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Numerical investigation on the regression rate of hybrid rocket motor with star swirl fuel grain
Acta Astronautica 127 (2016) 384–393
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Numerical investigation on the regression rate of hybrid rocket motor with star swirl fuel grain Shuai Zhang n, Fan Hu, Weihua Zhang College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, Hunan, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 November 2015 Accepted 3 June 2016 Available online 7 June 2016
Although hybrid rocket motor is prospected to have distinct advantages over liquid and solid rocket motor, low regression rate and insufficient efficiency are two major disadvantages which have prevented it from being commercially viable. In recent years, complex fuel grain configurations are attractive in overcoming the disadvantages with the help of Rapid Prototyping technology. In this work, an attempt has been made to numerically investigate the flow field characteristics and local regression rate distribution inside the hybrid rocket motor with complex star swirl grain. A propellant combination with GOX and HTPB has been chosen. The numerical model is established based on the three dimensional Navier–Stokes equations with turbulence, combustion, and coupled gas/solid phase formulations. The calculated fuel regression rate is compared with the experimental data to validate the accuracy of numerical model. The results indicate that, comparing the star swirl grain with the tube grain under the conditions of the same port area and the same grain length, the burning surface area rises about 200%, the spatially averaged regression rate rises as high as about 60%, and the oxidizer can combust sufficiently due to the big vortex around the axis in the aft-mixing chamber. The combustion efficiency of star swirl grain is better and more stable than that of tube grain. & 2016 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Numerical simulation Hybrid rocket motor Regression rate Star swirl fuel grain
1. Introduction Hybrid rocket motor, which is an attempt to exploit some advantages of liquid rocket engine and solid rocket motor technology, presents a number of advantages including safety, reliability, low cost, throttling capability and minimal environmental impact. These advantages make hybrid rocket motor attractive and prospective in many fields, such as target missiles, low-cost tactical missiles, upper stage motors, boosters for launch and sounding rockets [1–3]. Especially, the successful launchings of SpaceShipOne and SpaceShipTwo spacecrafts, which adopted hybrid rocket motor as their propulsion system to accomplish a commercial sub-orbital space tourism, have accelerated the development of hybrid rocket motor [4–6]. Nevertheless, hybrid rocket motor exhibits some drawbacks during the research period. The outstanding ones among these drawbacks are low-regression rate and insufficient efficiency. The fuel regression rate of hybrid rocket motor is obviously lower than that of the solid rocket motor for its non-premixed diffusion combustion [7]. As a result, to compensate for a low fuel surface regression rate, grain may need to go from a single port to multiple n
Corresponding author. E-mail address: [email protected] (S. Zhang).
http://dx.doi.org/10.1016/j.actaastro.2016.06.017 0094-5765/& 2016 IAA. Published by Elsevier Ltd. All rights reserved.
ports to increase the effective burning surface area to reach the desired chamber pressure [2]. The HyFlyer suborbital launch vehicle, developed under the project Hybrid Technology Option Project (HyTOP), was powered by a 250,000 pound thrust hybrid rocket motor [8]. This motor adopted a 15 ports wagon wheel fuel grain to gain the effective burning surface area. Another 250,000 pound thrust hybrid rocket motor under the project Hybrid Propulsion Demonstration Program (HPDP), adopted a 7 ports wagon wheel fuel grain [9]. Kim et al. [10] investigated combustion characteristics of the cylindrical multiport grain of a hybrid rocket motor experimentally. But this type of grain is not an efficiency way since large fuel slivers will remain at the end of burn [1]. Besides the multiple ports grain, star grain is another way to increase the effective burning surface area [11–13]. As for the other drawback of insufficient efficiency, it's due to insufficient mixing of unreacted fuel gas and oxidizer in mixing chamber. Tian hui et al. [13] have investigated on putting an aft mixing chamber diaphragm in a hybrid rocket motor to increase the combustion efficiency by both numerical and experimental methods. Other researchers [14–16] found that a diaphragm placed along grain length can both raise efficiency and regression rate as a result of vortex flow. Nevertheless, a diaphragm will increase motor structure mass at the same time. In 2005, Lee, et al. [17] added helical troughs to several PMMA tube grains with different pitches. This fuel grain configuration is
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
Nomenclature Variables
A a C C* c D E e G h k M ṁ n p R r T t u Y
ε η λ
Arrhenius pre-exponential constant regression rate leading coefficient molar concentration of species characteristic velocity specific heat capacity mass diffusion coefficient activation energy energy mass flux sensible enthalpy turbulence kinetic energy in turbulence model reaction rate in combustion model molecular weight mass flow rate flux exponent pressure universal gas constant fuel regression rate temperature time velocity mass fraction turbulence dissipation rate efficiency thermal conductivity
adopted to simply increase the burning surface area and to try to induce swirl flow. Experimental results indicated that this helical grain can lead to an increase in regression rate up to 50%. With the development and progress of science and technology, Rapid Prototyping (RP) became a technology used to generate 3-dimentional shapes by computer control. By this technique, the fuel grains of hybrid rocket motor can be fabricated to more complex ones than conventional fuel grains extending a 2-dimentional cross-section into the third dimension. In 2011, Fuller et al. [18], researchers of the Aerospace Corporation, successfully tested small-scale motors with a multi-port helix grain and a coaxial grain which are fabricated by the material epoxy-acrylate with RP technology. They also fabricated a straight star grain and a helical star grain with the same star cross-section. As a result, with the same volume and mass of grain, the burning surface area of the helical star grain is 25% higher than the straight star grain. Additionally, the twist of the helix is expected to promote mixing. In the next 2 years, researchers of the Pennsylvania State University [19–21] noticed the potential of RP fuel grains and began to collaborate with the Aerospace Corporation. Several samples of either printed pure acrylic, printed heterogeneous paraffin/acrylic matrix, or cast paraffin grains provided by collaborators were tested in the Long Grain Center Perforated (LGCP) hybrid rocket motor. It was found that, at the same gaseous-oxygen mass flux, regression rate increases by about 270% in 1/2-tpi star swirl acrylic samples over the result of the published correlation by Zilliac and Karabeyoglu [22]. Whitmore et al. [23] from Utah State University developed a semi-analytical engineering model to describe the effects of helical fuel ports on hybrid fuel regression rates. Although Armold et al. [19] investigated the performance of hybrid rocket motor with star swirl fuel grain experimentally and obtained the internal ballistic and regression rate; the flow field characteristics, combustion process and local regression rate distribution on burning surface inside the hybrid rocket motor with
μ ρ ω ν′ ν″
385
viscosity density net rate of production of species stoichiometric coefficient for reactant stoichiometric coefficient for product
Subscripts
ave b c eff f fuel g n o P p R ref s sim surf t th tot
average backward chamber effective forward fuel gas phase normal direction oxidizer product species pressure reactant species reference solid phase simulation surface turbulence theory total
the particular non-conventional star swirl grain is still unclear. Numerical method [11, 24–30] is an effective way to study these problems with which experimental method is hard to handle. However, previous studies are mostly focus on the single tube grain, then a few multi-port grain and star grain. Few numerical investigations on the internal flow of hybrid rocket motor with the complex star swirl fuel configuration type have been reported. The main objective of this paper is to numerically investigate the flow field characteristics, combustion process and local regression rate distribution on complex burning surface inside the hybrid rocket motor with star swirl grain. The numerical model including governing equations, turbulence model, combustion model and coupled gas/solid phase formulations are presented, and then verified through the comparison between computational results and experimental data corresponding to the geometry and test conditions employed in lab-scale experiments [29,31]. The fuel grain configuration and size in Ref. [19] are adopted to this paper directly. The propellant combination is gaseous-oxygen (GOX) and hydroxyl terminated polybutadiene (HTPB) for its wide application in numerical research field of HRM. The numerical simulations are performed for two kinds of fuel grain configurations including tube and star swirl shapes in order to make a contrast. The influences of oxidizer mass flux and fuel type on the regression rate are then presented and discussed based on the simulation results.
