3.1. Velocity Attenuation Characteristic
(1) Initial velocity
The conclusion that the carbon dioxide in the blasting tube is supercritical before the energy release plate broke was obtained [
6]. Supercritical carbon dioxide is a material state with special fluid characteristics. Its gas phase and liquid phase tend to be similar in nature. It can also be said that supercritical carbon dioxide is a homogeneous phase fluid, which not only has the compressibility and outflow of gas, but also has the fluidity of liquid. In the initial stage of boiling liquid expanding vapor explosion (BLEVE), after the destruction of energy release tablets, supercritical carbon dioxide two-phase flow spews out rapidly from the air outlet, and the rapid release of fluid belonged to the sonic or supersonic flow. In the outlet area, carbon dioxide vaporized rapidly, and the time was very short, where
; it was the critical flow, and the mass flow rate of the fluid was [
33]
where
; it was subcritical flow, and the mass flow rate of the fluid was [
33]
where
Pa was atmospheric pressure, Pa;
Pg was the pressure inside the blasting tube, Pa;
was the quality flow rate of the gas phase component, kg/s;
was gas mass, kg;
t was time, s;
Y was the expansion factor of gas;
Cd was the gas leakage coefficient,
Cd = 1 when the crack was round,
Cd = 0.95 when the crack was triangular or diamond, and
Cd = 0.90 when the crack was rectangular.
A was the crack area, m
2;
was the adiabatic index of gas;
M was the molecular mass of the gas, kg/mol;
;
R was the general gas constant, 8.314 J/ (mol ∙K);
T was fluid temperature, K.
The initial average velocity of discharge
was [
34]:
where
denoted the density of gas in the air outlet (kg/m
3).
The density of the air outlet fluid was the density of carbon dioxide in the blasting tube. Through Equations (2) and (3), the initial release velocity of the jet in the four tests was 469.93 m/s, 338.87 m/s, 496.32 m/s, and 419.81 m/s.
In order to verify the correctness of the calculation process, we simplified the model and estimated the liquid carbon dioxide jet velocity based on the Bernoulli equation. Within a very short time after the rupture of the energy release sheet, high-pressure carbon dioxide after gasification flowed out of the release hole to form high-pressure gas jet at high speed. The schematic diagram of high-pressure carbon dioxide jet was shown in
Figure 3.
In the process of high-pressure carbon dioxide movement, the energy loss caused by blasting tube, directional shear fragment resistance, turbulence, eddy flow, gravitational field and other conditions were not considered, and no energy loss occurred during high-speed carbon dioxide motion at the moment of fracture of the directional shear sheet until the release hole. In this case, the Bernoulli equation of the ideal fluid was used to estimate the jet velocity
v1 of carbon dioxide fluid at the release hole, which was:
where
was the pressure of carbon dioxide inside the blasting tube at the moment before the explosion,
was the density of carbon dioxide inside the blasting tube at the moment before the explosion,
was the velocity of carbon dioxide inside the blasting tube at the moment before the explosion,
was the pressure of high-speed carbon dioxide fluid released from the air outlet,
was the density of high-speed carbon dioxide fluid released from the release hole, and
was the velocity of high-speed carbon dioxide fluid released from the air outlet.
At the breaking moment of the energy release plate, it was considered that the carbon dioxide of blasting was in a static state, and the pressure was the maximum pressure before the breaking of the energy release plate. When the high-speed carbon dioxide fluid started to release from the air outlet, the gas carbon dioxide just started to contact the external atmosphere, so the pressure was atmospheric pressure at this time, while the density of carbon dioxide was relatively stable at the moment of release. The density measured by four experiments of liquid carbon dioxide under laboratory conditions was 0.103 MPa. By substituting the above data into Equation (4), the instantaneous velocity of carbon dioxide released from the release hole was 410.78 m/s, 343.14 m/s, 428.76 m/s, and 390.27 m/s.
