Structure of Shock Wave in a High Molecular Vapor

Hiroyuki Ogawa*, Yasunori Kobayashi*,Naruhito Nishikawa** and Takeshi Yoshida**

^Institute of Space and Astronautical Science** Tokyo University of Agriculture and Technology

Abstract. Shock waves in vapors of w-perfluropentane (C5F12) and w-perflurooctane (CgF18) are studied. The Rankine-Hugoniot analyses show that for conditions of experiments condensation occurs behind a shock in n-C^u vapor, andthat the saturation vapor pressure of w-C8F18 is lower than the frozen (no phase change) pressure behind the shock.Irregular (corrugated) shock fronts are observed for «-C8F18, while is not for n-C5F12. The linearized theory shows theshock in w-C5F12 vapor is in the region of strong stability, while the shock in w-C8F18 vapor is near the intermediate regionbetween a region of strong stability and a region of strong instability. The two-dimensional shock is simulatednumerically: an irregular shock appears in n-C8F18 vapor.

INTRODUCTION

The present work deals with shock-induced vapor-liquid phase changes in fluids with large molar heat capacities,i.e. fluids with many molecular degrees of freedom. The vapors of such fluids tend to condense on adiabaticcompression (retrograde behavior) because the work of compression produces a correspondingly small increase intemperature, while vapors of fluids of lower heat capacity, such as water, which tend to condense on adiabaticexpansion (regular behavior). Retrograde phase changes are usual in many two-phase systems: Rankine-cycle heatengines with multiatomic working fluids, hydrocarbon-fuel combustion engines, etc.

For liquid-vapor systems of fluids of retrograde fluids, new phenomena appear; the liquefaction shock wave, theshock front instabilities, rarefaction shock, for example1.

In the present, the shock waves propagating through the saturated vapors of «-perfluoropentane (CsF^) and n-perfluorooctane (CgFig) are investigated experimentally, theoretically and numerically.

EXPERIMENTAL ARRANGEMENT

The shock tube shown in Fig.l is used to study shock waves in high molecular vapors. The shock tube consists ofthe high-pressure chamber, the low-pressure section, the test section, and the dump tank. The nitrogen is used as adriver gas, which is filled in the high-pressure chamber. The high-pressure chamber is separated from the low-pressure section by a quick-valve. The low-pressure section and the dump tank are separated by the diaphragm film,which is ruptured by a shock wave. The low-pressure section and the test section are filled with a saturated testvapor. The flows are visualized by the Schlieren technique through the windows located at the test section. The twopressure transducers are located at the center of test section (PB), and at the 215mm upstream (PA).

PROPERTIES OF TEST VAPORS

The properties of test vapors are summarized in Table 1. The specific heats of vapor at constant pressure areestimated from the chemical formula2.

CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00

599

FIGURE 1. Sketch of shock tube.

TABLE 1. Properties of Test Vapors.Properties «-C5Fl: /i-CgF18Molecular WeightSpecific Gravity (25 degC)Vapor Pressure (25 deg.C)Specific Heat of LiquidBoiling PointHeat of Vaporization (Boiling Point)Specific Heat of Vapor at ConstantPressureSpecific Heat Ratio of VaporAntoine Parameter, A (Ref. 3)

B (Ref. 3)___________C (Ref. 3)

2881.63

81.4kPa1089J/kg/K30 deg.C

87293 J/kg698 J/kg/K

(Estimated from Chemical Formula2)1.037

4.38421257.8-13.231

4381.76

3.87kPa1047 J/kg/K102 deg.C

92110 J/kg707 J/kg/K

(Estimated from Chemical Formula2)1.043

3.98031200.0-77.078

RANKINE-HUGONIOT RELATION

The Rankine-Hugoniot relation for gas-vapor mixture consists of the mass, momentum, and energy conservationequations:

plul = p2u2 (1)

(2)Pluf+Pl=p2u22+p2

hl+-u2=h2+-u22 (3)

where p , u , p, and h are the density, the velocity, the pressure, and the specific enthalpy, respectively. Thesubscripts 1 and 2 denote upstream and downstream of shock, respectively. By assuming that the gas and the vaporare calorically perfect, the specific enthalpy is given by

h = (YgCp,g+YvCp,v+YlCl)T + YvLQ (4)where Y is the mass fraction, Cp is the specific heat at constant pressure, C7 is the specific heat of liquid, T is thetemperature, and LQ is the latent heat of evaporation at zero temperature. The subscripts g, v, and / denote theinert gas, the vapor and the liquid, respectively. The gas and the vapor are assumed thermally perfect gas.

