1.2) Considering the substrate to be a semi-infinite solid maintained at an initial temperature T i and is
6.4 Calibration of Shock Tube
6.4.2 Experimental procedure for shock tube operation
Initially, an aluminium diaphragm of 1.2 mm thickness separates the driver and driven section of the shock tube (Figs. 6.11 (a-b)). Since the strength of the shock increases with increase in the ratio of speeds of sound, it is desirable to have a driver gas with a low molecular weight.
Conversely, driven gas should have high molecular weight. So, the strongest shock wave is obtained by using a heavy driven gas and a light driver gas. While meeting these requirements, the present investigation is aimed for two driver gases (nitrogen and helium) with air in the driven section of the shock tube. At the beginning of the experiment, the pressure inside the driven section
p1 is maintained at 0.18 bar and all the valves are closed. The driver section
p4 is filled with nitrogen/helium through a high pressure cylinder and the diaphragm ruptures at a pressure of approximately 20 bar. The sudden rupture of the diaphragm (due to the pressure difference between the driver and driven section of the tube) creates a shock wave that propagates into the driven section. The critical factor in designing the V-groove on the diaphragm plays an important role during the rupture process as shown in Fig. 6.11. If the diaphragm does not rupture instantaneously, then it leads to the only formation of compression waves as observed in Figs.spreads to the edges [Takayama et al., 2014]. Therefore, the gas flow starts as a jet initially followed by a subsequent mass flow of driver gas after the petal like complete rupture of the
Fig. 6.11: Diaphragm rupture process in the shock tube
diaphragm (Fig. 6.11-f). Often this controlled nature of diaphragm rupture resembles the formation of shock wave as a consequence of coalescence of series of compression waves. The sudden rise
Fig. 6.12: Pressure rise across primary and reflected shock in the shock tube
in pressures across the shock wave induces mass motion of the driven gas (air). The primary shock gets reflected from the end plate, thus forming the reflected shock. The pressure jumps across the primary as well as a reflected shock are captured from the pressure transducers mounted at the last segment of the driven tubein the form of voltage signals (Fig. 6.5-b). The typical voltage signal is noted from the pressure sensor with ‘nitrogen/helium’ as driver gas and ‘air’ as driven gas. Based on the ‘sensitivity’ information as supplied by the manufacturer, the pressure rise across the primary and reflected shocks are measured (Fig. 6.12). With the knowledge of the distance between the pressure taps
S and the time taken by the shock waves
t to travel this distance (obtained from pressure signals) as shown in Fig. 6.12, the speed of sound in the “region 1”
a1 , the shock wave velocity
Vs and the experimental shock Mach number
Ms e,
can be calculated from Eq. (6.8).1 1 ,
1
; ; s
s s e
S V
V a RT M
t a
(6.8) Table 6.1: Comparison of shock Mach numbers between analytical calculations and experiments
Sl.
No.
Driver Gas: Nitrogen; Driven Gas: Air
1 0.18 bar; 1 1.4; 4 1.4; 1 287 J kg.K ; 4 297 J kg.K
p R R
Ms p2 p1 p5 p1 T T2 1 T T5 1
Theory
Ms t,Exp.
Ms e,Theory
p2 p1t
Exp.
p2 p1e
Theory
p5 p1t
Exp.
p5 p1e
Theory
T2 T1t
Exp.
T2 T1e
Theory
T5 T1t
Exp.