(2) Basic characteristics of the jet
When carbon dioxide jet exited from the orifice at
u0, it formed turbulent jet flow with the surrounding air, resulting in discontinuity surface with discontinuous velocity. As shown in
Figure 4, the jet flow was mainly composed of the initial segment, transition segment, and the main segment. The initial section included the shear layer or mixing layer extending from the orifice boundary to the inside and outside and the potential flow core whose central part was not affected and whose velocity
u0 was maintained. The main section of the jet was the jet after the full development of turbulence. The transition section of the jet was a very short fluid between the initial section and the main section, which was generally not considered.
(3) Attenuation law of axial flow velocity
The momentum flux of each section of the jet was conserved and equal to the momentum of the exit section, i.e.,
In the equation,
u0 and
r0 are the flow velocity and radius of the exit section, respectively, as shown in
Figure 5.
Considering the similarity of the flow velocity distribution of each section of the main body, namely:
Taking
be as the characteristic half thickness, where
r =
be,
u =
um/e, substituting the above equation
Assuming that the jet thickness extends linearly, namely:
In the equation,
c = 0.114,
D is the diameter of the circular orifice, and the above equation became
This equation shows that the circular turbulent jet axis flow velocity varies with x−1.
Let
um =
u0, and the initial length of the circular turbulent jet can be obtained [
27]:
Here, D is the diameter of the blasting tube 22 mm; therefore, the initial length of the circular turbulent jet of the liquid carbon dioxide blasting system was 0.136 m, and the pressure was also approximately equal within this range. In engineering applications, if the difference between the radius of the blast hole and the radius of the blast tube was greater than this value, the damage capability would be greatly reduced.
3.2. Positive Phase Pressure Function
Figure 6 shows the pressure–time history curves of the same distance outside the blast tube in the four explosion tests. The curves of the four tests have the same variation.
It was assumed that, under ideal conditions, an ideal pressure sensor was assumed for a given fixed distance R, which did not hinder the flow of the jet after impacting the pressure front and was able to perfectly record all pressure changes, as shown in
Figure 7. The ideal jet impact pressure–time history curve recorded by the instrument. At some time after the explosion of the liquid carbon dioxide blasting system, the instrument recorded the ambient pressure
P0. At the arrival time
t0, the pressure rises abruptly to a certain peak value
Ps+. Then, during time
t0+
t1, the pressure first decayed to a certain level value
Ps for a time
t2, then decayed to
P0 via time
t3, then continued to fall to
Ps−, and finally returns to
P0 during the experiencing time
t4. The pressure–time history curve of the pressure from the peak
Ps+ to the ambient pressure
P0 was called the positive phase. The pressure–time history curve of the pressure between the ambient pressures
P0 and
Ps− was a negative phase. The positive and negative phase impulses were defined by the following equation [
35]:
The damage effect of the jet impact pressure on the target can be measured by three parameters: (1) peak pressure; (2) positive phase zone action time; and (3) impulse density I+. It can be seen from the above analysis that the decay law and mathematical model of the positive phase are the keys to study the impact pressure curve of the liquid carbon dioxide blasting jet.
For traditional ideal blast or shock waves, there was a large amount of research work based on experimental tests, according to theoretical analysis, the following pressure–time history function forms are recommended [
35]:
(1) Linear attenuation function
This is the simplest form of function with two arguments. The shock wave pressure reaches a maximum at the beginning of the explosion, the form of the assumed linear decay of pressure when considering the blast wave load of the structure:
where
t+ is positive phase time (
). When fitting experimental data with this function, the actual value of
is usually reserved, and the positive phase duration
t+ is adjusted to maintain the actual positive impulse
I+. It is also possible to adjust the positive phase duration so that the initial decay rate of Equation (14) matches the initial decay rate of the test data. This will cause the positive impulse to be small which often used for response calculations.
