2.1 Detection Steps of Differential Pressure Method
The airtightness detection method of the differential pressure method mainly refers to the air pressure comparison between the master cavity and tested cavity through a specific structure. If the air pressure of the master cavity and tested cavity is balanced, it indicates that the tested body is in a good airtight state. A pressure difference between the master cavity and tested cavity indicates that the tested cavity has leaked. A structural diagram of the differentialpressure detection method is shown in Figure 1.
In this method, the air compressor fills highpressure gas in air tank 1 (representing the master cavity without leakage) and air tank 2 (representing the tested cavity with leakage) simultaneously. If the detected workpiece does not leak, the differential pressure sensor maintains a balanced state, whereas if the detected workpiece leaks, the air pressure in the workpiece measurement chamber changes, causing the differential pressure sensor to lose balance.
Air valves 1 and 2 represent the onoff valves of the air tank and air compressor, respectively. Air valve 3 was connected to the two air tanks. Air valves 4 and 5 were used for the exhaust in the experiment, a flow sensor was used for calibration to detect leakage, and temperature and pressure sensors were used to detect the gas state inside the air tank. All the sensors and solenoid valves were controlled using an acquisition card and an upper computer.
The specific test steps are as follows:

(1)
Charging both chambers simultaneously: Open the air compressor and valves 1 and 2 to keep the pressure filled into the cavity consistent with the master pressure of tank 1, which was used for the master chamber, and tank 2, which was used for the test chamber.

(2)
Balancing the pressure of the two air tanks: close valves 1 and 2 and open valve 3 to balance the pressure in the master tank and tested tank.

(3)
Detection step: Read the change in the pressure difference at both ends of the differential pressure sensor, which is caused by the leakage of the tested tank.

(4)
Judgment: The read pressure difference is judged according to the preset value to determine whether the tightness of the tested tank is qualified.
For each charging and testing, the pressures of the two chambers should exhibit the following trend: First, a large amount of air is rapidly poured into the air tank during charging to increase the internal pressure until charging is stopped. In the balance and measurement stage, if the air tank is well sealed, the two chambers are affected by cooling the air tank and reducing the thermal movement of the internal air molecules, leading to a reduction in the pressure in the gas tank. If there is leakage in the gas tank, the total amount of air in the gas tank is reduced, and the pressure in the gas tank is reduced after some air is leaked. In the discharging stage, it can be regarded as a sharp increase in the air leakage and a rapid decrease in the pressure in the gas tank.
2.2 Charging Process and Measurement Process
During charging, the gas in the master chamber is regarded as ideal, and the pressure change in the master chamber is determined by the ideal gas state equation. After differentiating the ideal gas state equation, it can be obtained that:
$$\frac{{\text{d}P_{{\text{m}}} }}{{\text{d}t}} = \frac{{P_{{\text{m}}} }}{{T_{{\text{m}}} }}\frac{{\text{d}T_{{\text{m}}} }}{{\text{d}t}} + \frac{{RT_{{\text{m}}} }}{{V_{{\text{m}}} }}G_{{{\text{im}}}} ,$$
(1)
where \(P_{{\text{m}}}\) represents the pressure in the master tank, \(T_{{\text{m}}}\) represents the temperature in the master tank,\({ }V_{{\text{m}}}\) represents the volume of the master tank, and \(G_{{{\text{im}}}}\) represents the charging flow rate in the master tank.
For the tested chamber, Eq. (2) can be obtained according to the energy conservation and ideal gas state equation. The difference is that there is also mass exchange \(G_{{{\text{ew}}}}\) caused by the leakage of the chamber during charging.
$$\frac{{\text{d}P_{{\text{w}}} }}{{\text{d}t}} = \frac{{P_{{\text{w}}} }}{{T_{{\text{w}}} }}\frac{{\text{d}T_{{\text{w}}} }}{{\text{d}t}} + \frac{{RT_{{\text{w}}} }}{{V_{{\text{w}}} }}\left( {G_{{{\text{iw}}}}  G_{{{\text{ew}}}} } \right),$$
(2)
where \(P_{{\text{w}}}\) represents the pressure in the tested tank, \(T_{{\text{w}}}\) represents the temperature in the tested tank,\({ }V_{{\text{w}}}\) represents the volume of the tank, and \(G_{{{\text{iw}}}}\) represents the charging flow rate in the tested tank.
