Effect of primary water injection angle on thermal propulsion performance of a water ramjet engine

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Effect of primary water injection angle on thermal propulsion performance of a water ramjet engine

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School of Marine Science and Technology, Northwestern Polytechnical University, Box 24, Xi’an, 710072, PR China

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a r t i c l e

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Article history: Received 23 September 2019 Received in revised form 12 November 2019 Accepted 7 December 2019 Available online xxxx

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Dechao Liu, Shulei Li ∗ , Gongnan Xie ∗

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Keywords: Water ramjet engine Hydroreactive metal fuel Primary injection angle Thermal propulsion Specific impulse

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Using high-performance hydroreactive metals as fuels, water ramjet engines (WREs) are regarded as the most competitive underwater power systems to meet high-speed and long voyage propulsion. However, its performances are strongly influenced by the water injection mode. Therefore, in this paper, the flow and thermal propulsion characteristics of WREs with different primary water injections were numerically investigated based on validated model. Firstly, the internal flow characteristics illustrated by streamlines, kinetic rate of the reaction and distribution of the components are discussed. It is found that there is a longitudinal vortex caused by sudden-expansion formed by annular inlet that can intensify mixing. Then, the effect of water injection angles on propulsion performance is investigated by considering six axial angles and two tangential angles. Results indicate that proper inclination of the primary water nozzles to the upstream makes the mixing occur earlier. Correspondingly, the specific impulse can be enhanced by approximately 8%. With the increase of tangential angle, more water congregates near wall, leading to incomplete reaction and the reduction of specific impulse. This paper is beneficial for the design of WRE water injection. © 2019 Elsevier Masson SAS. All rights reserved.

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1. Introduction

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Water ramjet engine (WRE) is a new conceptual underwater propulsion system which utilizes high-performance hydroreactive metal fuel (HMF) reacting with seawater to product working medium further to generate thrust [1]. Due to the absence of the oxidant onboard, WRE possesses a great potential to improve specific impulse, which is expected to reach about twice than solid rocket engine [2]. With its outstanding advantages of small size, light weight, simple structure and good safeties, WRE will become a competitive choice to satisfy the idea of the high speed and large voyage as underwater propulsion system. In terms of HMF, the researchers have been working on the oxygen-poor fuel that contain the active metals (aluminum (Al), magnesium (Mg), lithium (Li), et al.) [3–6]. In particular, Al has been specially focused on due to its potential chemical energies, storage requirements, and low toxicity [7]. As shown in Table 1, the volume energy density of Al is approximately three times than that of the Li, and is about twice than that of the Mg. Nevertheless, Al has a higher melting point (933 K) and a higher boiling point (2740 K) than that of other active metal, which would lead

60 61 62 63 64 65 66

*

Corresponding authors. E-mail addresses: [email protected] (S. Li), [email protected] (G. Xie).

https://doi.org/10.1016/j.ast.2019.105630 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

to the onset of Al/H2 O reaction more difficult. Moreover, fresh Al particle could be also easily oxidized to product a dense aluminum trioxide (Al2 O3 ) film covering the surface in the process of propellant production, and preventing further Al/H2 O reaction [8]. To against the Al2 O3 film, Miller et al. [9] came up with a concept vortex combustor, in which Al particles and liquid water were injected tangentially at the outer periphery of the device. The radially inward velocity of the water and subsequent steam fluidized the outer layer of particles, and shearing action and collisions in the centrifugally fluidized layer would reduce the particle diameter. Unfortunately, it still needs a lot of efforts to put into practical project application. Compared with the homogeneous and heterogeneous reaction of Al/H2 O system, the Mg/H2 O reaction will occur in gas phase with considerable kinetics rate due to a lower boiling point (1390 K). In order to choose oxygen-poor fuels with higher specific impulse and the lower onset temperature of reaction, Huang et al. [10] conducted a series of experiments to test the combustion performance of Al/H2 O, Mg/H2 O, and Mg-Al/H2 O by scanning electron microscopy, X-ray diffraction (XRD), and simultaneous thermogravimetric analysis (TGA). Results showed that the oxidation onset temperatures of magnalium powders were much lower than both Mg and Al powders, and the propellant samples that contained magnalium powders exhibited the best performances in terms of the burning rate, primary combustion heat, and Al reaction efficiency, which indicated that HMF based on mag-

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Nomenclature A d,Al Af A out Ar CD Cg c p,Al CS d D0 D 0,P E F Fx f v,0 I sp mAl,0 N Pe Pn P out q m,g q m,f r R Rc Re R kin R par R par R w,f T Al T bd,Al T vap,Al T Mi T∞ u

68 2

surface area of particle . . . . . . . . . . . . . . . . . . . . . . . . . . . [m ] burning area of HMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m2 ] area of the nozzle outlet . . . . . . . . . . . . . . . . . . . . . . . . . [m2 ] pre-exponential factor of reaction drag coefficient mean reacting gas species concentration in the bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [kg/m3 ] specific heat of Al particle . . . . . . . . . . . . . . . . . . [kJ/(kg·K)] mean reacting gas species concentration at the particle surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [kg/m3] diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [μm] bulk diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . [m/s] diffusion rate coefficient for reaction activation energy for the reaction . . . . . . . . . . . . [J/kmol] trust of WRE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [N] external force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [N] evaporation mass fraction of the Al particle specific impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [N·s/kg] initial mass of the Al particle . . . . . . . . . . . . . . . . . . . . . [kg] apparent reaction order (dimensionless) pressure of environment . . . . . . . . . . . . . . . . . . . . . . . . [Mpa] bulk partial pressure of the gas phase species . . [Mpa] the outlet pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Mpa] mass flow rate of gaseous phase . . . . . . . . . . . . . . . . [kg/s] mass flow rate of HMF . . . . . . . . . . . . . . . . . . . . . . . . . . [kg/s] burning velocity of HMF . . . . . . . . . . . . . . . . . . . . . . . [mm/s] universal gas constant . . . . . . . . . . . . . . . . . . . . . . [J/kmol·K] chemical reaction rate coefficient (units vary) relative Reynolds number (particles Reynolds number) kinetic rate of reaction rate of particle surface species reaction per unit area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [kg/m2 ·s] rate of particle surface species depletion . . . . . . . [kg/s] total water/fuel ration temperature of Al particle . . . . . . . . . . . . . . . . . . . . . . . . . . [K] boiling point of Al particle . . . . . . . . . . . . . . . . . . . . . . . . . [K] evaporation temperature of Al particle . . . . . . . . . . . . . [K] average wall temperature of Mi grids . . . . . . . . . . . . . . [K] temperature of continues phase . . . . . . . . . . . . . . . . . . . . [K] velocity of the fluid phase . . . . . . . . . . . . . . . . . . . . . . . [m/s]

