Dynamic modeling and simulation of the combustion of aluminized solid propellant with HMX and GAP using moving boundary approach

Combustion and Flame 213 (2020) 409–425

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Dynamic modeling and simulation of the combustion of aluminized solid propellant with HMX and GAP using moving boundary approach Thuan A. Vo a, Minyoung Jung a, Derrick Adams a, Hongmin Shim b, Hyunsoo Kim b, Raymoon Hwang c, Min Oh a,∗ a b c

Department of Chemical and Biological Engineering, Hanbat National University, Dongseo-daero 125, Yuseong-gu, Daejeon, Republic of Korea Agency for Defense Development, 462 Jochiwon-gil, Yuseong-gu, Daejeon 305-150, Republic of Korea Department of Chemical and Biomolecular Engineering, Yonsei University, Seodaemun-gu, Seoul 03722, Republic of Korea

a r t i c l e

i n f o

Article history: Received 6 September 2019 Revised 27 November 2019 Accepted 6 December 2019

Keywords: Aluminized solid propellant Dynamic simulation Moving boundary modeling Multi-phase combustion Parametric study

a b s t r a c t This research describes the influence of nano-sized aluminum with varying contents (0–20 wt%) on the combustion behaviors of HMX/GAP/Al in aspects such as burning rate, surface temperature, gas phase temperature, mole fraction of species, and specific impulse. A rigorous mathematical model is developed for three phases (solid phase, condensed phase, and gas phase) with detailed kinetics. This model consists of 507 gas phase reactions and 4 condensed phase reactions of HMX/GAP/Al combustion. The model also emphasizes the phase transitions and reactions of aluminum in the gas phase. Based on this model, dynamic simulation is carried out at various propellant compositions and operating pressures by means of the moving boundary approach using gPROMS software package. The simulation results are in close agreement with the experimental data at slight marginal errors for HMX/Al combustion, predicting the combustion behaviors for the HMX/GAP/Al system. Accordingly, the model predicts the gas phase temperature of 3061 K for the 20 wt% Al and the specific impulse of 258.53 s for the 15 wt% Al of HMX/GAP/Al propellant under an operating pressure of 100 atm. The increase in burning rate and specific impulse by increasing the pressure is also indicated. According to this study, the addition of aluminum particles with a content range of 15–20 wt% considerably improves combustion behaviors. By dynamic modeling and simulation, a detailed framework for studying the multi-phase combustion of aluminized solid propellant is introduced. © 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction Solid propellants are energy-filled materials, used to launch and propel rockets and missiles [1]. The basic components of propellants are oxidizer, binder, plasticizer, and metal particles. The compounds often used as oxidizers are ammonium perchlorate (AP) [2,3], ammonium nitrate (AN) [3], ammonium dinitramide (ADN) [3], nitroglycerin (NG) [4,5], nitrocellulose (NC) [4,6], cyclotrimethylenetrinitramine (RDX) [3,7,8], and cyclotetramethylenetetranitramine (HMX) [3,9–11]. The most commonly employed binders are either inert (e.g. hydroxyl–terminated polybutadiene [HTPB]), or active (e.g. glycidyl azide polymer [GAP], bis-azidomethyl oxetane [BAMO], 3-azidomethyl-3-methyl oxetane [AMMO]) [3]. Nitrate esters such as 1,2,4-Butanetriol trinitrate (BTTN), and Trimethylolethane trinitrate (TMETN) are the main plasticizers [3]. Magnesium, zinc, boron, and, especially, aluminum

∗

Corresponding author. E-mail address: [email protected] (M. Oh).

are also used as part of the metal-fuel in order to increase the performance of a solid rocket motor [12,13]. The high-heat energy released rapidly by exothermic reactions in the gas phase is the main energy source in the operating mechanism of rocket motor. Using dispersed metals in solid propellants improves the energy performance of the composite solid propellant due to the high temperature of combustion, specific impulse and combustion stability [12,14]. Aluminum is a widely used component in solid rocket propellants, pyrotechnics, and explosives owing to its outstanding characteristics, such as high reactivity and heat of combustion, low toxicity, and good stability. Aluminum acts as a metal fuel in solid propellant and thus changes the performance of motor. In aluminized solid propellant combustion (e.g. HMX/Al or HMX/GAP/Al), the burning surface temperature is lower than the melting point of aluminum; thereby, the phase transitions and reactions of aluminum take place in an environment of gas products which are created during the combustion of the oxidizer and binder. In the recent past, many experiments and numerical analyses have been studied to identify the combustion characteristics (e.g. burning

https://doi.org/10.1016/j.combustflame.2019.12.015 0010-2180/© 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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Nomenclature Symbol

ρ A V z t x y r k

S m n Q H  q q T

λ φ S



M R g

ν

P u h

σ d

density, kg/m3 surface area of solid propellant, m2 volume, m3 coordinate, m time, s mass fraction volume fraction (in solid phase) reaction rate reaction rate constant, specific heat ratio, (m3 /mol)n -1 /s mass source, kg/s solid propellant mass, number of reactions, kg number of components, heat rate from outside to solid phase, J/s enthalpy of reaction, J/mol heat flux of laser, W/m2 latent heat of phase transition, J/kg temperature, K thermal conductivity, W/m/K volume fraction mass (energy) source rate per volume, kg/m3 /s, mol/m3 /s, J/m3 /s molecular weight, kg/mol ideal gas constant, J/mol/K standard acceleration of gravity, m/s2 stoichiometric pressure, N/m2 velocity, m/s convective heat transfer coefficient, W/m2 /K Stefan-Boltzmann constant, diameter, m

Subscripts and superscripts I, II, III zones in gas phase i ith component j jth reaction s solid phase c condensed phase g gas phase b bubble (in condensed phase) l Liquid (in condensed phase) sc interface of solid and condensed phase cg interface of condensed and gas phase e nozzle exit m mass source E energy source con convection heat transfer rad radiation heat transfer rxn reaction Al aluminum melt melting evap evaporation mix mixture Alsol solid aluminum Alliq liquid aluminum Algas gas aluminum Alssol solid aluminum in solid phase Alcsol solid aluminum in condensed phase Algsol solid aluminum in gas phase liq

Alg

liquid aluminum in gas phase

rate, flame temperature, or specific impulse) of solid propellant systems using aluminum particles as metal fuel. A literature review on the combustion of aluminized solid propellant, as well as the combustion mechanism of aluminum in the presence of solidpropellant gaseous products is presented in Table 1. Most of the previous researches like Ecker et al. [17], Meda et al. [21], Kasztankiewicz et al. [22], and Deluca and Galfetti [23] mainly focused on the application of aluminum in AP/HTPB-based solid propellant. In the case of Muravyev et al. [9], the research did not consider the binder component (used GAP for HMX) – an important component that is helpful for connecting solid ingredients to create the propellant. Moreover, most of them simulated the combustion of aluminum that is independent of solid propellant, only considering reactions between aluminum and available gaseous products and not taking into account the whole phases in the combustion of the propellant mixture [16,18,19]. Thus, these simulations did not correspond to the real combustion of aluminized solid propellant. In the studies by Meda et al. [21] and Deluca and Galfetti [23], the main focus was on the combustion characteristics in an area such as burning rate while there are many other characteristics that need to be determined in aluminized solid propellant combustion. Thus far, not much research has been carried out to predict the combustion of aluminized solid propellant with the oxidizer component of HMX in both the simulation and experiment. The number of researches on the dynamic simulation of aluminized solid propellant combustion is also inadequate. Heretofore, there is no model for the combustion of ternary composite propellant employing a moving boundary approach. The moving boundary method is used to confirm feasibility and predict the dynamic combustion behaviors of AP (mono propellant) [2] and HMX/GAP (binary system) [24]. By applying the moving boundary method, the problem about changing the length of propellant during combustion is solved. In this study, a model using the moving boundary method for the ternary propellant system consisting of HMX (oxidizer), GAP (binder), and aluminum (metal particles) was developed. Moving boundary method is a strong approach to explain the moving boundary phenomena during propellant combustion, and minimize the impact of the numerical diffusion caused by numerical methods. Accordingly, the study on solid propellant combustion is highly practical, and suitable for real situations. A mathematical model which includes the conservation equations of mass, energy, and species concentration was established in one-dimensional domain, along with coordinate transformation. The reaction kinetics of the condensed phase and gas phase for HMX and GAP was considered, along with the gas phase reaction kinetics for Al. It should be noted that the developed HMX/GAP/Al model of the ternary propellant system became the HMX/Al model of the binary propellant system by removing the GAP binder component. Through the dynamic simulation, the model for HMX/Al combustion was validated with experimental data and combustion behaviors for the HMX/GAP/Al propellant system in a cylindrical combustion chamber was predicted. The combustion characteristics of HMX/GAP/Al propellant were categorized under burning rate, propellant mass, surface temperature of propellant, gas-phase temperature, composition of products in gas phase, and specific impulse. Therefore, the effect of aluminum particles on solid rocket propellants was also recorded. 2. Theoretical background 2.1. Phenomena and methodology Figure 1 describes the combustion process of HMX/GAP/Al propellant in a chamber of length L. At the initial state, the solid phase

