Adv. Studies Theor. Phys., Vol. 7, 2013, no. 5, 207 - 228

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Energetic Measures of Effectiveness

John F. Moxnes∗, Tomas L. Jensen and Erik Unneberg

Norwegian Defence Research Establishment (FFI) P. O. Box 25, NO-2027 Kjeller, Norway

Copyright © 2013 John F. Moxnes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this work we study different energetic measures such as calorimetric energy of explosion, work of explosion, work of Carnot, enthalpy of explosion and free energy of explosion of aluminized explosives. For highly aluminized explosives with around 50 % aluminum and 50 % RDX we find that the work of Carnot is of the same size as the work of explosion. In general, we find that due to irreversible processes, none of the commonly used energy concepts in the literature are good measures of effectiveness. Mathematical relations that can be used to calculate the mechanical work, the work of explosion and the work of Carnot are presented. Keywords: RDX, Aluminum, Reactions, Work of explosion, Mechanical work, Work of Carnot 1 Introduction The addition of metal particles to energetic materials is a well-known method to improve their efficiency. Aluminum (Al) powders are now widely used in pyrotechnics, rocket propellants, fuel-air and aluminized explosives. In propellants Al is used to increase thrust, while in military explosives to enhance air blast, incendiary effects, and bubble energies in underwater weapons. In general, cost and environmentally benign products are additional advantages of aluminized energetic materials.

* Corresponding author; e-mail: john-f.moxnes@ffi.no

208 J. F. Moxnes, T. L. Jensen and E. Unneberg

Even for a closed space (reaction chamber without air) the addition of Al to energetic materials such as RDX strongly influences the reaction gases. The temperature of the gases becomes higher, but the number of gas molecules decreases. After volume expansion of the reaction gases to the ambient pressure, the temperature of the reaction gases is typically much higher than in explosives without Al. For a semi open or open space additional energy can be achieved if Al reacts with air oxygen. In this article we perform a thermodynamical analysis for a closed space as well as for an open space. However, a thermodynamical analysis gives no more than an upper border for what can be achieved for different energies. In this way a thermodynamical analysis can only be used as a first step in a more advanced modeling including reaction kinetics and mass transport.

Energy analysis has been applied to the hydrodynamic theory of detonation of gases. Exergy analysis compensates loss due to irreversibility of the system [1]. Due to the slow reaction rates and irreversible processes during utilization of aluminized explosives, uncertainty exists to which thermodynamic quantity that should be used as a good measure of effectiveness. An overview of methods for determining explosives performance has been given by Scilly [2]. The calorimetric energy (heat) of explosion (detonation) is easy to calculate when the chemical composition of the explosion products is known, and this quantity has frequently been used as a measure of the blast energy. It is defined to be the difference in the internal energy at constant pressure and temperature (we call it –ΔUP,T ≥0). It can be found experimentally (although approximately) by performing measurements in a calorimetric bomb, which is a closed chamber, with the explosive within, placed in a heat bath.

The theoretical maximum work an explosive can yield during constant pressure and temperature is theoretically restricted from above by minus the change in Gibbs free energy (the change is defined as the Gibbs free energy of the explosive minus the Gibbs free energy of the explosive products, we designate it –ΔGP,T). It has for a long time been used as a measure of blast energy [3]. Experience has indicated that –ΔGP,T can be used as a somewhat adequate measure of the energy actually available for explosive yield [3]. For aluminized explosives, however, –ΔGP,T is neither experienced in practice nor theoretically.

In the first step of an explosive reaction the explosive reacts at approximately constant volume and internal energy to form reaction products. The entropy, however, increases significantly due to irreversible processes, but no external work is utilized. In the next step reaction products expand adiabatically and to a good approximation also isentropic. The work of explosion is defined as the line integral of the pressure with respect to volume during the adiabatic

Energetic measures of effectiveness 209 expansion to the state where the pressure of the reaction products equals the external (ambient) pressure. The brisance ability of the explosives is associated with the work of explosion. Theoretical calculations show that it should peak at around 30 % Al and 70 % RDX. The peak value is around 40 % larger than for pure RDX per unit mass. By comparing the expansion velocity (and energy) of rapidly expanded cylinders it was found that at a volume expansion of a factor nine the energy was lower than theoretically calculated. It was even lower than for pure RDX per unit mass [4].