2. Numerical model 2.1. Governing equations For the complex configuration of the star swirl grain, the three dimensional Navier–Stokes equations are applied to describe flow processes inside the hybrid rocket motor. The vector form of the equations are represented as follows:
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∂Q ∂E i ∂Vi + = +S ∂t ∂xi ∂xi
(1)
Table 1 Regression parameters of HTPB.
⎞ ⎛ 0 ⎛ ⎞ ρ ui ⎛ ρ ⎞ ⎟ ⎜ τij ⎜ ⎟ ⎜ ρ uj ⎟ ⎟ ⎜ ρ + δ u u p i j ij ⎟, V = ⎜ ∂T ∂Ym ⎟ Q = ⎜ ρe ⎟, E = ⎜⎜ ∑ i λ + ρ + τ h D u eff m m j ij m ( ρ e + p) u i ⎟ ∂xi ∂xi ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟⎟ ⎜⎜ ∂Ym ⎝ ρYm ⎠ ρui Ym ⎠ ⎝ ρDm ∂x ⎠ ⎝ i and the vector S contains the mass, momentum, energy and species source terms related to combustion and user defined. Ym , Dm and hm are the mass fraction, mass diffusion coefficient and sensible enthalpy for species m separately. λeff is the effective conductivity ( λ + λt , where λt is the turbulent thermal conductivity, defined according to the turbulence model).
Tsurf o 722 K
3.965
55881.1
Tsurf 4 722 K
0.01104
20552.5
by both the eddy dissipation model and the finite rate model, adopting the minimum of these two rates. The eddy dissipation model, which is developed on the hypothesis of fast fuel burning, has been applied in the prediction of diffusion flame in HRM widely and successfully [11,24,34]. The net rate of production of species i due to reaction r is controlled by turbulent mixing, and given by the expression below:
ωi, r = νi′, r Mi Aρ 2.2. Turbulence model The RNG k–ε turbulence model, which provides an analyticallyderived differential formula for effective viscosity considering low Reynolds number effects, is employed for modeling the turbulence, owing to the low Reynolds number flow near the burning surface of the grain. The two transport equations are represented as follows:
∂ ( ρk) + ∂∂x ( ρui k) ∂t i ∂ ⎛ ∂k ⎞ ⎜ αk μeff ⎟ + Gk + Gb − ρε − YM + Sk = ∂xj ⎝ ∂xj ⎠
⎛ ⎞ ⎛ Y ⎞ ∑ YP ⎟ ε R ⎟, B N P min ⎜ min ⎜ ⎜ R ⎝ νi′, r MR ⎠ ∑ ν″ M ⎟ k j, r j ⎠ ⎝ j
ωC4 H6 = − MC4 H6 kf 1CC4 H6 CO 2 ωO 2 = MO 2 ⎡⎣ −3.5kf 1CC4 H6 CO 2 − 0.5 kf 2 CCO CO0.5 − kb2 CCO 2 ⎤⎦ 2
(
ωCO (2)
(
)
ω H2 O = 3MH2 O kf 1CC4 H6 CO 2
)
(6)
The forward and backward reaction rates, kf and kb , for two reactions are expressed as Arrhenius functions in the form of k = AT n exp ( −E /RT ), and the values of the parameters are listed in Table 2 [35].
(3)
where the effective viscosity μeff = μ + μt , the turbulent viscosity
μt = ρCμ k2/ε , and the turbulent thermal conductivity λt = μt cp/Pr . The model constants are as follows: Cμ = 0.0845, C1ε = 1.42, C2ε = 1.68. 2.3. Combustion model The propellant combination in the simulation is GOX oxidizer and HTPB fuel. The major gaseous pyrolysis product of the solid HTPB fuel grain is 1,3-butadiene(C4H6) with the hypothesis that the solid HTPB decomposes into the gaseous 1,3-butadiene(C4H6) directly, neglecting the melting process of solid HTPB. Because the exact details of the pyrolysis process of HTPB have not been wellunderstood, an Arrhenius-type semi-empirical equation was applied by Chiaverini et al. [32] to describe the regression rate of the HTPB fuel through the experiment investigation. The equation is:
r = Ae−E / (RTsurf )
)
= MCO ⎡⎣ 4kf 1CC4 H6 CO 2 − kf 2 CCO CO0.5 − kb2 CCO 2 ⎤⎦ 2
(
∂ ( ρε) + ∂∂x ( ρui ε) ∂t i ε ε2 ∂ ⎛ ∂ε ⎞ ⎜ αε μeff ⎟ + C1ε ( Gk + C3ε Gb ) − C2ε ρ = k k ∂xj ⎝ ∂xj ⎠
(5)
where the constants A and B are 4 and 0.5 respectively. The finite rate model is also used to calculate the net rate of production of species in order to avoid the local excessive reaction rate caused by the strong turbulence. The net rate of production and destruction of species are represented as follows [29]:
ωCO 2 = MCO 2 kf 2 CCO CO0.5 − kb2 CCO 2 2
− Rε + Sε
E , J/mol
A , m/s
where,
(4)
where Tsurf is the burning surface temperature, the value of the parameters A and E are given in Table 1 [32]. A global two-step combustion model is applied to the gas phase reaction for simplification [29,33], given by
C4 H6 + 3.5O2 → 4CO + 3H2 O CO+0.5O2 ↔ CO2 The rate of production and destruction of species are controlled
2.4. Coupled gas/solid phase formulations In the hybrid rocket motor, the solid fuel absorbs the heat from the diffusion flame zone, then decomposes into gas phase and diffuses into the flame zone to react with the oxidizer. Thus the combustion process can sustain, and causes the regression of the burning surface. The regression rate is determined by the energy and mass balance on the burning surface. The mass balance equation is:
ρg un = − ρs r
(7)
The energy balance equation is:
⎛ ∂T ⎞ ⎛ ∂T ⎞ T Tsurf −λ eff ⎜ ⎟ = − λ s ⎜ ⎟ + ρs r hC4surf H6 − hHTPB ⎝ ∂n ⎠ g ⎝ ∂n ⎠s
(
)
(8)
where the term on the left side is the convection heat flux to the burning surface. The first term on the right side is the heat flux conducted into the solid fuel grain. The last term is the variation of Table 2 Reaction rate parameters for the finite rate model. Reaction rate
A
n
E /R , K
kf 1
1.3496 1010
0.0
kf 2
2.2400 106
0.0
5032.7
kb2
1.1000 1013
0.97
39456.5
15108
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
enthalpy from the solid HTPB fuel to the pyrolysis product C4H6, and this additional term makes the energy balance equation different from the normal solid wall conduction heat flux equation. The conduction heat flux term can be calculated as follows:
⎛ ∂T ⎞ −λ s ⎜ ⎟ = ρs rcs (Tsurf − Tref ) ⎝ ∂n ⎠s
(9)
Thus, substituting Eq. (9) into (8), then the energy balance equation is:
⎛ ∂T ⎞ T Tsurf −λ eff ⎜ ⎟ = ρs rcs (Tsurf − Tref ) + ρs r (hC4surf H6 − hHTPB ) ⎝ ∂n ⎠ g
(10)
Because that, T
T
surf ref ρs rhHTPB = ρs r (hHTPB + cs (Tsurf − Tref ))
(11)
So, the energy balance equation is simplified:
⎛ ∂T ⎞ T Tref −λ eff ⎜ ⎟ = ρs r (hC4surf H6 − hHTPB ) ⎝ ∂n ⎠ g
(12)
With the Eqs. (4) and (12), both the fuel surface temperature Tsurf and the fuel regression rate r can be calculated during the iteration of the simulation. After the calculation of regression rate, the produced mass, momentum, species of C4H6 and the absorbed energy due to the pyrolysis of HTPB can be determined. The mass, momentum, energy and species are treated as user defined source terms and added into the first layer cell mesh adjacent to the burning surface. 2.5. Validation of the numerical model In order to validate the numerical model, the experimental regression rates along with the axial direction of a 2D slab motor obtained by Chiaverini et al. [29,31] are compared with the simulation results, as shown in the Fig. 1. The experimental configuration and size are reported in Ref. [31]. In Fig. 1, the experimental data points for case 1 represent the regression rates of test 9 under the condition that the gaseousoxygen flow rate is 0.2 kg/s (the oxidizer flow flux is 208 kg/m2 s). Likewise, the experimental data points for case 2 represents the test 7, of which the gaseous-oxygen flow rate is 0.1 kg/s (the oxidizer flow flux is 104 kg/m2 s). Simulation 1 and simulation 2 are the regression rates predicted by this simulation model corresponding to case 1 and case 2 respectively. Table 3 gives the error analysis of validation. For the case 1, the average regression rate error of the simulation result is 0.9%, indicating that the calculated regression rates match well with the experimental ones. In regard to case 2, the error is 7.9%. The calculated result is also reasonable and acceptable.