u par V out xpar

the velocity of the discrete phase . . . . . . . . . . . . . . . [m/s] the outlet velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [m/s] displacement of the discrete phase . . . . . . . . . . . . . . . . [m]

Greek symbol

α β δ1

ε εd ξ

η λ

μ ρ ϕ

70 71 72 73 74

axial injection angle tangential injection angle percentage of the average wall temperature mass fraction of the condensed phase blackness of particle the temperature exponent of reaction effectiveness factor of reaction thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . [W/(mK)] dynamic viscosity of fluid . . . . . . . . . . . . . . . . . . . [kg/(m·s)] density of the fluid phase . . . . . . . . . . . . . . . . . . . . . [kg/m3 ] discharge coefficient of nozzle

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Subscripts Al C e exp f g kin m out P par ref sim w 0

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aluminum chemical reaction environment experiment fuel gas kinetic mass rate outlet pressure particle phase reference simulation water initial conditions

88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103

Abbreviations WRE HMF LISA LVZ

water ramjet engine hydroreactive metal fuel linearized instability sheet atomization low velocity zone

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nalium powders had greater potential and application prospect. Therefore, in this paper, HMF that contains magnalium powders was selected. The combustion progress of WRE with twice water injections shown in Fig. 1. HMF finished primary combustion at the propellant surface to products high temperature fuel gas containing metal particles, and seawater as oxidizer is atomized into micron-order droplets through spray nozzle, and then the droplets evaporate by absorbing heat from hot gas fuel and acutely combustion with gaseous metal or metal particles. Hence, the mixing effect of water with fuel gas directly affects the performance of the WRE. In order to improve the mixing effect, various numerical and experimental investigations were devoted to the structural design and combustion organization in recent years. Luo et al. [11] pointed out the advantage of dual water injection and put forward an actualization scheme. Huang et al. [12] established the design methods of the length of combustion chamber and the distance between primary and secondary water. Yang et al. [13] obtained the variation trends of specific impulse, nozzle outlet temperature and outlet velocity with water-fuel ratio by experimental study and numerical simulation. The results showed that there was the optimal

water-fuel ratio, leading to the specific impulse of WRE approaching peak value. Yang et al. [14] further optimized the distribution of primary and secondary water-fuel ratio for a WRE. Chao et al. [15] investigated the influence of different primary gas entrance angle on specific impulse of WRE. The results indicated that convergence of multi-strand gas significantly improved the working pressure and specific impulse. With respect to droplet-gas mixing process[16–18], the mixing process of a liquid jet in supersonic crossflow was investigated numerically using large eddy simulation, results revealed that the gas flow structures have a significant effect on the mixing process of the droplets. Besides, Bai et al. [19, 20], Zhang et al. [21–23] and Sun et al. [24] measured the droplet distribution and flow field structure of cross-flow in the rectangle tube under cold conditions by two-dimensional particle image velocimeter (PIV). The experiment results showed that the two-phase interaction was related to initial spray configuration. In short, the previous investigations were mainly focused on position of twice water injections, allocation of water-fuel ratio, injection angle of inlet fuel gas and the flow field distribution of water atomization in confined space. The study on the influence of water injection on performance of WRE, especially under actual

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Table 1 Physical properties of hydroreactive metals.

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Density (g·cm−3 )

Mass energy density (kJ·g−1 )

Volume energy density (kJ·cm−3 )

Melting point (K)

Boiling point (K)

Heat of vaporization (kJ·mol−1 )

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Be Al Mg Li Na

1.85 2.7 1.74 0.53 0.97

37.26 16.95 14.92 29.23 6.07

68.93 45.77 25.27 15.49 5.89

1558 933 923 452 371

3243 2740 1390 1609 1156

309.35 284.44 136.13 135.96 97.42

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Symbol

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84

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88

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89

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90

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91

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92

27

93

28

94

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Fig. 1. Schematic description of combustion progress of a WRE with twice water injections based on hydroreactive metal fuel.

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100

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101

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102

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103

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104

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105

40

106

41

107 108

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Fig. 2. Schematic of WRE Computational domain with twice water injections.

45 46 47 48 49 50 51 52 53 54 55 56 57 58

combustion conditions is still scanty. Therefore, aim at this interest WRE applied to underwater propulsion system, it is essential to research more details about the axial and tangential angles of water injection during the combustion process on the basis of numerical simulation. In this paper, a numerical model considering combustion and water atomization multiphase flow was established based on 3D computational domain of WRE, whose reliability was verified by ground tests of WRE and thermodynamic calculation. Numerical simulations were conducted based on magnalium powders HMF under different axial angles (15◦ , 30◦ , 50◦ , 65◦ , 75◦ and 90◦ ) and tangential angles (5◦ , 10◦ and 20◦ ) of primary water injection. Then, the internal flow field properties were analyzed and the variation trends of combustor temperature, component concentration and specific impulse of WRE were discussed.

