**1. Introduction**

Multistage radial turbines—usually referring to the multistage radial turbo expanders—are the key component of power generation systems in compressed air energy storage (CAES) and are also employed in the recycling of waste heat, residual pressure, and gas in the petrochemical industry. A schematic diagram of the conventional structure of a multistage radial turbine in CAES systems is shown in Figure 1. It consists of several single-stage radial turbines in separate casings, connected in series. In addition, heat exchangers are connected between stages to preheat the compressed air. Depending on the inlet pressure of the multistage radial turbine, which is generally from 3 to 10 MPa, the number of turbine stages is typically between three and five to ensure that the pressure ratio of each turbine stage falls in the range of two to five. The specific value of the pressure ratio of each turbine stage is related to the turbine inlet temperature. When cold energy production is not required, it is often necessary to meet the requirements of the turbine outlet temperature close to atmospheric temperature. Multistage radial turbines can realize the e fficient energy release of a large expansion ratio from the high pressure of compressed air storage to the atmospheric pressure, and have the

advantages of high efficiency, compact structure, and large power capacity. Thus, they are widely used in CAES systems [1,2].

In recent years, CAES technology has received increasing attention as one of the most promising solutions to the problems of intermittency and lack of control in renewable energy generation [3,4]. When multistage radial turbines in CAES systems are integrated with renewable energy generation, the fluctuating power output and inlet pressure make them able to operate under variable working conditions. Guide vane control is a mature and efficient mass flow regulation method that is widely used in turbomachinery. simulation studies have demonstrated the superiority of guide vane control for multistage radial turbines in CAES systems [5–7]. The challenge of multistage radial turbine design thus becomes to maintain high efficiency over a broader range of variable operating conditions involving guide vane opening changes.

**Figure 1.** Schematic diagram of a multistage radial turbine in a compressed air energy storage (CAES) system.

As shown in Figure 1, the intermediate turbine duct of the multistage radial turbine is usually a circular pipe in an inter-stage heat exchanger. The pressure change of the internal flow is small, and the flow characteristic is simple. Thus, understanding the internal flow pattern at each turbine stage is more important than that in the intermediate turbine duct between turbine stages. It is the primary factor affecting the performance of multistage radial turbines. Therefore, the design optimization of a single radial turbine stage is still the focus for the design of multistage radial turbines. Computational fluid dynamics (CFD) is the currently preferred method of turbine design optimization, as it enables accurate internal flow analysis to guide the detailed turbine design [8–11]. However, prior to its application, a reasonable one-dimensional preliminary design is the necessary first step in radial turbine design [12–14]. Having such a preliminary design is particularly important for the overall performance analysis of multistage radial turbines.

The loading-to-flow diagram first proposed by Chen and Baines is a classical preliminary design method which has several advantages [15,16]. It relates to the operating conditions of the radial turbine. More importantly, it allows for the creation of a contour map of the expected turbine efficiency, with the loading coefficient and flow coefficient as variables, based on a large amount of data from different radial turbine tests (Figure 2). Thus, it has been widely used for the preliminary design of radial turbines [17–19]. However, the loading-to-flow diagram was developed to achieve an efficient preliminary design of a radial turbine under a single operating condition with fixed geometry. It cannot guarantee an optimal radial turbine design for variable operating conditions, especially with guide vane opening changes. Thus, it is necessary to update the current preliminary design method to meet the needs of variable operating conditions. So far, there has been little public research on this issue apart from the work of Lauriau et al. [20]. They provide some theoretical bases for the preliminary design of variable guide vane geometry-based radial turbines, taking into consideration the need for multi-point specifications. They also show how the loading-to-flow map can be modified for different optimal target regions.

**Figure 2.** The loading-to-flow diagram [13].

By considering the off-design performance optimization, this paper updates the loading-to-flow diagram method for the preliminary design of radial turbines to accommodate variable operating conditions. The influence of the design value of guide vane outlet flow angle on the rotor loss characteristics was investigated in the continuity of the work presented by Lauriau et al. [20]. Subsequently, aiming at the preliminary design of multistage radial turbines in CAES systems, the optimal design of the guide vane outlet flow angle is discussed from the perspective of the matching of variable operating conditions with rotor loss characteristics. As far as the authors are aware, no similar studies have been performed.