When the work of explosion has been utilized, i.e. after the adiabatic expansion of gases to the ambient pressure, the temperature of the reaction products is usually much higher than the temperature of the surroundings. This temperature actually increases strongly with the Al content of the explosive and varies from 600 K to 2300 K. The Carnot process is in principle the optimum process that makes even more mechanical work available (work of Carnot) through a heat machine running between these warm reaction gases and the cold gases of the surroundings. Obviously, the ideal Carnot process is in practice not reached. However, in practice surrounding air could be heated up (thermobarically) by the warm reaction gases to higher pressures. In principle a part of the work of Carnot can thus be utilized as additional work on structures by the heated gases of the surroundings. The sum of the explosion work and the work of Carnot is the maximum mechanical work that can be achieved by the reaction products of the explosive. This work is less than –ΔGP,T due to irreversible processes, especially from the first to the second state mentioned above. However, when surrounding air in addition is allowed to react chemically with reaction products from the explosive (for instance with Al), the calorimetric energy, the work of explosion and the work of Carnot can increase even more per unit mass of explosive.

The work of explosion and the work of Carnot are difficult to calculate since both the equation of state and calorimetric equation of state are needed to compute the line integral. The explosion work is also difficult to measure but Gurney energies have been used [4]. As an estimate, the negative change in the explosion enthalpy (-ΔHP,T) is often used as a substitute. -ΔHP,T accounts for the explosion work. However, it also includes the change in the internal energy from the state reached after adiabatic expansion to the state where the reaction products have the same temperature and pressure as the surroundings) (the so-called thermal energies). Thus, the change in enthalpy includes a fraction of the work of Carnot. Thereby –ΔHP,T is numerically too large to give a good - although feasible - approximation to the explosion work. The change in enthalpy is also difficult to measure, and the calorimetric energy of explosion (detonation) is usually a good approximation of the change in enthalpy. Theoretically, different

210 J. F. Moxnes, T. L. Jensen and E. Unneberg results have been reported due to a diversity of chosen freezing temperatures of the reaction products. Below the freezing temperature the reaction products are assumed to be unaltered (i.e. frozen due to low kinetics). In general, temperatures from 1000 K to 2100 K have been chosen to fit data. An issue is the variation of the freezing temperature with different types of reaction products and weight percent of Al.

In Section 2 below we study the rules of thermodynamics fundamentally. We study energy concepts such as calorimetric energy (heat) of explosion, work of explosion, enthalpy of explosion and Gibbs free energy of explosion. Useful mathematical relations are revealed. In Section 3 we compare theoretical results for different mixtures of RDX and Al. Conclusions are drawn in Section 4. 2 Changes in energy during reaction

The Gibbs free energy G, the Helmholtz free energy A and the enthalpy H are well known and defined by

, ,def def def

G U PV T S A U T S H U PV= + − = − = + (2.1)

where “def” means definition. During explosion we assume as a first reaction step (state 1 to state 2) that the internal energy U and the volume V are constant while

0Q∆ = (the process is adiabatic but the entropy may still change due to irreversibility).

Assume that the second step (from state 2 to state 3) is an adiabatic expansion ( 0Q∆ = ) (irreversible changes are still possible) to the state where the total pressure of the reaction products equals the outer pressure 3 3,roomP P P T T= = = . Finally, assume cooling (from state 3 to state 4) in a way that the temperature and pressure become the same as in state 1:

4 4,room roomP P P T T T= = = = . We have from state 1 to state 4 that

, , , ,

, ,

P T P T room room P T P T room

P T P T room

G U P V T S A P VH U P V

∆ = ∆ + ∆ − ∆ = ∆ + ∆

∆ = ∆ + ∆ (2.2)

The calorimetric energy of the explosive is defined to be ,P TU . To a good approximation we have that

