## 1 Introduction  
A steam ejector is a fluid phase-change device, which can utilize low-grade energy, such as industrial waste heat, to generate vacuum environments. The first steam ejector was designed by [PERSON] in 1939 [1]. Ever since it has been developed, the steam ejector has been widely applied in multiple industrial sectors, including refrigeration, chemistry, metallurgy, petroleum, and food processing [2; 3; 4; 5]. Steam ejectors can also be applied to a fire suppression system in place of water mist to provide greater displacement of oxygen from a fire plume due to the evaporation of water droplets. Based on nozzlesand water mist, numerous studies on different characteristics of the system such as pressure, injection rate, cone angles, and droplet dimensions were conducted for different working conditions and were performed both experimentally and numerically in recent years [6; 7; 8]. A steam eject utilizes high-pressure steam to pump low-pressure fluids. It has many advantages over other commonly used devices, such as no moving parts, less-required maintenance and low cost in manufacturing and energy consumption. Although the structure of a steam eject is relatively simple, the flow behavior within the enclosure can be extremely complex, involving phenomena such as transonic flow, shockwaves, phase change etc. The configuration of a typical steam ejector [9] along with the corresponding pressure changes and velocity distribution is shown in Figure 1. In general, a steam ejector consists of four main components: (i) primary nozzle, (ii) mixing chamber, (iii) throat, and (iv) subsonic diffuser. The steam ejector work cycle starts with the acceleration of highly pressurized primary fluid to supersonic after passing through the nozzle. Approaching the end of the nozzle, a relative low-pressure region is generated due to the high velocity, which entrains the secondary fluid stream from the evaporator into the mixing chamber. The velocity of the secondary fluid increases within the mixing chamber due to the energy and momentum exchange between the two fluid streams. Subsequently, the speed of the mixed flow is reduced to subsonic range, due to the normal shock wave occurring in the throat. The mixed flow is further compressed after exiting via the subsonic diffuser. This entire entrainment process is very complex, and analysis requires a deep understanding of the underlying physical fluid behaviors and the change in water droplets within the steam ejector. One of the key parameters to assess the performance of the steam ejector is the entrainment ratio or mass flow ratio [10], which is defined as:  
\[EntrainmentRatio=\frac{Mass\ flow\ rate\ of\ primary\ flow}{Mass\ flow\ rate\ of\ secondary\ flow} \tag{1}\]  
As a characteristic index of efficiency, a stream ejector with a higher entrainment ratio is usually preferable. In addition, the performance of the steam ejector can also be described by the characteristic curve (see Figure 1) and herein the critical back pressure plays an important role. According to the critical back pressure, the steam ejector entrainment ratio can be separated into three modes [11]: (i) critical mode (double-choking), (ii) subcritical mode (single-choking) and (iii) back-flow mode  
Figure 1: Configuration of a typical steam ejector and flow characteristic along the axis of the steam ejector. (malfunction) as indicated in Figure 2. In the double-choking mode, the entrainment ratio of the stream ejector remains as a constant with the increase of back pressure up to a critical point. Afterwards, the entrainment ratio dramatically decreases to zero, and the ejector loses its function. Previous works have been carried out using both experimental and numerical approaches to evaluate the effect of key parameters on the performance of a steam ejector. These studies include the optimization of key components such as the mixing chamber, nozzle, throat etc. as well as the assessment of the system under various operating conditions have also been performed. For instance, [PERSON] et al.