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411

Table 1 Literature review on aluminized solid propellant combustion and aluminum combustion mechanism. Researchers

Major content and results

Babuk et al. [15]

- Both the experiment and mathematical model were presented for the burning of nano aluminum-based solid rocket propellants with HTPB and various oxidizers (AN, AP, and HMX). - It indicated that the agglomerate with metal-covered oxide influenced the propellant burning rate. - The decrease in the size of metal fuel particles increased the burning rate at the same operating pressure.

Huang et al. [16]

- The flame propagation behavior and structure of aluminum particle-laden flow in different oxidizer environments, including air, oxygen, and water vapor were assessed. - The effect of particle size on the burning characteristics of aluminum particle/oxidizer mixtures was determined by means of the SENKIN program. - A simple model with several assumptions especially focused on the detailed chemical kinetics.

Ecker et al. [17]

- Aluminum particle combustion in a solid rocket combustion chamber with a fuel composition of 71%AP/14%HTPB/15%Al by weight was modeled. - Gas phase reaction kinetics of aluminum, along with the evaporation and condensation of the Lagrangian phase were considered to comprehensibly model for the solid rocket motor (SRM) flow field.

Beckstead [18]

- Aluminum particle combustion was presented in a two-dimensional, unsteady-state, and kinetic-diffusion-vaporization controlled numerical model. - 15 reaction kinetics were used to account for species generation and destruction. - Calculations were made in conditions similar to what might occur in a solid propellant rocket motor where the major oxidizers are water and CO2 .

Sundaram et al. [19]

- Aluminum combustion mechanisms for different particle sizes and pressures were reviewed. - Physicochemical processes such as heat and mass transfer between the particle and the gas, phase transitions in the oxide layer, and exothermic chemical reactions were analyzed. - Calculations were performed using the NASA Chemical Equilibrium with Applications (CEA) program. - Adiabatic flame temperature and product composition for the combustion of aluminum with different oxidizers were also identified.

Wang et al. [20]

- A three-dimensional model of aluminized composite propellant combustion with heat conduction in solid and combustion stages in the gas phase was developed. - Dynamics of aluminum in the neighborhood of propellant surface such as time of ignition, particle temperature, and moving distance of aluminum particles from the surface because of surface tension effects were emphasized.

Meda et al. [21]

- The burning rate of AP/HTPB/Al solid propellant with a composition ratio of 68/17/15 by weight percent was measured. - The burning rate increased with increasing pressure inside the chamber. - Addition of nanosized-particles (0.17 μm) significantly increased the burning rate.

Kasztankiewicz et al. [22]

- AP/HTPB/Al propellant was examined for specific impulse. - The specific impulse increased with varying Al contents ranging from 0 to 18 wt%. However, the specific impulse decreased by increasing Al content ranging from 18 to 20 wt%.

Deluca and Galfetti [23]

- AP/HTPB/Al propellant (68/17/15 by weight percent) was tested to evaluate the application of aluminum in solid rocket propellant were conducted. - The burning rate was studied under a wide variety of rocket-operating conditions, covering the mass fraction of nano-Al, as well as particle size.

Muravyev et al. [9]

- The combustion behaviors of the HMX/Al system in fields such as gas phase temperature, composition of gaseous products, burning rate, and specific impulse at the different mass fractions of HMX/Al in the initial mixture, was determined through experimental work. - Combustion parameters and morphology of compositions based on HMX and aluminum were studied at different particle sizes and mixing techniques.

propellant which is a mixture of HMX powder, aluminum particles and GAP binder are physically mixed in a cylindrical combustor. As the solid propellant is heated by a laser heat source, the propellant surface temperature increases to the melting point of GAP (450 K) and HMX (530 K). The formation of condensed phase occurs immediately after the melting of HMX and GAP. The melting is followed very fast by the condensed phase reactions which are first-order decomposition reactions of HMX and GAP. The initiation of the chemical reactions of HMX (liquid) and GAP (liquid) results in gaseous products which react with each other. Thus, the ignition occurs. The various gaseous components (bubbles) formed from condensed phase reactions, react with each other, leading to high exothermic gas phase reactions. The laser heat source is still supplied until the heat flux from the gas phase to the propellant surface is high enough to maintain the combustion. When stable combustion is reached, the laser is turned off. Since the burning surface temperature is 70 0–80 0 K for the HMX or HMX/GAP propellant combustion [10,24,25], the alu-

minum particles of the melting point 933 K [3] are still in solid state in the condensed phase, leave the surface along with decomposed gaseous products as particle-laden flow. In the gas phase region of the propellant combustion, phase transitions (melting and evaporation) and combustion reactions of aluminum occur. Because temperature profile of gas phase is in the range of the surface temperature and flame temperature, state of aluminum is different in each zone of the gas phase. In zone GI , the gas phase temperature is lower than the melting point of aluminum, and thus aluminum is in the solid state. In zone GII , gas phase temperature is higher than the melting point of aluminum but lower than its evaporation temperature and, thus aluminum is in the liquid state (melting of aluminum occurs). In zone GIII , the gas phase temperature is higher than the evaporation temperature of aluminum, hence, vapor aluminum combustion occurs. The kinetic mechanism of the elementary reactions of vaporized aluminum with gaseous oxidizers has been demonstrated by Huang et al. [16] and Ecker et al. [17].

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Fig. 1. Conceptual flowsheet of HMX/GAP/Al propellant combustion procedure. Table 2 Characteristics of regions in 1-D model of HMX/GAP/Al propellant combustion. Region

Characteristics

Solid phase

Compositions: HMX, GAP, and Al Reaction: none Heat transfer: conduction

∀zs (t) ∈ [0, zsc (t)) Solid-condensed phase interface

∀z(t ) = zsc (t )

Condensed phase

∀zc (t) ∈ (zsc (t), zcg (t)) Condensed-gas phase interface

∀z(t ) = zcg (t )

Reaction: none Heat transfer: conduction and convection Compositions: HMX (liquid), GAP (liquid), bubbles, and Al (solid) Reaction: decomposition reactions of HMX liquid) and GAP (liquid) Heat transfer: convection Reaction: decomposition reactions of HMX liquid) and GAP (liquid), and reactions between bubble components Heat transfer: convection, radiation

Gas phase

∀zg (t) ∈ (zcg (t), L] Zone GI

∀zg (t ) ∈ (zcg (t ), zg (t )(Tg = Tmelt,Al )] Zone GII

∀zg (t ) ∈ (zg (t )(Tg = Tmelt,Al ), zg (t )(Tg = Tevap,Al )] Zone GIII

∀zg (t ) ∈ (zg (t )(Tg = Tevap,Al ), L]

Compositions: gaseous products from HMX and GAP combustion, and Al (solid) Reaction: reactions of gaseous products from the decomposition of HMX and GAP Heat transfer: convection and radiation Compositions: gaseous products from HMX and GAP combustion, and Al (liquid) Reaction: reactions of gaseous products from the decomposition of HMX and GAP Heat transfer: convection and radiation Compositions: gaseous products from HMX, GAP, and Al combustion Reaction: reactions of gaseous products from the decomposition of HMX, GAP and Al combustion reactions Heat transfer: convection and radiation