Energetic measures of effectiveness 211

, , , , , ,

, , ,

P T P T room room P T P T room P T P T

P T P T room P T

G U P V T S U T S AH U P V U

∆ = ∆ + ∆ − ∆ ≈ ∆ − ∆ = ∆

∆ = ∆ + ∆ ≈ ∆ (2.3)

The mechanical work is often of interest since it correlates with the ability of the explosive to damage structures. To study the mechanical work more fundamentally, consider a closed chamber (system) with volume V. The first fundamental law of thermodynamics states that for a closed system

mod

un unU W W Q W U W Q∆ = ∆ + ∆ + ∆ ⇒ −∆ = −∆ + ∆ + ∆ (2.4)

where “mod” means model assumption. W−∆ is the useful external work (we choose sign such that the external work done by the surroundings on the system is defined to be positive), unW∆ is the useless work, U∆ is the change in internal energy, and Q∆ is the heat change (defined to be positive during influx of heat and negative during outflux). It follows that the external useful work equals the change in the internal energy, plus a work due to the useless work, plus a term due to heat change. To fix the ideas we set un roomW P V∆ = − ∆ , where roomP is the external pressure (pressure in the room) and V∆ is the volume change.

Next, the second fundamental law of thermodynamics states that for a closed system (no mass flux)

modQ T S∆ ≤ ∆ (2.5)

where S is the entropy and T is the temperature of the closed system. For reversible changes the equality sign applies by definition. From (2.4) and (2.5) it follows that

roomW U P V T S−∆ ≤ −∆ − ∆ + ∆ (2.6)

At constant pressure and temperature the equations (2.4), (2.5) and (2.6) give

, , , ,

, , , , ,

0

, ,

( ) ( )

0

P T P T P T P T

room P T P T P T P T P T

P T P T

G U PV TS U P V T SP V W Q P V T S W Q T S W

W G≤

∆ = ∆ + ∆ −∆ = ∆ + ∆ − ∆

= − ∆ + ∆ + ∆ + ∆ − ∆ = ∆ + ∆ − ∆ ≤ ∆

⇒ ≤ −∆ ≤ −∆

(2.7)

The mechanical work is thus always less than or equal to –ΔGP,T . Notice that

212 J. F. Moxnes, T. L. Jensen and E. Unneberg

0

, , , ,P T P T P TH G T S Q T S≤

∆ = ∆ + ∆ ∆ ≤ ∆ (2.8)

The change in Gibbs free energy is negative and the heat flux is also typically negative (that means out of the closed system). Accordingly, it is impossible to decide whether the change in the enthalpy is larger or smaller than the change in Gibbs free energy. But for most practical cases , 0P TS∆ ≥ , which means that the change in the enthalpy is larger than the change in Gibbs free energy.

Next, we will calculate the mechanical work more explicitly. From state 1 to state 2 no work is performed. The volume of the reaction products expands adiabatically from state 2 to state 3. This work of explosion eW is by definition given by

( ) 0def

eroomW P P V∆ = − ∆ ≥ (2.9)

During adiabatic expansion we have that Q∆ =0, and we have from equation (2.4) when using that un roomW P V∆ = − ∆

room roomW U P V Q U P V∆ = ∆ + ∆ −∆ = ∆ + ∆ (2.10)

The work of explosion is the negative work, to read

( )eroom roomW P P V W U P V U P V∆ = − ∆ = −∆ = −∆ − ∆ ⇒ ∆ = − ∆ (2.11)

Assume that we have an equation of state P = ( , )P V T , and a calorimetric

equation of state ( , ) ( , )U U V T T T V U= ⇔ = . Equation (2.11) can be written as a differential equation for the adiabatic relation ( ), ( )ad adU U V P P V= = , to read

( , ( , ) ( , ( , )adad ad ad

ead

room

dUU P V T V U V P V T U VdV

dUdW PdV dV

∆ = − ∆ ⇒ = −

= − −

(2.12)