T5 T1e
1 2.42 2.24 6.66 6.38 27.53 28.31 2.04 1.90 3.34 2.98
2 2.38 2.17 6.44 6.31 26.16 29.51 2.02 1.84 3.25 2.84
3 2.39 2.26 6.51 5.85 26.50 22.92 2.03 1.91 3.28 3.01
4 2.42 2.23 6.69 6.01 27.71 29.06 2.06 1.89 3.35 2.96
5 2.43 2.26 6.72 6.18 27.85 24.92 2.07 1.91 3.36 3.0
Driver Gas: Helium; Driven Gas: Air
1 0.18 bar; 1 1.4; 4 1.66; 1 287 J kg.K ; 4 2077 J kg.K
p R R
6 3.68 3.49 15.63 11.53 89.65 82.84 3.56 3.30 6.78 6.17 7 3.65 3.34 15.43 8.96 88.14 67.99 3.53 3.10 6.70 5.71 8 3.52 3.35 14.26 12.50 79.62 76.03 3.33 3.12 6.25 5.76 9 3.62 3.42 15.11 11.36 85.83 75.92 3.48 3.20 6.58 5.95 10 3.67 3.22 15.59 16.73 89.34 76.03 3.56 2.95 6.76 5.37
Further, the theoretical shock Mach number
Ms t, is obtained from initial pressure ratios
p4 p1
across the diaphragm using Eq. (6.3). The detailed parametric analysis is available in Appendix B. With helium and nitrogen as driver gases (region ‘4’) and air as driven gas (region ‘1’), the
comparative assessment of shock Mach numbers
Ms t, andMs e,
is obtained from both the methods (Table 6.1 and Fig. 6.13). Using the values ofMs t, and Ms e, , one-dimensional shock tube relations (Eqs. (6.2-6.7)) have been used to compute the theoretical and experimental values of pressure and temperature ratios across both primary shock
p2 p1
and
T T2 1
and reflected shock
p5 p1
and
T T5 1
. These values are calculated for five set of experiments for both nitrogen (N2)and helium (He) as driver gas and the comparative behaviours are given in Table 6.1 and Figs. 6.14(a-b)).Fig. 6.13: Comparison of shock Mach numbers (experiment and theory) as a function of pressure ratio
p4 p1
Fig. 6.14: (a) Pressure rise across primary and reflected shocks as a function of shock Mach numbers
Upon reaching the end flange of the driven tube, the shock wave gets reflected at much lower speed than the primary shock. With the knowledge of primary shock Mach number
Ms t, andMs e,
, it is possible to calculate the reflected shock Mach numbers
MR t, and MR e,
Table 6.2: Calculation of shock tube parameters using Nitrogen and Helium as driven gas
Sl.
No.
Driver Gas: Nitrogen; Driven Gas: Air
1 0.18 bar; 1 1.4; 4 1.4; 1 287 J kg.K ; 4 297 J kg.K
p R R
P bar4 P4/P1 MS MR Theory Exp. Theory Exp.
1 19.65 109.17 2.42 2.25 1.97 1.37
2 17.51 97.28 2.38 2.18 1.91 1.35
3 18.13 100.72 2.39 2.26 1.93 1.39
4 19.99 111.05 2.425 2.24 1.98 1.43
5 20.2 112.22 2.429 2.26 1.99 1.41
Driver Gas: Helium; Driven Gas: Air
1 0.18 bar; 1 1.4; 4 1.66; 1 287 J kg.K ; 4 2077 J kg.K
p R R
6 19.856 110.31 3.68 3.49 2.25 1.95
7 19.167 106.48 3.65 3.34 2.24 2.29
8 15.995 88.861 3.52 3.36 2.22 1.85
9 18.271 101.51 3.62 3.42 2.24 2.50
10 19.65 109.17 3.67 3.22 2.25 1.76
Fig. 6.14: (b) Temperature rise across primary and reflected shocks as a function of shock Mach numbers by using Eq. (6.7). The values of MR t, andMR e, are given in Table 6.2, while the trend of the plot is shown in Fig. 6.15. In addition, the percentage of deviation for each of the measured and calculated parameter of the shock tube is illustrated in Table 6.3. All these calibration curves (Fig.
6.13, Figs. 6.14 (a-b) and Fig. 6.15) show a reasonably good agreement (within ± 12 %) between the theory and experiments for nitrogen driver. However, the deviation seems to be higher for
helium driver in certain test cases, which may be due to its lighter weight and higher shock Mach number [Persico et al., 2005]. Since most of the shock tube parameters depend on the square of the shock Mach number, the deviation seems to be amplified. It may also be emphasized that material of the diaphragm and its groove also plays a critical role in the calculation of shock Mach number from the pressure ratio
p4 p1
.Table 6.3: Percentage deviation of shock tube parameters during its calibration Sl.
No .
Driver Gas: Nitrogen Error % of deviation from Theory
, ,
, s t s e
s t
M M
M
2 1 2 1
2 1
t e
t
p p p p
p p
5 1 5 1
5 1
t e
t
p p p p
p p
2 1 2 1
2 1
t e
t
T T T T
T T
5 1 5 1
5 1
t e
t
T T T T
T T
1 7.02 4.2 -2.8 6.86 10.78
2 8.4 2.01 -12.8 8.91 12.61
3 5.44 10.13 13.5 5.91 8.23
4 7.63 10.16 -4.8 8.25 11.64
5 6.95 8.03 10.52 7.73 10.71
Driver Gas: Helium
6 5.16 26.2 7.59 7.45 8.95
7 8.49 41.9 22.86 12.22 14.71
8 4.54 12.34 4.5 6.47 7.88
9 5.52 24.81 11.54 7.9 9.53
10 12.26 -7.31 14.89 17.08 20.51
Fig. 6.15: Reflected shock Mach number as a function of primary shock Mach number
6.4.3 Stagnation heat flux measurement in the shock tube