P=p(YgRg+YvRv)T (5)

where Rg and Rv are the specific gas constants of the gas and the vapor, respectively. The mass fractions of inertgas do not change across the shock, and the sum of mass fractions is equal to 1:

Yv+Y,=l-Yg (6)The number of equations is 6, while the number of unknowns is 7 (p2 , u2, p2 , Yv 2, Yn , h2 , and T2 ) when

the upstream conditions of shock are given. The another equation is given separately depending on the conditionbehind the shock, as follows:

Case 1: no condensation occurs behind the shock.Y,,2=0 (7A)

Case 2: the condensation occurs behind the shock.The vapor is saturated behind the shock. The vapor pressure is given by the Antoine equation:

600

B

where pv 2 = p2Yv2RvT2. The parameters A,B, and C are given in Table 1.

RESULTS AND DISCUSSIONS

(IE)

Experimental Results

Figure 2 shows a typical Schlieren photograph of a shock through saturated n-C*Fu vapor. The shock front isregular: a plane shock front is observed. A sharp change in density (dark region) is observed behind the shock front.The dark region is considered to be resulted from the vapor condensation to the cold wall.

FIGURE 2. Typical Schlieren photograph of a shock in w-C5F12 saturated vapor (left figure), and corresponding pressure profilesat the pressure transducers (right figure). The upstream condition is pi=74.61kPa, Ti=23.1deg.C, and Ui=149.5m/s.

Figure 3 shows a typical Schlieren photograph of a shock through saturated «-C8Fi8 vapor. The shock front isirregular: a wavy shock front is observed. Small-scale density fluctuations are observed behind the shock front.Sharp pressure fluctuations are also observed behind the shock.

T i m e ( m s e c )

. ^ ^ •••• •••••J ^ ^ BFIGURE 3. Typical Schlieren photograph of a shock in w-C8Fi8 saturated vapor (left figure), and corresponding pressure profilesat the pressure transducers (right figure). The upstream condition is pi=4.831kPa, Ti=24.8deg.C, and Ui=289.2m/s.

Rankine-Hugoniot Analyses for Vapor-Liquid System

Figure 4 shows the comparison of analytical results with experimental ones for n-C^u vapor. They agree verywell. Figure 5 shows the mass fraction of liquid behind the shock, which is obtained as the solution of Rankine-Hugoniot equation. Ten to fifteen percent of n-C^u vapor condenses behind the shock. Figure 6 shows the relationbetween the pressure and the temperature behind frozen shock ("frozen9 means that no phase change occurs) and thesaturation vapor pressure at the corresponding temperature for «-C5Fi2. In the experiments, the ratios of pressure on

601

the two sides of shock, P2/Pi, are in the range 2.2<P2/Pi<3.3. The pressure behind the frozen shock is higher than thesaturation vapor pressure. Therefore the condensation occurs behind the shock. It should be noted that there is theregion where the downstream pressure of frozen shock is lower than the saturation vapor pressure (17<P2/Pi).

1Q.

3.2

3

2.8

2.6

2.4

2.2

213

K8

r#*

X TheoryO Experiment

0 140 150 160 170u1 (m/s)

FIGURE 4. Rankine-Hugoniot solutions and experimental results for «-C5F12.

!2

Mas

s fra

ctio

n of

\

0

0

0

0

0

0

.14

.12

0.1

.08

.06

.04

.02

0

4 [

f*4t*

i i i i i i

135 140 145 150 155 160 165 170

u1 (m/s)

FIGURE 5. Mass fraction of liquid behind the shock in «-C5F12 vapor, obtained as the solution of Rankine-Hugoniot equation.

Figure 7 shows the comparison of analytical results with experimental ones for w-CgFig vapor. They agree fairlywell. The discrepancies are considered due to the impurity of the vapor; it is contaminated by the leak of driver gas.Analytical results for w-CgFig vapor show that for the experimental condition the liquid -phase n-C^u does not existin the downstream of the shock. Figure 8 shows the relation between the pressure and the temperature behind frozenshock and the saturation vapor pressure at the corresponding temperature. In the experiments, ?2/Pi are in the range14<P2/Pi<25. The pressures behind shock are lower than the saturation vapor pressure. Therefore no condensationoccurs in the downstream of the shock. It should be noted that there is the region where the pressure is higher thanthe saturation vapor pressure. (l<P2/Pi<15.) This fact suggests that the condensation and the vaporization occurinside the shock wave.

Shock-Front Instability

Majda and Resales4 analyze a linearized mechanism of instability through radiating boundary waves, andpropose a stability condition of shock-front. A shock-front is strongly stable if

1G+l

(8)

602

45

40

35

3025

2015

10

270

-P2/P1 (theory,frozen)

- - - - .Pvs2/P1

P2/P1 (exp)

320 370T2

420 470

FIGURE 6. Pressure and the vapor saturation pressure behind frozen shock for ntemperature T2 is denoted by Pvs2.