As an example, assume an ideal gas with an ideal calorimetric equation of state. This means that ( )( , ) / , / /v v vU T V mc T T U mc P nRT V nRU mc V= ⇒ = = = , where m is the mass and vc is the specific heat at constant volume and n is the

Energetic measures of effectiveness 213 number of moles of gas molecules. The pressure of the expansion products reaches the pressure 3 roomP P= at state 3. The temperature of the reaction products at state 3 is called 3T . This temperature is found by using that 3 roomP P= .This gives from state 2 to state 3 for an ideal gas

( ) /( )2 3 2 3 2 3

( ) /( , ( , ))

/ / /v

ad adad

vnR mc

dU V nRU Vp V T U VdV mc

U U V V T T−

= − = −

⇒ = =

(2.13)

More generally, we achieve from (2.12)

( ) ( )

( )( ) ( )

( ) ( )( ) }{

2 3 3 2 1 3 3 1

3 3 3 3 3 3

3 3

( ) ,

, , ( ) , , ( ) ,

( ) ,

( ), , ( ) , ( ) , , ( )

eead

room room room room

room ad ad

ad

ad ad ad ad

dUdW P W P U U P V V U U P V VdV dV

P V T V U V T T V U V

U V U

U U V T T V U V P P V P V T V U V

= − − ⇒ = − − − = − − −

= =

=

= = = =

(2.14)

The temperature 3T is often larger than roomT . In that case a Carnot process can be used to calculate additional mechanical work from the reaction products by a heat machine. This is the second part of the mechanical work. We call it the work of Carnot cW .

To study the work of Carnot, we will first apply a simplified analysis. We consider two infinite reservoirs, one with temperature 3T (hot) and one infinite reservoir with a lower temperature roomT . Consider that the machine simply as a box with some stuff. Assume that a Carnot machine is working between these two reservoirs. During a single cyclic process of a Carnot machine the temperature and entropy of the box (machine) can be drawn in a T/S diagram as a closed path. Let S be along the horizontal axis and T along the vertical axis. Without loss of generality assume that the process is reversible. In principle this can be achieved since we are free to choose the stuff in the machine and also the time period for one cycle. During a reversible cycle the work is given by

214 J. F. Moxnes, T. L. Jensen and E. Unneberg

( )0 0

0

defc

dS dS

W W dU dQ dU TdS TdS TdS≥ ≤

=

= − = − + = − + = +∫ ∫ ∫ ∫ ∫

(2.15)

This integral is graphically given as the area enclosed of the path in the T/S diagram. The heat into the machine (box) from the hot reservoir is given by

0

c

dS

Q TdS≥

= ∫ (2.16)

Let us follow an arbitrary path in the T/S diagram. When dS is positive along the path it follows (due to reversibility) that the heat flux is positive, i.e. into the box. The mathematical problem is to find the form (configuration) of the path in the T/S diagram such that the efficiency is a maximum. That means that we seek the maximum of the work that the machine applies on the surroundings for a given amount of heat taken from the hot reservoir. A restriction imposed on the mathematical problem is that the maximum and minimum temperature traced out by the path is

3T and roomT , respectively. The efficiency η , which is to be

maximized, is given mathematically from equation (2.15) and (2.16) as

30 0 0 0 0

/ / 1 / ,def

c croom

dS dS dS dS dS

W Q TdS TdS TdS TdS TdS T T Tη≥ ≤ ≥ ≤ ≥

= = + = + ≤ ≤

∫ ∫ ∫ ∫ ∫

(2.17)

Thus, the maximum efficiency is achieved if 0dS

TdS≤∫ is as small as possible

while 0dS

TdS≥∫ is as large as possible. However, the two integrals are obviously

not independent. The integral along any arbitrary path in the T/S diagram may be described mathematically as a sum of integrals over many small loops of a given type: the small loops have vertical lines at the point where the change in entropy changes sign. Then it is quite clear that the maximum value of the efficiency for arbitrary loops is reached if the path is horizontal along

3T when 0dS ≥ , and horizontal along roomT when 0dS ≤ . Then also the final loop in the T/S diagram is a rectangular path. This closed curve, corresponding to a rectangular path in the T/S diagram is called the Carnot process (or cycle or path). Then it follows directly from the area of the rectangle that ( ) ( )3 3 3/ 1 /room roomT T T T Tη = − = − . Thus

Energetic measures of effectiveness 215 the work achieved due to heat extraction from the hot reservoir is simply given by the well known formula

3

1c c roomTW QT

= −

(2.18)

cQ is the heat going into the Carnot machine during one Carnot cycle. But this is also the heat leaving the hot reservoir of the reaction products during one cycle.

