Comparison of methods for calculating the shock ... - Springer Link

Shock Waves (2010) 20:73–83 DOI 10.1007/s00193-009-0230-x

ORIGINAL ARTICLE

Comparison of methods for calculating the shock hugoniot of mixtures Oren E. Petel · Francois X. Jetté

Received: 1 July 2008 / Revised: 14 November 2008 / Accepted: 24 August 2009 / Published online: 18 September 2009 © Springer-Verlag 2009

Abstract Various methods used to determine the shock Hugoniot of condensed phase multi-component mixtures are reviewed and compared to available experimental data. The assumptions inherent in the different models are presented in this overview and their implications are discussed. The comparisons of the various models demonstrate that the predicted shock Hugoniots are in good agreement with published data despite the simplifying assumptions that are associated with the models. Averaging models are shown to be among the simplest methods to implement and result in the closest agreement with experimental data. Keywords Shock Hugoniot · Multi-phase mixtures · Rankine–Hugoniot relations PACS 47.40-Nm · 47.40-Rs

1 Introduction The propagation of shock waves in condensed phase multicomponent mixtures has been a subject of considerable interest as it is related to the attenuation of blast waves and shock wave initiation of heterogeneous explosives. Thus, a number of methods have been developed to model wave propagation in such mixtures. These methods can be divided into three major categories: (a) unsteady flow models that attempt to resolve the interactions between the components of the mixture; (b) steady models of mixtures in mechanical Communicated by F. Zhang. O. E. Petel (B) · F. X. Jetté Department of Mechanical Engineering, McGill University, 817 Sherbrooke W., Montreal, QC H3A 2K6, Canada e-mail: [email protected]

equilibrium which do not explicitly model the component interactions; and (c) equilibrium-based averaging models. The first category of models are intended to solve the propagation of an unsteady wave through a multi-component mixture in which neither mechanical nor thermal equilibrium between the components is assumed a priori [1–4]. These models are based on the mixture theories of Truesdell and Toupin [5] which have been extended to explicitly consider multi-phase systems [6,7]. Since equilibrium is not assumed, the components are modeled as two separate interacting continua with a set of governing conservation equations for each component of the mixture. The accuracy of these models is difficult to verify as there is limited experimental data available for comparison and the models rely on intercomponent mass, momentum and energy exchange terms that are empirical in nature. The present work will not consider these models in detail since it is not necessary to model transient processes in order to calculate the Hugoniot of the mixture. For example, it is not necessary to know the kinetics of a chemical reaction in order to compute the thermodynamic equilibrium state of the products. In the case of mixtures with fine scales of heterogeneity (or large domains), the aforementioned unsteady models can be simplified significantly, as the mechanical relaxation time scale between the components of the system is relatively short [8], resulting in pressure and velocity equilibration between the mixture components. The assumption of mechanical equilibrium eliminates the need to explicitly consider the component interactions in the governing equations. For such systems, the conservation equations can be applied to the mixture as a whole, which leads to the Rankine–Hugoniot equations. However, these simplified models result in a system of governing equations which is under-constrained, requiring an additional criterion to close the system of equations and obtain the unique Hugoniot of a

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O. E. Petel, F. X. Jetté

given mixture. Models and closure criteria of this type, which will be referred to as “Closure Models,” have been proposed by a number of authors [9–13]. The third category of models, which will be referred to as “Averaging Models,” consist of those which assume pressure equilibrium between the components of the mixture and rely on averaging methods to determine a unique shock Hugoniot of the mixture [14–16]. These methods do not explicitly consider the governing conservation equations for the mixture to determine this Hugoniot, but only require knowledge of the Hugoniot curve for each individual component. At first consideration, the models from the second category may appear to provide more accuracy compared to the “Averaging Models” since a solution to the governing equations is explicitly sought. However, the added complexity of the models may not always lead to more accurate results than the simpler “Averaging Models” due to the assumptions that are made in determining the closure criterion. The objective of the present study is to consider the most common methods from each category for direct comparison to experimental data. For this comparison, the methods will first be reviewed individually in an overview of their key aspects. It should be noted that the Hugoniot of a mixture refers to a collection of equilibrium end states due to a shock wave process in a given material (or mixture) at a given initial thermodynamic state. Thus, a Hugoniot is not a thermodynamic path and should not be treated as such. The end state of the Hugoniots will be defined by the region of mechanical equilibrium behind a shock wave at the end of what is commonly referred to as the relaxation zone. Experimentally, this region is likely not in full equilibrium as thermal equilibrium requires longer timescales, however the thermal contribution to internal energy is negligible in systems containing only condensed phases. The Hugoniot of a mixture can therefore be defined by a collection of mechanical equilibrium end states or relaxed states. 1.1 Relations common to all equilibrium mixture models While there are significant differences in the approaches of the equilibrium models of interest, certain commonalities remain. The models all assume that the internal energy and density of the mixture must be related to the weighted sum of the individual component properties. If the two components of the mixture are referred to by the subscripts a and b, the mixture rules can be written as ρ = αa ρa + (1 − αa )ρb ρe = αa ρa ea + (1 − αa )ρb eb

