## 5 The One-Dimensional Analytical Model
### The Standard One-Dimensional Model  
To simplify the one-dimensional ejector model, several common assumptions are made as shown below:  
1. The flow is one-dimensional and steady all over the ejector. 2. The fluid is an ideal gas with constant specific heats (air is used in this work as the fluid). 3. Friction at the walls is negligible. 4. The kinetic energy at the inlet of primary flow, inlet of secondary flow and outlet of the ejector is negligible. 5. The flow at the outlet of the primary nozzle is supersonic, i.e., the flow through the primary nozzle is choked. This assumption is merely based on the practical point of view, i.e., an ejector is usually required to operate at the critical conditions at which both primary and secondary flow are choked. 6. The ejector is adiabatic, rigid (no boundary work involved), and impermeable (no flow entering or leaving the ejector besides flow through the inlets and outlet of the ejector). 7. The isentropic efficiency of the primary nozzle and the subsonic diffuser (\(\eta_{N}\) and \(\eta_{D}\)) is known beforehand. In this work, the values are assumed to be constant with values of 0.95 and 0.8 for the nozzle and diffuser, respectively. These values are based on the typical efficiency of the devices. However, it should be kept in mind that actually, these fixed values of efficiency cannot be applied in all operating conditions of an ejector, and they will be one of the sources of error in the model. Therefore, a study to estimate the efficiencies of the nozzle and especially, the diffuser should be conducted  
Figure 9: The Mach number contour plot for M \(\geq\) 1 showing the periodically constricted hypothetical annular duct (turbulence model: transition SST; \(P_{0}\) = 618 kPa). in the future. By doing so, their efficiencies can be estimated more accurately by just knowing the geometry and operating conditions of the ejector. * For an ejector that operates in the critical mode, a normal shock is assumed to happen at the exit of the mixing section (station 3). Therefore, the Mach number at that location, \(M_{3}\), will be an input and have a value of one. For an ejector that operates in the sub-critical mode, the outlet static pressure, \(P_{4}\), will be an input while the \(M_{3}\) will be an output. However, in this work, the focus will be on the critical mode operation and, therefore, \(M_{3}=1\) will act as an input and \(P_{4}\) as an output. It should be noted that this assumption (\(M_{3}=1\)) has been found to be inaccurate at certain operating conditions. Yet, it is simple to apply and is thought to be a good initial approximation. In fact, the value of \(M_{3}\) can either be greater or less than one, depending on where the shock starts. This is a challenge that remains to be solved. The main targets of this model are to obtain the \(\dot{m}_{s}\), \(\dot{m}_{p}\), and \(P_{4}\). The given/known parameters are as follows: 1. The primary and secondary stagnation conditions (i.e., properties at station 0 and 2 in Figure 1). 2. Ejector geometry:
* Primary nozzle: Throat diameter, \(D_{th}\), and outlet diameter, \(D_{1}\). * Converging mixing section: Inlet diameter, \(D_{CV,in}\). * Constant area mixing section (ejector throat): Diameter, \(D_{3}\). 3. Either the \(M_{3}=1\) (for critical mode operation) or the \(P_{4}\) (for sub-critical mode operation). However, the current work will focus on the critical mode operation of an ejector and, therefore, \(M_{3}=1\) will act as an input and \(P_{4}\) will be an output (assumption no. 8). 4. The values of \(\eta_{N}\) and \(\eta_{D}\) are 0.95 and 0.8, respectively (assumption no. 7). The governing equations (continuity, momentum, and energy balance equations), as well as equation of state of an ideal gas, will be applied whenever needed to construct the model. There are five control volumes required to develop the model and these control volumes will be shown in Figures 10-12.