The separation nut is achieved by the internal mechanism driven by gunpowder gas, which needs to be completed within a few milliseconds. The separation mechanism is complex, and it is very important to accurately describe the separation behaviour of the internal mechanism for predicting the reliability of the separation function of the separation nut. However, the separation behaviour of the inner sleeve, piston and nut flap is difficult to evaluate experimentally. Therefore, the propellant combustion model of the separating nut and the separating model of each mechanism are established.
2.2.1. Combustion Model
The initiator requires carbon black-potassium nitrate (CPN) as the main explosive charge. This charge consists of numerous minute granules and a binder mixture. The granules have an uneven surface. To simplify the combustion model, the combustion equation is constructed using Equations (1) and (2) [
28].
where
χ,
λ and
μ are the shape characteristic quantities of the propellant,
ψ is the combustible relative mass of the propellant,
σ is the relative surface area of the combustible powder,
Z =
e/el represents the relative thickness,
e is the arc thickness at any time and
el is the starting arc thickness.
The 0-D combustion model based on Saint Robert’s law is used to calculate the pressure of the gas inside the chamber, which is essential for obtaining the actuating force. The relationship between the burn rate,
rb, arc thickness,
e, and pressure is as follows [
29]:
where
P is the pressure,
n is the burn rate exponent and
u is a constant determined by the chemical composition and initial temperature of the propellant.
In conclusion, the mass generation rate of the gas from the CPN particles is estimated as follows [
30]:
where
ηg is the mass fraction of gas in the combustion product of the propellant and
mp is the loading mass of the propellant.
The aforementioned combustion model can be used to calculate the pressure inside the chamber. Hence, the law of conservation of mass in a control volume of the combustion chamber is given by the following equation [
31]:
where
ρg is the density of the propellant gas and
V is the volume of the combustion chamber.
Equation (6) can be obtained from Equation (5):
where
is the rate of change of the volume of the combustion chamber, as given by the following equation:
where
ρs is the density of the CPN particles and
ρcp is the density of the condensed phase in the powder product.
The energy conservation relation of the internal gas can be expressed by Equation (8) [
32]:
where
ηp is the non-ideal gas correction factor,
cv and
cp are the specific heats of constant volume and pressure of the gas, respectively,
Tf is the explosion temperature at constant pressure,
Asle and
Apiston are the compression areas of the inner sleeve and piston, respectively,
vsle and
vpiston are the separation speeds of the inner sleeve and piston, respectively,
is the heat dissipation rate and
T is the temperature of the gas.
For the high-temperature and high-pressure gas in the separation nut, the ideal gas state equations are given by Equations (9) and (10) [
33].
Here,
αg is the covolume and
R is the gas constant.
Here, k is the specific heat ratio of the fuel gas.
Equations (6) and (8) are combined and the ideal gas state equations (Equations (9) and (10)) are used to derive the rate of change of the pressure.
A simplified model of heat exchange between the propellant products and the outside through the cavity wall is established according to [
34], and the heat dissipation rate can be estimated using Equation (12):
where
h is a constant convective heat transfer coefficient,
Tw is the temperature of the vessel wall,
σs is the Stefan–Boltzmann constant,
αw is the absorption rate of the vessel wall,
ξ is the net emissivity of the product and
Aw is the instantaneous surface area of the vessel side wall in contact with the product.
The combustion parameters applied to this combustion model are listed in
Table 1.
2.2.2. Motion Modelling of the Separation Mechanism
The separation motion equations of the main separation mechanisms in the separation stage are established according to the different separation states of the separation nut.
- (1)
Inner sleeve separation model
The force analysis of the inner sleeve is shown in
Figure 3, and the separation equation of the inner sleeve is derived from Equations (13)–(15):
where
is the shear force of the shear pin,
is the friction force of the sealing ring,
is the friction force between the inner sleeve and nut flap,
Fpre is the preload applied by the mounting bolt,
is the friction coefficient between the inner sleeve and nut flap and
is the separation time of the inner sleeve.
- (2)
Piston separation model
A force analysis of the piston separation is shown in
Figure 4. The separation equation of the piston in the axial direction is as follows:
where
is the mass of the piston,
is the acceleration of the piston,
is the axial pressure between the nut flap and the piston,
is the friction between the nut flap and the piston and
α is the support angle between the piston and the nut flap.
- (3)
Nut flap separation model
When the inner sleeve releases the constraint on the nut flap, the latter begins to separate. A stress analysis of the nut flap is shown in
Figure 5. The axial and radial separation equations of the nut flap are obtained using Equations (17) and (18).
where
and
are the axial and radial displacements of the nut flap, respectively,
is the radial pressure between the nut flap and the piston,
and
are the axial and radial pressures of the bolt, respectively,
and
are the axial and radial pressures of the end cap, respectively,
is the friction between the bolt and nut flap,
is the friction between the end cap and the nut flap,
γ is the half angle of the tooth profile of the nut flap,
δ is the support angle between the nut flap and the end cap,
is the mass of the nut flap and
is the separation time of the nut flap.
In this study, the simulation model of the separation nut is calculated and analysed by using the MATLAB/Simulink. Simulink is a visual simulation tool based on block diagram design environment. It provides a large number of simulation design blocks, such as input/output, mathematical calculation, integral/differential, signal processing and other modules, which can avoid a large number of writing programs. Meanwhile, differential equation solvers such as ODE45, ODE23 and ODE113 are provided in Simulink, which provides convenience for solving many differential equations in the separation simulation model. The initial values of the parameters for the motion model of the separation nut are listed in
Table 2.
The pressure in the cavity and the motion curve of each mechanism are shown in
Figure 6.
As shown in
Figure 6, the shear pin is sheared at 0.26 ms under the action of high pressure, and the inner sleeve is started. At 4.45 ms, the inner sleeve moves by 5 mm, releasing the radial restraint on the nut flap. At the same time, the nut flap and piston are started. At 5.18 ms, the radial displacement of the nut flap reaches 1 mm, and the restraint on the bolt is released. Under the action of the preload, the bolt begins to separate from the separation nuts to realise separation, and the inner sleeve, piston and nut flap move continuously. At 5.70 ms, the piston and nut flap move to the designed maximum stroke and stop moving. At 9.96 ms, the inner sleeve moves to the designed maximum stroke.
To verify the simulation results of the motion of the separation nut mechanism, a pressure sensor is used to test the pressure of the separation chamber. The output pressure of the electric squib is measured at 15 MPa (5 mL). The selected sensors are listed in
Table 3.
To simulate the axial gravity-free state of the separation device of the inter-stage cabin and reduce the interference of the external environment on the sensor, the test plates (600 mm × 60 mm × 10 mm) are lifted by four flexible elastic ropes. The connecting plate is connected to the test plates to simulate the separation state. The test system is shown in
Figure 7.
The separation process of the separation nut is evaluated by employing the separation test system of the separation nuts, and the key separation parameters, such as cavity pressure, are measured. The test curve is shown in
Figure 8.
To analyse the pressure change process more intuitively, the values of the pressure and time of the first peak and second peak, as shown in the curve of
Figure 6, are compared and presented in
Table 4. The relative errors of the key pressure parameters are 1.30%, 1.96%, 3.48% and 2.97%, which are all less than 5%. This shows that the simulation model can accurately predict the variation law of the pressure in the cavity during the action of the separation nut and accurately describe the motion law of the separation nut mechanism.