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2. Physical models

61 62 63 64 65 66

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In this paper, an experiment model of WRE is selected as geometric model in numerical simulation. The combustion progress of WRE as shown in Fig. 1. WRE is always regarded as the double reaction zone model, i.e., the HMF finishes primary combustion adiabatically at grain head face under constant-pressure condi-

tions and produces high-temperature gases that carry the molten or gaseous metal to react with outer water in secondary combustion chamber. The component of primary combustion products and their thermodynamic properties can be calculated by the thermodynamic calculation program, hence, the numerical investigation work focuses on secondary combustion. Additionally, due to the periodicity of computational domain, the representative longitudinal section through X–Y plane of overall geometric layout is shown in Fig. 2, in which the primary combustion progress is omitted. The overall length of model is 665.7 mm, while the radius of the fuel inlet, combustion chamber, nozzle throat and outlet are 30 mm, 45 mm, 8 mm and 13 mm, respectively. Moreover, twice water injections severally adopt four pressure swirl atomizers which are arranged uniformly in combustor wall. Notably, two kinds of injection angle of pressure swirl atomizers are considered including axial injection angle (α ) and tangential injection angle (β ). As shown in Fig. 3(a), α is defined as the angle between the axis of pressure swirl nozzle and the upstream direction of the mainstream. In the same way, β is the angle that between the axis of the pressure swirl nozzle and normal direction of the cross-section of combustion chamber, as shows in Fig. 3(b).

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Fig. 3. Schematic diagram of axial angle and tangential angle of primary water injection in a WRE. (a) Axial angle of primary water injection; (b) Tangential angle of primary water injection.

3. Computational details

where Re is relative Reynolds number (particle Reynolds number), written as follows:

20 21

3.1. Overview

Re = ρ dpar

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

The CFD (computational fluid dynamics) software FLUENT version 15.0 was employed to conduct the simulation. The transition from internal injector flow of twice water injection to fully-developed spray was simulated using pressure-swirl atomizer model, i.e., linearized instability sheet atomization (LISA) model of Schmidt et al. [25]. Moreover, the discrete phase model (DPM) was used to characterize the movement of droplets, inert particle and Al particle under the Lagrange reference frame, whereas the gas phase was solved with the Eulerian approach. Discrete phase was coupled with gas flow field via source terms, and the stochastic model was applied considering the influence of turbulent fluctuation on particle motion. Besides, the variation progress that evaporation rate, droplet diameter and droplet temperature with time was described by droplet evaporation model. The geometric modeling adopted the three-dimensional design software SOLIDWORKS version 2015, and ICEM CFD software version 15.0 was applied to generate computational grids.

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44 45 46

49 50 51 52 53 54 55 56 57 58 59 60

dxpar dt du par dt

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FD =

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(1)

= F D (u − u par ) +

g x (ρpar − ρ )

ρpar

(2)

18μ

C D Re

ρpar d2par 24

μ

88 89 90 91

As reported by Salita [26], in this paper, it was assumed that 20% of the Al in the propellant result in formation of Al2 O3 caps due to surface oxidation, whereas 80% Al was considered to be part of the continuous phase in the form of smoke. The particle multiple surface reaction model was employed to simulate the combustion process of Al particle, which can be expressed as follows:

92 93 94 95 96 97 98 99

Volatilization stage [27–31]:

100

While T Al ≥ T vap,Al , T Al ≥ T bd,Al and mAl > (1 − f v,0 )mAl,0 , the transformation of Al from solid particle to gaseous is simulated by constant rate reaction model, i.e., homogeneous reaction of the AL. The volatilization rate can be written as follows:

1

dmAl

(6)

f v,0 (1 − f w,0 )mAl,0 dt



CD =

( 24 )(1 + 0.15Re0.687 ) (Re < 1000) Re 0.44 (Re ≥ 1000)

where T Al , T vap,Al and T bd,Al are the temperature, evaporation temperature and boiling point of Al particle, respectively; f v,0 are the evaporation mass fraction of the Al particle; mAl,0 is the initial mass of the Al particle; dmAl /dt is the gradient of mass over time. The gas components that volatile from Al particle enter the mainstream and react. The changing of the Al particle diameter can be written as follows:

dAl dAl,0

= 1 + (C sw − 1)

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dt

= h A d,Al ( T ∞ − T Al ) + hfg

dmAl dt

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the ratio of the mass that has been volatilized to the total volatile mass of the particle. In the volatilization stage, the heat transfer of particle can be described as:

dT Al

109

116

(7)

(1− f )m −m diameter at the start of volatilization; The term f (w1,−0 f Al,)0m Al is v, 0 w,0 Al,0

mAl c p,Al

(4)

(1 − f w,0 )mAl,0 − mAl f v,0 (1 − f w,0 )mAl,0

where dAl is current Al particle diameter; dAl,0 is the Al particle

(3)

where μ is the dynamic viscosity of the fluid phase; C D is drag coefficient, which can be calculated as follows:

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= u par

where xpar and u par are the displacement and velocity of the particles, respectively; F D (u − u par ) is the drag force of particle per unit mass; u is the velocity of the fluid phase; ρ is density of the fluid phase; ρ par is density of the discrete phase; dpar is the diameter of the particle; F D can be written as follows:

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The gas-phase governing equations are composed of mass, momentum and energy conservation equations, the detailed 3D RANS equations are same as those in Ref. [14]. Discrete phase equations [14]:

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85 87

(5)

Particle surface reaction model:

A=−

3.2. Governing equations

(u par − u )

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83 84

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82

+ A d,Al εd σ (θR4 − T d4 )

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(8)

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where c p,Al is the specific heat of Al particle; h is the convective heat transfer coefficient; A d,Al is the surface area of particle; T ∞ is the temperature of continues phase; εd is blackness of particle;

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Table 2 Viscosity

3

Species

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μ for Mg, Al(g) and CO.