(1)

or v = X a va + (1 − X a )vb e = X a ea + (1 − X a )eb

123

(2)

where ρ is the density, v is the specific volume (v = ρ1 ), e is the specific internal energy, αa is the volume fraction of component a, and X a is the mass fraction of component a. The mixture rules used in the unsteady flow models (the first category) differ slightly from those presented in equilibrium models. The mixture internal energy is often written containing a term representing the contribution of the species diffuu2

sion velocities, of the form (αi ρi 2ri ) to the mixture internal energy. The diffusion velocities of component i, u ri is defined as the difference between the local velocity of component i and the barocentric mixture velocity. Since the models under consideration assume mechanical equilibrium, the contribution of the diffusion velocities to the mixture internal energy is negligible and (1) or (2) are used. When considering an inert mixture that has reached postshock wave mechanical equilibrium, a common assumption is that the mass fraction of the mixture will remain constant. This assumption is made both implicitly and explicitly in all of the models which are being discussed. No such assumption is made concerning the volume fraction of the system and it will therefore be allowed to vary as a result of the relative compressibilities of the components in the mixture. It follows that, as in all work involving compressible substances, an equation of state (EOS) is required to relate the internal energy of a material to mechanical properties (e.g., e = e(P, ρ)). The Mie-Grüneisen EOS and Stiff Gas EOS are typically used to describe the materials of interest. The two EOSs can be expressed in the following manner P − Pref (ρ) + eref (ρ) ρ P − γ P∞ e(P, ρ) = ρ(γ − 1)

e(P, ρ) =

(3) (4)

where the subscript “ref” in the Mie-Grüneisen EOS refers to a reference curve (such as an isentrope or an isotherm), is known as the Grüneisen , and γ and P∞ are fitted parameters to the Stiff Gas EOS.

2 Equilibrium/closure models The major difference between the approaches of the “Equilibrium/Closure Models” and the “Averaging Models” is that in the former, the conservation equations are explicitly solved for the mixture in combination with the mixture rules and equations of state presented above. The governing conservation equations, the Euler equations, for a one-dimensional shock wave propagating in an inert, multi-component mixture with pressure and velocity equilibrium, neglecting material strength considerations (hydrodynamic assumption), can

Comparison of methods for calculating the shock hugoniot of mixtures

be written as: ∂ρ ∂ρu + =0 ∂t ∂x ∂ρu ∂(ρu 2 + P) + =0 ∂t ∂x 2 ∂ρe ∂u(ρe + ρ u2 + P) + =0 ∂t ∂x

(5)

where ρ is the density of the mixture, u is the material velocity, P is the pressure and e is the specific internal energy of the mixture. For the case of a steady shock wave, where the control volume of interest moves at the velocity of the steady shock wave, the conservation equations can be expressed as, ρ0 U = ρ(U − u) P0 + ρ0 U 2 = P + ρ(U − u)2 1 2 U ρ0 e0 + ρ0 U + P0 2 1 2 = (U − u) ρe + ρ(U − u) + P 2

2.2 Turbulent entropy criterion A recent study by Gavrilyuk and Saurel [13] invoked turbulent entropy (TE) considerations to develop a closure criterion, relating the TE to the micro-kinetic energy of the multi-phase mixture. The micro-kinetic energy of a mixture is a non-equilibrium concept related to a gradient in material velocity (u) between the different components of a mixture prior to the mixture reaching mechanical equilibrium. Therefore, the value of the TE along the mixture Hugoniot is zero since the system is, by definition, in mechanical equilibrium. This method attempts to address the shortcomings of the aforementioned SC criterion, which neglects momentum and energy exchange between the components of the mixture by considering the relaxation process of the mixture. The authors modify (7) as follows for each component, ei − e0i =

2.1 Single component criterion The first closure criterion that will be considered is based on the assumption that each component of the mixture must obey its respective single component (SC) Hugoniot constraints [9,10,12]. Therefore the closure criterion is that of a SC Hugoniot, where the three conservation equations (6) are reapplied to each component of the mixture separately. This criterion is represented by the expression (P + P0 ) (v0i − vi ) 2

criterion is questionable as it does not give a proper physical representation of the behavior of the system. When considering the validity of this closure criterion, it is important to consider the definition of a single material shock Hugoniot: a collection of equilibrium states that can be achieved through a single shock wave process referenced to an initial thermodynamic state in a given material. The nature of a heterogeneous system is such that the equilibrium state of each individual component will depend strongly on the presence of other components in the system and therefore each component should not be constrained to its respective individual Hugoniot. The use of the SC criterion (7) amounts to assuming that the momentum and energy of each component in the mixture can be conserved separately. However, in reality there must be an exchange of momentum and energy between the components of the mixture.

