ANALYSIS OF SHOCK-WAVE PROPAGATION IN THERMODYNAMICALLY

EQUILIBRIUM FOAM

N. M. Kuznetsov, E. I. Timofeev, and A. V. Gubanov

Because of the promise shown by the use of foams for shock-wave attenuation in open air explosive demolition tasks, a number of studies have been performed [I-7], dedicated to vari- ous aspects of wave dynamics in such media. Preliminary experimental results on physical properties of foams (density, liquid loss rate, etc.) and their effect on a shock wave creat- ed by detonation of an explosive charge of mass M 5 1 kg were presented in [i]. In [2], a shock tube was used to perform studies of the motion of long (wave action time on the order of 3 msec) shock waves in a foam at G 0 = 2-16 kg/m 3. Here o0 = g0@s where e 0 and ps are the volume concentration and density of water. The speed of the long shock waves did not change for traversal of distances up to 1.3 m [2].

The experimental study in [I] of shock-wave attenuation in a foam (M = 1-5 g and o0 = 15 kg/m 3) confirmed the results of [3], showing a more rapid pressure drop in the wave in the two-phase medium as compared to air. The individual experiments of [3] were performed at M = 1 kg. In [4], results were presented concerning the velocity of motion of long waves and their interaction with a rigid wall l~oated iF foam. It was shown that the shock-wave velocity was independent of the thermophysical properties of the gas contained in the foam cells, while the pressure of the reflected wave was the same as in the pure gas (nitrogen).

Theoretical studies of shock-wave motion in a foam were performed in [5-7] on the basis of the model concepts of [8], according to which the two-phase medium is considered as a pseudogas with effective adiabatic index F:

v = ? . ( l + n ~ ) / ( l + ~q6). (1 )

Here ~ = o4/(I - ~0)Pg0; ~ = C/Cp is the ratio of the specific heats of the condensed phase and the gas at constant pressure; Pg0 is the initial gas density; 7 is the adiabatic index of the gas. In [5] the problem of the strong stage of a point charge explosion in a foam was solved on the basis of a modification of Eq. (i)

r = ~ .(1 + qox ) / ( l + ~ x ) (2)

(where ~ is a function of the temperatures of the gas T and the condensed phase z beyond the the wave). According to the data of [5], replacement of the nitrogen in the foam cells by helium leads to more intense damping of the wave, which was explained by the authors as due to increased interphase heat exchange. The same problem was considered in [6], but with the assumption of exponential dependence of F on time behind the shock-wave front.

The interaction of long shock waves with a foam screen situated in air was studied in [7], and the problem of a point explosion in foam was analyzed with consideration of resis- tance. We will note the coincidence of results of [5, 6] and [7] regarding the effect of foam density on the intensity of shock-wave damping (with increase in 90 the damping intensi- fies), and the satisfactory agreement of these results with experiment.

Analysis of experimental and theoretical data permits the conclusion that it is possible to apply the model of [8] for description of a number of cases of shock-wave motion in a foam In particular, the concepts of [8] give velocity D values for long shock waves which agree satisfactorily with experimental values at a 0 ~ 5 kg/m 3 [2]. It was proposed in [2] that mass exchange affected the value of D. Shock-wave damping was also described satisfactorily. (With regard to short waves, [7] suggested the impossibility of heating the water to the boil- ing point.) However, the model of [8] does not agree with experiment for a wave reflected from a rigid wall. In connection with this, [4] concluded the absence of thermal equilibrium between the phases. We note that at a 0 = i0 kg/m 3 it follows from Eq. (i) that F = i. In

Moscow. Translated from Fizika Goreniya i Vzryva, Vol. 22, No. 5, pp. 126-132, September October, 1986. Original article submitted January 3, 1985; revision submitted July 29, 1985.

0010-5082/86/2205-0615512.50 �9 1987 Plenum Publishing Corporation 615

this case, calculation with the model of [8] gives identical intensification coefficients in the incident and reflected waves P2/Pl = Pl/P0, which corresponds to isothermal compres- sion of the gas (here P0, Pl, and P2 are the initial pressure and pressures in the incident and reflected shock waves).