Mg Al(g) CO

6 7

68

μ = a + bT + cT 2 + dT 3 + eT 4 /kg · (ms)−1 a × 10

5

5

6

1.772 4.821 4.118

b × 10

8

1.968 1.583 4.715

c × 10

11

0.372 0.412 −1.311

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d × 10

15

−1.205 −0.846 2.655

e × 10

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20

71

10.180 5.333 −20.712

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8 9 11

Species

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Mg Al(g) CO

14 15 16 17 18 19 20 21

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

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g × 105

h × 109

i × 1012

j × 1016

1.889 6.402 4.087

2.581 1.645 7.639

5.118 6.258 −13.243

−2.127 −1.481

2.799 1.217 −1.154

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While mAl ≤ (1 − f v,0 )(1 − f w,0 )mAl,0 , i.e., homogeneous reaction of Al is completed, the Al particle begin to react on the surface. The particle reaction rate can be expressed as follows:

R = D 0 (C g − C S ) = R c (C S ) N

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(9)

where D 0 is the bulk diffusion coefficient; C g is the mean reacting gas species concentration in the bulk; C S is the mean reacting gas species concentration at the particle surface; R c is the chemical reaction rate coefficient; N is the apparent reaction order. The concentration C S at the particle surface is not known, so it should be eliminated, and the expression is recast as follows:

 R = Rc Cg −

R

N (10)

D0

Besides, the following equations are used to describe the reaction rate of a particle surface with the water vapor:

R par = A d,Al ηr R par



R par = R kin P n −

R par

N

(11) (12)

D 0,P

where R par is the rate of particle surface species depletion; ηr is the effectiveness factor; R par is the rate of particle surface species reaction per unit area; R kin is the kinetic rate of reaction; P n is the bulk partial pressure of the gas phase species; D 0,P is the diffusion rate coefficient for reaction. The effectiveness factor ηr is related to the surface area, and can be used in each reaction in the case of multiple reactions. D 0,P is given as:

 D 0,P = C 1

( T par + T ∞ )/2

0.75

dpar

53 54

1.907

78 79 80 81 82 83

50 51

77

f × 103

Surface reaction stage:

41 42

76

λ = f + g T + hT 2 + iT 3 + jT 4 /W · (mK)−1

22 23

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Table 3 Thermal conductivity λ for Mg, Al(g) and CO.

10

(13)

The kinetic rate of reaction is expressed by Arrhenius equation, as follows:

R kin = A r T Al e −( E / R T Al ) ζ

(14)

The rate of the particle surface species depletion for reaction order N = 1 is given as:

R Par = A d,Al ηr P n

D 0,P R kin D 0,P + R kin

(15)

where A r is the pre-exponential factor; ζ is the temperature exponent; E is the activation energy for the reaction; R is the universal gas constant.

Table 4 HMF propellant data.

84 85

Property Composition

Value Mass%

86

AP Al HTPB Mg

35.3 25 21.7 18

88

Density Burn rate

1638.3 kg/m3 10.55 mm/s

87 89 90 91 92 93 94

3.3. Thermophysical parameters and chemical reaction model

95 96

Thermophysical parameters model

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Due to higher adiabatic flame temperature for primary combustion, the variations of viscosity μ, thermal conductivity λ and isobaric heat capacity c p with combustor temperature which should be taken into account. In terms of the μ and λ, Fluent database provides the piecewise-polynomial forms for N2 , H2 , H2 O and C(s). In contrast, Mg, Al(g) and CO employ the fitting correlation formulas which were established on the basis of the NASA Technical Report [27], as listed in Table 2 and Table 3. In addition, basic thermophysical properties are applied for Al(s), MgO and Al2 O3 . The relationship of c p connecting temperature can be expressed by linear fitting formula (16), where the constant A, B, C, D and E can be calculated according to Ref. [32].

cp = A + B T + C T 2 + D T 3 + E T 4

(16)

Chemical reaction model An experimental HMF is adopted in simulation, whose formula and performance data are as shown in Table 4. Based on above, by virtue of the thermodynamic calculation program, the component content of primary combustion products and thermal properties of the fuel gas can be obtained. The results show that Mg will participate in secondary combustion in the gas phase, and carbon particle also will be regarded as gas phase because of their nanoscale-diameters (average diameter no more than 70 nm [33]). In addition, the endothermic reaction (C + H2 O → CO + H2 ) of C/H2 O and the exothermic reaction (Mg + H2 O → MgO + H2 ) of Mg/H2 O would be solved by eddy dissipation model (EDM) instead of costly Arrhenius law. 3.4. Boundary and initial conditions Boundary conditions Thermodynamic calculation results [34] indicate that equilibrium adiabatic flame temperature of primary combustion is 1930 K

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Table 5 Mass fraction of the primary combustion products.

2 3

Gaseous phase

Mass%

Condensed phase

Mass%

CO MgCl2 Mg H2 N2

23.02 11.18 8.41 3.57 7.94

AL C MgO AL2 O3

21.26 8.90 10.43 5.29

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Table 6 Average wall temperature variation percentage δ1 .

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Mesh

M1 (coarse)

M2 (fine)

M3 (dense)

Elements Average wall temperature (K) δ1 (%)

314,936 1669.7

537,346 1770.9

766,654 1780.3

12 13 14 15

0.53

18 19 20 21 22 23 24 25 26 27 28

74 75 76

which is served as the inlet temperature, and the concentration of the primary combustion products is listed in Table 5, omitting the small quantity (concentration less than 1%). The mass flow rate of gaseous phase (q m,g ) are calculated by mass conservation, written as:

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n f A f × ap c

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× (1 − ε )

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where ρf and A f are density and burning area of HMF, respectively; ap nc means burning velocity r of HMF; ε is the mass fraction of the condensed phase. Besides, the no-slip, adiabatic and stationary wall were applied. The normal gradients of pressure and component mass fraction were assumed as zero. The discrete phase lost a 15% momentum and reflected off while hitting against the wall. Since the flow at outlet was supersonic, there was no needed for appoint analytic conditions at outlet, and the second-order extrapolation method was adopted.