(6)

where U is the steady shock wave velocity, the subscript 0 refers to the initial mixture properties and the absence of a subscript refers to the post-shock relaxed state, where the mixture is in mechanical equilibrium. Combining (2) and (6) with an EOS for each component of the mixture results in a system that is under-constrained, meaning that a closure criterion would be required for the mixture to have a unique shock Hugoniot. This closure criterion could have a number of forms provided that it relates two of the variables from the governing equations in a physical manner. In the following discussion, three such criteria will be used as examples.

ei − e0i =

75

(7)

where the subscript i refers to an individual component in the mixture (component a or b) and vi is the specific volume of component i. Equation (7), expressed for each component, replaces the equation of conservation of energy (6c) in the governing equations. While this constraint has been employed by a number of authors, the validity of this

(P + P0 ) (v0i − vi ) − χ v −t vi 2

(8)

where t is the turbulent polytropic exponent defined by the space geometry (i.e., value of 3, 2 or 5/3 for velocity fluctuations in one, two or three dimensions, respectively) and χ is the TE of the mixture. The subscript added to the TE exponent, t , was added to differentiate this coefficient from the Grüneisen defined earlier and does not appear in the work of the original authors. Assuming pressure equilibrium between the components, the authors combine these Hugoniot equations (8) using a mixture rule (2) and an EOS (4) to generate a locus of intermediate states for the mixture on the P − v plane, dependent on χ . Technically, this curve is not a Hugoniot since it does not represent equilibrium states of the mixture. For various shock strengths, critical values of χ can be found such that the resulting locus of intermediate states for the mixture are tangent to the Rayleigh line of a shock wave of given strength. This is demonstrated graphically for two

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approximated using the Mie-Grüneisen EOS. This criterion was expressed by the following set of equations

point on mixture Hugoniot

P = PT a + PEa = PT b + PEb e0i PT i ∼ = v0i v0i βsi βsi PE = −1 βsi vi

Pressure (P)

tangency point (P∗ , v∗ , χ ∗)

Rayleigh line χ=χ χ=0 mixture Hugoniot

∗

χ=χ

∗∗

Specific Volume (v) Fig. 1 Illustration of the Gavrilyuk and Saurel [13] turbulent entropy criterion

different shock wave strengths in Fig. 1. Although the tangency condition is used to define a unique solution to the equations, there is no physical basis for this choice other than the desire for a unique solution. The mixture is assumed to relax from the tangency point to the actual mixture Hugoniot (χ = 0) according to a relationship derived from entropy considerations for constant specific heats and using the stiff gas EOS (4): γ γ ln P∗ + P∞,a va∗a ln P + P∞,a va a γ = γ (9) ln P + P∞,b vb b ln P∗ + P∞,b vb∗b where the asterisk (∗) denotes the pressure and specific volume at the tangency point, the lack of a subscript refers to the mixture Hugoniot and the subscripts a and b refer to the mixture components. Expression (9) represents the closure criterion for this method. By varying the shock strength (slope of the Rayleigh line), the entire mixture Hugoniot can be constructed. It should be noted that this theory attempts to describe a shock wave process using a methodology reminiscent of Chapman–Jouguet detonation theory. While the tangential constraint of the Chapman–Jouguet point has a physical explanation related to energy release and the propagation limits of detonations, the physical validity of this similar constraint on a shock wave in an inert mixture is not clear.

(10)

where 0i is the Grüneisen of component i, βsi is the isen tropic bulk modulus of component i, βsi is the first derivative of the isentropic bulk modulus of component i with respect to pressure at constant entropy, and the subscript T and E refer to the thermal and elastic contribution to pressure, respectively. For a system in mechanical equilibrium, the equality in (10a) used in combination with the mixture rules (2) provides a closure criterion for the governing equations (6). While βsi can be found for most materials, it is not trivial to calculate βsi , which means that βsi must be determined empirically by fitting the Murnaghan equation to the individual component Hugoniot in the P − v plane for each component. However the use of a Murnaghan equation for the elastic pressure component in this model will lead to difficulties in modeling soft materials such as liquids, for which the Murnaghan equation is not appropriate. The Kreuger and Vreeland [11] approach is similar to an earlier mixture method developed by Duvall and Taylor [16]. Duvall and Taylor determined a mixture Hugoniot for an approximate polyethylene-quartz mixture by combining the conservation of energy equation for a mixture (6c) with a Mie-Grüneisen EOS for each mixture component using an isothermal Murnaghan Equation as the reference curve such that, voi βT i i 1 − 1 + (ei − e0i ) (11) P= βT i βT i vi vi where βT i is the coefficient of isothermal compressibility of component i and βT i is the first derivative of the coefficient of isothermal compressibility of component i with respect to pressure at constant temperature. Once again, for a system in mechanical equilibrium, equating (11) for the two components of a mixture, combined with a mixture rule (2), provides a closure criterion for the governing equations (6). Similar to the Kreuger and Vreeland method [11], the use of an isothermal Murnaghan equation will limit the application of this method to single phase (solid–solid) mixtures.