In the propellant combustion, the interfaces between solidcondensed phase and condensed -gas phase continuously moved. To simulate this, the moving boundary model was converted to a fixed boundary model by coordinate transformation [2,24]. The physio-chemical characteristics of all the regions of HMX/GAP/Al propellant combustion are elucidated in Table 2. Here, the concept of phase is based on propellant combustion comprised of HMX (oxidizer) and GAP (binder), while that of the zone is based on the phase state of the aluminum in the gas phase of the HMX/GAP

propellant combustion. In this study approach, the initial propellant mixture of HMX, GAP, and aluminum particles is divided into two parts: the first part is HMX/GAP and second part is aluminum particles because the melting point of aluminum is higher than the surface temperature of HMX/GAP propellant. The definition of phase is based on the combustion of a typical propellant, which creates three phases including solid phase, condensed phase (liquid and bubble), and gas phase (pure gasses) [2,4,10]. Considering the effect of aluminum particles on HMX/GAP propellant combus-

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413

Table 3 Chemical kinetics for the condensed phase reactions in HMX/GAP/Al propellant combustion. Reaction

A (s−1 )

E (J/mol)

H (kJ/mol)

HMX → 4CH2 O + 4N2 O HMX → 4H2 CN + 4NO2 C78 H131 O27 N66 → C78 H131 O27 N24 + 21N2 C78 H131 O27 N24 → 21C(s) + 10H2 + 18HCN + 3NH3 + 13CO + 4CH2 O + 7H2 O + 12CH4 + 3CNO + 3CH2 + 2C2 H4

1E13 3E16 1E17 2E13

143,929 184,514 167,360 125,520

−295.8 909.5 −732 −2934.6

A = pre-exponential factor, E = activation energy, H = heat of reaction.

tion, we described the state of aluminum particles in three phases by comparing the melting point of aluminum to the surface burning temperature of HMX/GAP combustion. As mentioned above, because the melting point of aluminum is higher than the surface temperature of HMX/GAP propellant combustion, aluminum would be in the solid state in condensed phase. To make clearer the state of aluminum particles in the gas phase, we introduced the concept of zones. With Zone GI , gas phase temperature is lower than melting point of aluminum and thus aluminum is in the solid state. With zone GII , gas phase temperature is higher than the melting point of aluminum but lower than its evaporation temperature so that aluminum is in liquid state. For zone GIII , the gas phase temperature is higher than the evaporation temperature of aluminum and as a result, aluminum is in the gaseous state. Heat transfer mechanisms playing an important role in the combustion of an aluminized solid propellant is explained in detail as follows: - In the domain of solid phase, the heat transfer is by conduction. - Solid-condensed phase interface: Heat transfer is in two forms, e.g. conduction and convection. Accordingly, the energy accumulation rate at in this interface is balanced by input heat energy transferred from the condensed phase to the interface by convection and output heat energy transferred from the interface to the solid phase by conduction. The energy accumulation at the interface is the energy source which results in the melting of solid propellant. - In the condensed phase: The dominant heat transfer is convection by the fluid flows (liquid and gas). Actually, conduction and radiation can exist in condensed phase, but they are inconsiderably small compared to convection. - Condensed-gas phase interface: interface between condensed phase and gas phase receives heat energy from gas phase by convection and radiation. Conduction is inconsiderable compared to convection and radiation. - Gas phase and aluminum particle: In combustion of HMX/GAP/Al propellant or HMX/Al propellant, aluminum particles leave burning surface and go into gas phase (which is created from HMX/GAP combustion). Aluminum particles receive heat energy from the exothermic gas phase reactions by the two dominant forms of heat transfer: convection and radiation, for phase transitions (melting and evaporation) and combustion of aluminum. 2.2. Chemical kinetics of aluminized solid propellant combustion For the HMX/GAP/Al or HMX/Al propellant combustion, the condensed phase reactions are first-order decomposition reactions of HMX and GAP. In this study, two global condensed phase reactions of HMX [26] and two global condensed phase reactions of GAP [27] are broken down in Table 3. For the gas phase, 230 and 253 (224 to 439 and 442 to 478) elementary reactions were used for HMX [26] and GAP [28], respectively. Table 4 shows the gas-phase reaction kinetics of aluminum. In this study, 24 reactions [17] were used by replacing the 3rd reaction with the 8th reaction in Table 1 [29]. Therefore, the gas phase reactions kinetics of 507 reactions were used to predict the combustion behaviors of the HMX/GAP/Al propellant.

Table 4 Reaction kinetics of aluminum combustion in the gas phase [17,29]. No.

Reaction

A

b

E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Al(g) + O2  AlO + O Al(g) + O  AlO AlO + AlO2  Al2 O3 (g) Al2 O3 (g)  Al2 O2 + O Al2 O3 (g)  AlO2 + AlO Al2 O2  AlO + AlO Al2 O2  Al(g) + AlO2 Al2 O2  Al2 O + O AlO2  AlO + O Al2 O  AlO + Al(g) AlOH  AlO + H AlOH  Al(g) + OH Al(g) + H2 O  H + AlOH Al(g) + H2 O  ppp  AlO + H2 AlO + CO2  AlO2 + CO Al(g) + CO2  AlO + CO AlH3 + H  AlH2 + H2 AlH2  AlH + H AlH3  AlH + H2 Al(g) + H  AlH AlH + H  Al(g) + H2 AlH2 + H  AlH + H2 Al(g)  Al(l) Al2 O3 (g)  Al2 O3 (l)

9.72E13 3E17 1.72E11 3E15 3E15 1E15 1E15 1E15 1E15 1E15 1E15 1E15 1.14E12 9.6E13 1.5E10 1.74E14 4.75E9 1.46E15 1.48E13 1.6E17 1E13 2E13 1E14 1E14

0 −1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.34 0 0 0 0

1.5995E2 0 −6.551E3 9.765E4 1.27E5 1.179E5 1.489E5 1.0425E5 8.855E4 1.332E5 1.147E5 1.32E5 8.798E2 5.70E3 −7.948E2 6.40E3 0 4.6448E4 6.1112E4 0 0 0 0 0

A = pre-exponential factor, b = temperature power factor, E = activation energy. Units are in cm, mol, s, K, and cal.

Table 5 Physical properties of HMX, GAP, and Al. Property

HMX

GAP

Al

Molecular formula Molecular weight (kg/mol) Density (kg/m3 ) Heat capacity (J/kg/K) Thermal conductivity (W/m/K) Melting temperature (K) Evaporation temperature (K) Heat of fusion (kJ/kg) Heat of evaporation (kJ/kg)

C4 H8 N8 O8 0.2962 1900 231.79 + 2.61T 0.251 530 – 252.7 –

C78 H131 O27 N66 2.423 1300 1882.8 0.146 450 – 957 –

Al 0.027 2700 879 205 933 2194 392.2 11,831.1

Formation enthalpies of the suboxides of aluminum during aluminum combustion were calculated based on CBS-Q method – an ab initio model [30]. Based on the values of these formation enthalpies, the heat of the reactions as shown in Table 4 was calculated. Table 5 shows the physical properties of HMX [26], GAP [26], and aluminum [21,31]. 2.3. Mathematical model By analyzing the phenomena in aluminized solid propellant combustion along with the validated models for monopropellant system [2] and binary propellant system [24], a rigorous mathematical model was developed for aluminized solid propellant (ternary propellant system) combustion consisting of three phases: solid phase, condensed phase and gas phase. In the gas phase, three zones were formed based on the phase state of aluminum. The main variables such as mole fraction (mass conservation),

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velocity (momentum conservation), and temperature (energy conservation) were calculated. Conservation equations are partial differential algebraic equations (PDAEs) with two independent variables of time and space while boundary conditions were determined based on the balance calculation at interfaces between solid – condensed phase or condensed-gas phase. The following assumptions are introduced according to previous works.

indicated that there is a contradiction between the diffusion mechanism and experimental data in the study of aluminum combustion because of the difference in thermal expansion coefficients of Al and alumina, which results in the combustion of nano-aluminum which is not limited by diffusion through the initial oxide shell [32]. • Two equilibrium transition reactions were presumed with high rate constants for Al and Al2 O3 [14,16]. • In the flame zone of the gas phase, aluminum vapor reacted with oxidizers O2 , H2 O and CO2. The aluminum/nitrogen reactions are not considered here [16,17]. • The diffusion phenomena in the condensed and gas phases is ignored [2,10].