(2) Exponential decay function
where
is attenuation index, and the function form is a commonly used function expression in the pressure decay law; many experimental data results indicate that the expression will accurately fit the records of many instruments on most positive phases. If someone gets relevant experimental results, he can obtain related pending parameters to perform regression analysis by using the functional expressions (14) and (15).
(3) Three parameter function
This function expression is often referred to as the “modified Friedlander’s equation [
35]”, which can describe the details of pressure changes well in many cases.
(4) Four parameter function
Many scholars have noticed that the data of the exponential decay rate decrease with time in the experiment, who put forward a four-parameter function expression that allows for more degrees of freedom:
The characteristics of Ps+、t+、I+ and exponential decay rate can be fitted and analyzed by use according to the equation.
(5) Five parameter function
According to the computing theory that shock wave generated by blasting point, Broad has put forward a function expression consisting of a five-parameter model to match the time history of positive phase pressure:
where
g and
h are constant. Up to now, the most complex expression for fitting positive phase time history data has also been proposed by Broad, which also contains five parameters:
where
a,
b, and
s are constant. This equation can fit with test data very well.
All of the above equations are empirical expressions. Equations (14) and (15) are simple, but the equations are quite different from the properties of some observed ideal waveforms. Equation (14) for linear attenuation is inaccurate, and Equation (15) cannot return to ambient pressure and is also inaccurate. Equation (16) is still simple and can match the observed parameters more precisely. Equations (17) through (19) are more and more complex, but they can be adjusted to approximate experimental or theoretical values. The author thinks that scholars should adopt the simplest form that is compatible with the accuracy required for any given analysis. The best solution might adopt Equation (16) of “modified Friedlander’s equation [
35]”. Because it allows adjustment to meet the most important blast wave properties, but not too complicated.
According to the above analysis and discussion, the regression analysis of the experimental data shows that the general function expression of the pressure–time history is (20).
Table 2 gives the fourth test pressure–time history parameter tables.
Figure 8 is a graph of the pressure–time history curve of four trials:
It can be seen from
Figure 8 and
Table 2 that the peak pressure of the impact pressure curve shows the same regularity as the peak pressure in the blasting tube in the four tests, and the pressure function model (20) fits well with the pressure curve obtained by the tests, indicating that the pressure function model can be used to express the phase change jet impact pressure curve of a liquid carbon dioxide blasting system. The time from
to
was 0.008894 s, 0.00668 s, 0.00673 s and 0.00816 s, respectively. The rate of pressure rise directly reflects the rate of high-pressure fluids working on the outside world. The pressure increase rates at 0.23 m in the four tests was 1.07 GPa/s, 0.82 GPa/s, 1.38 GPa/s, and 1.12 GPa/s, respectively. For
in the pressure curve, it can be smaller than the increasing peak pressure
, or equal to
, which is related to the thermodynamic characteristics and phase transition process in the blast tube. It can be seen from
Figure 8 that the pressure of
is longer, which is 0.021 s, 0.009 s, 0.013 s and 0.016 s, respectively. This process can be regarded as a stable loading process of jet impact, which represents that the phase changed jet impact of the liquid carbon dioxide blasting system. In the pressure quasi-static process, the impulse
of the jet represents the energy of the jet impact and has the same regularity as the energy in the burst.
From the above analysis, the process of phase change of liquid carbon dioxide blasting system includes two main processes: dynamic loading process and quasi-static loading process.
3.3. Pressure Decay Characteristic
The damage of the liquid carbon dioxide blasting system to the target was mainly achieved by the jet impact pressure and the quasi-static action of the high-pressure gas under the sealed condition. The jet impact and impulse are the main factors. The pressure function and impulse variation of the jet have been discussed in the previous section. Combined with the TNT equivalent of the liquid carbon dioxide blasting system, the authors will discuss the impact pressure characteristics and attenuation characteristics at different distances.