2.3 Pressure equilibrium method Kreuger and Vreeland [11] took an alternate approach to determining a closure criterion for the system. In their study, pressure was defined as the sum of an elastic and a thermal (plastic) component. The elastic portion was expressed by a Murnaghan equation [17] and the thermal portion was

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3 Averaging models The two models that will be considered are the multicomponent models proposed by McQueen et al. [15] and Batsanov [14]. These models are simple in their approach and


3.1 Isothermal averaging method The method proposed by McQueen et al. [15] for determining the shock Hugoniot of a mixture is more involved than the model of Batsanov [14]. The former method involves first determining the 0K isotherm of each component in the mixture using the relation 0 dP − · P0K dv 0K v0 dP 0 2v0 (12) = · PH + + v − v0 2v0 0 dv H where the subscripts “0K ” and “H ” refer to the 0K isotherm and the shock Hugoniot, respectively. The individual component Hugoniot in the P − v plane that is used in (12) can be expressed by PH =

2 (v − v ) C0i 0i i [v0i − Si (v0i − vi )]2

(13)

where C0i and Si are parameters from a linear fit to Hugoniot data in the U − u plane of the form U = C0i + Si · u for mixture component i. The 0K isotherm of the mixture in the P − v plane is determined by combining the 0K isotherms of the individual components in the P − v plane according to the mixture rule in (2a), assuming that pressure is in equilibrium between the two components. The value of the Grüneisen for the mixture is found according to the mixture rule v0a v0b v0 = Xa + (1 − X a ) (14) 0 0a 0b This mixture isotherm is then used to determine the Hugoniot of the mixture via (12), solving the ODE for the Hugoniot pressure term. While the derivation of this method might seem rigorous, it treats the Hugoniot as a thermodynamic path that can be integrated, differentiated and reconstructed. A Hugoniot is not a thermodynamic path, but rather a collection of equilibrium states associated with a shock wave process in a material referenced to a specific initial state. Therefore, physically it is not correct to name the curve determined by integrating (12) using the mixture 0K isotherm a Hugoniot. This curve, while it may be a sufficient approximation to the actual mixture Hugoniot curve in some cases, is not a Hugoniot curve in the true physical meaning of the term. It can be shown for many mixtures that if this same mixture rule (2a) is simply applied directly to the individual compo-

10 0 K Isotherm Mixture Averaging Method Direct Averaging of P−v Hugoniots

9

Shock Velocity (km/s)

do not explicitly involve solving the conservation equations (6) to determine the shock Hugoniot for the mixture, but rather make use of mixture rules directly involving the individual component Hugoniots of each material to determine the Hugoniot of the mixture.

77

8 7 6 5 4 3 2 1

0

1

2

3

4

Material Velocity (km/s)

Fig. 2 A comparison of the 0K isotherm averaging method of McQueen et al. [15] and a direct pressure equilibrium specific volume averaging method to experimental data for a mixture of Epoxy (59.5%) and Spinel (40.5%) by volume [18] in the U − u plane

nent Hugoniots in the P − v plane (13) under an assumption of pressure equilibrium, without first calculating the 0K isotherm as in the method of McQueen et al. [15], the resulting mixture Hugoniot curves rarely deviate considerably from the results of the 0K isotherm mixing method (see Fig. 2). This simpler method amounts to assuming that the influence of temperature on the internal energy is a second order effect in condensed materials and does not need to be accounted for explicitly. From the comparison of these two methods, this simplification seems to be an accurate assumption for condensed materials in general. 3.2 Kinetic energy averaging method The realization of the practical equivalence of these two methods brings the discussion to the last model that will be considered, proposed by Batsanov [14]. The premise of the model is that the mixture is under pressure equilibrium and that an equilibrium material velocity of the mixture can therefore be determined by averaging the material velocities on the individual component Hugoniots according to a velocity based mixture rule. The mixture velocity is obtained for a given pressure from the following relation, u 2mix = X a u a2 + (1 − X a )u 2b

