# Analysis of the Energy Efficiency of a Pneumatic Booster Regulator with Energy Recovery
## 3 Energy Efficiency Evaluation Method for Booster Regulator with Energy Recovery (VBA-R)
### Pressure Response of VBA-R
#### 3.3.5 Pressure Response in Tank  
A booster regulator is always used with a tank to restrain the pressure fluctuations. The flow rate output of a booster system depends on the pressure in the tank, which is also affected by the boost ratio of the booster regulator. Whenever the system is working, air flows out the tank to drive the load, but the tank is charged by the booster regulator at the same time. According to Equation (7), the output flow rate rises with increasing pressure in the tank, then the pressure in the tank decreases. Meanwhile, the output flow rate from the booster regulator dynamically decreases or increases with the changing pressure in the tank. Thus, the equilibrium pressure develops in the tank. This is a simple self-balance system, and if the air consumption grows with the downstream load, the pressure in the tank will dynamically adjust and will tend to balance after fluctuating for a period of time. The charge and discharge model of the tank is simplified, as shown in Figure 3. The inlet is a check valve; its opening depends on the pressure difference between the booster regulator and tank, and the flow rate changesdynamically as it opens. The outlet is a throttle valve, and we simulated different loads by adjusting its opening.  
The tank was regarded as an isothermal chamber (ITC) because the wall area was large enough to exchange heat sufficiently with the atmosphere. Then, the gas state equation could be simplified, as the change of temperature could be neglected. Thus, the pressure response in the tank could be obtained as Equation (15) by differentiating the gas state equation:  
\[\frac{dp_{t}}{dt}=\frac{1}{V_{t}}G^{\prime}R\theta_{a} \tag{15}\]  
where \(G^{\prime}\) is the net flow rate of the tank, which is affected by the booster regulator, tank and downstream load. Equation (15) also indicates that the pressure differential is proportional to the net flow rate for an ITC.  
The check valve (AKH10-00, SMC, Tokyo, Japan) was used in the experiment, and the air flowing into the tank depended on its flow-rate characteristics. According to the supplier catalogue, the cracking pressure of the valve is 5 kPa, its sonic conductance is 4.8 \(\times\) 10\({}^{-8}\) (m\({}^{3}\)/(s Pa)), and its critical pressure ratio is 0.5. We assume that the check valve will fully open once the pressure reaches the cracking pressure. The flow rate characteristics changes with the pressure difference and can then be expressed as:  
\[C_{ch}=\left\{\begin{array}{ll}0&(p_{br}-p_{t}<5\text{kPa})\\ 4.8\times 10^{-8}&(p_{br}-p_{t}\geq 5\text{kPa})\end{array}\right.\]  
Although the critical pressure ratio is also affected by the pressure difference [23], it is very small, so we set it as constant in the simulation. By substituting these parameters into Equation (7), we can obtain the flow rate into or out of the tank. Therefore, the net flow rate can be calculated by:  
\[G^{\prime}=G_{ch}-G_{th} \tag{16}\]  
When we calculated \(G_{th}\), the downstream pressure was set as constant (atmosphere) because we used a throttle valve to simulate the load. However, in practical application, it should be set by considering the pressure of the load. By analysing Equations (15) and (16), we recognised that the pressure in the tank would fluctuate with variations of the net flow rate.