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Table 3 Error analysis of validation.
Spatially averaged regression rate for case1 Spatially averaged regression rate for case2
Exp (mm/ s)
Simulation (mm/s) Error (%)
1.12
1.11
0.9
0.76
0.82
7.9
In conclusion, the numerical model established in this work can be used to simulate the quasi-steady working process and predict the regression rate in the hybrid rocket motor. Besides the acceptable accuracy, this model can be adapted to the variation of the oxidizer flow flux. Overall, the comparisons make a confidence that the model will provide meaningful results.
3. Simulation results and discussion 3.1. Physical configuration The star swirl grain configuration in the work of Armold et al. [19] is taken as a reference in this work. The cross section of the grain port is a star, and the star number is 6. The whole port is formed by the way that the star cross section rotates around the axis as it moves along the axis direction. The grain length is 101.6 mm (4 in.), and the pitch is 1/2 tpi (turns per inch), which means that the star cross section turn half circle when it moves 1 in. along the axis direction. Besides the star swirl grain, a tube grain with the same area of cross section is presented in this work to make a comparison to the star swirl grain. The main parameters of the physical configuration are listed in Table 4. 3.2. Meshes of computational domains The computational domains for the two kinds of grains are three-dimensional shown in Fig. 2 and Fig. 3. Since the domain of the tube grain is simple, the mesh uses hexahedral structured elements with a total number of 586 thousands. As far as the star swirl grain, the hybrid mesh is used in order to generate mesh in this complex domain. The hexahedral structured elements are used for the boundary layer zone, and the tetrahedral and pyramidal unstructured elements are used for the interior zone, which can be shown in Fig. 4(b). The whole quantity of the mesh elements is 2.4 million. The height of the first layer mesh over the burning surface for two grains are all refined to ensure a value of y þ of the order of one, which can guarantee the heat transfer process to be predicted precisely. Table 4 Main parameters of the physical configuration.
Fig. 1. Comparisons of regression rate between simulation and experimental value.
Parameter
value
Inner diameter of chamber, mm Length of precombustion chamber, mm Length of aft-mixing chamber, mm Length of fuel grain, mm Inner diameter of tube grain, mm Diameter of nozzle throat, mm Expansion area ratio of nozzle Star number of the star swirl grain Port cross section area of tube grain, mm2 Port cross section area of star swirl grain, mm2
30.4 25.4 25.4 101.6 11.2 6.0 3.42 6 98.1 98.1
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Fig. 2. Mesh of the tube grain domain. Fig. 5. Temperature contours of the main sections of tube grain.
Fig. 3. Mesh of the star swirl grain domain.
3.3. Initial and boundary conditions Fig. 6. Temperature contours of the main sections of star swirl grain.
The whole circle plane on the front of the pre-chamber is mass flow rate inlet, at which the GOX mass flow rate ranges from 8 to 40 g/s (mass flux from about 80 to 400 kg/m2 s) with temperature of 300 K. The nozzle outlet is pressure outlet, the conditions of which are extrapolated from the interior domain under supersonic condition. The rest surfaces are all no-slip wall boundary conditions. In particular, the wall temperature and regression rate of the burning surface are calculated by the coupled gas/solid phase formulation during the iteration process of the simulation. Except for the temperature, the initial values of the whole domain are initialized by the inlet boundary condition before the iteration. The initial temperature is set at 2000 K to give an initial regression rate and ignite the motor. 3.4. Flow field characteristics The results of the cases in which the oxidizer mass flow rate is 16 g/s for two kinds of grains are represented here to show the
flow characteristics. Fig. 5 gives the classical temperature contours of the tube grain. There is a thin flame layer above the burning surface with temperature about 3500 K, and the distance from flame to grain surface becomes larger as the increase of the axis coordinate due to the consumption of GOX. The vortexes in the aft-mixing chamber shown in Fig. 7 can enhance the mixing of species, making a more sufficient combustion. Consequently, the temperature in the vortex flow zone is higher than the core straight flow zone in aftmixing chamber. The temperature contours of the star swirl grain are very different from the tube grain as a result of the intricate grain configuration shown in Fig. 6. As the oxygen flows into the grain port, it reacts with the pyrolysis product of the fuel grain forming a thin flame layer above the burning surface. Because of the swirl port, the flame layer in the star slot is asymmetrical and closer to the windward surface, shown in Fig. 9. Same as the tube grain, the
Fig. 4. Mesh of the cross section of port. (a) tube grain (b) star swirl grain.
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
Fig. 7. 3D streamlines of tube grain.
Fig. 8. 3D streamlines of star swirl grain.
distance from flame to grain surface becomes larger as the increase of the axis coordinate due to the consumption of GOX. But the area of star swirl grain surface per unit length of axis is larger than that of the tube grain, and more oxygen will be depleted. As a result, the flame layers in the star slot converge at the distance of about 0.04 m away from the grain head with a peak temperature of 3830 K. Then, as the increase of the axis coordinate, the flame in the slot degenerates to the vicinity of the root of star. The flame above the root of star is always closer to the burning surface. In the aft-mixing chamber, the temperature distribution is different from that in tube grain case. The temperature in axial core zone is higher than that in round zone, illustrating that sufficient combustion occurs in this zone. As for the reason, the 3D streamlines in Fig. 8 shows that a big vortex around the axis is formed in the aft-mixing chamber caused by the swirl flow during the star swirl grain port. Due to the vortex, species can mix well in core zone. Besides, the centrifugal force on the oxidizer stream which is anticipated by Armold et al. [19] can be verified from the streamlines in Fig. 8. The streamlines is equally distributed when entrancing the grain port, but as the gas flows downstream, the streamlines become closer to the cusp of star. From Fig. 10, it can be seen that the mass fraction of O2 of the tube grain along the axis keeps 1 until it reaches the nozzle, then decreases to about 0.94. Moreover, temperature keeps 300 K from inlet to the entrance of nozzle. Then, it reaches the peak of about 615 K and decreases as the nozzle supersonic flow. It is demonstrated that the oxygen near the axis does not react sufficiently. However, the distribution of temperature and mass fraction of O2 along the axis of star swirl grain are different from the tube grain. In the grain port, the mass fraction of O2 decreases and the temperature increases. In the aft-mixing chamber and nozzle, the mass fraction of O2 decreases to 0, and temperature reaches the peak of about 3750 K where the sufficient combustion occurs. These analysis illustrate that the utilization coefficient of O2 of the star swirl grain is better than the tube grain. 3.5. Regression characterization In order to analyze the fuel regression rates of different positions on the burning surface of the star swirl grain, four characteristic points are selected as shown in Fig. 11(a). Point A represents the root of star, Point C represents the cusp of star, Point D represents the windward grain surface, and Point B represents the leeward grain surface. In 3D view, these four points in 2D grain
389
