# A New Windage Loss Model for S-CO\({}_{2}\) Turbomachinery Design
## 1 Introduction  
Supercritical CO\({}_{2}\) (S-CO\({}_{2}\)) power cycle technology has received interest from a wide range of energy industries because of its identified advantages. One of the advantages is that the S-CO\({}_{2}\) Brayton cycle can achieve competitive efficiency (\(\sim\)50%) even with moderate temperature heat sources (450-750 \({}^{\circ}\)C) [1]. High efficiency can be achieved by compressing CO\({}_{2}\) near the critical point (30.98 \({}^{\circ}\)C, 7377 kPa) because the high density of CO\({}_{2}\) near the critical point makes compression work small and maximizes heat recuperation [2]. The system also has compact components and a simple configuration due to the high average density and having no phase change processes in the cycle. These advantages make the S-CO\({}_{2}\) power cycle suitable for distributed power generation [3]. In order to successfully commercialize the S-CO\({}_{2}\) power conversion system, the design of components involves optimization. However, the strong real gas effects of S-CO\({}_{2}\) near the critical point make the design of an S-CO\({}_{2}\) cycle component challenging. Turbomachinery is the key component for expansion and compression processes. It has a significant impact on the efficiency of an S-CO\({}_{2}\) power system. Therefore, turbomachinery design should be able to predict the performance of a real component with small uncertainty to have minimal impact on the design of an S-CO\({}_{2}\) power cycle. Generally, utilizing appropriate loss models increases the prediction capability of turbomachinery design since losses in turbomachinery affect geometry, operation point, and the associated design of a motor or generator. Good loss models for the S-CO\({}_{2}\) turbomachinery design should be able to predict the losses even under the strong real gas effects of S-CO\({}_{2}\). For this reason, many studies have focused on developing loss models of S-CO\({}_{2}\) turbomachinery [4; 5; 6]. The turbomachinery losses can be classified into two types: internal and external losses. The internal loss mainly affects the design of the primary flow path, while the external loss determines losses occurring in the secondary flow path, which typically translates to additional work loss in turbomachinery. Thus, auxiliary cooling systems for turbomachinery are mainly affected by external losses. To minimize the size of the auxiliary system, the external loss should first be estimated with good precision. Windage loss is one of the most significant external losses in S-CO\({}_{2}\) turbomachinery. It is by far the dominant loss for the S-CO\({}_{2}\) conditions which contrasts with turbomachinery operating with air. This is due to S-CO\({}_{2}\) having a higher density and S-CO\({}_{2}\) turbomachinery having a higher rotational speed for the same capacity. To demonstrate how S-CO\({}_{2}\) conditions increase windage loss in turbomachinery, the windage loss model proposed by [PERSON] is used to compare the windage losses between the S-CO\({}_{2}\) condition and air condition, which is shown in Figure 1.  
The S-CO\({}_{2}\) recompression test conducted by Sandia National Lab (SNL) [7] provides a useful illustration of how windage loss is handled in practice. A Turbo-Alternator-Compressor (TAC) used in this test features a single shaft that connects compressor and turbine impellers. SNL constructed a turbomachine with rotor cavity depressurization to reduce its windage loss by utilizing a scavenging pump. However, the minimization of an auxiliary system such as a scavenging pump or a cooling system is necessary to utilize S-CO\({}_{2}\) for the distributed power generation because the auxiliary systems can overwhelm the advantage of an S-CO\({}_{2}\) power system being compact. Therefore, accurately predicting windage loss not only reduces the prediction uncertainty of cycle performances, but also impacts the economy of the system by reducing uncertainty in the design of auxiliary systems.