Correction of the quantity F with the aid of some formal correction parameter, as at- tempted in [5, 6], permits improved agreement of calculation with experiment for attenuation of a shock wave in foam, but appears somewhat artificial.

The use of the model concepts of [8] and the approximations of [5, 6] based thereon was related to the absence of a satisfactory equation of state for the medium. Calculations were then performed of the shock compressibility of foam on the basis of the thermodynamic equa- tion of state of a ternary water-vapor-ideal gas system, which was constructed with use of the analytical equation of state of the water-vapor system [9]. According to [9], the de- pendence of the internal energy Ews of the water-vapor system on vapor pressure pw and spe- cific system volume vws (caloric equation of state) can be expressed in the form

E~, = I, + r162 (kg " c m ~ ) , ( 3 )

ZO~Q 0,216 = |9 ~no 7/~ I t = ~ " ~ P , o , 1~1 . , o u o p w

(Pw and Vws being expressed in kg/cm 2 and cma/g). Equation (3) approximates the tables of [I0] sufficiently well at temperatures from 360~ tO the critical point. The error in Ews calculation over this range comprises -1%, while the vapor pressure can be calculated to an accuracy of 0.1% by the expression [9]

P~ = [ ( T / 3 , 3 5 6 4 �9 I 0 - ' ) ' / ' - - i2 ,5085] ~ (kg/em'~, ( 4 )

As a rule, foams are used for shock-wave attenuation under conditions close to normal (T = 280-300~ In this case, as a result of the anomaly in the thermodynamic properties of water, the relatively simple Eq. (3) leads to large errors. In the range 280 ~ T E 360~ the dependence of Ews on Pw and VwE can be approximated by the expression

w h e r e f = = f l - 1 . 0 0 9 . 2 a ; += = (+1 - 0 . i 2 8 ) ~ 1 , a = e x p ( 0 . 0 0 9 5 / p w ~ - 3 . 5 2 8 p w ~ X z = l + (O.O098/pw) =. The interpolation error of Eq. (5) relative to the thermodynamic tables of [I0] does not exceed 1.2% over the temperature range in question.

To construct the equation of state of the ternary system, we introduce the notation: ~s ~w, =g, mass concentrations of water, vapor, and gas, respectively; and vs Vw, Vg, spe- cific volumes of the same components. The mass concentrations are normalized to unity (~s + ~w + ~g = i). The specific volume of the mixture is defined by the expression

v = ~ v , + ~L,,~ = ~,~ ' ,+ ~ : ~ . ( 6 )

For the internal energy E and pressure p of the ternary system we have

E - ~E=., + ( t - - ea)E~, ( 7 )

p = p~ + RT/t, tvg. ( 8 )

Here ~ = ~s + ~w is the mass concentration of water and vapor in the mixture; Eg and ~ are the specific internal energy and molecular weight of the gas. Substituting into Eqs. (7), (8) Eqs. (3), (4) with use of Eq. (6) and considering that v = ~Vws we obtain

E ----- = / + ~ u + (l - - ~x) 84,78'/' ( 9 ) ~.t ( ? - - i ) '

( i - ~)84,787 P = P,,~ + ~(~_=,, ,~) , ( i 0 )

where f = fl, ~ = ~i or f = fa, ~ = ~2 for 280 ~ T $ 360~ or 360~ < T < Tcr.

The numerical values of =, T, and v have definite limits. The temperature is limited above by Tcr, the critical point. From Eq. (6) and the relationships ~s g O, Vw g vs it follows that

616

o., MPa

IB It \

Fig. i.

i /

! 'S' \ i I \,,

aos ~, m~ /kg so p, /~

Fig. l Fig. 2

Shock adiabats of nitrogen (1-3) and helium (2', 3') systems: a = 0.8 (i), 0.6 (2), 0.4 (3); P0 = i/vo; P0 = 5.66 (I), 2.84 (2, 2'), 1.89kg/m ~ (3,3');vo = 0.177 (I), 0.352 (2, 2'), 0.529 m3/kg (3, 3'); 4, 4') total evaporation of water in nitrogen and helium foams, v 0 values for all shock adiabats lie to the right of v shown on graph.