section are four helix curves on grain surface shown in Fig. 11(b). Fig. 12 shows the local regression rate distribution of the tube grain and the four characteristic points of the star swirl grain along the axial direction. The tube grain result shows the typical changing trend with an initial decrease due to the boundary layer growth, and then an increase due to the mass and heat addition. The initial peak value of regression rate at the front edge of the fuel grain is caused by the starting of the thermal boundary layer. At this position, the flame is very close to the grain surface, which brings a large temperature gradient and convective heat flux. The last peak value of regression rate at the tail edge of fuel grain is due to vortex enhancement to convective heat flux. Because of the starting of the thermal boundary layer, there are initial peak values on all the regression rate distributions of the four characteristic points of star swirl grain, then the regression rate decreases to a minimum value and then increases. At the length of about 0.024 m, the regression rate of Point C begins to decrease because the flame begins to degenerate from the cusp of star shown in Fig. 9(b). At about Z ¼0.038 m, the flame layer in the star slot has merged completely and continues to degenerate, then the regression rates of Point B and D begin to decrease. Because the flame keeps close to the root of star, the regression rate of Point A keeps a relative high value of about 2.5 mm/s. Since flame is closer to the windward surface than to the leeward surface, regression rate of Point D keeps a little higher than that of Point B. Obviously, most of the regression rate of star swirl grain is much higher than that of the tube grain. In order to compare the regression rate of different grains conveniently, the spatially averaged regression rates and the fuel mass flow rates under different oxygen flow flux are listed in Table 5. The spatially averaged regression rate is defined as follows:
rave =
∫surf rdA Asurf
(13)
where Asurf is the burning surface area. As the oxygen flow flux rise, the spatially averaged regression rates of all the two fuel grain types increase. At each oxygen flow flux, the spatially averaged regression rate of star swirl grain is higher than that of tube grain, and the growth rate ranges from 30.1% to 60.3%. As the oxygen flow flux rise, the growth rate increase. These regression rate points are shown clearly in Fig. 13. The averaged fuel regression rate is expressed in the form of power law rave = aGon , where Go is oxidizer mass flux. The exponent of the power law for star swirl grain is 0.5608, a little higher than 0.4373 for tube grain. Beside the higher averaged regression rate, the burning surface area of star swirl grain is 106.29 cm2, about three times that 35.64 cm2 of tube grain with the condition of same port area and same grain length. As the result of higher regression rate and larger burning surface area, the fuel mass flow rate of star swirl grain is much higher than that of tube grain. The growth rate can reach up to 377.7%. In order to verify that the three-dimensional simulation results are reasonable and acceptable, the regression rate points of the tube grain are compared with the experimental regression rate results of studies performed by several of the leading researchers. The propellant combination is also GOX and pure HTPB, and grain configuration is tube. As shown in Fig. 14, the tube grain simulation points of this paper are located in a reasonable range. 3.6. Combustion performance For the hybrid rocket motor system, the oxidizer to fuel ratio O/ F, the characteristic velocity C * and the efficiency of characteristic velocity ηC * are used to measure the combustion performance. They are defined as follows:
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Fig. 9. Temperature contours of sections. (a) Z¼ 0.017 m (b) Z¼ 0.024 m (c) Z¼ 0.038 m (d) Z ¼0.063 m.
O/F =
Fig. 10. Distribution of temperature and mass fraction of O2 along the axis of motor.
ṁ o ṁ fuel
(14)
C* =
pc ⋅Athroat ṁ tot
(15)
ηC * =
* Csim * Cth
(16)
where Athroat is the area of nozzle throat, ṁ tot is the total mass flow * is the characteristic velocity calculated by the simulation rate, Csim * is theoretical characteristic velocity obtained by therresults, Cth modynamic calculation with the same oxidizer to fuel ratio, chamber pressure, nozzle expansion area ratio under the condition of vacuum. These results are summarized in Table 6. O/F of both tube grain and star swirl grain increase as the oxygen flow flux rises. O/F of
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
391
Fig. 11. Characteristic positions for local regression rates analysis. (a) 2D grain section (b) 3D grain surface.
Fig. 12. Distribution of local regression rate along axial direction. Table 5 Spatially averaged regression rates and fuel mass flow rates of different grain types.
ṁ o , g/s
8 10 12 16 20 30 40
Go , kg/m2 s Tube grain
Star swirl grain
rave , mm/s
rave , mm/s
0.747 0.818 0.880 0.985 1.076 1.331 1.506
0.972 1.136 1.264 1.495 1.692 2.097 2.415
81.5 101.9 122.3 163.1 203.9 305.8 407.7
Growth rate (%)
30.1 38.9 43.6 51.8 57.2 57.6 60.3
Tube grain
Star swirl grain
ṁ fuel ,
ṁ fuel ,
g/s
g/s
2.47 2.70 2.91 3.26 3.55 4.39 4.98
9.58 11.19 12.46 14.73 16.67 20.66 23.79
Growth rate (%)
287.9 314.4 328.2 351.8 369.6 370.6 377.7
Fig. 14. Regression rate comparison with various hybrid rocket studies using GOX/ HTPB [1,36–41].
tube grain is about 4–5 times that of star swirl grain for its lower * , Cth * and ηC * of tube grain are all decrease as fuel mass flow rate. Csim the oxygen flow flux rises. It illustrates that, as the oxygen flow flux rises, the combustion performance of tube grain becomes worse, and more and more oxidizer is wasted without sufficient * and Cth * are both increase as combustion. As for star swirl grain, Csim the oxygen flow flux rises, and ηC * keeps a relative high value of about 90%. It illustrates that the combustion performance of star swirl grain is better and more stable than that of tube grain.
4. Conclusions Based on GOX/HTPB propellant combination, the three dimensional numerical model for hybrid rocket motor internal flow field simulation has been established with coupled gas/solid phase formulations. And this model is validated to be accurate through the comparison between simulation results and the experiment data from the work of Chiaverini et al. Then numerical simulation investigations of hybrid rocket motor with star swirl fuel grain and tube fuel grain are conducted in this work. Temperature contours in the motor and regression rate distributions of characteristic points are presented and discussed. The influences of the oxidizer mass flux and the fuel grain type on spatially averaged regression rate are analyzed. The combustion performances of both grain types are compared. Several main conclusions can be deduced as follows:
Fig. 13. Spatially averaged regression rates of different grain types.
(1) Under the conditions of the same grain length and the same port area, the star swirl grain can enlarge burning surface area comparing to tube grain. In this work, the burning surface area of the six-pointed star swirl grain is about three times that of the tube grain.
392
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Table 6 Combustion performance of HRM with different grain types.
Go , kg/m2 s
81.5 101.9 122.3 163.1 203.9 305.8 407.7 Average
Tube grain
Star swirl grain
O/F
pc , MPa
* , m/s Csim
* , m/s Cth
ηC * (%)
O/F
pc , MPa
* , m/s Csim
* , m/s Cth
ηC * (%)
3.239 3.704 4.124 4.908 5.634 6.834 8.032 –
0.572 0.669 0.759 0.920 1.070 1.407 1.668 –
1544.7 1489.4 1439.3 1350.6 1284.7 1156.8 1048.5 –
1673.6 1640.6 1612.0 1567.9 1533.7 1484.6 1439.5 –
92.3 90.8 89.3 86.1 83.8 77.9 72.8 84.7
0.835 0.894 0.963 1.086 1.200 1.452 1.681 –
0.781 0.971 1.160 1.537 1.911 2.823 3.705 –
1256.1 1295.6 1340.9 1414.2 1473.5 1575.6 1642.2 –
1420.9 1450.3 1483.2 1536.6 1591.1 1730.4 1796.1 –
88.4 89.3 90.4 92.0 92.6 91.1 91.4 90.8
(2) At the same oxidizer mass flux level, most of the local regression rate distribution of the four characteristic points of the star swirl grain along the axial direction are higher than that of the tube grain. Beside higher regression rate, the swirl flow through the star swirl grain port can form a big vortex around the axis in the aft-mixing chamber. The vortex flow can make the oxidizer react sufficiently. The combustion efficiency of star swirl grain is better and more stable than that of tube grain. (3) For both of the two fuel grain types, spatially averaged regression rate increases as the oxidizer mass flux rises. The star swirl grain shows a higher influence of mass flux than the tube grain, as demonstrated by a bigger exponent of the power law (0.5608 vs 0.4373). In the mass flux range from about 80 to 400 kg/m2 s, the averaged regression rate of star swirl grain is higher than that of tube grain at each mass flux level. Growth rate of regression rate can reach up to 60.3%, which is meaningful for hybrid rocket motor.