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Initial conditions In this paper, the Al particles were put into the computational domain with the initial diameter of 20 μm, MgO and Al2 O3 are injected into secondary combustor as inert particles with a 5 μm diameter. The above particles all kept the consistent initial velocity and temperature with the local gas stream. Deserve to be mentioned, the initial velocity and diameter of the droplets can be calculated in virtue of the spray model, i.e., LISA model. The water was injected with 298 K, its mass flow rate depended on the water/fuel ration R w,f , which could be expressed as:

R w,f =

qm,w qm,f

(18)

where q m,w , q m,f are the mass flow rate of the water and HMF, respectively. The stoichiometric water/fuel ratio R wf,sto means the mass of water needed to exactly complete reaction for a unit mass magnesium-based fuel; the value of R wf,sto is 0.408 in this paper.

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3.5. Grid independence verification

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The overall computational domain was discretized utilizing hexahedral elements, with refined meshes near the water inlet and nozzle throat. In consideration of the infeasible of high Re standard k-ε turbulence model in laminar sub layer, the non-equilibrium wall function was employed to solve the complex flow near the wall. Therefore, the thickness of the first layer grid near wall was arranged to satisfy 30 < y + < 300. To verify the meshindependent, three mesh systems M1 , M2 , and M3 were severally generated with 314,936, 537,346 and 766,654 elements. The comparison of static wall temperature under M1 , M2 , and M3 grid systems with the same boundary conditions is shown in Fig. 4. It can be found that the temperature distributions for M2 and M3 grids are extremely similar. The difference on average wall temperature between Mi and Mi+1 grids is represented by the value of δ1 , the specific formula is displayed as:

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Fig. 4. A comparison of static wall temperature under M1 , M2 , and M3 mesh systems, R w,f = 0.45.

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δ1 =

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3.6. Model validation

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In order to verify the reliability of the numerical model, the simulated combustor wall temperature and combustor pressure are compared with direct-connect experiments on ground and theoretical results as presented in Fig. 5 (the percentage in the figure indicates the relative error). The validated range of the R w,f is 0.28–0.45. The results show that within the R w,f range (0.28–0.45), the relative deviations between simulated temperature and experimental results are all less than 7.85%, but the relative deviation on chamber pressure between simulated and theoretical values goes up to 14.8% when R w,f is 0.28. This is due to the fact that the Al/H2 O reaction was assumed as an overall reaction in simulation, hence, while R w,f is 0.28, less than R wf,sto (0.408), some of Al remain to be solid phase due to incomplete reaction. However, actually the reaction mechanism consists of multistep reactions, those Al will react with others to product gaseous products, which is going to contribute a part of combustor pressure. Based on the above analysis, the following work will be conducted under R w,f > R wf,sto . Besides, specific impulse I sp is the central performance indicator of a WRE, and it can be expressed as follows:

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I sp = F /qm,f = (1 + R w,f ) V out + ( P out − P e ) A out /qm,f − R w,f V 0 (19)

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where F is the trust of WRE; P out and P e are the pressure of the nozzle outlet and that of environment, respectively; A out is the area of the nozzle outlet; V 0 is the injection velocity of the droplets, written as [25]:

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Fig. 5. Comparison of chamber wall temperature of X = 0 mm and chamber equilibrium pressure by numerical simulation against experimental and theoretical results by Tian et al. [35].

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Fig. 6. Model validation in terms of specific impulse with R w,f by this study and the theoretical data (Tian et al. [36]).



V 0 = ϕ 2 P /ρw

(20)

where the ϕ and  P are separately discharge coefficient and differential pressure of the nozzle, ρw is the density of water. To further validate the reliability of the model under the wider range of R w,f , Fig. 6 shows the comparison of the specific impulse I sp between numerical and theoretical values obtained by thermodynamic calculation program. The R w,f changes from 0.1–2.0. It can be seen that the numerical results show good agreement with the theoretical data in Ref. [36]. In short, As the actual combustion progress is complex and the reaction mechanism is ambiguous. Therefore, although there are slight differences between the simulation results and the experimental, theoretical data, the trends and error are acceptable.

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4. Results and discussion

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4.1. Flow field parameters distribution of WRE

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First, flow parameters are analyzed to expose the characters of internal flow. The primary water/fuel ratio R w,f1 and secondary water/fuel ratio R w,f2 are separately given as 0.4 and 0.8 based on the water/fuel ratio range that calculated by thermodynamic program, when the other boundary conditions are the same as 3.4. Fig. 7 reveals the temperature distribution for X–Y plane which through