• All aluminum particles underwent phase changes before they became aluminum vapor and reacted with oxidizers [16]. • Aluminum combustion was controlled by reaction kinetics (not diffusion), and the oxide layer surrounding a solid aluminum core was not considered. The analysis indicated that mass diffusion is faster than chemical reactions for particle diameters smaller than the critical value, which is 100 μm at p = 1 atm and 1 μm at p = 100 atm. It is concluded that combustion of nano aluminum particles is kinetically controlled over the pressure range of 1–100 atm [19]. In addition, the study of Levitas

With the above assumptions, a rigorous and idealized mathematical model could be developed. Table 6 shows the mathematical model with moving boundaries for the combustion of the HMX/GAP/Al propellant system.

Table 6 The three-phase mathematical model of HMX/GAP/Al combustion. Phase

Conservation

Detailed equation/Eq. no.

Solid phase

Mass conservation [kg/s]

A

d (ρs (t )zs (t )) = −S m,s , dt



∀zs (t ) ∈ [0, zsc (t )]

(1)

Q

, zs (t ) = zsc (t ) ∀zs (t ) ∈ [0, zsc (t ))     dTs (zsc (t ) )   Q  = A qcs − qs = A hcs (Tc (zsc (t )|+ ) − Ts (zsc (t ) ) ) − λs S m,s =

melt Hmix

0,

d zs

melt Hmix = 

i

ms (t ) = ρs (t )Vs (t ) = ρs (t )Azs (t ), ∀zs (t ) ∈ [0, zsc (t )]

ρs (t )c p,s (t )

 ∂ Ts ∂ ∂T λs (t ) s , ∀zs (t ) ∈ (0, zsc (t )) = ∂t ∂ zs ∂ zs

Boundary condition Ts (0, t ) = Ts,0



Condensed phase

Bubble mass conservation: [ mkg3 s ]

i f Ts (zsc (t ) ) < Tmelt otherwise

∂ ∂  (φ ρ ) = − (φ ρ u ) + Sm,b , ∀zc (t ) ∈ (zsc (t ), zcg (t )] ∂t b b ∂ zc b b b 

n



Sm,bi =

i=1

(4)

(5)

(6)

(7)

λs (t ) ∂ Ts (∂zzscs(t )) = −q laser , Ts (zsc (t ) ) = Tmelt ,

Sm,b =

(3)

  xi Himelt

i = GAP, i f Tmelt,GAP ≤ Tc < Tmelt,HMX i = GAP or HMX , i f Tc ≥ Tmelt,HMX

Energy conservation [ mJ3 s ]

(2)

n

m

ν ji r j

(8)

(9)

(10)

i=1 j=1

r j = k j Cli

(11)

 b

where, k j = A j Tc j exp





Cli = 1 − φb − φAlcsol ρli

−E j RTc



1 φ Mli li

(12)

(13)

⎧ ⎪ ⎨i = GAP, i f Tmelt,GAP ≤ Tc < Tmelt,HMX ⇒ φli = φlGAP = 1 i = GAP or HMX, i f Tc ≥ Tmelt,HMX ⇒ φlGAP = yGAPy+GAPyHMX ⎪ ⎩ φ = yHMX lHMX yGAP +yHMX Boundary condition φb (zsc (t )) = 0

(14)

(continued on next page)

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415

Table 6 (continued) Phase

Conservation

Detailed equation/Eq. no.

mol Bubble species conservation: [ m 3s ]

 ∂ ∂  φ C =− φ u C + Sm,bi , ∀zc (t ) ∈ (zsc (t ), zcg (t )] ∂ t b bi ∂ zc b b bi 

Sm,bi =

m

(15)

ν ji r j

(16)

j=1

Boundary condition Cbi (zsc (t ) ) = 0 Energy conservation: [ mJ3 s ]

ρ¯ c c¯ p,c

(17)

∂ Tc ∂ Pc ∂ Tc  = − ρ¯ c uc c¯ p,c + SE,c , ∀zc (t ) ∈ (zsc (t ), zcg (t )] ∂t ∂t ∂ zc n   b 1 − φb − φAlcsol ρlα φlα A j Tc j exp



SE,c =



j=1

−E j RTc



1 Mlα

(18)

H j ,

(19)

∀zc (t ) ∈ (zsc (t ), zcg (t )) 

S E,c (zcg (t ) ) =

q laser Sarea Vcondense

,

if

S E,c (zcg (t ) ) = Q,

if

q laser Sarea Vcondense q laser Sarea Vcondense

>Q

(20)

≤Q

with Q = Qcon + Qrad + Hrxn

Qcon =

Qrad =

(21)

[hg−c (Tg (zcg (t )|+ ) − Tg (zcg (t )) )]Sarea Vcondense

(22)

   4  σ Tg (zcg (t )+ ) − Tg4 (zcg (t )) Sarea

(23)

Vcondense

Hrxn =

n 





b

1 − φb (zcg (t ) ) − φAlcsol (zcg (t ) ) ρlα (zcg (t ) )φlα (zcg (t ) )A j Tc j exp

j=1

−E j RTc (zcg (t ) )

Boundary condition Tc (zsc (t )) = Tmelt,GAP Gas phase

mol Mass conservation [ m 3s ]

ρ¯ g c¯ p,g

(27)

RTg

(28)

∂ Tg ∂ Pg ∂ Tg  = − ρ¯ g u¯ g c¯ p,g + SE,g , ∀zg (t ) ∈ (zcg (t ), L] ∂t ∂t ∂ zg

(29)

ρ¯ g = φg ρg + φAlgsol ρAl sol + φAlgliq ρAl liq

(30)

c¯ p,g = xg c p,g + xAlgsol c p,Algsol + xAl liq c p,Al liq g



SE,g = − =−

n 

j=1 n 

j=1

Momentum conservation: [ mkg2 s ]

(24)

(26)

Boundary condition Cgi (zcg (t ) ) = Cbi (zcg (t ) ) Energy conservation: [ mJ3 s ]

1 H j Mlα

(25)

∂ (φgCgi ) ∂ (φg ugCgi )  =− + Sm,gi , ∀zg (t ) ∈ (zcg (t ), L] ∂t ∂ zg  n

E  Sm,gi = νji A j Tgb j exp − j Cgni j=1



(31)

g

r j H j − rmelt,Al qmelt,Al − revap,Al qevap,Al

r j H j − φAlgsol

4 σ (Tg4 −Tmelt,Al )+hg (Tg −Tmelt,Al )

dAl

− φAl liq

σ (Tg4 −Te4vap,Al )+hg (Tg −Tevap,Al )

g

(32)

dAl

Boundary condition Tg (zcg (t ) ) = Tc (zcg (t ) )

(33)

∂ (ρg ug ) ∂ Pg ∂ (ρg ug ug ) ∂ 2 ug =− − + μg , ∀zg (t ) ∈ (zcg (t ), L ) ∂t ∂ zg ∂ zg ∂ zg2

(34)

Boundary condition ug (zcg (t ) ) = ub (zcg (t ) )

(35)

∂ ug ( L ) =0 ∂ zg

(36)

416

T.A. Vo, M. Jung and D. Adams et al. / Combustion and Flame 213 (2020) 409–425 Table 7 Equations for calculating the mole fraction of solid and liquid aluminum in condensed and gas phase. Phase Condensed phase

Equation/Eq. no.