Fig. 2. Dependence of ~l ~ aw/at on Pl/P0 in nitrogen (i, 2) and helium (I') foams; P0 = 1.89 (i, i') and 5.66 kg/m 3 (2).

~ u/v,o. (ii)

At a = v/vw the system transforms to binary (dry vapor-~as) and is no longer described by Eq. (9). The limitation on v below is determined from Eq. (I0) by the inequality v > Vmin = as163 For v = as163 the pressure of the undissolved ideal gas becomes infinitely great. In the vi- cinity of the limit v = ~s163 the properties of the vapor-water subsystem change significantly as a consequence of, for example, solution of the gas in the water, and the gas becomes non- ideal. These limitations appear at mixture pressures significantly exceeding the critical pressure for the liquid-vapor subsystem.

The Hugoniot equation together with Eqs. (3)-(I0) form a complete system for calculation of the thermodynamic equilibrium state of the foam behind the shock wave. Concrete calcula- tions were performed for two ternary systems, differing significantly in the thermophysical properties of the gas: water-vapor--nitrogen (which we shall term the nitrogen system) and water-vapor-helium (the helium system). In the calculations the values T0 = 293.16~ P0 = 0.1013 MPa were used and a series of initial values were specified for the specific volume of the system v 0.

Shock adiabats of the nitrogen and helium systems are shown in Fig. i in the coordinates p~-v. Comparison of curves 3', 3 and 2, 2' shows that the helium system has higher shock-wave compressibility than nitrogen. In the case of sufficiently high P0 shock compression can lead to total evaporation of the liquid. Thus, for example, at P0 = 1.89 kg/m ~ the boundary of the total evaporation region corresponds to Pl = 4.1 MPa, while at P0 = 2.84 kg/m 3, Pl = 9.5 MPa~ Equation (ii) was used to construct lines 4, 4' which separate the ternary system from binary (dry vapor--gas) in the coordinates pl-v. These states are located below and above the curves 4, 4'. ~

For complete evaporation of the water in the helium system it is necessary to increase the pressure in the wave as compared to the nitrogen system. This is true because at a fixed initial density of the ternary medium the initial mass concentration of water in the helium system (~0) is greater than in the nitrogen system. For example, at P0 = 2.84 kg/m 3 in the nitrogen foam =s = 0.594, and in the helium ~s = 0.937.

It is evident from analysis of the curves that, in a number of cases (for sufficiently small v0), the derivative dpz/dv along the shock adiabats changes sign. On curves 3', 2, i the points corresponding to this change of sign are denoted by letters B' and B (helium and

617

r MPa /

t,:, /// / ' , 1 //

I

! , m / s e e ! / ~

i ' i I ~ / / / / " i

4 i //

/

Fig. 3 Fig. 4

Fig. 3. Boundary states of nitrogen (i) and helium (2) foams.

Fig. 4. Quantity D vs Pl/P0; = = 0.4 (i), 0.5 (2), 0.6 (3), 0.8 (5); P0 = 1.89 (I), 2.27 (2), 2.84 (3), 5.66 kg/m ~ (5). 4, 6) After model of [8] with F = > = 1.4 and F = 1.02, respec- tively (P0 = 5.66 kg/m3).

nitrogen systems, respectively). In the general case this fact is related to the character of mass exchange behind the shock wave. While with increase in Pl the liquid mass behind the shock-wave front increases (vapor condensation), dpl/dv is always negative. If increase in pressure leads to evaporation, then at certain v~ the derivative changes sign~

The direction of mass exchange can be characterized conveniently by the quantity

p ~ ~t" I ~l 1 ~ ~ w ; ' ~ l ~ ~ .