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the water injection nozzle. To all appearances, the temperature field could be divided into four regions: entrance region (i), combustion region (ii), evaporation region (ii) and nozzle region (iv). In detail, the region (i) is located before the primary water injection. Due to the sudden-expansion formed by annular inlet, there is a backflow zone formed in the head area of the combustor, which causes an increase in temperature near the wall; The region (ii) is located in the area between twice water injections, where the average temperature first decreases slightly then rises sharply to the highest point (about 2520 K). This is principally because droplets and Al particles firstly absorb heat to evaporate, then the Al/H2 O and Mg/H2 O reactions release lot of heat; The region (iii) is described as the section that begins at secondary water injection and ends at the convergent section of the combustor. In this area, a mass of droplets evaporate into hot vapor as working medium, accordingly, the combustor temperature drops to approximately 1200 K; The region (iv) is the nozzle where the internal energy is converted into the kinetic energy and the average temperature reaches the lowest (about 877 K). Fig. 8 gives out the gas streamlines and velocity distribution of X–Y plane in region (i) and region (ii). It can be found that a longitudinal vortex which caused by sudden-expansion appears near inlet wall in the region (i). Meanwhile, a higher temperature zone is corresponding to the low velocity zone (LVZ) formed by the vortex, which means that the entrainment effect of the vortex enhances the mixing effect. Therefore, the annular inlet, sudden-expansion and other structures that can form vortices could be considered in physical design of the WRE. Moreover, the turbulence is more obvious nearby two nozzles; the airstream is obstructed by two water jet flow, resulting in the streamlines bent. Fig. 9(a) and Fig. 9(b) describe the kinetic reaction rate of the Mg/H2 O system and Al/H2 O system, respectively. As shown in Fig. 9(a), the reaction rate of Mg remains high value at the interface of mainstream and droplets in the region (i). This is because that Mg engages in reaction in form of gas phase, and then the Mg/H2 O system conducts the homogeneous reaction. Furthermore, Mg is almost burned out before primary water injection, and the heat released by Mg/H2 O reaction supplies for the evaporation of the droplets and Al particles. On the contrary, the combustion progress of the Al particle begins at the region (i), and till to region (ii) nearby the secondary water injection. The evaporation of secondary water makes that the fluid temperature is too low (less than onset temperature 1700 K of Al/H2 O reaction) to continue the reactions, as shown in Fig. 9(b). The mass fraction distributions of each component are presented in Fig. 10. It is distinctly seen that the most of H2 O concentrates upon the region (iii), while a small amount of H2 O distributes the backflow zone and nearby combustor axis. It can be interpreted as follows: 1) on one hand, primary droplets are accelerated in the axial direction due to stronger transverse flow, which leads to that droplets have been dramatically evaporated and reacted before arriving at axis in region (ii); 2) on the other hand, because of the less reductant and weaker transverse flow, the secondary water peacefully evaporates into water vapor as working medium toward downstream, hence, the concentration of H2 O holds the highest value (about 0.4) nearby outlet. In terms of the pseudo-fluid Al2 O3 and MgO, the concentration of MgO approaches peak value 0.134 nearby combustor wall in region (ii) and keeps higher value in backflow zone, but sharply reduces to 0.062 in region (iii). In contrast, the Al2 O3 smoke only concentrates upon the flame surface in region (ii) instead of the backflow zone, owing to the fact that the discrete phase particle are not involved in the backflow because of stronger inertia. In the meantime, the flame surface contains the highest concentration H2 (above 0.0625), and that in the backflow region also holds higher value. The above distributions can be used to better confirm the reaction zones of the

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Mg/H2 O and Al/H2 O systems. Finally, the distribution regularities of CO identify with those of H2 . In a word, the backflow near the inlet is beneficial to the mixing of gas phase reactants and makes the reactions occur in advance.

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4.2. Effect of axial angle of primary water injection

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In this part, the total water/fuel ratio R w,f is set as 2.0 when the primary water/fuel ratio R w,f1 is given as 0.4. In view of the mixing effect with the injection of water toward the upstream is better than that to the downstream, six axial angles (15◦ , 30◦ , 50◦ , 65◦ , 75◦ , and 90◦ ) of primary water to the upstream are considered. The temperature fields of six cases on X–Y plane are illustrated in Fig. 11, while the variations of the area-weighted average temperature on isometric cross sections with axial distance are described in Fig. 12. Fig. 11 indicates that in entrance region (i), as the α is not more than 30◦ , a low temperature zone is found at the backflow zone in region (i). The smaller α is, the larger the range of low temperature zone is and the lower temperature is. As shown in Fig. 11,

in the backflow zone, the temperature difference of inlet between 15◦ and 90◦ accesses 450 K. This indicates that there are part of primary water droplets gathered in the sheltered area formed by the sudden-expansion formed by annular inlet, and most of them absorb heat to evaporate over there. With α further growing, the lower temperature zone disappears; a higher temperature (about 2100 K) zone near wall is formed. In the combustion region (ii), when the α rises from 15◦ to 90◦ , the high temperature area gradually radial expands and approaches to the axis, meanwhile the temperature rises significantly. As the α is 90◦ , the temperature reaches about 2400 K in a large range near the axis; As regards evaporation region (iii) where most of secondary water evaporates into working medium, the lower upstream temperature and the uneven temperature distribution will dramatically lessen the amount of the heat absorbed by the secondary droplets, which brings about the fall of the average temperature in region (iii) and the drop of the outlet pressure. Besides, when α varies from 15◦ to 65◦ , the higher temperature zone in region (iii) water gradually moves towards the nozzle. However, further increasing α to

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Fig. 10. Each gas-phase component concentration distribution of X–Y longitudinal section of WRE, R w,f1 = 0.4 and R w,f2 = 0.8.

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90◦ , the change is towards the opposite direction. In general, it is clearly seen from Fig. 12 that the whole temperature distribution of evaporation region (iii) is more uniform and higher than α is 65◦ . In terms of the nozzle region (iv) where the heat energy of the working medium is converted into the kinetic energy of the WRE, the variation regular in region (iv) identifies with that in the previous region. Based on Eq. (19), the outlet velocity V out and pressure P out are displayed in Fig. 13. It is clearly observed that as α increases, both V out and P out present typical parabolic changes. When α is 65◦ , V out and P out simultaneously approach peak values of 1230.8 m/s and 0.12 Mpa, respectively. Fig. 14 describes the variations of the area-weighted average concentration of H2 O on isometric cross section with axial distance while Fig. 15 presents streamlines and velocity distributions in region (i) and region (ii) under three typical axial angles (α = 30◦ , 65◦ and 90◦ ). An important phenomenon is observed that the distribution of H2 O nearly concentrates near wall in the first two regions, and when α is under 50◦ , the concentrations of H2 O in the head area of combustor have higher values, which corresponds to the backflow zone in region (i). It can be explained that the distribution of H2 O is closely related to the initial configuration of spray and the vortex formed by gaseous phase. As known, the initial droplet velocity of spray cone can be divided into axial velocity and radial velocity, i.e., upstream-wise direction