Our hot reservoir of reaction products is not infinite, and we cannot apply equation (2.18) directly since the temperature of the hot reservoir would decrease during every finite Carnot cycle. Thus we find it mathematically correct to define what we would call an infinitesimal Carnot cycle. For an infinitesimal Carnot cycle (process) we have

3

1c c roomTW QT

∆ = ∆ −

(2.19)

The temperature in the hot reservoir (reaction products) will decrease for each cycle. We can apply the infinitesimal cycle an infinite number of times.

cQ∆ is the heat soaked out of the hot reservoir of reaction products for each cycle. We assume that this is done by heat conduction only. Then we can write

cU Q Q∆ = ∆ = −∆ , where U∆ now is the change of the internal energy of the reservoir of reaction products at constant volume. Thus we have that

( )3 3

3 31 ( ) ( ) ( ) ( )'room room

T Tc c croomroom room roomT T

TW dW dQ U T U T T S T S TT

= = − = − − − ∫ ∫ (2.20)

Consider again as an example the ideal calorimetric equation of state for

the reaction products: vU mc T= . Inserting into (2.20) gives

( )

( ) ( )

33

1 1/ 1/3 2 2 3 2 2/ , /

cv room v room

room

room room

TW mc T T mc T LnT

T T P P V V P Pγ γ− −

= − −

= =

(2.21)

The mechanical work becomes more generally from equation (2.14) and equation

216 J. F. Moxnes, T. L. Jensen and E. Unneberg (2.20)

( ) 2 3 3 2 3 4 3 4

1 3

e

e room room

c

e

Thermal energyWe c

room roomMechanical work Work of explosion Work of Carnot T T T TWork of explosion W

Work of Carnot W

W

W W W U U P V V U U T S S

U U

≈

= ==

=

≈

= + = − − − + − − −

= −

( )

( )

13

,

3 1 3 4 3 4

1 4 3 2

e room room

c

P T

Thermal energy

room roomT T T TWork of explosion W H

Work of Carnot W

Calorimetric energy of explosion

room

H

P V V U U T S S

U U P V V

= == =−∆

=

−∆

− − + − − −

= − − −

( ) ( ) ( )3 4 1 4 3 1 3 4room room roomT S S U U P V V T S S− − = − − − − −

(2.22)

Notice that the mechanical work W is lower than the change in internal energy by two terms. One of them is due to the work against the outer pressure. This term is usually small, and can be neglected. A second term is due to the entropy change and may be significant.

Due the nature of the explosive and the reaction steps 1-4, the change in entropy from state 1 to state 3 is called

13 3 1S S S∆ = − . Ideally this term is zero. Equation (2.22) then becomes

( ) ( )1 3 1 1 13 , 13

00

( ) ( ) ( ) ( )room room room room P T roomW U T U T P V V T S T S S T G T S≥≥

= − − − − + ∆ − = −∆ − ∆

(2.23)

Usually 13 3 1 0S S S∆ = − ≥ . (2.23) shows that the mechanical work W (explosive

work + work of Carnot) is lower than ,P TG−∆ by the term

13roomT S∆ . The reason for this is that from state 1 to state 3 0 Q T S= ∆ < ∆ .