### **2. Preliminary Design Method Based on the Loading-to-Flow Diagram**

According to the definition of the loading-to-flow diagram [17], the loading and flow coefficient can be explained by using the velocity triangles as shown in Figure 3, where the absolute velocity *c* is a vector addition of the circumferential velocity *u* and the relative velocity *w* in the direction of the blade (as a formula: →*c* = →*u* + →*w*). The absolute velocity *c* can be split into a circumferential component *cu* and a meridian component *cm*. According to Chen and Baines [16], the meridional component of flow velocity at the rotor inlet and outlet can be considered approximately equal. Thus, from Equations (1) and (2), the loading coefficient (ψ) and flow coefficient (φ) can be expressed as a function of the guide vane outlet flow angle α4 and the relative flow angle β4 at rotor inlet, respectively:

$$\psi = \frac{c\_{\mathfrak{u}4}}{\mathfrak{u}\_4} = \frac{\tan(\alpha\_4)}{\tan(\alpha\_4) - \tan(\beta\_4)},\tag{1}$$

$$\phi = \frac{c\_{m6}}{u\_4} \approx \frac{c\_{m4}}{u\_4} = \frac{1}{\tan(\alpha\_4) - \tan(\beta\_4)},\tag{2}$$

where subscripts 4 and 6 denote the rotor inlet and outlet, respectively.

**Figure 3.** The velocity triangles of the radial turbine rotor inlet and outlet.

The existing literature shows that the optimal values for β4 are in the range of −40◦ to −20◦ [21,22], and the recommended values for α4 are in the range of 60◦ to 80◦ [14]. Thus, by substituting the recommended values of α4 and β4 into Equations (1) and (2), the optimum loading-to-flow coefficient range with high expected turbine efficiency could be obtained for the radial turbine preliminary design.

On the other hand, rotor loss is a major factor affecting turbine efficiency, and is directly related to the flow velocity of the fluid in the rotor [23,24]. According to Lauriau et al. [20], the rotor loss can be minimized by reducing the mean velocity of the fluid relative to the rotor passage, which can be expressed as a function of α4, φ, and the mean blade angle at rotor outlet (β<sup>6</sup>*m*), as shown in Equation (3). Furthermore, the optimal value of the flow coefficient corresponding to the minimum rotor loss can be expressed by Equation (4).

$$\text{Minimize:} \overline{\mathcal{W}}\_r^2 = \frac{w\_4^2 + w\_6^2}{2u\_4^2} = \frac{1}{2}\phi^2(\frac{1}{\cos^2(\alpha\_4)} + \frac{1}{\cos^2(\beta\_{6m})}) + \frac{1}{2} - \phi\tan(\alpha\_4),\tag{3}$$

$$\phi\_{opt,r} = \frac{\tan(\alpha\_4)}{\left(\frac{1}{\cos^2(\alpha\_4)} + \frac{1}{\cos^2(\beta\_{\ell m,opt})}\right)},\tag{4}$$

where *Wr* denotes the ratio of the mean velocity of the fluid relative to the rotor passage with respect to the circumferential velocity of the rotor, β<sup>6</sup>*<sup>m</sup>*,*op<sup>t</sup>* denotes the recommended values for the mean blade angle at rotor outlet in the range of −60◦ to −45◦ [16,25], and *w*4 and *w*6 denote the relative velocity of air flow at the rotor inlet and outlet, respectively.

Figure 4 depicts the variation of the flow coefficient corresponding to the minimum total loss (corresponding to the optimal values for β4) and minimum rotor loss (corresponding to the optimal values for β<sup>6</sup>*<sup>m</sup>*,*op<sup>t</sup>*) of the radial turbine as a function of the guide vane outlet flow angle. The former is generally used as the design flow coefficient. It can be seen that the two tended to coincide above a relatively large guide vane outlet angle range (<sup>α</sup>4 ≥ 80◦). However, the design flow coefficient significantly increased with the decrease of the guide vane outlet flow angle, but the change of the flow coefficient corresponding to minimum rotor loss was relatively small. The difference between the two gradually increased with the decrease of the guide vane outlet flow angle. Experimental studies have indicated that changes in rotor losses are the most critical factors affecting turbine efficiency under variable operating conditions including pressure ratio change and guide vane opening change [26,27]. It can be inferred that the deviation between the flow coefficient corresponding to minimum rotor loss and the design flow coefficient caused by the different guide vane outlet flow angles will result in different turbine loss characteristics. Therefore, the value of the guide vane outlet flow angle

becomes the key to the preliminary design that aims at optimizing the off-design performance of the radial turbine.

**Figure 4.** Effect of guide vane outlet flow angle on the optimal flow coefficient value.

### **3. Analysis of Rotor Loss Characteristics**

In this section, the relationship between the design value of guide vane outlet flow angle (<sup>α</sup>4,*d*) and the rotor loss of the radial turbine is investigated for two typical operating conditions (i.e., variation in pressure ratio and variation in guide vane opening change from the designed value). The ratio of the mean velocity of the fluid relative to the rotor passage with respect to the circumferential velocity of the rotor (*Wr*) was determined to infer rotor loss in the early phase of preliminary design without the need of detailed turbine parameters. Since the rotor loss plays a key role in the off-design efficiency of radial turbines, we assumed that *Wr* is a reasonable indictor to qualitatively estimate the rotor loss and turbine efficiency in the preliminary design phase.
