3}{2}{T}^2+\frac{a_4}{3}{T}^3+\frac{a_5}{4}{T}^4+a_7$$

(30)

4. Blade Optimization for Velocity Triangles

4.1. Blade Optimization Procedure

Blade optimization was performed to satisfy the velocity triangles, and target performance of each point suggested in the preliminary design by adjusting the appropriate parameters of the blade. This optimization was performed sequentially from the first to the last stage.

In a handful of studies on radial outflow turbines, no models or criteria exist for the deviation angle and thickness of the blades of the turbine. In the preliminary design of this study, the deviation angle model of the axial flow turbine was used, and $t_{max}/c$ was set to 0.20 as an initial value. Meanwhile, Kim et al. [3,8] revealed that the blade deviation angle is an important factor in the design of the radial outflow turbine, and Ainley and Mathieson [39] mentioned that $t_{max}/c$ is related to blade loss. That is, the deviation angle and $t_{max}/c$ of the blade have a significant influence on the performance of the turbine. Therefore, for the geometry of the radial outflow turbine calculated using the preliminary design algorithm, it is necessary to examine the 3D viscous flow analysis result according to the blade exit angle (${\alpha }_{2b},{\beta }_{3b}$) related to the deviation angle and $t_{max}/c$.

For the optimization in this study, the exit angle of the blade (${\alpha }_{2b},{\beta }_{3b}$) and $t_{max}/c$ were selected as the optimization variables. Because the incidence angle ($i$) of the blade was not considered in the preliminary design, the rotor inlet blade angle (${\beta }_{2b})$ was adjusted to minimize the incidence angle ($i$) that occurred during the optimization process. Figure 9 shows a flow chart of the optimization for one stage of the radial outflow turbine using these variables. In the blade optimization process, slightly coarse meshes were applied for fast CFD calculation, and a mesh independence test was performed on the final geometry.

4.2. Blade Optimization of the Radial Outflow Turbine

The radial outflow turbine was first performed with the first-stage nozzle blade. Figure 10 shows the turbine inlet total pressure ($P_{01}$) and nozzle exit flow angle (${\alpha }_2)$ according to the nozzle exit blade angle (${\alpha }_{2b})$ and the blade maximum thickness-true chord ratio $\left(t_{max}/c\right)$ in the first-stage nozzle blade. The reason for confirming $P_{01}$ was to obtain the target pressure ratio. The flow angle was checked to optimize the velocity triangle of the blade. It can be seen that $P_{01}$ and ${\alpha }_2$ increase as ${\alpha }_{2b}$ and $t_{max}/c$ increase. $P_{01}$ and ${\alpha }_2$ suggested in the preliminary design are 14.00 MPa and 63.9°, respectively. When ${\alpha }_{2b}$ is 67.1° and $t_{max}/c$ is 0.25, $P_{01}$ and ${\alpha }_2$ have values close to the target values.

The first-stage rotor blade was optimized by positioning it downstream from the optimized first-stage nozzle blade. Figure 11 shows the turbine inlet total pressure ($P_{01}$) and rotor exit flow angle (${\beta }_3)$ according to the rotor exit blade angle (${\beta }_{3b})$ and the blade maximum thickness-true chord ratio $\left(t_{max}/c\right)$ in the first-stage rotor blade. As ${\beta }_{3b}$ bends in the negative direction and $t_{max}/c$ increases, $P_{01}$ increases and ${\beta }_3$ decreases. ${\beta }_3$ suggested by the preliminary design is −62.5°. When ${\beta }_{3b}$ is −65.8° and $t_{max}/c$ is 0.25, $P_{01}$ and ${\beta }_3$ are similar to the target values.

The reasons for these results are as follows: When the exit angle of each blade (${\alpha }_{2b},{\beta }_{3b}$) is adjusted, the flow angle of the blade (${\alpha }_2,{\beta }_3$) changes naturally. During this process, the flow velocity on the exit side increases, which causes an increase in the pressure difference between the inlet and exit of the blade. In the CFD analysis, the static pressure on the exit side of the blade was regarded as a constant boundary condition. As $t_{max}/c$ increases, the blade becomes thicker, and the length of the throat between the blades becomes narrower. This also resulted in an increase in the flow velocity on the exit side, which increased the turbine inlet total pressure ($P_{01}$).