and perpendicular to stream-wise direction, and the radial velocity is proportional to the α . Hence, the smaller α will lead to a lower droplet radial velocity, which could not penetrate the mainstream; meanwhile it also will make the axial velocity increase to upstream, which promotes more droplets being easily involved in vortex and gathering in the backflow zone. It can be well demonstrated by Fig. 15 (a), when α is 30◦ , there is a large-scale vortex formed in the head area while the central mainstream is almost not disturbed. As α rises from 30◦ to 90◦ , the concentration distribution of H2 O presents sharply decline near wall and gradually radial expands from the wall to the axis. Meanwhile, the turbulence near primary water injection is also stronger. It means that the higher radial velocity makes droplet have the ability to through the gas flow, less influenced by mainstream. Moreover, it is obviously seen from Fig. 15 (b) and Fig. 15 (c) that the moderate axial angle 65◦ shows more vortices and a wider range of LVZ nearby primary water injection than that of 90◦ , which can further extend the residence time of droplet and intensify the mixing with fuel gas. This is because that as α grows from 65◦ to 90◦ , the spray cone leads to the initial velocity of more droplets towards downstream, as well as the superposition of the mainstream velocity will also accelerate droplets to downstream, resulting in no enough time for those droplets to complete gasification and engage in reaction in region (ii). Therefore, they will finish evaporating

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Fig. 11. Static temperature distribution of X–Y longitudinal section of WRE under six different axial injection angles (c) α = 50 ◦ ; (d) α = 65◦ ; (e) α = 75◦ ; (f) α = 90 ◦ .

α , R w,f1 = 0.4 and R w,f2 = 1.6. (a) α

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Fig. 12. The variations of area-weighted average temperature on isometric cross sections with axial distance under six different axial injection angles α , R w,f1 = 0.4 and R w,f2 = 1.6.

Fig. 13. The variations of outlet velocity V out and outlet pressure P out with axial injection angles α , R w,f1 = 0.4 and R w,f2 = 1.6.

in the mixing region (iii) and make the combustor average temperature reduce, as shown in Fig. 12. To sum up, the smaller α is, the weaker mixing effect of the H2 O with gases is; The moderate α (about 65◦ ) makes the two-phase reaction occur ahead of time and is conducive for the uniform mixing of H2 O and fuel gas; The larger α (close to 90◦ ) makes some primary droplets concentrate

upon downstream, which affects the evaporation of secondary water. As stated above, the specific impulse I sp is defined to weigh the trust generated by per unit mass fuel under the given operating condition. Fig. 16 reveals the I sp changes with α . Clearly, the overall trend of I sp identifies with those of outlet velocity and pressure. When α are 65◦ and 15◦ , I sp reaches the maximal value

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of 3659.9 N·s/kg and the minimal value of 3255.8 N·s/kg, respectively. Notably, compared with vertical injection of primary water, the specific impulse I sp have improvements of 8.13% when α is 65◦ . In a word, as α is 65◦ , the WRE possesses the best specific impulse performance.

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4.3. Effect of tangential angle of primary water injection

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(5◦ , 10◦ and

In this section, the effect of tangential angles β 20◦ ) are discussed, when α is 90◦ and the other operation conditions are identified with those in previous section. Fig. 17 and Fig. 18 illustrate that the streamlines in cross section of X = 150 mm and X = 200 mm under three different β , respectively. Obviously, there are several clockwise vortices appearing nearby nozzles and a larger-scale anticlockwise vortex formed in the center of combustor. Another interesting phenomenon is

11

observed that with increasing of β , the size of vortices grows while the quantity of that reduces, at the same time, the vortex core region gradually moves towards the combustor center. It can be well understood in this way. In atomization progress of water, the droplets have significant perturbation on the movement of the mainstream. According to the initial state of spray cone, the droplets resultant velocity in the cross section can be divided into radial and tangential velocities. The tangential velocity makes the droplets do circular motion by continually reversal the velocity directions, while the radial velocity pushes droplets to move toward the combustor center. Furthermore, the tangential velocity is in direct proportion to β and the centripetal force of circular motion is proportional to the square of the tangential velocity. Hence, the growth of the β brings about that too small resultant force for droplets to provide the centripetal force, which further makes the more and more droplets keep away from the combustor center to do circle motion with a larger radius. Figs. 17 and 18 indicate that

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α in region (i) and (ii), R w,f1 = 0.4 and R w,f2 = 1.6.

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Fig. 18. Streamlines distribution of three tangential angles β in cross section of X = 200 mm, R w,f1 = 0.4 and R w,f2 = 1.6. (a) β = 5◦ ; (b) β = 10 ◦ ; (c) β = 20 ◦ .

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for the given position, with the increase of β , the streamlines rotation behavior around the combustor internal wall is more obvious and the range of that is also wider. In the meantime, with the increasing β , the reducing radial velocity makes vortices be elongated along with peripheral direction and smaller-scale vortices be merged into large vortices. Besides, comparing Fig. 17 with Fig. 18 under given β , it can be found that the size of vortices gradually shrink and the quantity of that also declines to the downstream. This is because that as the distance from the nozzle increases, the perturbation of the droplets on gas phase decreases, and both energy attenuation and merge of vortices occur. Nevertheless, the rotation behavior around the combustor internal wall hardly changes under larger β . It is clearly seen that the small-scale vortices enhance the turbulence intensity, which makes better mixing effect. In contrast, the larger the vortices are, the bigger the centrifugal force is and the more evident the droplets aggregation effect is. All in all, the zero inclination of primary water injection in tangential direction show the best mixing effect. To better comprehend the influence of tangential angle on flow field, the temperature distribution of cross section at X = 200 mm is given as Fig. 19, and the variations of temperature with Y or Z coordinate axis position are shown as Fig. 20. Notably, according to Fig. 19, the entire temperature field could be divided into three vivid annular zones, including middle higher temperature region, interior and outer lower temperature region, accordingly, the temperature on two axes appears as a saddle distribution, as