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Numerical investigation on the regression rate of hybrid rocket motor with star swirl fuel grain Shuai Zhang n, Fan Hu, Weihua Zhang College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, Hunan, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 November 2015 Accepted 3 June 2016 Available online 7 June 2016
Although hybrid rocket motor is prospected to have distinct advantages over liquid and solid rocket motor, low regression rate and insufficient efficiency are two major disadvantages which have prevented it from being commercially viable. In recent years, complex fuel grain configurations are attractive in overcoming the disadvantages with the help of Rapid Prototyping technology. In this work, an attempt has been made to numerically investigate the flow field characteristics and local regression rate distribution inside the hybrid rocket motor with complex star swirl grain. A propellant combination with GOX and HTPB has been chosen. The numerical model is established based on the three dimensional Navier–Stokes equations with turbulence, combustion, and coupled gas/solid phase formulations. The calculated fuel regression rate is compared with the experimental data to validate the accuracy of numerical model. The results indicate that, comparing the star swirl grain with the tube grain under the conditions of the same port area and the same grain length, the burning surface area rises about 200%, the spatially averaged regression rate rises as high as about 60%, and the oxidizer can combust sufficiently due to the big vortex around the axis in the aft-mixing chamber. The combustion efficiency of star swirl grain is better and more stable than that of tube grain. & 2016 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Numerical simulation Hybrid rocket motor Regression rate Star swirl fuel grain
1. Introduction Hybrid rocket motor, which is an attempt to exploit some advantages of liquid rocket engine and solid rocket motor technology, presents a number of advantages including safety, reliability, low cost, throttling capability and minimal environmental impact. These advantages make hybrid rocket motor attractive and prospective in many fields, such as target missiles, low-cost tactical missiles, upper stage motors, boosters for launch and sounding rockets [1–3]. Especially, the successful launchings of SpaceShipOne and SpaceShipTwo spacecrafts, which adopted hybrid rocket motor as their propulsion system to accomplish a commercial sub-orbital space tourism, have accelerated the development of hybrid rocket motor [4–6]. Nevertheless, hybrid rocket motor exhibits some drawbacks during the research period. The outstanding ones among these drawbacks are low-regression rate and insufficient efficiency. The fuel regression rate of hybrid rocket motor is obviously lower than that of the solid rocket motor for its non-premixed diffusion combustion [7]. As a result, to compensate for a low fuel surface regression rate, grain may need to go from a single port to multiple n
Corresponding author. E-mail address: [email protected] (S. Zhang).
http://dx.doi.org/10.1016/j.actaastro.2016.06.017 0094-5765/& 2016 IAA. Published by Elsevier Ltd. All rights reserved.
ports to increase the effective burning surface area to reach the desired chamber pressure [2]. The HyFlyer suborbital launch vehicle, developed under the project Hybrid Technology Option Project (HyTOP), was powered by a 250,000 pound thrust hybrid rocket motor [8]. This motor adopted a 15 ports wagon wheel fuel grain to gain the effective burning surface area. Another 250,000 pound thrust hybrid rocket motor under the project Hybrid Propulsion Demonstration Program (HPDP), adopted a 7 ports wagon wheel fuel grain [9]. Kim et al. [10] investigated combustion characteristics of the cylindrical multiport grain of a hybrid rocket motor experimentally. But this type of grain is not an efficiency way since large fuel slivers will remain at the end of burn [1]. Besides the multiple ports grain, star grain is another way to increase the effective burning surface area [11–13]. As for the other drawback of insufficient efficiency, it's due to insufficient mixing of unreacted fuel gas and oxidizer in mixing chamber. Tian hui et al. [13] have investigated on putting an aft mixing chamber diaphragm in a hybrid rocket motor to increase the combustion efficiency by both numerical and experimental methods. Other researchers [14–16] found that a diaphragm placed along grain length can both raise efficiency and regression rate as a result of vortex flow. Nevertheless, a diaphragm will increase motor structure mass at the same time. In 2005, Lee, et al. [17] added helical troughs to several PMMA tube grains with different pitches. This fuel grain configuration is
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
Nomenclature Variables
A a C C* c D E e G h k M ṁ n p R r T t u Y
ε η λ
Arrhenius pre-exponential constant regression rate leading coefficient molar concentration of species characteristic velocity specific heat capacity mass diffusion coefficient activation energy energy mass flux sensible enthalpy turbulence kinetic energy in turbulence model reaction rate in combustion model molecular weight mass flow rate flux exponent pressure universal gas constant fuel regression rate temperature time velocity mass fraction turbulence dissipation rate efficiency thermal conductivity
adopted to simply increase the burning surface area and to try to induce swirl flow. Experimental results indicated that this helical grain can lead to an increase in regression rate up to 50%. With the development and progress of science and technology, Rapid Prototyping (RP) became a technology used to generate 3-dimentional shapes by computer control. By this technique, the fuel grains of hybrid rocket motor can be fabricated to more complex ones than conventional fuel grains extending a 2-dimentional cross-section into the third dimension. In 2011, Fuller et al. [18], researchers of the Aerospace Corporation, successfully tested small-scale motors with a multi-port helix grain and a coaxial grain which are fabricated by the material epoxy-acrylate with RP technology. They also fabricated a straight star grain and a helical star grain with the same star cross-section. As a result, with the same volume and mass of grain, the burning surface area of the helical star grain is 25% higher than the straight star grain. Additionally, the twist of the helix is expected to promote mixing. In the next 2 years, researchers of the Pennsylvania State University [19–21] noticed the potential of RP fuel grains and began to collaborate with the Aerospace Corporation. Several samples of either printed pure acrylic, printed heterogeneous paraffin/acrylic matrix, or cast paraffin grains provided by collaborators were tested in the Long Grain Center Perforated (LGCP) hybrid rocket motor. It was found that, at the same gaseous-oxygen mass flux, regression rate increases by about 270% in 1/2-tpi star swirl acrylic samples over the result of the published correlation by Zilliac and Karabeyoglu [22]. Whitmore et al. [23] from Utah State University developed a semi-analytical engineering model to describe the effects of helical fuel ports on hybrid fuel regression rates. Although Armold et al. [19] investigated the performance of hybrid rocket motor with star swirl fuel grain experimentally and obtained the internal ballistic and regression rate; the flow field characteristics, combustion process and local regression rate distribution on burning surface inside the hybrid rocket motor with
μ ρ ω ν′ ν″
385
viscosity density net rate of production of species stoichiometric coefficient for reactant stoichiometric coefficient for product
Subscripts
ave b c eff f fuel g n o P p R ref s sim surf t th tot
average backward chamber effective forward fuel gas phase normal direction oxidizer product species pressure reactant species reference solid phase simulation surface turbulence theory total
the particular non-conventional star swirl grain is still unclear. Numerical method [11, 24–30] is an effective way to study these problems with which experimental method is hard to handle. However, previous studies are mostly focus on the single tube grain, then a few multi-port grain and star grain. Few numerical investigations on the internal flow of hybrid rocket motor with the complex star swirl fuel configuration type have been reported. The main objective of this paper is to numerically investigate the flow field characteristics, combustion process and local regression rate distribution on complex burning surface inside the hybrid rocket motor with star swirl grain. The numerical model including governing equations, turbulence model, combustion model and coupled gas/solid phase formulations are presented, and then verified through the comparison between computational results and experimental data corresponding to the geometry and test conditions employed in lab-scale experiments [29,31]. The fuel grain configuration and size in Ref. [19] are adopted to this paper directly. The propellant combination is gaseous-oxygen (GOX) and hydroxyl terminated polybutadiene (HTPB) for its wide application in numerical research field of HRM. The numerical simulations are performed for two kinds of fuel grain configurations including tube and star swirl shapes in order to make a contrast. The influences of oxidizer mass flux and fuel type on the regression rate are then presented and discussed based on the simulation results.