∂ (φAlcsol ρAl sol ) ∂ (φAlcsol uAlcsol ρAl sol )  =− + SAl sol , ∀zc (t ) ∈ (zsc (t ), zcg (t )] c ∂t ∂ zc

(37)



with SAl sol = 0

(38)

φAlcsol (zsc (t )) = φAlssol (zsc (t ))

(39)

φAlcliq = 0

(40)

c

Gas phase





Zone GI : ∀zg (t ) ∈ zcg (t ), zg (t ) Tg = Tmelt,Al



(41)

∂ (φAlgsol ρAl sol ) ∂ (φAlgsol uAlgsol ρAl sol )  =− + SAl sol g ∂t ∂ zg

(42)



with SAl sol = 0

(43)

Boundary condition : φAlgsol (zcg (t ) ) = φAlcsol (zcg (t ) )

(44)

g

φAlgliq (zg (t )) = 0

(45)









Zone GII : ∀zg (t ) ∈ zg (t ) Tg = Tmelt,Al , zg (t ) Tg = Tevap,Al



(46)

∂ (φAlgsol ρAl sol ) ∂ (φAlgsol uAlgsol ρAl sol )  =− + SAl sol g ∂t ∂ zg 

with SAl sol = −φAlgsol

(47)

4 σ (Tg4 − Tmelt,Al ) + hg (Tg − Tmelt,Al )

(48)

dAl qmelt,Al

g

∂ (φAlgliq ρAl liq ) ∂ (φAlgliq uAlgliq ρAl liq )  + = S liq Alg ∂t ∂ zg with S



Algliq

(49)

4 σ (Tg4 − Tmelt,Al ) + hg (Tg − Tmelt,Al )

= φAlgsol

(50)

dAl qmelt,Al









Zone GIII : ∀zg (t ) ∈ zg (t ) Tg = Tevap,Al , L

(51)

φAlgsol (zg (t )) = 0

(52)

∂ (φAlgliq ρAl liq ) ∂ (φAlgliq uAlgliq ρAl liq )  + = S liq Alg ∂t ∂ zg

(53)

with S



Algliq

= −φAl liq

σ (Tg4 − Te4vap,Al ) + hg (Tg − Tevap,Al )

g

Due to the presence of solid and liquid aluminum in the condensed and gas phases, it is necessary to calculate the mole fraction of these components to elucidate some terms in the equations of Table 6. The mole fraction of aluminum at solid and liquid states in condensed and gas phase are shown in Table 7. In the domain of gas phase, there are three zone divisions, GI , GII , and GIII , (Fig. 2) based on the phase state of aluminum.

2.3.1. Moving interface During combustion, the phase interfaces (solid–condensed phase and condensed–gas phase) move continuously. The burning rate is defined by the moving velocity of the condensed – gas phase interface [2].

(54)

dAl qevap,Al

 Moving speed of interface between solid and condensed phases

hcs (Tc (zsc (t )|+ ) − Tc (zsc (t ))) − λs dzsc =− dt ρs Hmelt

dTs (zsc (t )) d zs

(55)

 Moving speed of interface between condensed and gas phases

dzcg (t ) = −(zcg (t ) − zsc (t ) ) dt

Burning rate : ul = −



dzcg (t ) dt

rc (zcg (t ))

ρl

(56)

(57)

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417

Fig. 2. Domain of gas phase.

2.3.2. The formula for specific impulse (Isp ) In this study, a reference [33] for calculating specific impulse was used. It was necessary to calculate exhaust gas velocity:

     k−1 k  2k RTg P e Ve =  1− k − 1 Mg

Pg

(58)

Specific impulse value can be calculated directly from the exhaust gas velocity as follows:

Isp =

Ve g

(59)

2.3.3. The heat transfer coefficient The convective heat transfer coefficient, h is calculated based on Ref. [2]. 3. Solution strategy 3.1. Simulation strategy Numerical method based on gPROMS software – a modeling and simulation package [34–37] – is applied for the solution strategy. The overview of the simulation strategy in this study is presented in Fig. 3. Originally, the mathematical model in Section 2.3 is constructed with moving boundaries during the combustion, serving as the moving boundary model. Accordingly, the domains of the solid phase, condensed phase, and gas phase are [0, zsc (t)], [zsc (t), zcg (t)], and [zcg (t), L]. Through the coordinate transformation presented in Fig. 4, the domain of all the phases during combustion is replaced by [0, 1] to obtain the fixed boundary model. Also, the model variables and their derivatives have been modified in the fixed boundary mathematical model [2]. Concerning partial differential algebraic equations (PDAEs), the finite difference method (here, backward finite difference method or BFDM is specifically used) is utilized for the approximation of partial derivatives. The mathematical model in this study comprises of many stiff equations that make the values of variables to change suddenly during combustion (e.g. burning rate, gas phase temperature), the centered finite difference method cannot properly handle this problem due to serious oscillations at the phase interfaces. Using the BFDM with various number of intervals along with the moving boundary approach minimizes the effect of numerical diffusion problems, and thus employed as the effective numerical method to solve PDAEs. The linear algebraic and nonlinear algebraic equations are solved by the quasi-Newton method and LA solver. Numerical methods integrated with gPROMS are approximated, and thus, the accuracy of solution depends on absolute and relative tolerances. In this study, the absolute and relative tolerance were set to a value of 10−5 per convergence standard. In the simulation strategy for the combustion of the ternary solid propellant system (HMX/GAP/Al), the model for the combustion of the binary solid propellant system (HMX/Al) was validated as a prerequisite step.

Table 8 Simulation basis of aluminized solid propellant. Parameter

Value

Combustor length Combustor diameter Initial propellant length Initial temperature Combustor pressure Laser flux

2.0 m 0.2 m 1.6 m 300 K Case study 100 (W/cm2 )

3.2. Simulation basis In this study, we carried out dynamic simulation for the combustion of HMX/Al binary propellant system for validation and combustion of HMX/GAP/Al ternary propellant system for prediction of dynamic behaviors. The composition of the initial propellant mixture was adjusted to establish a case study for research. In both the binary system (HMX/Al) and ternary system (HMX/GAP/Al), aluminum particles with a nano-size of 0.18 μmwere used as a metal fuel having 0, 5, 10, 15, and 20 wt%. For the HMX/GAP/Al system, the mass ratio of the oxidizer (HMX) and binder (GAP) was always 80:20. Various operating pressures at 40, 60, 80, and 100 atm were established for case study. The specific design and simulation conditions are shown in Table 8. Design parameters are chosen based on reference values [38–40]. 4. Results and discussion The results mainly consist of two parts. The first part (4.1) validates the developed mathematical model governing the dynamic behavior of the binary solid propellant system with HMX and Al particles. The second part (4.2–4.4) covers the dynamic behavior prediction of the ternary solid propellant (HMX, GAP and Al particles) combustion. The reason for this two-step strategy is inevitable since there is no reported experiment or result from theoretical researches. However, it should be noted that the mathematical model for the binary and ternary solid propellant systems is developed within the same framework, and this eventually helps to model the binary system as one trivial case by eliminating the binder from ternary model without any further modification. 4.1. Validation of the model with the combustion of HMX/Al propellant In this study, the results of the HMX/Al propellant combustion simulation were compared with the experimental data [9] to validate the model. Figure 5 represents the effect of the Al content on the gas phase temperature at a chamber pressure of 60 atm. The results were compared with experimental data [9]. A slight deviation (lower than 1%) of the simulation result from the experimental data indicates a good agreement. Generally, gas phase temperature increases with a corresponding increase in aluminum

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Fig. 3. Simulation strategy with moving boundary model.

Fig. 4. Coordinate transformation in aluminized solid propellant combustion.

content in the range of 0–20 wt%. However, no significant increase is observed from 15 to 20 wt% Al. The simulation results reveal that the composition with 15 wt% Al produces a gas phase temperature of 3758 K, which is about 15% higher than HMX monopropellant (0 wt% Al). This rise in gas phase temperature as the Al content escalates in the propellant compo-

sition can be explained by the higher rate of exothermic reactions of aluminum (vapor) with gaseous oxidizers (O2 , H2 O, and CO2 ) and subsequent reactions, especially the condensation reaction of alumina (Al2 O3 ). It should be noted that the phase transition from Al2 O3 (gas) to Al2 O3 (liquid) releases an extremely high heat energy (approximately 1076.6 kJ/mol). The

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Fig. 5. The dependence of gas phase temperature on Al content at a chamber pressure of 60 atm.

amount of aluminum reacting in the gas phase increases as Al content increases in propellant composition which results in a higher rate of subsequent reactions and condensation reaction of Al2 O3 and thus larger heat energy is released in the gas phase during the combustion of aluminized solid propellant. Mole fraction of the major products in the gas phase of HMX/Al propellant combustion at 60 atm is shown in Fig. 6. The simulation results deviate from the experimental data by 16% on average. The results denote that the mole fraction of N2 slightly decreases from 0.347 to 0.301 as Al content increases from 0–20 wt%. Meanwhile, the mole fractions of CO and H2 show a tendency to increase with increasing Al content. Consequently, the mole fractions of CO2 and H2 O considerably decrease with increasing aluminum content, with CO2 being least (0.0 0 078) for 20 wt% aluminum in propellant. The mole fractions of CO2 and H2 O decreased while those of H2 and CO increased due to the mole balance of Al (g) reactions with CO2 and H2 O in the gas phase, which produce CO and H2 as the products (Table 4). It can be seen that the mole

