# A Method of Two-Stage Pressure Control Based on Multistage Orifices
## 2 Theoretical Analysis  
Can high- and low-pressure control be realized at a fixed pressure point on the basis of multistage orifices? This section focuses on the theoretical feasibility of pressure control based on multistage orifices in series. The relationship between flow rate and pressure drop through a short orifice is shown in Equation (1) [11; 12; 26; 27; 28]. \[Q=C_{d}A\sqrt{\frac{2\Delta p}{\rho}}=kd^{2}\sqrt{\Delta p} \tag{1}\]  
where \(Q\) is volume flow rate, \(C_{d}\) is the discharge coefficient, \(A\) is the area of orifice flow section, \(\rho\) is fluid density, \(\Delta p\) is pressure drop across the short orifice, \(d\) is the orifice diameter, and \(k\) is a coefficient defined as  
\[k=\frac{\pi}{4}C_{d}\sqrt{\frac{2}{\rho}}.\]  
All losses due to friction are considered in discharge coefficient \(C_{d}\), which can be regarded as a constant when the flow through the orifice is in the turbulent regime [11; 12; 29]. [PERSON] proposed the hydroelectric analogy method to analyze a hydraulic resistance network [27]. Referring to the definition of nonlinear resistance in electronics, the concept of static liquid resistance \(R\) of the orifice was introduced [4; 30]. The definition is as in Equation (2). \[R=\frac{\Delta p}{Q} \tag{2}\]  
On the basis of Equations (1) and (2), \(R\) can be expressed as follows:  
\[R=\frac{\sqrt{\Delta p}}{kd^{2}}=\frac{Q}{k^{2}d^{4}} \tag{3}\]  
Equation (3) shows that the larger the diameter of the orifice is, the less the static liquid resistance. The flow through orifices in series is equal, that is,  
\[Q_{1}=Q_{2}=\cdots=Q_{n}=Q \tag{4}\]  
where terms \(Q_{1}\), \(Q_{2}\),..., \(Q_{n}\) represent the volume flow rate of each orifice, respectively, and suffix letter n indicates the number of orifices in series. If the effect of pipe resistance is ignored, the total pressure drop of orifices in series can be expressed as follows:  
\[\Delta p_{z}=\Delta p_{1}+\Delta p_{2}+\cdots+\Delta p_{n} \tag{5}\]  
where \(\Delta p_{z}\) represents the total pressure drop of series orifice circuit and \(\Delta p_{1}\), \(\Delta p_{2}\), \(\cdots\), \(\Delta p_{n}\) represent the pressure drop of each orifice. Combined with Equations (2)-(5), the total pressure drop of orifices in series can also be expressed as follows:  
\[\Delta p_{z}=\frac{Q^{2}}{k^{2}}\left(\frac{1}{d_{1}^{4}}+\frac{1}{d_{2}^{4} }+\cdots+\frac{1}{d_{n}^{4}}\right) \tag{6}\]  
where \(d_{1}\), \(d_{2}\), \(\cdots\), \(d_{n}\) represent the diameter of each orifice. According to Equation (6), if two orifices are connected in series, the pressure drop of each orifice is shown as Equations (7) and (8). If three orifices are connected in series, thepressure drops of each orifice are given by Equations (9)-(11) respectively. \[\Delta p_{2-1} =\frac{1}{1+\left(\frac{d_{1}}{d_{2}}\right)^{4}}\Delta p_{2-z} \tag{7}\] \[\Delta p_{2-2} =\frac{1}{1+\left(\frac{d_{2}}{d_{1}}\right)^{4}}\Delta p_{2-z}\] (8) \[\Delta p_{3-1} =\frac{1}{1+\left(\frac{d_{1}}{d_{2}}\right)^{4}+\left(\frac{d_{ 1}}{d_{3}}\right)^{4}}\Delta p_{3-z}\] (9) \[k=\frac{\pi}{4}C_{d}\sqrt{\frac{2}{\rho}}.\]0 (10) \[k=\frac{\pi}{4}C_{d}\sqrt{\frac{2}{\rho}}.\]1  
where \(\Delta p_{2-1}\) and \(\Delta p_{2-2}\) represent the pressure drop of each orifice when two orifices are connected in series, and \(\Delta p_{3-1}\), \(\Delta p_{3-2}\), \(\Delta p_{3-3}\) represent the pressure drop of each orifice when three orifices are connected in series; \(\Delta p_{2-z}\) and \(\Delta p_{3-z}\) represent the total pressure drop value in the two cases of series connection, respectively. On the basis of Formulas (7)-(11), some conclusions can be drawn:  
1. If the diameters of orifices connected in series are equal, the pressure-drop value of each orifice is also equal. 2. If the diameter difference between the orifices in series is large and satisfies prerequisites \(d_{1}\geq 2d_{2}\), \(d_{2}=d_{3}\), the pressure drop of the orifice with the largest diameter is very small. There is a difference, however, between the two cases.