2. Numerical model 2.1. Governing equations For the complex configuration of the star swirl grain, the three dimensional Navier–Stokes equations are applied to describe flow processes inside the hybrid rocket motor. The vector form of the equations are represented as follows:
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S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
∂Q ∂E i ∂Vi + = +S ∂t ∂xi ∂xi
(1)
Table 1 Regression parameters of HTPB.
⎞ ⎛ 0 ⎛ ⎞ ρ ui ⎛ ρ ⎞ ⎟ ⎜ τij ⎜ ⎟ ⎜ ρ uj ⎟ ⎟ ⎜ ρ + δ u u p i j ij ⎟, V = ⎜ ∂T ∂Ym ⎟ Q = ⎜ ρe ⎟, E = ⎜⎜ ∑ i λ + ρ + τ h D u eff m m j ij m ( ρ e + p) u i ⎟ ∂xi ∂xi ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟⎟ ⎜⎜ ∂Ym ⎝ ρYm ⎠ ρui Ym ⎠ ⎝ ρDm ∂x ⎠ ⎝ i and the vector S contains the mass, momentum, energy and species source terms related to combustion and user defined. Ym , Dm and hm are the mass fraction, mass diffusion coefficient and sensible enthalpy for species m separately. λeff is the effective conductivity ( λ + λt , where λt is the turbulent thermal conductivity, defined according to the turbulence model).
Tsurf o 722 K
3.965
55881.1
Tsurf 4 722 K
0.01104
20552.5
by both the eddy dissipation model and the finite rate model, adopting the minimum of these two rates. The eddy dissipation model, which is developed on the hypothesis of fast fuel burning, has been applied in the prediction of diffusion flame in HRM widely and successfully [11,24,34]. The net rate of production of species i due to reaction r is controlled by turbulent mixing, and given by the expression below:
ωi, r = νi′, r Mi Aρ 2.2. Turbulence model The RNG k–ε turbulence model, which provides an analyticallyderived differential formula for effective viscosity considering low Reynolds number effects, is employed for modeling the turbulence, owing to the low Reynolds number flow near the burning surface of the grain. The two transport equations are represented as follows:
∂ ( ρk) + ∂∂x ( ρui k) ∂t i ∂ ⎛ ∂k ⎞ ⎜ αk μeff ⎟ + Gk + Gb − ρε − YM + Sk = ∂xj ⎝ ∂xj ⎠
⎛ ⎞ ⎛ Y ⎞ ∑ YP ⎟ ε R ⎟, B N P min ⎜ min ⎜ ⎜ R ⎝ νi′, r MR ⎠ ∑ ν″ M ⎟ k j, r j ⎠ ⎝ j
ωC4 H6 = − MC4 H6 kf 1CC4 H6 CO 2 ωO 2 = MO 2 ⎡⎣ −3.5kf 1CC4 H6 CO 2 − 0.5 kf 2 CCO CO0.5 − kb2 CCO 2 ⎤⎦ 2
(
ωCO (2)
(
)
ω H2 O = 3MH2 O kf 1CC4 H6 CO 2
)
(6)
The forward and backward reaction rates, kf and kb , for two reactions are expressed as Arrhenius functions in the form of k = AT n exp ( −E /RT ), and the values of the parameters are listed in Table 2 [35].
(3)
where the effective viscosity μeff = μ + μt , the turbulent viscosity
μt = ρCμ k2/ε , and the turbulent thermal conductivity λt = μt cp/Pr . The model constants are as follows: Cμ = 0.0845, C1ε = 1.42, C2ε = 1.68. 2.3. Combustion model The propellant combination in the simulation is GOX oxidizer and HTPB fuel. The major gaseous pyrolysis product of the solid HTPB fuel grain is 1,3-butadiene(C4H6) with the hypothesis that the solid HTPB decomposes into the gaseous 1,3-butadiene(C4H6) directly, neglecting the melting process of solid HTPB. Because the exact details of the pyrolysis process of HTPB have not been wellunderstood, an Arrhenius-type semi-empirical equation was applied by Chiaverini et al. [32] to describe the regression rate of the HTPB fuel through the experiment investigation. The equation is:
r = Ae−E / (RTsurf )
)
= MCO ⎡⎣ 4kf 1CC4 H6 CO 2 − kf 2 CCO CO0.5 − kb2 CCO 2 ⎤⎦ 2
(
∂ ( ρε) + ∂∂x ( ρui ε) ∂t i ε ε2 ∂ ⎛ ∂ε ⎞ ⎜ αε μeff ⎟ + C1ε ( Gk + C3ε Gb ) − C2ε ρ = k k ∂xj ⎝ ∂xj ⎠
(5)
where the constants A and B are 4 and 0.5 respectively. The finite rate model is also used to calculate the net rate of production of species in order to avoid the local excessive reaction rate caused by the strong turbulence. The net rate of production and destruction of species are represented as follows [29]:
ωCO 2 = MCO 2 kf 2 CCO CO0.5 − kb2 CCO 2 2
− Rε + Sε
E , J/mol
A , m/s
where,
(4)
where Tsurf is the burning surface temperature, the value of the parameters A and E are given in Table 1 [32]. A global two-step combustion model is applied to the gas phase reaction for simplification [29,33], given by
C4 H6 + 3.5O2 → 4CO + 3H2 O CO+0.5O2 ↔ CO2 The rate of production and destruction of species are controlled
2.4. Coupled gas/solid phase formulations In the hybrid rocket motor, the solid fuel absorbs the heat from the diffusion flame zone, then decomposes into gas phase and diffuses into the flame zone to react with the oxidizer. Thus the combustion process can sustain, and causes the regression of the burning surface. The regression rate is determined by the energy and mass balance on the burning surface. The mass balance equation is:
ρg un = − ρs r
(7)
The energy balance equation is:
⎛ ∂T ⎞ ⎛ ∂T ⎞ T Tsurf −λ eff ⎜ ⎟ = − λ s ⎜ ⎟ + ρs r hC4surf H6 − hHTPB ⎝ ∂n ⎠ g ⎝ ∂n ⎠s
(
)
(8)
where the term on the left side is the convection heat flux to the burning surface. The first term on the right side is the heat flux conducted into the solid fuel grain. The last term is the variation of Table 2 Reaction rate parameters for the finite rate model. Reaction rate
A
n
E /R , K
kf 1
1.3496 1010
0.0
kf 2
2.2400 106
0.0
5032.7
kb2
1.1000 1013
0.97
39456.5
15108
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
enthalpy from the solid HTPB fuel to the pyrolysis product C4H6, and this additional term makes the energy balance equation different from the normal solid wall conduction heat flux equation. The conduction heat flux term can be calculated as follows:
⎛ ∂T ⎞ −λ s ⎜ ⎟ = ρs rcs (Tsurf − Tref ) ⎝ ∂n ⎠s
(9)
Thus, substituting Eq. (9) into (8), then the energy balance equation is:
⎛ ∂T ⎞ T Tsurf −λ eff ⎜ ⎟ = ρs rcs (Tsurf − Tref ) + ρs r (hC4surf H6 − hHTPB ) ⎝ ∂n ⎠ g
(10)
Because that, T
T
surf ref ρs rhHTPB = ρs r (hHTPB + cs (Tsurf − Tref ))
(11)
So, the energy balance equation is simplified:
⎛ ∂T ⎞ T Tref −λ eff ⎜ ⎟ = ρs r (hC4surf H6 − hHTPB ) ⎝ ∂n ⎠ g
(12)
With the Eqs. (4) and (12), both the fuel surface temperature Tsurf and the fuel regression rate r can be calculated during the iteration of the simulation. After the calculation of regression rate, the produced mass, momentum, species of C4H6 and the absorbed energy due to the pyrolysis of HTPB can be determined. The mass, momentum, energy and species are treated as user defined source terms and added into the first layer cell mesh adjacent to the burning surface. 2.5. Validation of the numerical model In order to validate the numerical model, the experimental regression rates along with the axial direction of a 2D slab motor obtained by Chiaverini et al. [29,31] are compared with the simulation results, as shown in the Fig. 1. The experimental configuration and size are reported in Ref. [31]. In Fig. 1, the experimental data points for case 1 represent the regression rates of test 9 under the condition that the gaseousoxygen flow rate is 0.2 kg/s (the oxidizer flow flux is 208 kg/m2 s). Likewise, the experimental data points for case 2 represents the test 7, of which the gaseous-oxygen flow rate is 0.1 kg/s (the oxidizer flow flux is 104 kg/m2 s). Simulation 1 and simulation 2 are the regression rates predicted by this simulation model corresponding to case 1 and case 2 respectively. Table 3 gives the error analysis of validation. For the case 1, the average regression rate error of the simulation result is 0.9%, indicating that the calculated regression rates match well with the experimental ones. In regard to case 2, the error is 7.9%. The calculated result is also reasonable and acceptable.