Figure 12 shows the incidence angle ($i$) and flow angle with respect to the rotor exit absolute velocity $\left({\alpha }_3\right)$ according to the rotor inlet blade angle (${\beta }_{2b})$ in the first-stage rotor blade. It can be seen that as ${\beta }_{2b}$ increases, $i$ decreases and ${\alpha }_3$ increases. If the rotor inlet blade angle (${\beta }_{2b}$) is increased, the incidence angle ($i$) can be minimized; however, the inlet flow angle (${\beta }_2$) increases. This increased the overall flow angle of the fluid passing through the blade, including the exit flow angle. Therefore, when the rotor inlet blade angle (${\beta }_{2b})$ increases, the flow angle $\left({\alpha }_3\right)$ increases. Meanwhile, when the incidence angle ($i$) becomes negative, the rotor inlet flow is directed toward the suction surface and not the pressure surface. According to Kim et al. [3,12], when the inlet flow of the rotor blade is directed toward the suction surface, turbine efficiency is significantly reduced. Therefore, it is preferable that the incidence angle ($i$) has a positive value, such that the rotor inlet flow is not directed toward the suction surface of the blade. ${\alpha }_3$ is related to the flow angle that enters the next nozzle stage. Therefore, ${\alpha }_3$ should be as close as possible to zero. Considering these conditions, satisfactory $i$ and ${\alpha }_3$ were obtained when ${\beta }_{2b}$ was 22.1°.

The second-stage nozzle blade was optimized by placing it downstream from the optimized first-stage blades. Figure 13 shows the turbine inlet total pressure ($P_{01}$) and nozzle exit flow angle (${\alpha }_2)$ according to the nozzle exit blade angle (${\alpha }_{2b})$ and the blade maximum thickness-true chord ratio $\left(t_{max}/c\right)$ in the second-stage nozzle blade. Similar to Figure 10, $P_{01}$ and ${\alpha }_2$ increase as ${\alpha }_{2b}$ and $t_{max}/c$ increase. The ${\alpha }_2$ value suggested by the preliminary design was 61.6°. When ${\alpha }_{2b}$ and $t_{max}/c$ are 65.0° and 0.25, respectively, $P_{01}$ and ${\alpha }_2$ have values similar to the target values.

The second-stage rotor blade was optimized by locating it downstream of the optimized first-stage blades and second-stage nozzle. Figure 14 shows the turbine inlet total pressure ($P_{01}$) and rotor exit flow angle (${\beta }_3)$ according to the rotor exit blade angle (${\beta }_{3b})$ and the blade maximum thickness-true chord ratio $\left(t_{max}/c\right)$ in the second-stage rotor blade. Similar to Figure 11, $P_{01}$ increases and ${\beta }_3$ decreases as ${\beta }_{3b}$ bends in the negative direction, and $t_{max}/c$ increases. The ${\beta }_3$ value suggested by the preliminary design was −67.6°. When ${\beta }_{3b}$ and $t_{max}/c$ are −70.7° and 0.20, respectively, $P_{01}$ and ${\beta }_3$ have values that are similar to the target.

Figure 15 shows the incidence angle ($i$) and flow angle with respect to the rotor exit absolute velocity $\left({\alpha }_3\right)$ according to the rotor inlet blade angle (${\beta }_{2b})$ in the second-stage rotor blade. Similar to Figure 12, it can be seen that as ${\beta }_{2b}$ increases, $i$ decreases and ${\alpha }_3$ increases. The optimization was performed so that both parameters had small values at the same time, and appropriate values were obtained when ${\beta }_{2b}$ was −15.1°. Here, it can be seen that the change in the incidence angle ($i$) according to ${\beta }_{2b}$ is nonlinear. The cause of this trend was considered to be the influence of the Mach number. In the case of −18.1° to −16.1° in Figure 15, a region with a high Mach number of 0.9 or more exists, and in other cases, only a relatively low Mach number of less than 0.9 is observed. Looking at the literature of Moustapha et al. [29], there are many studies on the Mach number of axial flow turbines, but not of radial outflow turbines.