- The saturated vapor pressure at the

&

24

22

20

18

1 6

1 4

1 2

10

25

X

>%r^x£ -°

- - * °o*8o ~X o0<?

x ,

X T h e o r y

O E x p e r i m e n t

0 300 350 400 450

u1(m/s)

FIGURE 7. Rankine-Hugoniot solutions and experimental results for w-

where G is the Gruneisen coefficient, and is related to the specific heat ratio, 7, as G = 7—!, for an ideal gas. Ifthe inequality (8) is violated the possibility of an instability exists. Figures 9 and 10 show the values of K in theexperiments and the regions of strong stability for w-CsF^ and w-QFig, respectively. The experimental conditionsare in the region of strong stability in case of n-CsF^, while they are very close to the boundary of the region ofstrong stability in case of «-C8Fi8. This fact suggests that the shock-front in «-C5Fi2 vapor is stable and therefore theregular shock appears, while the shock-front in «-C8Fi 8 vapor could be unstable and the irregular shock may appear.

603

60

50

40

30

20

10x

-P2 /P1 ( f rozen )

! P v s 2 / P 1

X P2/P1(exp)

300 320 340 360 380T2

400 420

FIGURE 8. Pressure and the vapor saturation pressure behind frozen shock for w-CgFi8. The saturated vapor pressure at thetemperature T2 is denoted by Pvs2.

0.1

———— 1/(G + 1)

• E x p e r i m e n t

1.4 1.5 1.6

Shock Mach n u m b e r , M1

FIGURE 9. Shock-front stability for w-C5Fi2. The parameter K is defined in Eq. (8). Experimental conditions are in the region ofstrong stability.

604

0.1

• E x p e r i m e n t

4.1 1 4.31 4 .51 4.71 4.91Shock Mach n u m b e r , M 1

FIGURE 10. Shock- front stability for «-CgF18. Experimental conditions are close to the region of instability.

Numerical Simulation of Two -Dimensional Shock in n- Vapor

A two-dimensional unsteady shock in w-CgFig vapor is numerically simulated to show the shock-front is unstable.To make the problem simple, it is assumed that 1) no inert gas is in the vapor, 2) no condensation occurs, 3) thevapor is saturated in the upstream of the shock, 4) the shock is infinitely thin, 5) viscosity and thermal conductionare neglected, and 6) vapor behaves as ideally perfect gas. The unsteady, two-dimensional Euler equation is solvednumerically using the third -order shock- capturing scheme. The calculated domain is a rectangle 1.0m in length andO.lm in width. The 500 x 50 grid points are used. The slightly disturbed shock is initially located at the center of thecalculation domain.

FIGURE 11. Numerical simulation of unsteady 2-dimensional shock, propagating a saturated pure w-C8F18 vapor. The shock islocated near the center of the calculation domain, and propagating from the left to the right. The density distribution is shown byshades. The upstream condition is pl = 0.5054 kg/m3, ul =388.3 m/s, 7J = 294.1 K.

605

Figure 11 shows the density distributions at several instants. The disturbances do not disappear: the shock-frontis not stable. The shock-front is irregular and changing with time. The density fluctuation resulting from the shock-front instability is observed in the downstream of the shock. The irregularity of shock-front and the densityfluctuation behind the shock are consistent with the experimental results.

SUMMARY

Shock waves in n-C5F12 and n-C8F18 vapors, which have high heat capacities and show retrograde behavior,are investigated experimentally, theoretically and numerically. We summarize the present work as follows:

Shock in n-CsFn Vapor

Regular plane shocks are observed in the experiments. The theoretical results based on Rankine-Hugoniotrelation agree very well with the experiments. The Rankine-Hugoniot solutions show that the 10~15 percent ofvapor condenses behind the shock. The shock conditions in the experiments are in the region of strong stability.

Shock in w-CsFis Vapor

Irregular shocks appear in the experiments. The microscopic density and pressure fluctuation are observedbehind the shock. The theoretical results agree fairly well with the experiments. It is shown theoretically that nocondensation occurs behind the shock. The shock conditions in the experiments are very close to the stability limit,which suggests that the shock-front is unstable. The two-dimensional numerical simulation shows the instability ofthe shock-front, and the density and pressure fluctuat ions in the downstream of the shock, which are consistent withthe experiments.

REFERENCES

1. Thompson, P. A., Carofano, G. C, and Kirn, Y., J. Fluid Mech 166, 57-92 (1986).2. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., The properties of gases and liquids, McGraw-Hill, 1977, pp.225-226.3. NIST Standard Reference Database 69, NIST Chemistry WebBook, http://webbook.nist.gov/chemistry/.4. Majda, A., and Resales, R, SIAM J. Appl. Math 43, 1310-1334 (1983).

606