387
Table 3 Error analysis of validation.
Spatially averaged regression rate for case1 Spatially averaged regression rate for case2
Exp (mm/ s)
Simulation (mm/s) Error (%)
1.12
1.11
0.9
0.76
0.82
7.9
In conclusion, the numerical model established in this work can be used to simulate the quasi-steady working process and predict the regression rate in the hybrid rocket motor. Besides the acceptable accuracy, this model can be adapted to the variation of the oxidizer flow flux. Overall, the comparisons make a confidence that the model will provide meaningful results.
3. Simulation results and discussion 3.1. Physical configuration The star swirl grain configuration in the work of Armold et al. [19] is taken as a reference in this work. The cross section of the grain port is a star, and the star number is 6. The whole port is formed by the way that the star cross section rotates around the axis as it moves along the axis direction. The grain length is 101.6 mm (4 in.), and the pitch is 1/2 tpi (turns per inch), which means that the star cross section turn half circle when it moves 1 in. along the axis direction. Besides the star swirl grain, a tube grain with the same area of cross section is presented in this work to make a comparison to the star swirl grain. The main parameters of the physical configuration are listed in Table 4. 3.2. Meshes of computational domains The computational domains for the two kinds of grains are three-dimensional shown in Fig. 2 and Fig. 3. Since the domain of the tube grain is simple, the mesh uses hexahedral structured elements with a total number of 586 thousands. As far as the star swirl grain, the hybrid mesh is used in order to generate mesh in this complex domain. The hexahedral structured elements are used for the boundary layer zone, and the tetrahedral and pyramidal unstructured elements are used for the interior zone, which can be shown in Fig. 4(b). The whole quantity of the mesh elements is 2.4 million. The height of the first layer mesh over the burning surface for two grains are all refined to ensure a value of y þ of the order of one, which can guarantee the heat transfer process to be predicted precisely. Table 4 Main parameters of the physical configuration.
Fig. 1. Comparisons of regression rate between simulation and experimental value.
Parameter
value
Inner diameter of chamber, mm Length of precombustion chamber, mm Length of aft-mixing chamber, mm Length of fuel grain, mm Inner diameter of tube grain, mm Diameter of nozzle throat, mm Expansion area ratio of nozzle Star number of the star swirl grain Port cross section area of tube grain, mm2 Port cross section area of star swirl grain, mm2
30.4 25.4 25.4 101.6 11.2 6.0 3.42 6 98.1 98.1
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Fig. 2. Mesh of the tube grain domain. Fig. 5. Temperature contours of the main sections of tube grain.
Fig. 3. Mesh of the star swirl grain domain.
3.3. Initial and boundary conditions Fig. 6. Temperature contours of the main sections of star swirl grain.
The whole circle plane on the front of the pre-chamber is mass flow rate inlet, at which the GOX mass flow rate ranges from 8 to 40 g/s (mass flux from about 80 to 400 kg/m2 s) with temperature of 300 K. The nozzle outlet is pressure outlet, the conditions of which are extrapolated from the interior domain under supersonic condition. The rest surfaces are all no-slip wall boundary conditions. In particular, the wall temperature and regression rate of the burning surface are calculated by the coupled gas/solid phase formulation during the iteration process of the simulation. Except for the temperature, the initial values of the whole domain are initialized by the inlet boundary condition before the iteration. The initial temperature is set at 2000 K to give an initial regression rate and ignite the motor. 3.4. Flow field characteristics The results of the cases in which the oxidizer mass flow rate is 16 g/s for two kinds of grains are represented here to show the
flow characteristics. Fig. 5 gives the classical temperature contours of the tube grain. There is a thin flame layer above the burning surface with temperature about 3500 K, and the distance from flame to grain surface becomes larger as the increase of the axis coordinate due to the consumption of GOX. The vortexes in the aft-mixing chamber shown in Fig. 7 can enhance the mixing of species, making a more sufficient combustion. Consequently, the temperature in the vortex flow zone is higher than the core straight flow zone in aftmixing chamber. The temperature contours of the star swirl grain are very different from the tube grain as a result of the intricate grain configuration shown in Fig. 6. As the oxygen flows into the grain port, it reacts with the pyrolysis product of the fuel grain forming a thin flame layer above the burning surface. Because of the swirl port, the flame layer in the star slot is asymmetrical and closer to the windward surface, shown in Fig. 9. Same as the tube grain, the
Fig. 4. Mesh of the cross section of port. (a) tube grain (b) star swirl grain.
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
Fig. 7. 3D streamlines of tube grain.
Fig. 8. 3D streamlines of star swirl grain.
distance from flame to grain surface becomes larger as the increase of the axis coordinate due to the consumption of GOX. But the area of star swirl grain surface per unit length of axis is larger than that of the tube grain, and more oxygen will be depleted. As a result, the flame layers in the star slot converge at the distance of about 0.04 m away from the grain head with a peak temperature of 3830 K. Then, as the increase of the axis coordinate, the flame in the slot degenerates to the vicinity of the root of star. The flame above the root of star is always closer to the burning surface. In the aft-mixing chamber, the temperature distribution is different from that in tube grain case. The temperature in axial core zone is higher than that in round zone, illustrating that sufficient combustion occurs in this zone. As for the reason, the 3D streamlines in Fig. 8 shows that a big vortex around the axis is formed in the aft-mixing chamber caused by the swirl flow during the star swirl grain port. Due to the vortex, species can mix well in core zone. Besides, the centrifugal force on the oxidizer stream which is anticipated by Armold et al. [19] can be verified from the streamlines in Fig. 8. The streamlines is equally distributed when entrancing the grain port, but as the gas flows downstream, the streamlines become closer to the cusp of star. From Fig. 10, it can be seen that the mass fraction of O2 of the tube grain along the axis keeps 1 until it reaches the nozzle, then decreases to about 0.94. Moreover, temperature keeps 300 K from inlet to the entrance of nozzle. Then, it reaches the peak of about 615 K and decreases as the nozzle supersonic flow. It is demonstrated that the oxygen near the axis does not react sufficiently. However, the distribution of temperature and mass fraction of O2 along the axis of star swirl grain are different from the tube grain. In the grain port, the mass fraction of O2 decreases and the temperature increases. In the aft-mixing chamber and nozzle, the mass fraction of O2 decreases to 0, and temperature reaches the peak of about 3750 K where the sufficient combustion occurs. These analysis illustrate that the utilization coefficient of O2 of the star swirl grain is better than the tube grain. 3.5. Regression characterization In order to analyze the fuel regression rates of different positions on the burning surface of the star swirl grain, four characteristic points are selected as shown in Fig. 11(a). Point A represents the root of star, Point C represents the cusp of star, Point D represents the windward grain surface, and Point B represents the leeward grain surface. In 3D view, these four points in 2D grain
389