Considering that the leakage mass flow is less than the charging mass flow, the influence of leakage can be ignored in the charging stage. Thus, Eqs. (1) and (2) can be simplified into the following equation:
$$\frac{{\text{d}P_{{\text{w}}} }}{{\text{d}t}} = \frac{{P_{{\text{w}}} }}{{T_{{\text{w}}} }}\frac{{\text{d}T_{{\text{w}}} }}{{\text{d}t}} + \frac{{RT_{{\text{w}}} }}{{V_{{\text{w}}} }}G_{{{\text{iw}}}} .$$
(3)
During the measurement, the air source interrupted the charging to the tested chamber and master chamber, and the two chambers were isolated from each other. At this stage, the master cavity only undergoes heat exchange with the environment, whereas the tested cavity undergoes heat exchange with the environment, and the quality is reduced owing to leakage.
For the master cavity, taking \(G_{{{\text{im}}}} = 0\) in charging Eq. (1), the following equation of state for the measurement process can be obtained, as shown in Eq. (4).
$$\frac{{\text{d}P_{{\text{m}}} }}{{\text{d}t}} = \frac{{P_{{\text{m}}} }}{{T_{{\text{m}}} }}\frac{{{\rm{d}}T_{{\text{m}}} }}{{\text{d}t}}.$$
(4)
For the tested cavity, taking \(G_{{{\text{iw}}}} = 0\) in the charging Eq. (2), the following equation of state for the measurement process can be obtained, as shown in Eq. (5).
$$\frac{{\text{d}P_{{\text{w}}} }}{{\text{d}t}} = \frac{{P_{{\text{w}}} }}{{T_{{\text{w}}} }}\frac{{\text{d}T_{{\text{w}}} }}{{\text{d}t}}  \frac{{RT_{{\text{w}}} }}{{V_{{\text{w}}} }}G_{{{\text{ew}}}} .$$
(5)
According to Eqs. (4) and (5), the relationship between the differential pressure and leakage can be obtained as shown in Eq. (6).
$$\frac{{\text{d}\Delta P}}{{\text{d}t}} = \frac{{P_{{\text{w}}} }}{{T_{{\text{w}}} }}\frac{{\text{d}T_{{\text{w}}} }}{{\text{d}t}}  \frac{{RT_{{\text{w}}} }}{{V_{{\text{w}}} }}G_{{{\text{ew}}}}  \frac{{P_{{\text{m}}} }}{{T_{{\text{m}}} }}\frac{{\text{d}T_{{\text{m}}} }}{{\text{d}t}}.$$
(6)
Theoretically, when the equilibrium time is sufficiently long, the differential of \(T_{{\text{w}}}\) and \(T_{{\text{m}}}\) is equal to zero, and Eq. (6) can be simplified as follows:
$$\frac{{\text{d}\Delta P}}{{\text{d}t}} =  \frac{{RT_{{\text{w}}} }}{{V_{{\text{w}}} }}G_{{{\text{ew}}}} .$$
(7)
That is,
$$G_{{{\text{ew}}}} =  \frac{{V_{{\text{w}}} \text{d}\Delta P}}{{RT_{{\text{w}}} \text{d}t}}.$$
(8)
2.3 Pressure Difference Substitute Equation
In the actual measurement process, because the pressure measurement inevitably produces measurement fluctuations, resulting in large fluctuations in the differential of the pressure, affecting the measurement results and prolonging the measurement time, this section analyzes the substitute equation of the pressure difference from the perspective of the mechanism and calculates the leakage through the substitute equation, which can obtain the results quickly and accurately. The differential pressure between the tested chamber and master chamber satisfies Eq. (9).
$$\Delta P = P_{{\text{w}}}  P_{{\text{m}}} .$$
(9)
To solve the expression of \(\Delta P\), it can be completed in two steps: first, the differential pressure of the leakfree tank is analyzed, and then the leakage differential pressure is superimposed. The following expression is adopted:
$$\Delta P = \Delta P_{T} + \Delta P_{L} .$$
(10)
When the tank is qualified, \(G_{{{\text{ew}}}} = 0\). It is then introduced in Eqs. (9) and (10), as shown in Eq. (11).