~U w -- U

Knowing the relationship v(pl) along the shock adiabat and the values of vs and Vw on the saturation line [i0], we can consider the dependence of ~i on Pl and P0 (Fig. 2). It is evident from comparison of curves 1 and I' that replacement of nitrogen in the foam cells by helium leads to an increase in the mass of unevaporated water behind the front of a shock wave of fixed intensity. A growth in the initial density of the system leads to the same result. On the segments of the shock adiabats considered (Fig. 2) the mass of the vapor in- creases with increase in pl/p0. In the same range of Pl/P0 variation, on the shock adiabats with other po values we find points in the vicinity of which the derivative dpl/dv along the shock adiabat changes sign (see, for example, curve l of Fig. 1).

Transferring the data on maximum total water evaporation pressures on the shock adiabats to the plane Pl-0~, we obtain a diagram of boundary states for the ternary system (Fig. 3), which permits graphical determination of the pressure at which total evaporation of water occur in a foam of given initial density.

In experimental study of shock-wave motion in a foam the velocity and pressure of the wave are measured. The value of D is found from shock-wave compressibility data (Fig. I) and the relationship

2 Pl - - Po D" ----- vo

U O ~ U "

Figure 4 shows the dependence of D on Pl/P0 for the nitrogen system. At Pl/P0 = const, the wave velocity is lower, the higher the initial density. As a rule, in experiments wave velocity and pressure are measured to accuracies of about 15 and 25%, respectively. Hence, and from comparison of curves 4-6, it is evident that it is impossible to obtain reliable information on the degree of liquid heating by comparing calculation with the experimental dependence D(pl) [2]. Shock-wave velocities in the helium system are somewhat lower than corresponding values in the nitrogen system, but the difference is small.

618

r,K !

J 45~, i

J &,p, :I! e/~'w,~

I .......... !

0 E ~ &, MPa 2 S &/~

Fig. 5 Fig. 6

Fig. 5. Function T(p~) in nitrogen (i, 2) and helium (i') foams: pQ = 1.89 (i, i') and 5.66 kg/m 3 (2).

Fig. 6. Reflection coefficient Pa/Pl vs intensity P~/Po: i) model of [ii]; 2, 5) model of [8] with P0 = 1.42 and r = 1.065 (helium foam) and 1.23 (nitrogen foam), respectively; 3') helium foam, P0 = 1.42 kg/m3; 3, 4) nitrogen foam, P0 = 1.42 and 3.79 kg/m 3, respectively; 6, 7) pure nitrogen (~ = 1.4) and helium (y = 1.67), respectively; 3, ~, o) nitrogen, helium, and hydrogen foams from data of [4].

Thus, the value of D in the thermodynamic equilibrium case depends weakly on the thermo- physical properties of the gas contained in the cells. Calculations show that the difference between curves of the type of 4-6 in Fig. 4 decrease with decrease in P0. To explain the disagreement of experimental data indicated in [2] with calculations of [8] it is necessary to seek other causes, since in the range Pl/P0 5 20 mass exchange has practically no effect on the velocity of motion of long shock waves. One such cause, for example, may be inexact control of the foam density.

The dependence of temperature on pressure in the wave, thermophysical properties of the gas contained in the foam cells, and initial density is illustrated by Fig. 5. Increase in P0 leads to a reduction in T at fixed Pl. Replacement of nitrogen by helium in the foam also leads to decrease in T for fixed Pl. Both results are the consequence of an increase in the fraction of energy expended in heating the large mass of water.

The degree of perfection of heat exchange between the phases can be evaluated by comparir the calculated reflection pressures (P2) on a rigid wall with those measured experimentally. From comparison of curves 3, 3' and 4 in Fig. 6, it is evident that the reflection pressure is higher in the thermodynamic equilibrium case. For helium systems (lines 3', 2) this dif- ference is significantly smaller. Curve 1 was constructed with use of the relationship P2/ P~ = Pl/P0 [ii] (F = 7 = l, isothermal gas), while curves 6, 7 correspond to pure nitrogen and helium. In the thermodynamic equilibrium case (seeFig. 6, curves 3, 3', 4) the reflec- tion pressure on a rigid wall is higher than in the gas (curves 6 and 7), but below its value in an isothermal gas (curve i). Figure 6 also shows the experimental data of [4], which when compared with curves 3, 3', 4, 6, and 7 show that thermodynamic equilibrium in the foam is not achieved in the reflected wave. This conclusion was confirmed by the experimental results of [12], according to which at P0 = 300 kg/m 3 the reflection pressure coincides with its value in a gas (for example, nitrogen) at Pl/P0 = i0, while the reflected wave has the form of a step with action time of 500 to 1000 Hsec. We recall that the value of P2 in a foam at Pa = 5-15 kg/m 3 decreases with time (triangular profile).