Theoretically, the mechanical work can be as high as –ΔGP,T. However,

often the work of explosion is more of interest for engines that use a power stroke. Say that we calculate the change in free energy and enthalpy during constant pressure and entropy. We have that

Energetic measures of effectiveness 217

, , ,

, , ,

P S P S room P S room

P S P S room P S

G U P V S T A P VH U P V G S T

∆ = ∆ + ∆ − ∆ = ∆ + ∆

∆ = ∆ + ∆ = ∆ + ∆ (2.24)

It follows that

roomW U P V−∆ ≤ −∆ − ∆ (2.25)

and that

, , ,

, , , , ,0

, , ,

( ) ( )

0

P S P T P T

room P S P S P S P S P S

P S P S P S

G U PV TS U P V S TP V W Q P V S T W Q S T W S T

W G S T H≤

∆ = ∆ + ∆ −∆ = ∆ + ∆ − ∆

= − ∆ + ∆ + ∆ + ∆ − ∆ = ∆ + ∆ − ∆ ≤ ∆ − ∆

⇒ ≤ −∆ ≤ −∆ − ∆ = −∆

(2.26)

Therefore the work is always less than or equal to minus the change in enthalpy at constant pressure and entropy. However, we see from equation (2.22) that the change in enthalpy during constant pressure and entropy is the work of explosion during reversible changes. Then we can (ideally) calculate the mechanical work by the change in the Gibbs free energy during constant pressure and temperature, and the work of explosion by the change in the enthalpy during constant pressure and entropy, to read

, , , ,, , , ( )e c eP T P S P T P SW G W H W W W G H reversible processes= −∆ = −∆ = − = −∆ + ∆ (2.27)

In practice heat leakage, friction and irreversible changes are involved. For processes without friction and heat leakage we have found that

, 13 13, eP T roomW G T S W H= −∆ − ∆ = −∆ . However, the calculations at state 3 are

somewhat difficult to carry out, as the line integral of the pressure is involved. We believe that for most situations with explosive materials 2 3S S≈ . By applying this approximation we reach some simple relations that make it easy to calculate the work of explosion, the work of Carnot and the mechanical work, to read

( ) ( )

, 13 , 12

13 12' 1 2

, ( )

, , , ( )

, ( )

P T room P T room

defe

room room

c e

W G T S G T S a

W H H H P P S S H P P S S b

W W W c

= −∆ − ∆ ≈ −∆ − ∆

= −∆ ≈ −∆ = = = − = =

= −

(2.28)

Firstly, the entropy in state 2 is easily calculated as neither the volume nor

218 J. F. Moxnes, T. L. Jensen and E. Unneberg the internal energy will change from state 1 to state 2. Next, the enthalpy 12'H−∆ is found and inserted into the equation for the work of explosion in (2.28b). However, this “virtual” state is not reached during the explosive process. Notice that state 3 is not found.

The work of explosion depends on the equation of state through the line integral of the pressure. It follows that it depends on the equation state. However, what is lost in calculations for the work of explosion, for instance due to an ideal gas assumption, will more or less be gained for the work of Carnot. Then the mechanical work which is the sum of the work of explosion and the work of Carnot, will not be very dependent on the equation of state.

Assume that we have two different types of engines for power production. One of them utilizes both the work of explosion and the work of Carnot, and the other one utilizes only the work of explosion through a power stroke. The efficiencies of the engines become

, , 13 , 12

, , , , ,

, 13 12'

, , , , ,

1: , :

: , : (

def defP T P T room P T room

P T P T P T P T P T

e edef defP S

p pP T P T P T P T P T

G G T S G T SW Wideally realisticG G G G G

H H HW Wideally realistic Power strokG G G G G

η η

η η

−∆ −∆ − ∆ −∆ − ∆= = = = = ≈−∆ −∆ −∆ −∆ −∆

−∆ −∆ −∆= = = = ≈ −−∆ ∆ −∆ −∆ −∆

)e

(2.29)

3 Theoretical results with RDX /Al mixtures

The reactions are studied by application of the NASA Glenn’s computer program Chemical Equilibrium with Application, version 2 (CEA2) [5].