$$\frac{{\text{d}P_{{\text{w}}} }}{{\text{d}t}} = \frac{{Rh_{{\text{w}}} S_{{\text{w}}} \left( {T_{a}  T_{{\text{w}}} } \right)}}{{C_{v} V_{{\text{w}}} }}.$$
(11)
Ideal gas equation of state \(P_{{\text{w}}} = \rho_{{\text{w}}} RT_{{\text{w}}}\). After replacing \(T_{{\text{w}}}\), we introduce it into Eq. (11) and sort it as follows:
$$\frac{{\text{d}P_{{\text{w}}} }}{{\text{d}t}} + \frac{{h_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} \rho_{{\text{w}}} }}P_{{\text{w}}}  \frac{{Rh_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} }}T_{a} = 0,$$
(12)
where \(\rho_{{\text{w}}}\) is the density of the compressed air in the tested chamber. Eq. (12) is the nonhomogeneous firstorder differential equation of the tested cavity pressure, and the solution to the equation is
$$P_{{\text{w}}} = \rho_{{\text{w}}} R\left( {T_{0}  T_{a} } \right)\text{exp}\left( {  \frac{{h_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} \rho_{{\text{w}}} }}t} \right) + \rho_{{\text{w}}} RT_{a} .$$
(13)
After replacing each constant parameter with a parameter such as Eq. (14), it obtains \(P_{{\text{w}}} = Ae^{  Bt} + C\), where the boundary condition is \(P_{{\text{w}}} \left( 0 \right) = \rho_{{\text{w}}} RT_{0}\).
$$A = \rho_{{\text{w}}} R\left( {T_{0}  T_{a} } \right),\;B = \frac{{h_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} \rho_{{\text{w}}} }},\;C = \rho_{{\text{w}}} RT_{a} ,$$
(14)
where \(T_{0}\) is the temperature of the tested chamber when valve 2 is closed, which is the starting temperature of the measurement; \({ }T_{a}\) is the ambient temperature, which is the final temperature of the tested cavity. Subsequently, the two temperatures satisfy the following relationship:
$$P_{s} = \rho_{{\text{w}}} RT_{0} ,$$
(15)
$$P_{\infty } = \rho_{{\text{w}}} RT_{a} .$$
(16)
The pressure change in the test chamber is the difference between the current pressure value and initial pressure value, and the initial pressure is the value \(P_{s}\) at the end of the inflation balance. By substituting Eq. (13) into Eq. (9), the following expression can be obtained:
$$\Delta P_{T} = \rho_{{\text{w}}} R\left( {T_{0}  T_{a} } \right)\text{exp}\left( {  \frac{{h_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} \rho_{{\text{w}}} }}t} \right) + \rho_{{\text{w}}} RT_{a}  P_{s} .$$
(17)
The initial pressure and initial temperature satisfy the ideal gas equation of state
$$\Delta P_{T} = \rho_{{\text{w}}} R\left( {T_{0}  T_{a} } \right)\text{exp}\left( {  \frac{{h_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} \rho_{{\text{w}}} }}t} \right)  \rho_{{\text{w}}} R\left( {T_{0}  T_{a} } \right).$$
(18)
After replacing the temperature value with Eqs. (15) and (16), we can deduce the following:
$$\Delta P_{T} = \left( {P_{s}  P_{\infty } } \right)\text{exp}\left( {  \frac{{h_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} \rho_{{\text{w}}} }}t} \right)  \left( {P_{s}  P_{\infty } } \right).$$
(19)
Therefore, if the tank has no leakage, the differential pressure after sealing should be Eq. (20), which is an exponential function with a constant parameter.
$$\Delta P_{T} = \alpha e^{  \beta t}  \alpha ,$$
(20)
where,
$$\alpha = P_{s}  P_{\infty } = \rho_{{\text{w}}} R\left( {T_{0}  T_{a} } \right),$$
(21)
$$\beta = \frac{{h_{{\text{w}}} S_{{\text{w}}} }}{{C_{v} V_{{\text{w}}} \rho_{{\text{w}}} }}.$$
(22)
Consider that the leakage mass flow \(G_{{{\text{ew}}}}\) is very small and has a slight impact on the cavity pressure; it can be approximately regarded as a fixed value, and the change in differential pressure caused by leakage is linear and meets the following requirements:
$$\Delta P_{L} =  \frac{{R\theta_{{\text{w}}} G_{{{\text{ew}}}} }}{{V_{{\text{w}}} }}t.$$
(23)
By the superposition of Eqs. (20) and (23), the expression of the variation of the differential pressure with time under leakage can be obtained, as shown in Eq. (24), in which the linear parameter is the differential pressure component caused by leakage, and the exponential parameter is the differential pressure caused by temperature recovery.
$$\Delta P = \Delta P_{T} + \Delta P_{L} = \alpha e^{  \beta t}  \lambda t  \alpha ,$$
(24)
where, \(\lambda = \frac{{RT_{{\text{w}}} G_{{{\text{ew}}}} }}{{V_{{\text{w}}} }}\).
Thus, the parameters can be identified according to the collected pressure value for the leakage calculation to avoid the influence of a large pressure differential on the measurement results.