shown in Fig. 20. Furthermore, with the increasing β , the ranges of two lower temperature regions continuously extend and that of the middle region shrinks. Meanwhile, the temperature near the wall also dramatically decreases, in detail, the temperature difference near wall changes from 260 K to 600 K as β changes from 5◦ to 20◦ . The phenomenon means that the larger β is, the more inadequate the Al/H2 O and Mg/H2 O reactions are. This distribution and variation can be explained in virtue of vortex effect. Firstly, the fuel gas of mainstream is radially divergence and involved into the smaller-scale vortices, due to the effect of the centrifugal force of the central larger-scale vortex. After that, the droplets are also sucked up and rotated into the periphery of the smaller-scale vortex on account of the droplets congregating in high strain and low vortex regions. Therefore, the mixing and reaction of fuel gas and H2 O mainly happen in the periphery of the small-scale vortices, as expressed in the Fig. 17 and Fig. 18, where the reaction zones are corresponding to the small-scale vortexes. Moreover, according to the above analysis of gas phase streamlines, it considered that the larger β is, the narrower the smaller-scale vortices are. Thus, as β increases, the range of the reaction zone also shrinks. For the outer lower temperature annular zone, the larger β will cause more droplets are thrown into the outside of reaction zone near wall under the stronger centrifugal force. And then these droplets have no opportunity to involve in the reaction and only gasify near wall by absorbing reaction heat. Hence, the lower temperature regions are more evident with the increasing β . In short, the larger

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Fig. 19. Static temperature distribution of three tangential angles β in cross section of X = 200 mm, R w,f1 = 0.4 and R w,f2 = 1.6. (a) β

= 5◦ ; (b)

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Fig. 20. The variations of static temperature in cross section of X = 200 mm with Y or Z coordinate axis position under three tangential angles β , R w,f1 = 0.4 and R w,f2 = 1.6. (a) Y-axis; (b) Z-axis.

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Fig. 21. The variations of H2 O concentration in cross section of X = 200 mm with Y or Z coordinate axis position under three tangential angles β , R w,f1 = 0.4 and R w,f2 = 1.6. (a) Y-axis; (b) Z-axis.

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the β is, the worse the initial mixing effect is, and the narrower the reaction region is. The mass fraction distribution of H2 O in Y or Z coordinate axis at cross section of X = 200 mm is demonstrated as Fig. 21. The results show that it show a U-type distribution along the axis, and the higher concentration (more than 0.025) is mainly located within 15 mm near wall. In addition, with the increase of β , the concentration maximum values near wall also rises and the difference value of that reaches up to 0.15 between 5◦ and 20◦ . Moreover, the range of higher concentration region is the widest

is 5◦ , that is to say, the smaller

when β β is, the more uniform H2 O distribution is. Further, due to the radial divergence effect of the central larger-scale vortex, the H2 O concentration in the region within about 25 mm of combustor center is below 0.02. All in all, the above distributions identify with the vortices effect. Therefore, in order to avoid congregation of H2 O near the wall, the primary water should inject with no inclination in tangential direction. Similarly, as shown in Fig. 22, the specific impulse I sp is calculated to reveal the variation of the overall performance of WRE under different tangential angles, in which that under β = 0◦ is

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Declaration of competing interest

1

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Acknowledgements

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This research was supported by the National 111 Project of China (B18041).

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References

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Fig. 22. Comparison of specific impulse I sp for four different tangential injection angles β , R w,f1 = 0.4 and R w,f2 = 1.6.

taken as a reference. It suggests that as β increases from 0 to 20◦ , the I sp declines progressively from 3384.8 N·s/kg to 3016.0 N·s/kg, the maximum decreasing amplitude is about 12.2% for β = 20◦ , compared with that for β = 0◦ . In conclusion, the specific impulse performance of WRE is then optimal when the primary water injection with no inclination in tangential direction; the larger β is, the poorer the performance is.

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5. Conclusions

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In this paper, the effects of axial angle and tangential angle of the primary water injection on flow and thermal propulsion characteristics in a WRE were numerically investigated based on three-dimensional model. Six axial angles (a = 15◦ , 30◦ , 50◦ , 65◦ , 75◦ and 90◦ ) and three tangential angles (β = 5◦ , 10◦ and 20◦ ) were considered. The flow field detail parameters and specific impulse were discussed. Some conclusions are drawn as follows.

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(1) The combustion progress of the WRE demonstrates vivid partitioning features. Due to the disparate initial phase state of Al and Mg, the reaction zones of the Mg/H2 O and Al/H2 O systems are region (i) and region (ii), respectively. Furthermore, a structure that can produce the vortices should be considered in application. (2) The axial angle of the primary water injection has significant influence on the specific impulse performance. For the injection of primary water toward upstream, the smaller α results in lower temperature forming in the entrance region, and the larger α (close to 90◦ ) leads to primary water not fully mixed with fuel gas. Compared with vertical injection of primary water, the specific impulse of moderate axial angel (about 65◦ ) have improvements of 8.13%. (3) The increasing of tangential angle brings about the rise in centrifugal force and the narrowing of outer vortices, which gives rise to that more H2 O congregates near the wall and the shrink of the combustion annular zone. Therefore, the larger β is worse for mixing effect. When β are 10◦ and 20◦ , the specific impulse performance will reduce 7.8% and 12.2%, respectively. By contrast, the specific impulse is the optimal when primary water injection is no inclination in tangential.

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In the future study, a combined influence of the axial angle, tangential angle and spray cone angle of primary water injection remains to be discussed.

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