419

Fig. 7. The dependence of specific impulse (Isp ) on Al content at 60 atm for HMX/Al propellant.

fraction of components and temperature in the gas phase have a close relationship. For example, the mole fraction of components does not change considerably as Al content increases from 15 to 20 wt% (Fig. 6) due to no significant increase in gas phase temperature (Fig. 5). Figure 7 shows the dependence of the specific impulse of HMX/Al propellants on Al content at 60 atm in comparison to the experimental data [9]. The specific impulse is affected directly by exhaust gas velocity through equation No. 59. The correlation of the exhaust gas velocity and proportion of Tg /Mg is elucidated in equation No. 58. It should be noted that Tg /Mg represents the thermodynamic energy state of an energetic material, a high Tg and a low Mg are required for propellants in order to increase the specific impulse [4]. The simulation results show a noticeably small deviation (lower than 1%) from the experimental data. In fact, the combustion of propellant produces many gas products, here we only considered the main components (Fig. 6) and thus the calculated values are slightly higher than that of experimental data. The specific impulse increases with increasing aluminum content in the range of 0–15 wt% due to an increase in Tg /Mg ratio. However, the

Fig. 6. Mole fractions of the main products in the gas phase of HMX/Al propellant combustion at 60 atm.

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Fig. 8. Burning rate profile of combustion for different propellant compositions at 100 atm.

value of Isp decreases slightly as aluminum content increases from 15 to 20 wt%. The reason here is that although gas phase temperature increases, the molar mass of combustion products increases more dominantly resulting in Tg /Mg ratio decrease. The calculation indicates that specific impulse of the propellant with 15 wt% Al is about 1.8% higher than for HMX monopropellant (0 wt% Al). Therefore, the HMX propellant with 15 wt% Al is the most prominent in terms of specific impulse among the aluminum ratios given. In fact, a propellant with 15 wt% Al was commonly studied in previous works [17,21,23]. Based on the analyzed results above for HMX/Al combustion, we have shown how accurate our methodology is. As mentioned earlier, our approach can be used for the ternary propellant system (HMX/GAP/Al). In the following Sections 4.2–4.4, the results of HMX/GAP/Al propellant combustion are analyzed. 4.2. Burning rate and propellant mass of HMX/GAP/Al propellant combustion Propellant mass change and burning rate are two important characteristics. They have a close relationship in terms of studying the dynamics of propellant combustion because propellant mass change depends on burning rate. The burning rate profile of HMX/GAP/Al propellant combustion at various compositions is shown in Fig. 8. Initially, the burning rate is zero because the laser source starts to supply heat energy for the propellant surface. As HMX and GAP start to melt, the condensed phase reactions also begin (after 0.033–0.034 s for five cases) and, consequently, the burning rate slightly increases (about 0.023 cm/s for five cases). The laser heat source continues to be supplied for the propellant. Heat transfer from the gas phase to the condensed phase involves convection and radiation. Between 7–8 s, heat flux from high exothermic gas phase reactions start to prevail over the laser heat source, and burning surface temperature and burning rate abruptly increases. When convection and radiation energy from the gas phase to condensed -gas phase interface is dominant compared to the laser heat source, the laser is turned off. Meanwhile, combustion reaches steady state whereas burning rate rapidly increases to a stable value. The burning rate of HMX/GAP/Al propellant at steady state increases with an increase in Al content in the initial

Fig. 9. Propellant mass profile for the combustion of different HMX/GAP/Al propellant compositions at 100 atm.

mixture. An increase in Al content in the composition of propellant raises the aluminum amount in the gas phase, and, thus the tendency of creating a higher reaction rate. As mentioned earlier in Section 4.1, the reaction kinetics of aluminum in the gas phase is comprised of several exothermic reactions (especially Al2 O3 (g)  Al2 O3 (l)); therefore, the higher the heat energy released in the gas phase is, the greater the heat flux from gas phase to condensed– gas phase interface becomes, and the higher the burning rate is, too (heat flux from gas phase to condensed – gas phase interface is the controlling factor for the burning rate [2,7,10]). Figure 9 shows the propellant mass profile of combustion of the HMX/GAP/Al propellant at various compositions. In general, propellant mass reasonably responds to burning rate change. At the initial state, the solid propellant mass of each case is different because the propellant density changes with different compositions. Aluminum density is higher than that of HMX and GAP; thus, adding aluminum makes propellant density increase which results in greater propellant mass with higher aluminum content at the same size.

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Fig. 10. The burning rate of HMX/GAP/Al propellant combustion varies with chamber pressure and Al content.

There seem to be no changes in the mass of the solid propellant until a stable combustion is obtained because the burning rate is insignificant (0.023 cm/s as mentioned above). When the combustion is stable, propellant mass rapidly decreases as presented through the slope of oblique lines. The propellant mass decreases faster at the higher aluminum content in propellant composition. This is based on the idea of increasing the burning rate when the increasing aluminum content is as shown in Fig. 8. Apparently, burning rate is the moving velocity of the propellant surface during combustion. Hence, the higher the burning rate is, the faster the propellant mass decreases because the length of the propellant decreases faster. Because of the difference in initial propellant masses (caused by the difference in density) and slope of the oblique lines which represent propellant mass change during the stable combustion (due to the difference in burning rate), profile of propellant mass change for all the various compositions of propellants crossover between 20 and 30 s. The effect of propellant composition and pressure on the burning rate of HMX/GAP/Al propellant is shown in Fig. 10. Increasing pressure reduces the flame distance from the surface and results in the rise of heat flux from the gas phase to condensed-gas phase interface. The increase of heat flux enhances the rate of decomposition reactions in the condensed phase (HMX and GAP in this study) and results in the rise in burning rate [4,41]. Otherwise as explained earlier in Section 4.1, increasing Al content in the initial propellant mixture results in the increase of the reaction rate in the gas phase of the aluminum (most of them are exothermic), with heat release thus increasing and, as a result, burning rate increasing. 4.3. Mole fraction of components for HMX/GAP/Al propellant combustion The mole fraction of components for HMX/GAP/Al propellant combustion is analyzed, with the composition being HMX:GAP:Al = 68:17:15 wt%, a common ratio of aluminized solid propellant in previous works [17,21,23]. Figure 11 represents the mole fraction profile of the main components in the condensed phase at ignition time for propellant with composition being HMX: GAP: Al = 68:17:15 wt%, at 100 atm. It represents products which are formed from the decomposition reaction of HMX (liquid) and GAP (liquid), along with solid aluminum. The bold solid line corresponds to temperature, and the points B and C represent the melt-

421

Fig. 11. Mole fraction profile of the main components in the condensed phase at ignition time of HMX/GAP/Al propellant combustion (HMX:GAP:Al = 68:17:15 wt%).

ing temperature of GAP and HMX, respectively. Accordingly, the melting starts at 0.031 s (for GAP) and 0.0314 s (for HMX), and the following are their reactions. After about 0.0314 s and 0.0333 s, reactions in the condensed phase of GAP and HMX occur respectively (as discussed in part of the burning rate), while the composition of products is consistent with the mole balance in reactions and content of HMX and GAP in the initial mixture (ratio HMX:GAP = 8:2). The result indicates that CH2 O and N2 O are dominant products in the condensed phase at ignition time, with mole fractions being 0.309 and 0.302. Furthermore, solid aluminum particles still exist in the condensed phase because a condensed phase temperature at ignition time is lower than the melting point of the aluminum metal (933 K). Figure 12 represents the mole fraction profiles of products in the gas phase combustion of HMX/GAP/Al propellant with 0 and 15 wt% Al content. Farther away from the burning surface (z = 0), the mole fractions of the components reach stable values which represent the final mole fraction of the products in HMX/GAP/Al propellant combustion. It can be seen that there is a difference in the composition of the gas phase products between the combustion of non-aluminized solid propellant and aluminized solid propellant. Also, the mole fraction of CO increases from 0.301 to 0.341 while that of H2 increases from 0.180 to 0.240 with the addition of 15 wt% Al into the solid propellant. Meanwhile, the mole fractions of CO2 and H2 O decreased from 0.035 to 0.0063 and from 0.140 to 0.043, respectively. This alteration in mole fraction is explained based on the reaction kinetics of aluminum in the gas phase. From the reactions 14, 15 and 16 (Table 4), one can see that CO2 and H2 O are reactants while CO and H2 are products, thus explaining the outcome. The results suggest that Al2 O3 is the dominant product of aluminum combustion (approximately 1% of the products in gas phase) as the consequence of a series of consecutive reactions of aluminum in the gas phase. The previous researches [16,18] also revealed that Al2 O3 is the main product in the combustion mechanism of aluminum. Therefore, the result of this study is satisfactory. 4.4. Surface temperature, gas phase temperature, and specific impulse for HMX/GAP/Al propellant combustion This section covers the burning surface temperature, gas phase, and specific impulse for HMX/GAP/Al propellant. Burning surface temperature is an important factor which determines the aluminum particles state during propellant combustion and is closely related to the gas phase temperature, while the gas phase

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Fig. 12. Mole fraction profile of the main gas phase products of HMX/GAP/Al propellant combustion with Al content 0 (a) and 15 wt% (b) at 100 atm.