(15)

where u a and u b are the material velocities for components a and b, respectively, at a given pressure on their respective individual Hugoniots. For materials whose P − v Hugoniot is given by (13), the expression for the material velocity, u i , on the Hugoniot of component i as a function of pressure is

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78


given by the expression

C0i 4Si v0i P ui = 1+ −1 2 2Si C0i

(16)

Although this averaging method is not thermodynamically rigorous, it will be shown in the following section that it is fairly accurate nonetheless in reproducing experimental Hugoniot data for multi-component mixtures. Interestingly, this method gives results identical to the direct average taken of the individual component Hugoniots in the P − v plane as discussed in the previous section and shown in Fig. 2. Note that the averaging methods are used to determine a mixture Hugoniot in a given plane (P − v for the McQueen method [15], P − u for the Batsanov method [14]). In order to determine the mixture Hugoniots in another plane (i.e.: the U −u plane), it is necessary to use the governing conservation equations for a mixture given by (6). 4 Comparison of models In this section, a comparison of the SC criterion (7), the TE Method (9), and the kinetic energy averaging (KEA) method (15) will be made. The method of McQueen et al. [15] will not be discussed further since there is excellent agreement between this method and the KEA method (i.e., the results of the two methods would be indistinguishable on the subsequent figures). The KEA method is the simpler of the two methods to implement, as it does not involve either knowledge of the Grüneisen or integration. The closure criterion proposed by Kreuger and Vreeland [11], as well as the method of Duvall and Taylor [16], will not be considered further either, due to the fact that the criteria require fitting constants which are difficult to obtain for soft materials.

Therefore these methods are labor intensive and would not be accurate for mixtures involving liquids or soft materials due to the use of a Murnaghan EOS. The comparison of the methods will be made in both the U − u and P − v Hugoniot planes. There are several compilations of experimental shock Hugoniot data that are available for reference [18,19] however, a large number of these data sets are available without proper documentation of the scales of heterogeneities and the methods used to measure the Hugoniot data. For instance, some data sets indicate both the mass fraction (or volume fraction) with which they were mixed as well as the initial density of the mixture. However, using the mixture rule (1) with the individual component densities in those mass or volume fractions does not always correspond to the quoted mixture density. For instance, the epoxy-enstatite mixture is mixed nominally at a 60% volume fraction of epoxy, but the quoted initial mixture density is 2.017 g/cc, which suggests a volume fraction of 54.3% epoxy. Therefore, the comparison can only be made by assuming that either the quoted mixing fraction or the initial density is correct. Saurel et al. [12] assumed that the initial densities were more reliable and altered the volume fraction accordingly. This methodology has been used in these calculations as well for consistency. This demonstrates that a thorough data set varying the length scales and volume fraction loading for mixtures is needed in the literature as a basis for future studies. The parameters used for the model comparisons, whose values were taken from Saurel et al. [12], are shown in Tables 1 and 2. The comparison of the various models to the collection of data sets can be found in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22, where the results of the three methods are plotted against the data sets.

Table 1 The tabulated values of the parameters used in the calculations of the Hugoniots of the mixtures of interest Material

ρ0 (g/cc)

γ

P∞ (GPa)

C0 ( km/s)

S 1.44

Epoxy

1.185

1.43

2.43

5.30

2.80

Periclase

3.584

2.49

3.49

45.7

6.60

1.37

Spinel

3.622

0.62

1.62

141

7.867

0.78

Paraffin 1

0.917

1.87

2.87

3.427

3.12

1.47

Paraffin 2

0.917

1.2

2.2

4.0

3.12

1.33

1.2

2.2

141

4.04

1.23

3.007

1.14

2.14

40.6

5.21

1.03

Chlorine trifluoride

1.885

2.0

3.0

1.41

1.50

1.52

Aluminum

2.712

2.8

3.8

14

5.20

1.40

Water

1.000

1.5

2.5

1.3

1.40

1.43

Calcite

2.665

2.3

3.3

11

3.70

1.43

Fieldspar

2.550

0.8

1.8

25

4.20

0.80

Tungsten Enstatite

123

19.20


79

Table 2 The tabulated root mean squared error values for the comparison of experimental data sets to three methods used to determine the U − u Hugoniot of a mixture Mixture

Volume fraction (α)

Density (g/cc)

εrms SC

εrms TE

εrms KEA

Paraffin 2–tungsten

0.803

4.510

0.49

0.40

0.23

3

Alekseev et al. [10]


0.917

2.440

0.73

0.33

0.33

5


Paraffin 1–periclase

0.462

2.353

0.12

0.68

0.16

7

Marsh [18]

