## 5 The One-Dimensional Analytical Model
### The Standard One-Dimensional Model
Figure 11: The dotted line is showing control volume C.  
Figure 10: The dashed (blue) line and dotted (red) lines are showing control volumes A and B, respectively. For control volume A shown in Figure 10, the mass and energy balance will be applied. The mass balance will result in Equation (5). \[\dot{m}_{4}=\dot{m}_{3}=\dot{m}_{p}+\dot{m}_{s}=(1+ER)\dot{m}_{p} \tag{5}\]  
where \(ER\) stands for entrainment ratio and defined as  
\[ER=\dot{m}_{s}/\dot{m}_{p} \tag{6}\]  
In order to obtain \(\dot{m}_{p}\), the compressible flow equation for a choked flow (assumption no. 5) is used as presented in Equation (7) below. \[\dot{m}_{p}=A_{th}\,P_{0}\sqrt{\frac{\gamma}{RT_{0}}}\bigg{[}\frac{2}{\gamma+ 1}\bigg{]}^{\frac{\gamma+1}{2(\gamma-1)}} \tag{7}\]  
where \(\gamma\) and \(R\) are the specific heat ratio and gas constant of air and have values of 1.4 and 287 J/kg-K, respectively. The \(A_{th}\) denotes the primary nozzle throat area. The energy balance applied to the control volume A in conjunction with assumptions no. 2 and 4 gives Equation (8). \[(T_{0}+\ ER\ T_{2})=(1+ER)T_{4} \tag{8}\]  
Up to now, there are five unknowns, i.e., \(ER\), \(\dot{m}_{p}\), \(\dot{m}_{s}\), \(\dot{m}_{4}\), and \(T_{4}\) and four equations which are Equations (5) to (8). Thus, another equation is needed to solve for these five unknowns. This equation will be obtained later via control volumes C and D.  
Next, control volume B, also shown in Figure 10, will be considered. Through this control volume, the velocity, pressure, and Mach number at the outlet of the primary nozzle, denoted by \(v_{1}\), \(P_{1}\), and \(M_{1}\), respectively, will be found. Applying energy balance to control volume B along with assumptions no. 2, 4, and 7, we will obtain Equation (9). \[v_{1}=\sqrt{2\,\eta_{N}\,c_{p}\,\ T_{0}\left[1-\bigg{(}\frac{P_{1}}{P_{0}} \bigg{)}^{\frac{\gamma-1}{\gamma}}\right]} \tag{9}\]  
As stated in assumption no. 7, the isentropic efficiency of the primary nozzle, \(\eta_{N}\), is given which is assumed to be 0.95. However, there are two unknowns (\(v_{1}\) and \(P_{1}\)) and only one equation, i.e., Equation (9). Thus, we need another equation. To obtain another equation, the area ratio relation of isentropic compressible flow will be used as follows. Mach number at the outlet of the primary nozzle (station 1), \(M_{1}\), can be calculated from (assuming there is no boundary layer separation between the throat and primary nozzle outlet):  
\[\frac{A_{1}}{A_{th}}=\frac{1}{M_{1}}\bigg{[}\frac{2}{\gamma+1}\bigg{(}1+ \frac{\gamma-1}{2}\ M_{1}^{2}\bigg{)}\bigg{]}^{\frac{\gamma+1}{2(\gamma-1)}} \tag{10}\]  
where \(A_{1}\) is the area at station 1 (the cross sectional area of the primary nozzle outlet). Hence, the pressure at the nozzle outlet, \(P_{1}\), can be obtained from  
Figure 12: The dotted (red) and dashed (blue) lines are showing control volumes D and E, respectively. Figure 11:0  
Now, there are three unknowns and three equations which are \(v_{1}\), \(P_{1}\), and \(M_{1}\) and Equations (9) to (11). Therefore, these parameters can be found. For control volume C (see Figure 11 above and Figure 13 below), momentum equation will be applied to obtain a useful equation. It should be noted that the area at station 1, \(A_{1}\), and station 2, \(A_{2}\), are not overlapping. Area at station 1 is the outlet area of the primary nozzle and station 2 is the annulus area of the inlet of the converging mixing section. Thus, the total area at the inlet plane of control volume C is \(A_{1}+A_{2}\) which is also referred to as the inlet area of the converging mixing section, \(A_{CV,in}\). The [PERSON] second law states that the sum of forces acting on an object should be equal to the time rate of change of momentum of that object. This statement transforms to the following equation:  
Figure 11:1  
where  
Figure 11:2  
In this equation, \(R_{x}\) and \(P_{CV,in,avg}\) are the \(x\)-direction of the reaction force and average static pressure on control volume C, respectively, done by the wall of the converging mixing section.