section are four helix curves on grain surface shown in Fig. 11(b). Fig. 12 shows the local regression rate distribution of the tube grain and the four characteristic points of the star swirl grain along the axial direction. The tube grain result shows the typical changing trend with an initial decrease due to the boundary layer growth, and then an increase due to the mass and heat addition. The initial peak value of regression rate at the front edge of the fuel grain is caused by the starting of the thermal boundary layer. At this position, the flame is very close to the grain surface, which brings a large temperature gradient and convective heat flux. The last peak value of regression rate at the tail edge of fuel grain is due to vortex enhancement to convective heat flux. Because of the starting of the thermal boundary layer, there are initial peak values on all the regression rate distributions of the four characteristic points of star swirl grain, then the regression rate decreases to a minimum value and then increases. At the length of about 0.024 m, the regression rate of Point C begins to decrease because the flame begins to degenerate from the cusp of star shown in Fig. 9(b). At about Z ¼0.038 m, the flame layer in the star slot has merged completely and continues to degenerate, then the regression rates of Point B and D begin to decrease. Because the flame keeps close to the root of star, the regression rate of Point A keeps a relative high value of about 2.5 mm/s. Since flame is closer to the windward surface than to the leeward surface, regression rate of Point D keeps a little higher than that of Point B. Obviously, most of the regression rate of star swirl grain is much higher than that of the tube grain. In order to compare the regression rate of different grains conveniently, the spatially averaged regression rates and the fuel mass flow rates under different oxygen flow flux are listed in Table 5. The spatially averaged regression rate is defined as follows:
rave =
∫surf rdA Asurf
(13)
where Asurf is the burning surface area. As the oxygen flow flux rise, the spatially averaged regression rates of all the two fuel grain types increase. At each oxygen flow flux, the spatially averaged regression rate of star swirl grain is higher than that of tube grain, and the growth rate ranges from 30.1% to 60.3%. As the oxygen flow flux rise, the growth rate increase. These regression rate points are shown clearly in Fig. 13. The averaged fuel regression rate is expressed in the form of power law rave = aGon , where Go is oxidizer mass flux. The exponent of the power law for star swirl grain is 0.5608, a little higher than 0.4373 for tube grain. Beside the higher averaged regression rate, the burning surface area of star swirl grain is 106.29 cm2, about three times that 35.64 cm2 of tube grain with the condition of same port area and same grain length. As the result of higher regression rate and larger burning surface area, the fuel mass flow rate of star swirl grain is much higher than that of tube grain. The growth rate can reach up to 377.7%. In order to verify that the three-dimensional simulation results are reasonable and acceptable, the regression rate points of the tube grain are compared with the experimental regression rate results of studies performed by several of the leading researchers. The propellant combination is also GOX and pure HTPB, and grain configuration is tube. As shown in Fig. 14, the tube grain simulation points of this paper are located in a reasonable range. 3.6. Combustion performance For the hybrid rocket motor system, the oxidizer to fuel ratio O/ F, the characteristic velocity C * and the efficiency of characteristic velocity ηC * are used to measure the combustion performance. They are defined as follows:
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Fig. 9. Temperature contours of sections. (a) Z¼ 0.017 m (b) Z¼ 0.024 m (c) Z¼ 0.038 m (d) Z ¼0.063 m.
O/F =
Fig. 10. Distribution of temperature and mass fraction of O2 along the axis of motor.
ṁ o ṁ fuel
(14)
C* =
pc ⋅Athroat ṁ tot
(15)
ηC * =
* Csim * Cth
(16)
where Athroat is the area of nozzle throat, ṁ tot is the total mass flow * is the characteristic velocity calculated by the simulation rate, Csim * is theoretical characteristic velocity obtained by therresults, Cth modynamic calculation with the same oxidizer to fuel ratio, chamber pressure, nozzle expansion area ratio under the condition of vacuum. These results are summarized in Table 6. O/F of both tube grain and star swirl grain increase as the oxygen flow flux rises. O/F of
S. Zhang et al. / Acta Astronautica 127 (2016) 384–393
391
Fig. 11. Characteristic positions for local regression rates analysis. (a) 2D grain section (b) 3D grain surface.
Fig. 12. Distribution of local regression rate along axial direction. Table 5 Spatially averaged regression rates and fuel mass flow rates of different grain types.
ṁ o , g/s
8 10 12 16 20 30 40
Go , kg/m2 s Tube grain
Star swirl grain
rave , mm/s
rave , mm/s
0.747 0.818 0.880 0.985 1.076 1.331 1.506
0.972 1.136 1.264 1.495 1.692 2.097 2.415
81.5 101.9 122.3 163.1 203.9 305.8 407.7
Growth rate (%)
30.1 38.9 43.6 51.8 57.2 57.6 60.3
Tube grain
Star swirl grain
ṁ fuel ,
ṁ fuel ,
g/s
g/s
2.47 2.70 2.91 3.26 3.55 4.39 4.98
9.58 11.19 12.46 14.73 16.67 20.66 23.79
Growth rate (%)
287.9 314.4 328.2 351.8 369.6 370.6 377.7
Fig. 14. Regression rate comparison with various hybrid rocket studies using GOX/ HTPB [1,36–41].
tube grain is about 4–5 times that of star swirl grain for its lower * , Cth * and ηC * of tube grain are all decrease as fuel mass flow rate. Csim the oxygen flow flux rises. It illustrates that, as the oxygen flow flux rises, the combustion performance of tube grain becomes worse, and more and more oxidizer is wasted without sufficient * and Cth * are both increase as combustion. As for star swirl grain, Csim the oxygen flow flux rises, and ηC * keeps a relative high value of about 90%. It illustrates that the combustion performance of star swirl grain is better and more stable than that of tube grain.
4. Conclusions Based on GOX/HTPB propellant combination, the three dimensional numerical model for hybrid rocket motor internal flow field simulation has been established with coupled gas/solid phase formulations. And this model is validated to be accurate through the comparison between simulation results and the experiment data from the work of Chiaverini et al. Then numerical simulation investigations of hybrid rocket motor with star swirl fuel grain and tube fuel grain are conducted in this work. Temperature contours in the motor and regression rate distributions of characteristic points are presented and discussed. The influences of the oxidizer mass flux and the fuel grain type on spatially averaged regression rate are analyzed. The combustion performances of both grain types are compared. Several main conclusions can be deduced as follows:
Fig. 13. Spatially averaged regression rates of different grain types.
(1) Under the conditions of the same grain length and the same port area, the star swirl grain can enlarge burning surface area comparing to tube grain. In this work, the burning surface area of the six-pointed star swirl grain is about three times that of the tube grain.
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Table 6 Combustion performance of HRM with different grain types.
Go , kg/m2 s
81.5 101.9 122.3 163.1 203.9 305.8 407.7 Average
Tube grain
Star swirl grain
O/F
pc , MPa
* , m/s Csim
* , m/s Cth
ηC * (%)
O/F
pc , MPa
* , m/s Csim
* , m/s Cth
ηC * (%)
3.239 3.704 4.124 4.908 5.634 6.834 8.032 –
0.572 0.669 0.759 0.920 1.070 1.407 1.668 –
1544.7 1489.4 1439.3 1350.6 1284.7 1156.8 1048.5 –
1673.6 1640.6 1612.0 1567.9 1533.7 1484.6 1439.5 –
92.3 90.8 89.3 86.1 83.8 77.9 72.8 84.7
0.835 0.894 0.963 1.086 1.200 1.452 1.681 –
0.781 0.971 1.160 1.537 1.911 2.823 3.705 –
1256.1 1295.6 1340.9 1414.2 1473.5 1575.6 1642.2 –
1420.9 1450.3 1483.2 1536.6 1591.1 1730.4 1796.1 –
88.4 89.3 90.4 92.0 92.6 91.1 91.4 90.8
(2) At the same oxidizer mass flux level, most of the local regression rate distribution of the four characteristic points of the star swirl grain along the axial direction are higher than that of the tube grain. Beside higher regression rate, the swirl flow through the star swirl grain port can form a big vortex around the axis in the aft-mixing chamber. The vortex flow can make the oxidizer react sufficiently. The combustion efficiency of star swirl grain is better and more stable than that of tube grain. (3) For both of the two fuel grain types, spatially averaged regression rate increases as the oxidizer mass flux rises. The star swirl grain shows a higher influence of mass flux than the tube grain, as demonstrated by a bigger exponent of the power law (0.5608 vs 0.4373). In the mass flux range from about 80 to 400 kg/m2 s, the averaged regression rate of star swirl grain is higher than that of tube grain at each mass flux level. Growth rate of regression rate can reach up to 60.3%, which is meaningful for hybrid rocket motor.
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