A possible cause for the foam in the reflected wave being in a state further removed from thermodynamic equilibrium than in the incident wave is thickening of the foam cell walls upon shock compression. Moreover, the experimentally measured action time of the reflected wave (wavelength) is significantly less than that of the incident wave.

The characteristic heating time of the liquid film is defined by the expression

t* ~ A=tx, (12)

where a is the film thickness; X is the thermal diffusivity of the liquid (X = 1.4"10-3 cm2/ sec). The thickness of cell walls parallel to the flow increases in the incident shock wave

619

in proportion to the degree of compression of the foam 8. For @ = i0 and A 0 = (3-30).i0 -~ cm [2, 13] ahead of the wave we obtain from Eq. (12) t* = 7"(i0-3-10 -l) sec. The observed action time of the reflected wave is significantly less than t*.

Thus, the results presented herein and a comparison of the same with experiment permit the conclusion that it is possible to use the thermodynamic approximation for calculation of incident (traveling) long shock waves in a foam, at least for o 0 ~ 5 kg/m 3. In calcula- tions of propagation of attenuation of short reflected or sufficiently short incident waves those approximations are more suitable in which heat and mass exchange between phases are not considered. To do this for o0 > 1 the condition t << t* is insufficient. The more rigid condition t < t** must be satisfied, where t** = t*/a02 is the heating (cooling) time of the surface layer of the film forming the foam cells, the mass of which, ml, is comparable to the mass of the gas mg filling the cells. If we consider the difference in the heat capaci- ties of the liquid cs and the gas cg, then not the masses, but rather the products ms163 and mgcg should be compared (where a 0 is measured in kg/m3).

In conclusion, we note that the equation of state of the ternary system can be used for solution of other hydrodynamics proolems, for example for calculation of the thermodynamic equilibrium state of a liquid film located on the walls of a shock tube.

LITERATURE CITED

i. F. H. Winfield and D. A. Hill, DRES-TN-389 (1977). 2. V. Mo Kudinov, B. I. Palamarchuk, B. E. Gel'land, et al., Priklo Mekh., 13, No. 3, 92

(1977). 3. V. Mo Kudinov, B. Io Palamarchuk, B. E. Gel'land, et alo, Doklo Akad. Nauk SSSE, 228,

No. 3, 555 (1976)o 4. A. A. Borisov et al., Acta Astronaut., 5, 1027 (1978)o 5. B. I. Palamarchuk, V. A. Vakhnenko, A. V. Cherkashin, et al., Reports of the 4th Int.

Symposium on Explosive Materials Processing [in Russian], Gotwaldow, Czechoslovakia (1979 6. V. A. Vakhnenko, V. M. Kudinov, and B. I. Palamarchuk, Prikl. Mekh., 18, No. 12, 91 (1982 7. Bo Eo Gel'fand, A. V. Gubanov, and E. I. Timofeev, Fiz. Goreniya Vzryva, 17, No. 4, 129

(1981). 8. Go Rudinger, Raket. Tekh. Kosmon., No. 7, 3 (1965). 9. N. M. Kuznetsov, Dokl. Akad. Nauk SSSR, 257, No. 4, 858 (1981).

i0. M. P. Vukalovich, Thermodynamic Properties of Water and Water Vapor [in Russian], Mash- giz, Moscow (1955).

ii. L. Campbell and A. Pitcher, Proc. R. Sot., A243, No. 1235, 534 (1958). 12. E. I. Timofeev, B. E. Gel'fand, et al., Dokl. Akad. Nauk SSSR, 268, No. I, 81 (1982). 13. A. A. Berlin and F. A. Shutov, Chemistry and Technology of Gas-Filled Higher Polymers

[in Russian], Nauka, Moscow (1980).

620