First of all we study the chemical composition of the gases in state 2, 3 and 4 at various contents (weight percent) of Al. In state 2 the reaction is changed from state 1 during constant volume and energy. The temperature is around 4000 K as seen in Figure 1. The maximum is achieved at around 15-30 % Al. From state 2 to state 3 the volume expansion of the gases is adiabatic but not completely isentropic. The temperature decreases to 500-2500 K. Now an Al content in the region 45-60 % gives the highest temperature. During the transition from state 3 to state 4 the pressure is constant. The temperature in state 4 is the room temperature.

Energetic measures of effectiveness 219

Figure 1. T/S-diagram. The entropy and temperature during the states 2, 3, and 4 for various RDX/Al weight ratios.

In Figure 2 we see the reaction products (from RDX/Al compositions) in state 4 as a function of the Al content. When pure RDX (i.e. 0 wt % Al) reacts, nitrogen (N2), water (H2O), carbon dioxide (CO2) and solid carbon (C) are formed in a molar distribution which is in stoichiometric accordance with the total reaction:

RDX (C3H6O6N6) → 3 N2 + 3 H2O + 3/2 CO2 + 3/2 C

When Al is introduced, the available oxygen atoms originating from RDX

will be consumed in the oxidation of Al to aluminum hydroxide (Al(OH)3) and above all aluminum oxide (Al2O3) at higher Al contents. At high Al concentrations Al2O3 will be the only oxygenate present in the product mixture. In addition, aluminum nitride (AlN) and aluminum carbide (Al4C3) will be produced as the amount of oxygen atoms is too low for a complete oxidation of Al.

220 J. F. Moxnes, T. L. Jensen and E. Unneberg

Figure 2. The molar distribution of reaction products as a function of the Al content. The temperature is 298 K (room temperature). The pressure is 1 bar.

Figure 3 to Figure 7 display product molecule distributions as a function of the temperature for different amounts of Al. The distributions shown in each of these figures represent state 4 (lowest temperature), state 3 and state 2 (highest temperature). Pure RDX (Figure 3) gives a constant number of moles N2 regardless of the state (2, 3 or 4). However, the amount of carbon monoxide (CO) is a major product at the highest temperature (state 2) due to the fact that the reaction between CO2 and C to produce CO will be thermodynamic feasible at higher temperatures due to increased entropy. As the Al content is increased from 15 % to 30 % (and also to higher values) the amount of N2 is no longer constant going from state 2 to state 3 to state 4. At the highest temperatures (state 2) AlN is a major product component, which accounts for the reduced N2 concentration.

Energetic measures of effectiveness 221

Figure 3. Molar distribution of reaction products from pure RDX (0 wt% Al) as a function of the temperature.

Figure 4. Molar distribution of reaction products as a function of the temperature. 15 wt% Al.

222 J. F. Moxnes, T. L. Jensen and E. Unneberg

Figure 5. Molar distribution of reaction products as a function of the temperature. 30 wt% Al.

Figure 6. Molar distribution of reaction products as a function of the temperature. 45 wt% Al.

Energetic measures of effectiveness 223

Figure 7. Molar distribution of reaction products as a function of the temperature. 60 wt% Al.

Figure 8. Different thermodynamic properties as a function of the Al content. , 12P T roomG T S∆ − : ideal gas. Mechanical work (P’CEA): mechanical work based on

line integral and non-ideal gas. Mechanical work (PCEA): mechanical work based on line integral and ideal gas.

224 J. F. Moxnes, T. L. Jensen and E. Unneberg

Figure 8 shows different energetic measures for various weight percents of

Al in RDX/Al compositions. The negative change in Gibbs free energy, enthalpy, Helmholtz free energy and the internal energy are seen. The work of explosion, the work of Carnot and the mechanical work are also displayed in the figure. We observe that for small contents of Al the work of Carnot is negligible compared to the work of explosion. But from around 50 % Al and upwards the work of Carnot is larger than the work of explosion. We find that , 12P T roomG T S∆ − matches the mechanical work very well. , 12P T roomG T S∆ − is here calculated in accordance with an ideal equation of state. We find that: a) the mechanical work is not very dependent of the equation of state and b) 2 3S S≈ .

Figure 9. The work of explosion as a function of the Al content. 12'H∆ : ideal gas. P’CEA: work of explosion based on a line integral for non-ideal gas. PCEA: work of explosion based on a line integral for ideal gas.

