Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1717095, 9 pages

https://doi.org/10.1155/2017/1717095

## One-Dimension Nonisentropic Model for the Flow of Aluminized Explosive Products

^{1}College of Electron-Mechanics Engineering, Beijing Institute of Technology, Beijing 100081, China^{2}State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Cheng Wu

Received 3 January 2017; Accepted 22 February 2017; Published 4 April 2017

Academic Editor: Babak Shotorban

Copyright © 2017 Ji Duan 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

A new analytic model of aluminized explosive products based on the method of characteristics for planar isentropic flow is proposed herein. The contribution of Al oxidation in the explosion products is investigated analytically. The flow behind the detonation front cannot be treated as isentropic due to the Al oxidation in the products. To solve the nonisentropic flow field of aluminized explosives products analytically, the assumption of local isentropic process is proposed. Based on this assumption, the flow field behind the detonation front of aluminized explosive is a function of only the reacted aluminum mass fraction in each time range. The metal plate test was conducted with the metal plate driven by RDX/Al/wax (76/20/4) and RDX/LiF/wax (76/20/4). The reacted aluminum mass can be obtained indirectly from the experiment results. The reacted aluminum mass was then applied to the analytic model, and the velocity of metal plate driven by RDX/Al/wax (76/20/4) and RDX/LiF/wax (76/20/4) was calculated. The final velocity of the metal plate driven by RDX/Al/wax was 7.8% higher than that driven by RDX/LiF/wax.

#### 1. Introduction

The addition of aluminum to condensed explosives to increase the total energy release of the explosive is common practice. Aluminum in its powdered form generally does not react quickly enough to contribute to the detonation front itself [1]; however, it can react in the products of the condensed phase explosive or in the surrounding atmosphere, significantly contributing to the strength and acceleration of the blast wave.

Often, a multiphase computational fluid dynamics (CFD) model is used (Frost et al. [2, 3], Zhang et al. [4], Ripley et al. [5], Milne et al. [6], Cooper et al. [7], Massoni et al. [8], Kim et al. [9], etc.) to predict the conditions for metal particle reaction in explosives. Mesoscale modeling with hydrocodes is also widely used (Frost et al. [10], Ripley et al. [11], Zhang et al. [12], Milne et al. [6], etc.). Although these models can, in principle, describe a great amount of detail about the phenomena of interest, developing an analytic model that captures the key elements of the problem in a way that makes the dominant features easily discernible would be preferred. Just like the classical Seshadri formulation [13] analytically describes the structure of premixed flames propagating in a uniform cloud of fuel particles. To establish the analytic model, the author simplified the problem and made some assumptions (gravitational effects, diffusion caused by pressure gradient, and heat transport by radiation is negligible). The Seshadri formulation is used to analyze the flame propagating under normal pressure and temperature. It is not suitable to analyze the combustion of Al and detonation products under high pressure and temperature. In this paper, we have developed a new model based on the classical theory for ideal explosive products (Taylor [14]) that incorporates the oxidation of Al in the products, allowing us to analytically investigate the contribution of Al oxidation in the detonation product.

The flow behind the detonation front of the aluminized explosive is more complex than the flow behind the detonation front of an ideal explosive. To analytically solve the flow behind the detonation front of the aluminized explosive, it is necessary to make some assumptions. For example, we propose an assumption called local isentropic process, which enables the conclusion that the flow field behind the detonation front of aluminized explosive is only a function of the reacted aluminum mass fraction at each time. A metal plate test was conducted to obtain the velocity of the plate and, indirectly, the mass fraction of reacted aluminum powder. Applying the model and the described assumptions, we calculated the velocity of metal plate driven by the RDX/Al/wax (76/20/4) and RDX/LiF/wax (76/20/4). The final velocity driven by RDX/Al/wax is 7.8% higher than the final velocity driven by RDX/LiF/wax. We then compared the test result with the result calculated by our analytic model and found that the final velocity of metal plate driven by the RDX/Al/wax (76/20/4) calculated by the analytic model is 4.5% higher than the test result.

#### 2. The Change of Entropy of the Detonation Products

The flow of the expanding detonation products is in general a highly complex problem. In the classic theory for the flow of detonation products, it is assumed that the flow behind the detonation front of explosive contained in a tube strong enough to confine the motion to one dimension and detonation products expand in a vacuum with (Stanyukovich [15]). The flow of detonation products is isentropic approximately. As a result, the flow of the expanding detonation products can be solved analytically. Due to the reaction of Al particles with the products, however, the classic theory for the flow of detonation products is not suitable for the nonisentropic flow of detonation products of aluminized explosive. In this section, the change of entropy of the detonation products caused by chemical reaction will be discussed.

In the present study, it is assumed that the flow of detonation products in the tube is adiabatic. In the second law of thermodynamics, entropy is an extensive state function under a reversible process: . However, the expansion of the detonation products with chemical reactions is irreversible process; thuswhere is the heat released by chemical reaction and is the temperature of the products.

According to (1), the entropy of detonation products behind the detonation front increases with mass fraction of reacted aluminum powder. Actually, the change of entropy caused by the chemical reaction contains two parts: entropy flow and entropy generation. The entropy flow is caused by the change of heat in the thermodynamic system and the entropy generation is caused by the irreversible factors. In this section, the heat released by chemical reaction is the main reason for the change of entropy in the products. So the entropy generation caused by the irreversible factors will not be discussed. In order to model the flow behind the detonation front of the aluminized explosive, the contribution of the Al oxidation in the products must be considered. However, the flow of detonation products of aluminized explosive is more complex than the flow of detonation products of ideal explosive. Some assumptions are necessary to solve the model analytically and will be discussed in the next section.