Table 6. Results of the preliminary design and CFD after blade optimization.

Variables	Unit	Preliminary Design	CFD
$\dot{W}$	MW	3.80	3.99
${\eta }_{ts}$	%	84.60	89.44
$P{R}_{ts}$	-	2.18	2.14

shows a comparison between the preliminary design results and the CFD results of the optimized geometry. Compared with Table 5, it can be seen that the values of the output ($\dot{W}$) and the pressure ratio ($P{R}_{ts}$) are closer to the target value. Thus, it can be seen that blade optimization considering the deviation angle, blade maximum thickness-true chord ratio, and incidence angle of the blade is an effective method.

$\dot{W}$${\eta }_{ts}$$P{R}_{ts}$

5. Performance of the Final Geometry

5.1. Mesh Independence Test

A mesh independence test was performed on the optimized final geometry. Figure 16 shows the test results. The final grid was approximately 9.6 million, and the grid information is shown in

Table 7. Information of the final grid.

Domains	No. Element	Max. y+
First-nozzle	1,771,264	25.09
First-rotor	1,793,532	22.84
Second-nozzle	2,785,140	24.19
Second-rotor	2,783,880	26.41

. Referring to the ANSYS CFX-solver modeling guide [37], in general, the recommended y+ for scalable wall function and automatic wall treatment is up to 300 [40]. Therefore, y+ specified in Table 7 was appropriate.

5.2. Analysis of the Performance and Flow Characteristics

Table 8. Main results of the initial geometry and final geometry.

Variables	Unit	Preliminary Design	Initial Geometry	Final Geometry
$\dot{W}$	MW	3.80	3.60	3.99
${\eta }_{ts}$	%	84.60	88.25	89.41
$P{R}_{ts}$	-	2.18	1.95	2.14
$T{R}_{ts}$	-	1.19	1.17	1.19
${\alpha }_2$ (First-stage)	°	63.9	60.5	64.2
${\beta }_3$ (First-stage)	°	−62.5	−59.0	−62.0
${\alpha }_2$ (Second-stage)	°	61.6	58.1	61.9
${\beta }_3$ (Second-stage)	°	−67.6	−65.3	−67.3

compares the main results of the initial geometry and the final geometry after blade optimization and the mesh independence test of the two-stage radial outflow turbine. It can be seen that the CFD results of output ($\dot{W}$), pressure ratio ($P{R}_{ts}$), temperature ratio ($T{R}_{ts}$), and flow angles in the final geometry are more similar to the preliminary design results than in the initial geometry. Here, the error rates of power ($\dot{W}$), efficiency (${\eta }_{ts}$), and pressure ratio ($P{R}_{ts}$) between target performance and CFD results were improved to 5.0%, 4.8%, and 1.8%, respectively. That is, more satisfactory results were obtained through blade optimization.

$\dot{W}$${\eta }_{ts}$$P{R}_{ts}$$T{R}_{ts}$${\alpha }_2$${\beta }_3$${\alpha }_2$${\beta }_3$ However, the efficiency (${\eta }_{ts}$) of the radial outflow turbine somewhat exceeds the target design efficiency, although the pressure ratio and temperature ratio match well. Based on these results, Kim and Kim [40] and Sauret and Gu [41] pointed out the problem of enthalpy and entropy models of ANSYS CFX. Meanwhile, they could be recalculated through NIST REFPROP V9.1 [31] based on the CFX results, such as the temperature and pressure of the turbine inlet and exit. The recalculated output ($\dot{W}$, Equation (1)) and efficiency (${\eta }_{ts}$, Equation (6)) using the NIST REFPROP database were 3.90 MW and 88.28%, with their errors of only 2.63% and 3.68%.

Figure 17 and Figure 18 show the streamlines of the initial and final geometries, respectively. In both geometries, no dominant separation point or recirculation region is observed.

Table 9. Incidence angles of the rotor blades.