TNT equivalent ratio is an important parameter to measure the power of explosives. It has important research significance for anti-explosive structure design, safety distance, and explosion accident. At present, through theoretical calculations and regression analysis of laboratory data, the peak pressure of TNT explosives’ shock wave with the attenuation law of propagation distance has the following analytical expression [
36]:
where:
is the peak pressure, MPa;
is the contrast distance;
is the TNT dose, kg; and
is the distance from the blast, m.
For the non-ideal explosion source of liquid carbon dioxide blasting system, a TNT equivalent can be used instead of TNT dose ω to calculate the peak pressure of different contrast distance
. By comparing the actual data of the same contrast distance
under laboratory conditions, the pressure decay characteristics of the liquid carbon dioxide blasting system were analyzed.
Table 3 gives the average distribution of the incident peak pressures at different contrast distances for the above four tests. The explosive energies of the four tests were 142.233 kJ, 124.372 kJ, 147.723 kJ and 136.725 kJ, respectively. TNT equivalents were 30.85 g TNT, 26.98g TNT, 32.04 g TNT, and 29.66g TNT, with an average of 29.88g TNT.
Figure 9 shows that the peak pressure test value fit curve and theoretical calculated value curve at different distances and contrast distances (since the liquid carbon dioxide blasting system is a symmetrical jet exit, the measured peak pressure is ideally 1/2 of the total pressure, so it corresponds to the peak pressure of TNT, which is also taken as 1/2 of the theoretical calculation).
It can be seen from
Table 3 and
Figure 9 that the peak pressure of the liquid carbon dioxide blasting system is significantly larger than the theoretical pressure peak of the TNT overpressure, which is due to the influence of the jet pressure at a certain distance after the explosion.
The peak pressure from 0.23 m to 0.6 m was reduced from 8.653 MPa to 1.515 MPa. This rapid decrease was due to the rapid decrease of the jet velocity of the high-pressure fluid, which caused the peak pressure to decrease. It can be seen from the analysis of the velocity decay law in
Section 3.1. The initial length of the circular turbulent jet was 0.136 m, and the distance has the same jet velocity. The jet with a centering distance of 0.23–0.6m was in the transition section and the main section of the circular turbulent jet. When the jet exceeded the main section, the influence of the jet gradually decreased, and the speed decreasing trend tended to be gentle. However, the peak pressure was also affected by the explosion shock wave. Therefore, the core distance was 0.6 m to 2.5 m, and the peak pressure decreased. The error in theoretical calculations was reduced. When the distance exceeded 2.5 m, the jet had no effect on the pressure, and, at this time, only the overpressure generated by the shock wave action. Based on the above analysis, it can be considered that the impact pressure field of the liquid carbon dioxide blasting system can be divided into three regions, namely, the explosion jet impact zone, the jet edge zone, and the shock wave action zone.
In order to visually and qualitatively describe the attenuation characteristics of the impact pressure of liquid carbon dioxide blasting system, based on the TNT equivalent, the TNT reference equation function relationship was used to compare and evaluate the impact pressure attenuation relationship of the liquid carbon dioxide blasting system impact pressure reference function. According to the theory of explosion similarity, when TNT explodes in the air, it has the same peak pressure at the same contrast distance and satisfies a certain functional relationship [
29]:
and let ∆P be the nth degree polynomial of ∛ω/r; then, it can be expanded to:
where
are undetermined coefficients, and a
0 = 0 is determined by the boundary condition, and the TNT explosive shock wave peak pressure equation.
Using the test peak pressure data values in
Table 3, using the least-squares principle to fit the data, the pressure–contrast distance equations and curves of the liquid carbon dioxide blasting system were obtained, and the pressure–contrast distance equation curve of the theoretical calculation equation was obtained—for comparison, as shown in
Figure 10.
The pressure–contrast distance fitting equation for the liquid carbon dioxide blasting system was shown in the following equation, with R
2 being 0.96:
It can be seen from the above fitting equation that the jet impact pressure of the liquid carbon dioxide blasting system accords with the functional relationship.