#### 3. The Assumption of Analytic Model

To solve the flow behind the detonation front and the contribution of Al oxidation in the products analytically, we simplified the problem. Consider the detonation of aluminized explosive contained in a tube so strong that the motion is confined to one dimension. The flow field generated by the high-explosive detonation products is modeled in a highly idealized manner: the products are treated as a perfect gas expanding into vacuum, with planar geometry. This simplification is based on the classical one-dimension model for ideal explosive. The Al particles are uniformly distributed in the products.

##### 3.1. The Assumption of Al Particles in Detonation Products

In this paper, it is assumed that none of the aluminum particles reacted during the detonation and the Al oxidation occurs in the products behind the detonation front [16, 17]. For detonation of aluminized explosives, the duration for energy release of the reaction of explosive components is generally less than 0.1 *μ*s, while the energy release for Al oxidation is in the order of microseconds to several milliseconds [18, 19]. Therefore, the reaction rate of explosive components is much quicker than the rate of Al oxidation. So the assumption mentioned above is valid for aluminized explosive.

It is assumed that Al particles in the flow field do not affect the flow before Al particles oxidation. This assumption is valid for the small particle loading in the high explosive (e.g., Rudinger [20]). The small particles could be viewed as small perturbation to the flow field of detonation products.

The afterburning of the aluminum is modeled following the work of [21], which was originally applied to the combustion of aluminum in a gas phase. We adapted the Noble-Abel equation to determine the EOS of gas phase aluminum (Kim et al. [22]).

##### 3.2. The Assumption of Local Isentropic Process

The flow behind the detonation front of aluminized explosive cannot be treated as an isentropic process approximately due to the chemical reaction of Al particles in the detonation products. We propose an assumption of local isentropic process to analyze the nonisentropic flow behind the detonation front of aluminized explosive.

The rate of aluminum oxidation is relatively slow, so we divided the products expansion process into many small ranges along the time axis. In each short time range, we assume that the process is approximately isentropic (the mass fraction of reacted aluminum is approximately constant). Using this method simplifies the flow behind the detonation front of aluminized explosive and enables analytical evaluation. We call this assumption local isentropic process.

Note that the detonation of aluminized explosive is extremely violent and fast; it is difficult to obtain detailed quantitative data related to this phenomenon directly. In this paper, the comparison of the velocities determined experimentally and obtained through the model is conducted. The model correctly described the contribution of Al oxidation in the detonation products of aluminized explosive. The assumption of local isentropic process is indirectly validated by this result.

#### 4. Model of Aluminized Explosive Products

In this study, it is assumed that the detonation of aluminized explosive contained in a tube is strong enough to confine the flow of the detonation products to one dimension. As mentioned in the assumption of aluminized explosive products, the high-explosive detonation products are treated as a perfect gas and the flow field generated by the high-explosive detonation products is not affected by Al particles before their oxidation. The state equation of high-explosive detonation products iswhere is the initial high-explosive mass fraction of aluminized explosive.

It is also assumed that the Al particles are uniformly distributed in the explosive and the Al particles exist in gas phase in the products. To determine the EOS of gas phase aluminum, we adapted the Noble-Abel equation of the form [22]where denotes gas constant, is the reacted aluminum mass fraction, is the density of aluminized explosive products, is the initial aluminum mass fraction of aluminized explosive, is the number of moles per unit volume, and is an empirical constant. The value of is obtained from [22], due to the fact that aluminized explosive studied in [22] is similar to our study. (Note that the influence of value of on our model is weak.)

The standard mixture rule applies to the aluminized explosive products. The state equation of aluminized explosive products is

Based on the first law of thermodynamics, the equation can be obtained as follows:where is specific heat of products at constant pressure, is specific heat of products at constant volume, is pressure of products, and is specific volume.

By inserting (4) in (5), it can be found that

Following the assumption of local isentropic process, the expansion of products in the tube was divided into several small time ranges. The rate of Al oxidation is slow relative to the short time ranges and the mass fraction of reacted aluminum is approximately constant. In each time range, (6) can be simplified as follows:

The adiabatic index of the products is

By inserting (8) in (7), it can be found that

Based on the first law of thermodynamics, the following equation can be obtained:

Equation (10) can be simplified to

The state equation of aluminized explosive products (4) can be inserted into (11), and the following equation can be obtained:

By inserting (9) in (12), it can be found that

Based on the assumption of local isentropic process, the flow of detonation products is approximately isentropic in each time range. Thus

By inserting (4) in (14), it can be found thatwhere is constant and is the mass fraction of reacted aluminum in time range .

The equations which represent the conservation of mass and momentum behind the detonation front of the aluminized explosive are

In each time range, using the method of characteristics for planar isentropic flow, (16) can be converted to ordinary differential equation. In time range , the Riemann invariant along a right-running characteristic is

When , . Therefore, the right-running wave has a constant slope in time range :

The same can be shown for left-running characteristics. However, the slope of the characteristic is different in every time range due to the reaction of Al oxidation in the detonation products. Therefore, the characteristics of aluminized explosive and ideal explosive differ, as shown by the sketch in Figure 1.