Variables	Unit	Initial Geometry	Final Geometry
$i$ (First-stage)	°	−2.2	1.6
$i$ (Second-stage)	°	0.1	2.7

lists the incidence angle ($i$) of the rotor blades for each geometry. In the initial geometry, the first-rotor blade inlet flow was directed toward the suction surface, and the second-rotor blade inlet flow was almost toward the stagnation point. In the final geometry, both the first- and second-rotor blade inlet flows were directed toward the pressure surface. According to the study results of Kim et al. [3,12], even though the inlet flow of the rotor blade is directed toward the pressure side, high-efficiency performance can be expected if it is not excessive. However, the turbine efficiency is significantly reduced when the rotor blade inlet flow is directed toward the suction surface. Therefore, in the initial geometry, although the incidence angle of the second-rotor blade was almost zero, the turbine efficiency decreased because the inlet flow in the first-rotor blade was directed toward the suction surface. As a result, it can be seen that the efficiency of the final geometry is greater than that of the initial geometry, as specified in Table 8.

$i$$i$ Looking at the velocity legends in Figure 17 and Figure 18, it can be seen that the overall velocity increased in the final geometry compared to the initial geometry. During the optimization process, the flow path between the blades was narrowed by adjusting the blades’ exit angle or increasing the blades’ thickness. This increased the velocity of the fluid passing through the blade, similar to the preliminary design result.

Figure 19 and Figure 20 show the pressure contours of the initial and final geometries, respectively. The increase in the fluid velocity according to the optimization increased the pressure difference between the inlet and exit of the turbine and played a positive role in securing the pressure ratio, which is the design goal. Therefore, from the pressure legends in Figure 19 and Figure 20, it can be seen that the inlet pressure of the final geometry is larger than that of the initial geometry. This is also indicated by the pressure ratios ($P{R}_{ts}$) specified in Table 8.

Figure 21 and Figure 22 show the Mach number contours of the initial and final geometries, respectively. Looking at the Mach number legends in Figure 21 and Figure 22, it can be seen that the Mach number increased in the final geometry compared to the initial geometry. In particular, a high Mach number was found at the throat and exit sides of the second-rotor blade. That is, geometry optimization can ensure a high-pressure ratio and high performance but may cause an increase in the Mach number at the end of the turbine. Therefore, attention must be paid to avoid choking when optimizing the blade.

6. Conclusions

In this study, a preliminary design algorithm for a radial outflow turbine that can be configured in multiple stages was presented. The preliminary design algorithm allows the turbine to be designed within the range of the velocity ratio ($\upsilon$) and loading coefficient ($\psi$), achieving high performance. Within these ranges, the stage load ratio ($\gamma$) range was given such that the entire stage had high efficiency, and the efficiencies between stages were made similar. The deviation angle model and blade maximum thickness-true chord ratio of the axial flow turbine were used, and the number of blades was determined by applying the Zweifel criterion. The proposed algorithm uses a GA to preliminarily design a turbine to achieve the target performance required in the thermodynamic cycle while satisfying the above design objectives.

Using the developed preliminary design algorithm, a two-stage radial outflow turbine using CO₂, which has recently attracted attention, was designed, and CFD was performed to verify the algorithm. As a result, it was found to be effective in the development of a CO₂ multi-stage turbine close to the target performance despite utilizing the results of the existing axial flow turbine and radial outflow turbine for the organic Rankine cycle. More specifically, the error rates for power ($\dot{W}$), efficiency (${\eta }_{ts}$), and pressure ratio ($P{R}_{ts}$) between target performance and CFD results were 5.3%, 3.7%, and 10.6%, respectively. This error can be improved by optimizing the blade. In this study, the parameters applied in the optimization process for the blade geometry were revealed. They are the blade exit angle related to the deviation angle, blade maximum thickness-true chord ratio, and incidence angle. Using these, the optimization process for blade geometry was shown in detail. Through the blade optimization process, the error rates of power ($\dot{W}$), efficiency (${\eta }_{ts}$), and pressure ratio ($P{R}_{ts}$) between target performance and CFD results were improved to 5.0%, 4.8%, and 1.8%, respectively. The results of this study are expected to be useful in the development of design technologies in the field of radial outflow turbines for CO₂ power cycles.