Fig. 13. Surface temperature of HMX/GAP/Al propellant combustion in various Al contents at 100 atm.

temperature influences the specific impulse through equations No. 58 and 59. The profile of the surface temperature for HMX/GAP/Al propellant combustion at 100 atm is shown in Fig. 13. It should be noted that the surface temperature in Fig. 13, the burning rate in Fig. 8, and propellant mass in Fig. 9 suggest intercorrelations between them in the combustion process. The surface temperature rapidly increases from the initial temperature of 300 K (point A) to 654 K (point D) with the laser heat source in a brief period of time (about 0.036 s). Points B and C represent the melting point of GAP (450 K) and HMX (530 K), respectively. Points B, C, and D in Fig. 13 correspond to those in Fig. 11. As the Al content increases, the times for reaching the melting point of GAP and HMX are expected to extend since propellant density increases while heat transfer from the laser heat source remains unchanged. Based on the results, the time taken in seconds to reach the melting point of GAP for the

various cases of Al content, 0, 5, 10, 15, and 20 wt% are 0.0305, 0.0306, 0.0308, 0.0310, and 0.0312 respectively while those of HMX are 0.0309, 0.0311, 0.0313, 0.0314, 0.0316. While GAP and HMX melt at points B and C respectively, condensed phase reactions start simultaneously. From point D, the surface temperature of propellant slightly decreases to 623 K in 0.964 s and then remains constant for a short while. The reduction can be attributed to the endothermic reactions dominating in the condensed phase, after that, the heat of endothermic reaction gets balanced by the heat of exothermic reactions, resulting in a constant value. As shown in Table 3, among the condensed phase reactions, there is one endothermic reaction with enthalpy of 909.5 kJ/mol. The decrease in surface temperature after point D proves that the absorbed heat rate for endothermic reaction is higher than the released heat rate of exothermic reactions, and thus heat energy is absorbed from burning surface by reactant (here is HMX). The surface is then stabilized because of the heat balance between endothermic and exothermic reactions (Table 3). Finally, the surface temperature of the propellant sharply increases and reaches a steady state. At this point, the laser heat source is put off and the heat flux for combustion is supplied by the exothermic reactions in the gas phase. Also, increasing the aluminum concentration from 0 to 20 wt% causes an increase in the steady-state surface temperature from 820 K to 838 K. The explanation to this is similar to the burning rate: an increase in aluminum concentration increases heat flux from the gas phase to condensed–gas phase interface, leading to increased burning surface temperature. The surface temperature recorded in all the cases is lower than the melting point of aluminum (933 K) which implies that it is still in solid-state at the condensed phase. The gas phase temperature of HMX/GAP/Al propellant combustion for different Al content at 100 atm is shown in Fig. 14. The farther away from the burning surface (z = 0), the higher temperature, and the steady-state gas phase temperature is obtained. All the cases of the Al content show similar trends but different magnitudes of steady-state flame temperature. For 0, 5, 10, 15, and 20 wt% of Al content in HMX/GAP/Al propellant, the steady-state flame temperatures of 2730 K, 2872 K, 2933 K, 2988 K, and 3061 K are reached respectively which shows a sharp increase as Al

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423

Fig. 14. Profile of the gas phase temperature for HMX/GAP/Al propellant combustion with different Al contents at 100 atm.

content increases. This increase concurs with the trends of burning rate and surface temperature for the aluminized solid propellant as explained in Figs. 8 and 13. Furthermore, the results propose increasing the initial Al content in the propellant, consequently raising the aluminum concentration in the gas phase thereby causing the rate of exothermic reaction (such as condensation reaction of Al2 O3 ) to increase and the heat released in the gas phase to become larger. The dependence of a specific impulse on Al content and chamber pressure of HMX/GAP/Al propellant is presented in Fig. 15. Generally, the Isp of HMX/GAP/Al propellant increases as the Al content increases from 0 to 15 wt% which is as a result of the increase in the proportion of Tg /Mg that decreases slightly as Al content increases to 20 wt%. This trend is similar to the results obtained by Kasztankiewicz et al. [22] for the AP/HTPB/Al propellant. The reduction in specific impulse from the 15 wt% Al is

consistent with the case of HMX/Al propellant as explained earlier. The combustion of aluminized solid propellant creates not only high combustion temperature due to the high heat of combustion of metal particles but also high molecular mass of the combustion products due to the metal oxides [4]. Thus, Tg /Mg is not always higher when Al content is increased in propellant. From the results, it can be seen that the operating pressure of 100 atm produces the highest specific impulse values for the various Al contents. Comparing the results to reference data [4], the specific impulse value of 253.1 s obtained for HMX/GAP propellant with no aluminum at 100 atm deviates by 2%. For any Al content, increasing pressure causes a sharp rise in specific impulse; for example, Isp increases by approximately 9% when pressure is increased from 40 to 100 atm for HMX/GAP/Al propellant containing 15 wt% Al. This can be explained with Equations No. 58 and 59.

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Acknowledgment This research was supported by the Agency for Defense Development (Project No. 20180 060 0 0 01). References

Fig. 15. The specific impulse of HMX/GAP/Al propellant combustion varies with chamber pressure and Al content.

5. Conclusion The combustion of HMX/GAP/Al propellant takes place in three phases: solid, condensed, and gas phases. The phase transitions and reactions of aluminum occur in the gas phase domain. This study developed a rigorous mathematical model through the use of the moving boundary method for multiphase combustion in order to predict its dynamic behaviors. Reaction kinetics mechanisms were presented for HMX and GAP (in the condensed phase and the gas phase) and aluminum (in the gas phase). The simulation results, such as gas-phase temperature, mole fraction of main gas products, and specific impulse showed good agreement with the experimental data for the HMX/Al combustion. By validating the model, combustion characteristics namely burning rate, surface temperature, gas phase temperature, mole fraction of gasphase products, and specific impulse were successfully predicted for HMX/GAP/Al propellant combustion at a laser intensity of 100 W/cm2 and initial temperature of 300 K. This study also carried out parametric studies on propellant composition and operating pressure to analyze dynamic behaviors. In this light, the difference in combustion behaviors between aluminized solid propellant and non-aluminized solid propellant was analyzed. Based on the results, the gas phase temperature increased from 2730 to 3061 K while the burning rate increased from 2.08 to 2.86 cm/s for HMX/GAP/Al propellant combustion with increasing Al content from 0 to 20 wt% at 100 atm. For the various pressures analyzed, the specific impulse increased with increasing Al content until it reached 15 wt% where it slightly decreased to 20 wt%. For example, specific impulse increased from 253.09 s to 258.53 s as Al content increased from 0 to 15 wt% and decreased to 257.94 s for the 20 wt% Al at 100 atm. The results also showed that the burning rate and specific impulse increased when increasing operating pressure. Altogether, the dynamic modeling and simulation provided a detailed framework for a comprehensive understanding of the aluminized solid propellant multi-phase combustion. From this study, it can be concluded that the addition of aluminum can improve the combustion characteristics of a solid propellant. Hence, it is possible to use aluminum as a metal fuel in solid propellants in order to increase the efficiency of rocket motors.

Declaration of Competing Interest None.