In Figure 9 the work of explosion is shown. A small but significant difference is seen between the work of explosion for an ideal gas calculated by the line integral and 12'H−∆ (ideal gas). The reason for this is the approximation 2 3S S≈ . However, a significant difference is shown between the work of explosion for a non-ideal and an ideal gas. The pressure in state 2 was around 2.5 GPa, which is significantly above the pressure level for which ideal gas conditions may be assumed (around 0.4 GPa).

In Figure 10 we compare the work of explosion with the work of Carnot,

and we compare the thermal heat with the energy lost due to entropy change from state 3 to state 4. We see that the thermal energy is always larger than the energy

Energetic measures of effectiveness 225 loss due to entropy change from state 3 to 4.

Figure 11 shows the efficiencies given in equation (2.20). We find that for low Al contents the two different efficiencies are comparable. This finding is in full agreement with the fact that the work of Carnot is negligible.

Figure 10. Different ratios as a function of the Al content.

Figure 11. Two different efficiencies as a function of the Al content.

226 J. F. Moxnes, T. L. Jensen and E. Unneberg

Figure 12. The change in entalpy as a function of the Al content when O2 can be consumed from the air.

Figure 13. The change in work of explosion as a function of the Al content when O2 can be consumed from the air.

The change in free enthalpy as a function of the Al content is shown in Figure 12. We vary the amount of external oxygen (O2) that can be used. Notice that we use Joule per unit mass of explosive on the Y-axis. When the amount of O2 increases, the peak value increases. When no external O2 is available, a maximum is reached at approximately 50 % Al. As the amount of O2 is

Energetic measures of effectiveness 227 successively increased, the value of the optimum Al content is also increased. It may be seen from the figure that an amount of 1 kg O2 can oxidize 1 kg pure Al. In fact, according to the oxidation reaction (towards the most thermodynamic stable product):

4 Al + 3 O2 → 2 Al2O3

889 grams of O2 is sufficient.

In Figure 13 the work of explosion is shown when external O2 can be used. The work of explosion increases when pure RDX can consume 0.1 kg O2 from the air, as a more complete combustion of the solid carbon is achieved. The work of explosion is reduced when 1.0 kg O2 can be consumed, as more energy is required to heat the excess O2. The same tendency is observed for an Al content up to approximately 35 %. Above this level the work of explosion is larger when 1.0 kg O2 can be consumed since more O2 is required to give a more complete combustion of Al and solid carbon. Figure 13 illustrates that the mixing ratio between RDX/Al and air is important for optimal performance.

4 Conclusion

In this article we have studied various energetic measures of aluminized RDX theoretically by applying the rules of thermodynamics. We have presented equations that can be applied to calculate the work of explosion, the work of Carnot or the mechanical work. Thereafter we applied a well-known thermodynamic computer code to calculate different energetic quantities for various content of Al in RDX/Al compositions. We found that for highly aluminized explosives (e.g. 50 %), the work of Carnot is of the same size as the work of explosion. We conclude that neither of the quantities free energy, enthalpy or internal energy of explosion is a good measure of effectiveness of aluminized energetic materials. References [1] R. Petela, Application of exergy analysis to the hydrodynamic theory of

detonation gases, Fuel Processing Technology, 67 (2000), 131-145.

[2] N.F. Scilly, Measurement of the explosive performance of high explosives, J. Loss Prev. Process Ind. 8(5) (1995), 265-273.

[3] G.F. Kinney, Explosive shocks in air, Chapter 1, The MacMillan Company, New York, 1962.

228 J. F. Moxnes, T. L. Jensen and E. Unneberg

[4] W.A. Trzcinski, S. Cudzilo, L. Szymanczyk, Studies of detonation characteristics of aluminum enriched RDX compositions, Propellants, Explos., Pyrotech., 32(5) (2007), 392-400.

[5] G. Sanford, B.J. McBride, Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications I. Analysis, vol. 1331, NASA Reference Publication, 1994.

Received: December, 2012