Paraffin 1–enstatite

0.447

2.073

0.05

0.34

0.10

9

Marsh [18]

Paraffin 1–feldspar

0.404

1.890

0.13

0.12

0.22

11

Figure

Reference

Adadurov et al. [20]

Epoxy–periclase

0.569

2.219

0.18

0.31

0.18

13

Marsh [18]

Epoxy–enstatite

0.543

2.017

0.17

0.22

0.10

15

Marsh [18]

Water–calcite

0.279

2.200

0.27

0.14

0.11

17

Kalashnikov et al. [21]

Epoxy–spinel

0.595

2.171

0.14

0.26

0.22

19

Marsh [18]

Chlorine trifluoride–aluminum

0.787

2.061

0.24

0.57

0.08

21

van Thiel [22]

The RMS error values are given in terms of the error in the shock velocity (U ) for each data point. The initial mixture densities quoted in the studies were assumed correct and the volume fractions were adjusted accordingly. The best fit to each data set is highlighted in bold letters

11

11

8 7 6 5 4 3 2 1

0

1

2

3

SC Criterion KEA Method Turbulent Entropy Method Experimental Data

10


9



10

4

5

6

9 8 7 6 5 4 3

1

2

Fig. 3 A comparison of experimental U − u Hugoniot data for a mixture of Tungsten (19.7%) and Paraffin (80.3%) by volume [10] and the calculated results of the SC Criterion, KEA Method, and TE Method

250

Pressure (GPa)

4

5

6

7

Fig. 5 A comparison of experimental U − u Hugoniot data for a mixture of Tungsten (8.3%) and Paraffin (91.7%) by volume [10] and the calculated results of the SC Criterion, KEA Method, and TE Method

4.1 Root mean squared error analysis

300 SC Criterion KEA Method Turbulent Entropy Method Experimental Data

200 150 100 50 0 0.05

3



0.10

0.15

0.20

0.25

Specific Volume (kg/m3)

Fig. 4 A comparison of experimental P − v Hugoniot data for a mixture of Tungsten (19.7%) and Paraffin (80.3%) by volume [10] and the calculated results of the SC Criterion, KEA Method, and TE Method

A root mean squared error analysis was performed for each model and data set as a means of comparison of the models. The root mean squared error was calculated with the formulas

n

1 εrms = [U M (u i ) − U E (u i )]2 n i=1 (17)

n

1 εrms = [v M (Pi ) − v E (Pi )]2 n i=1

where εrms is the root mean squared (RMS) error, n is the number of available data points for a given mixture, u i is the material velocity corresponding to experimental data point i, Pi is the pressure corresponding to experimental data point

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80

O. E. Petel, F. X. Jetté 8

180

140


Pressure (GPa)


120 100 80 60 40 20 0 0.15

0.20

0.25

0.30

0.35

0.40

0.45

6 5 4 3


1

3


7


50

Pressure (GPa)

8

6 5 4

40 30 20 10

0

1

2

0 0.25

3

0.30

Fig. 7 A comparison of experimental U − u Hugoniot data for a mixture of Periclase (53.8%) and Paraffin (46.2%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method

0.40

0.45

0.50

Fig. 10 A comparison of experimental P − v Hugoniot data for a mixture of Enstatite (55.3%) and Paraffin (44.7%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method 7

60

40



50

30 20 10 0 0.25

0.35



Pressure (GPa)

2

Fig. 9 A comparison of experimental U − u Hugoniot data for a mixture of Enstatite (55.3%) and Paraffin (44.7%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method

9


0


Fig. 6 A comparison of experimental P − v Hugoniot data for a mixture of Tungsten (8.3%) and Paraffin (91.7%) by volume [10] and the calculated results of the SC Criterion, KEA Method, and TE Method

3


7

0.30

0.35

Specific Volume

0.40

0.45

(kg/m3)

6


5

4

3 0.0

0.5

1.0

1.5

2.0

2.5

3.0


Fig. 8 A comparison of experimental P − v Hugoniot data for a mixture of Periclase (53.8%) and Paraffin (46.2%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method

Fig. 11 A comparison of experimental U −u Hugoniot data for a mixture of Feldspar (59.6%) and Paraffin (40.4%) by volume [20] and the calculated results of the SC Criterion, KEA Method, and TE Method

i, U is the shock velocity, and v is the specific volume, where the subscripts M and E refer to values calculated with the model of interest and values measured in experiment, respectively. The RMS error values are listed in Tables 2 and 3 for

the comparison of the three methods to the experimental data sets in the U − u and P − v plane, respectively. All three methods provide reasonable agreements with the experimental data, however the SC criterion and the KEA