[1] B.P. Mason, C.M. Roland, Solid propellants, Rubber Chem. Technol. 92 (2019) 1–24. [2] N.D. Vo, M.Y. Jung, D.H. Oh, J.S. Park, I. Moon, M. Oh, Moving boundary modeling for solid propellant combustion, Combust. Flame 189 (2018) 12–23. [3] M.W. Beckstead, K. Puduppakkam, P. Thakre, V. Yang, Modeling of combustion and ignition of solid-propellant ingredients, Prog. Energy Combust. Sci. 33 (2007) 497–551. [4] N. Kubota, Propellants and Explosives – Thermochemical Aspects of Combustion, Wiley VCH, 2002. [5] V.N. Marshakov, B.V. Novozhilov, Combustion of a propellant and its extinction upon rapid depressurization: a comparison of theory and experiment, Russ. J. Phys. Chem. B 5 (2011) 474–481. [6] S. Jain, W. Park, Y.P. Chen, L. Qiao, Flame speed enhancement of a nitrocellulose monopropellant using graphene microstructures, J. Appl. Phys. (2016) 120. [7] Y. Liau, V. Yang, S.T. Thynell, Modeling of RDX/GAP propellent combustion with detailed chemical kinetics, Solid Propellant Chem. Combust. Mot. Inter. Ballist. (20 0 0) 477–50 0. [8] A.M.A. Elghafour, M.A. Radwan, H.E. Mostafa, A. Fahd, S. Elbasuney, Highly energetic nitramines: a novel platonizing agent for double-base propellants with superior combustion characteristics, Fuel 227 (2018) 478–484. [9] N. Muravyev, Y. Frolov, A. Pivkina, K. Monogarov, D. Ivanov, D. Meerov, et al., Combustion of energetic systems based on HMX and aluminum: infuence of particle size and mixing technology, Cent. Eur. J. Energ. Mater. 6 (2009) 195–210. [10] E.S. Kim, V. Yang, Y.C. Liau, Modeling of HMX/GAP pseudo-propellant combustion, Combust. Flame 131 (2002) 227–245. [11] K. Prasad, R.A. Yetter, M.D. Smooke, An eigenvalue method for computing the burning rates of HMX propellants, Combust. Flame 115 (1998) 406–416. [12] L.L. Minkov, E.R. Shrager, E.V. Pikushchak, Optimum disposition of metal particles in the propellant grain, Int. J. Aerosp. Eng. 2014 (2014). [13] L.T. Deluca, Innovative solid formulations for rocket propulsion, Eurasian Chem. J. 18 (2016) 181–196. [14] C. Griego, N. Yilmaz, A. Atmanli, Analysis of aluminum particle combustion in a downward burning solid rocket propellant, Fuel 237 (2019) 405–412. [15] V. Babuk, I. Dolotkazin, A. Gamsov, A. Glebov, L.T. DeLuca, L. Galfetti, Nanoaluminum as a solid propellant fuel, J. Propuls. Power 25 (2009) 482–489. [16] Y. Huang, G.A. Risha, V. Yang, R.A. Yetter, Analysis of nano-aluminum particle dust cloud combustion in different oxidizer environments, 43rd AIAA Aerospace Sciences Meeting and Exhibit (2005). [17] T. Ecker, S. Karl, K. Hannemann, Modeling of aluminum particle combustion in solid rocket combustion chambers, 53rd AIAA/SAE/ASEE Joint Propulsion Conference (2017). [18] M.W. Beckstead, A Summary of Aluminum Combustion, Report on: Internal Aerodynamics in Solid Rocket Propulsion, Rhode-Saint-Genèse, Belgium, 2002. [19] D.S. Sundaram, V. Yang, V.E. Zarko, Combustion of nano aluminum particles (Review), Combust. Explos. Shock Waves 51 (2015) 173–196. [20] X. Wang, T.L. Jackson, J. Buckmaster, L. Massa, K. Hossain K, Three-dimensional modeling of aluminized composite solid propellant combustion, 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit (2006), p. 9. [21] L. Meda, G. Marra, L. Galfetti, S. Inchingalo, F. Severini, L.T. Deluca, Nano-composites for rocket solid propellants, Compos. Sci. Technol. 65 (2005) 769–773. ´ [22] A. Kasztankiewicz, K. Ganczyk-Specjalska , A. Zygmunt, K. Cies´ lak, B. Zakos´ cielny, T. Gołofit, Application and properties of aluminum in rocket propellants and pyrotechnics, J. Elem. 23 (2018) 321–331. [23] L.T. Deluca, L. Galfetti, Pre- and post-burning analysis of aluminized propellants. Comparison of four Russian nano-Al powders, Technical Report, London, England, December 2003–December 2004. [24] G.H. Lee, M.Y. Jung, J.C. Yoo, B.S. Min, H.M. Shim, M. Oh, Dynamic simulation of ignition, combustion, and extinguishment processes of HMX/GAP solid propellant in rocket motor using moving boundary approach, Combust. Flame 201 (2019) 129–139. [25] V.P. Sinditskii, V.Y. Egorshev, M.V. Berezin, V.V. Serushkin, Mechanism of HMX combustion in a wide range of pressures, Combust. Explos. Shock Waves 45 (2009) 461–477. [26] J.E. Davidson, Combustion Modeling of RDX, HMX and GAP With Detailed Kinetics, Brigham Young University, 1996 The Dissertation of Doctor of Philosophy. [27] K.V. Puduppakkam, M.W. Beckstead, Combustion modeling of glycidyl azide polymer with detailed kinetics, Combust. Sci. Technol. 177 (2005) 1661–1697. [28] M.W. Tanner, Multidimensional Modeling of Solid Propellant Burning Rates and Aluminium Agglomeration and One-Dimensional Modeling of RDX/GAP and AP/HTPB, Brigham Young University, 2008 The Dissertation of Doctor of Philosophy. [29] V.B. Storozhev, A.N. Yermakov, Combustion of nano-sized aluminum particles in steam: numerical modeling, Combust. Flame 162 (2015) 1–9. [30] M.T. Swihart, L. Catoire, Thermochemistry of aluminum species for combustion modeling from ab initio molecular orbital calculations, Combust. Flame 121 (20 0 0) 210–222.

T.A. Vo, M. Jung and D. Adams et al. / Combustion and Flame 213 (2020) 409–425 [31] Y. Chen, D.R. Guildenbecher, K.N.G. Hoffmeister, M.A. Cooper, H.L. Stauffacher, M.S. Oliver, et al., Study of aluminum particle combustion in solid propellant plumes using digital in-line holography and imaging pyrometry, Combust. Flame 182 (2017) 225–237. [32] V.I. Levitas, Mechanochemical mechanism for reaction of aluminium nano- and micrometre-scale particles, Phil. Trans. R. Soc. A (2017) 1–14. [33] https://www.thespacerace.com/forum/index.php?topic=1481.0 (last accessed: 19/06/15). [34] Process system engineering. Model developer guide. https://www.psenterprise. com/ (last accessed: 19/05/05). [35] S.H. Kim, B.W. Nyande, H.S. Kim, J.S. Park, W.J. Lee, M. Oh, Numerical analysis of thermal decomposition for RDX, TNT, and composition B, J. Hazard. Mater. 308 (2016) 120–130.

425

[36] S.H. Kim, S.W. Baek, J.C. Lee, W.J. Lee, S.U. Hong, M. Oh, Dynamic simulation of liquid polymerization reactors in Sheripol process for polypropylene, J. Ind. Eng. Chem. 33 (2016) 298–306. [37] N.D. Vo, D.H. Oh, S.H. Hong, M. Oh, C.H. Lee, Combined approach using mathematical modelling and artificial neural network for chemical industries: steam methane reformer, Appl. Energ. 255 (2019) 1–18. [38] http://risacher.org/rocket/eqns.html (last accessed: 19/08/25). [39] http://www.aerospacengineering.net/1255/ (last accessed: 19/09/01). [40] K.R.A. Kumar, K.N. Lakshmisha, Modeling re-ignition and chuffing in solid rocket motors, Proc. Combust. Inst. 29 (2002) 2905–2912. [41] M. Pandey, S. Jha, R. Kumar, S. Mishra, R.R. Jha, The pressure effect study on the burning rate of ammonium nitrate-HTPB-based propellant with the influence catalysts, J. Therm. Anal. Calorim. 107 (2012) 135–140.