123


81 10


20


9


Pressure (GPa)

25

15 10 5

8 7 6 5 4 3

0 0.30

0.35

0.40

0.45

0.50

2

0.55

0

0.5

1

1.5

Specific Volume (kg/m3) Fig. 12 A comparison of experimental P − v Hugoniot data for a mixture of Feldspar (59.6%) and Paraffin (40.4%) by volume [20] and the calculated results of the SC Criterion, KEA Method, and TE Method

70


9 8

Pressure (GPa)


3

3.5

4

4.5

80

10

7 6 5 4


60 50 40 30 20 10

3 0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.25

4.5

0.30

0.35

Fig. 13 A comparison of experimental U −u Hugoniot data for a mixture of Epoxy (56.9%) and Periclase (43.1%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method

0.45

0.50

Fig. 16 A comparison of experimental P − v Hugoniot data for a mixture of Epoxy (54.3%) and Enstatite (45.7%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method 10

100 90



80 70 60 50 40 30 20


8 7 6 5 4 3

10 0 0.25

0.40



Pressure (GPa)

2.5

Fig. 15 A comparison of experimental U −u Hugoniot data for a mixture of Epoxy (54.3%) and Enstatite (45.7%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method

11

2

2


0.30

0.35

0.40

0.45

0.50


2

0

1

2

3

4


Fig. 14 A comparison of experimental P − v Hugoniot data for a mixture of Epoxy (56.9%) and Periclase (43.1%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method

Fig. 17 A comparison of experimental U − u Hugoniot data for a mixture of Water (27.9%) and Calcite (72.1%) by volume [21] and the calculated results of the SC Criterion, KEA Method, and TE Method

method each provide the best fit to experimental data for roughly half of the data set. The TE method rarely provides the best fit to a data set in either plane. While the SC criterion and KEA method are the two superior methods, it seems

that the KEA method has the tightest bounds on RMS errors among the methods considered, as it rarely provides the worst approximation to an experimental data set. The upper bounds of the RMS error in the U −u Hugoniot for the three methods

123

82

O. E. Petel, F. X. Jetté 8

70

7


50


Pressure (GPa)

60

40 30 20 10 0 0.20

0.25

0.30

0.35

0.40

Specific Volume

0.45

6 5 4 3 2 1 0.0

0.50


0.5

1.0

(kg/m3)

1.5

2.0

2.5

3.0

3.5


Fig. 18 A comparison of experimental P − v Hugoniot data for a mixture of Water (27.9%) and Calcite (72.1%) by volume [21] and the calculated results of the SC Criterion, KEA Method, and TE Method

Fig. 21 A comparison of experimental U −u Hugoniot data for a mixture of Chlorine trifluoride (78.7%) and Aluminum (21.3%) by volume [22] and the calculated results of the SC Criterion, KEA Method, and TE Method

10

8

40

7 6 5 4

30 25 20 15 10

3 2


35

Pressure (GPa)


45


9

5

0

1

2

3

4


0 0.25

0.30

0.35

0.40

0.45

0.50


Fig. 19 A comparison of experimental U − u Hugoniot data for a mixture of Epoxy (59.5%) and Spinel (40.5%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method 90

Fig. 22 A comparison of experimental P − v Hugoniot data for a mixture of Chlorine trifluoride (78.7%) and Aluminum (21.3%) by volume [22] and the calculated results of the SC Criterion, KEA Method, and TE Method

80

Pressure (GPa)

error values for multiple data sets is not quite meaningful, the bounds of the RMS error indicate that the KEA method provides a slightly better approximation to the data sets.


70 60 50 40 30

5 Conclusion

20 10 0 0.25

0.30

0.35

0.40

0.45

0.50


Fig. 20 A comparison of experimental P − v Hugoniot data for a mixture of Epoxy (59.5%) and Spinel (40.5%) by volume [18] and the calculated results of the SC Criterion, KEA Method, and TE Method

are 0.73, 0.68, and 0.33 for the SC criterion, the TE method, and the KEA method, respectively. The upper bounds of the RMS error in the P − v Hugoniot for the three methods are 0.025, 0.029, and 0.012 for the SC criterion, the TE method, and the KEA method, respectively. Although averaging RMS

123

Determining a closure criterion for the governing conservation equations in equilibrium models of multi-component systems has been the subject of a number of studies from which a variety of criteria have been proposed. However, the difficulties associated with solving the model equations, determining a suitable closure criterion, and obtaining accurate EOS parameters seem to render their use quite laborious for a simple determination of the shock Hugoniot of multicomponent mixtures in light of the evidence presented in the current study. It has been shown that a simple model, the KEA method of Batsanov [14], provides a simpler and often more accurate solution to determining the shock Hugoniots of mixtures


83

Table 3 The tabulated root mean squared error values for the comparison of experimental data sets to three methods used to determine the P − v Hugoniot of a mixture Mixture

Volume fraction (α)

Density (g/cc)

εrms SC

εrms TE

εrms KEA


0.803

4.510

0.008

0.011

0.004

4



0.917

2.440

0.025

0.016

0.011

6


Paraffin 1–periclase

0.462

2.353

0.003

0.018

0.005

8

Marsh [18]

Paraffin 1–enstatite

0.447

2.073

0.009

0.008

0.012

10

Marsh [18]

Paraffin 1–feldspar

0.404

1.890

0.003

0.009

0.006

12

Adadurov et al. [20]

Epoxy–periclase

0.569

2.219

0.005

0.008

0.006

14

Marsh [18]

Epoxy–enstatite

0.543

2.017

0.006

0.008

0.006

16

Marsh [18]

Water–calcite

0.279

2.200

0.008

0.004

0.005

18

Kalashnikov et al. [21]

Epoxy–spinel

0.595

2.171

0.005

0.010

0.008

20

Marsh [18]

Chlorine trifluoride–aluminum

0.787

2.061

0.010

0.029

0.006

22

van Thiel [22]

Figure

Reference

The RMS error values are given in terms of the error in the specific volume (v) for each data point. The initial mixture densities quoted in the studies were assumed correct and the volume fractions were adjusted accordingly. The best fit to each data set is highlighted in bold letters

than equilibrium closure models. The overall accuracy of the collection of these models, in spite of their physical shortcomings, is evidence of the insensitivity of the shock Hugoniot of a mixture to small variations in internal energy for condensed phase materials. Since solving the system of governing equations for the mixture does not provide an improvement over much simpler averaging methods, the more complicated approaches are much less attractive. In the case of steady systems in mechanical equilibrium, it has been demonstrated that it is not necessary to explicitly solve the governing conservation equations of the mixture to accurately determine its shock Hugoniot. Acknowledgments The authors would like to thank Andrew Higgins for constructive discussions and useful comments.

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7. Kenyon, D.E.: Thermostatics of solid–fluid mixtures. Arch. Ration. Mech. Anal. 62, 117–129 (1976) 8. Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F., Stewart, D.S.: Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids 13(10), 3002–3024 (2001) 9. Dremin, A.N., Karpukhin, I.A.: Method of determination of shock adiabat of the dispersed substances (in russian). Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 1(3), 184–188 (1960) 10. Alekseev, Yu.F., Al’tshuler, L.V., Krupnikova, V.P.: Shock compression of two-component paraffin–tungsten mixtures. J. Appl. Mech. Tech. Phys. 12(4), 624–627 (1971) 11. Krueger, B.R., Vreeland, T.: A Hugoniot theory for solid and powder mixtures. J. Appl. Phys. 69(2), 710–716 (1991) 12. Saurel, R., Le Metayer, O., Massoni, J., Gavrilyuk, S.: Shock jump relations for multiphase mixtures with stiff mechanical relaxation. Shock Waves 16(3), 209–232 (2007) 13. Gavrilyuk, S.L., Saurel, R.: Rankine–Hugoniot relations for shocks in heterogeneous mixtures. J. Fluid Mech. 575(1), 495–507 (2007) 14. Batsanov, S.S.: Effects of explosions on materials: modification and synthesis under high-pressure shock compression. Springer, Berlin (1994) 15. McQueen, R.G., Marsh, S.P., Fritz, J.N.: Hugoniot equation of state of twelve rocks. J. Geophys. Res. 72, 4999–5036 (1967) 16. Duvall, G.E., Taylor, S.M.: Shock parameters in a two component mixture. J. Compos. Mater. 5(2), 130–139 (1971) 17. Murnaghan, F.D.: The compressibility of media under extreme pressures. Proc. Natl. Acad. Sci. 30, 244–247 (1944) 18. Marsh, S.P.: LASL Shock Hugoniot Data. University of California Press, California (1980) 19. Trunin, R.F.: Shock Compression of Condensed Materials. Cambridge University Press, Cambridge (1998) 20. Adadurov, G.A., Dremin, A.N., Ryabinin, Yu.N.: About behaviour of some substances at shock compression. Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki 4, 115–119 (1964) 21. Kalashnikov, N.G., Pavlovskii, M.N., Simakov, G.V., Trunin, R.F.: Dynamic compressibility of the minerals of calcite group. Izvestiya Akademii Nauk SSSR, Fizika Zemli 2, 23–29 (1973) 22. van Thiel, M.: Compendium of shock wave data. Technical report, Lawrence Livermore Laboratory Report UCRL-50108 (1977)

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