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Theoretical study of a solid fuel scramjet combustor
PII:
Acta Astronautica Vol. 45, No. 3, pp. 155±166, 1999 # 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765/99 $ - see front matter S0094-5765(99)00113-7
THEORETICAL STUDY OF A SOLID FUEL SCRAMJET COMBUSTOR RACHEL BEN-AROSH, BENVENISTE NATAN{, ELYESER SPIEGLER and ALON GANY Faculty of Aerospace Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel (Received 25 September 1998) AbstractÐThe combustion of a solid fuel under supersonic cross ¯ow conditions was investigated theoretically. A two-dimensional, axisymmetric, turbulent (k±e ), two-reaction, six-species reactive ¯ow model was developed and solved numerically. The nominal case investigated simulated stagnation conditions resulting from ¯ight Mach number of 5 at 15 km for a designed combustor entrance Mach number of 1.5. The results demonstrate the feasibility of solid fuel combustion under supersonic cross ¯ow where the diusion ¯ame supplies a substantial heat addition to the ¯ow. The sensitivity of the combustor performance to ¯ight Mach number, combustor inlet Mach number, combustor size, step size and length of fuel grain was studied. The eects on ¯ow pattern, regression rate pro®les as well as ¯ame temperature and location were evaluated. Typical computed combustion eciencies of the order of 0.7±0.9 support potential use of solid fuel scramjet. # 1999 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The demand for cost reduction and increased dependability of transporting payload to orbit has led to a constantly increasing interest in development of modern air breathing propulsion systems for hypersonic vehicles. Consequently, attention is being focused on the supersonic combustion ramjet (commonly known as scramjet). Ramjet engines operate at supersonic ¯ight Mach numbers. In the conventional ramjet, the air¯ow is slowed down to subsonic ¯ow velocities throughout the combustion chamber in order to achieve better ¯ame stabilization and combustion eciencies. However, for ¯ight Mach numbers above 5, better performance (higher speci®c impulse) can be achieved if the combustor ¯ow Mach number remains supersonic [1,2]. The scramjet engine is usually powered by liquid fuels. For certain applications, however, one can see an advantage in employing solid fuels. The system design is greatly simpli®ed, storage is very convenient, and a feeding system is not required. Hence, low cost propulsion system is enabled. However, unlike the case of liquid fuel combustion, the use of a solid fuel gives no direct control on fuel ¯ow rate and injection velocity. The solid fuel undergoes degradation and gasi®cation because of heat feedback from the hot gas ¯ow, resulting in regression of the solid wall and establishment of diusion ¯ame within the boundary layer above the wall. Flame holding is achieved by the inlet step.
{Corresponding author.
In contrast to the liquid fuel scramjets, only a few publications are available on solid fuel supersonic combustion. Preliminary experimental eorts aimed at operating such combustors were made by Witt [3] and Angus [4], who conducted initial screening tests to determine geometry that could provide ignition and ¯ame stabilization. These tests resulted in selection of a speci®c con®guration for further study performed by Angus et al. [5]. Their experiments yielded combustion eciencies between 48% and 87% and stagnation pressure losses of 50±70% for inlet Mach number of 1.5 and exit Mach number of 1.1±1.4. However, in all of those experiments some injection of hydrogen was required to sustain combustion. Following these works, an experimental investigation was performed by Ben-Yakar et al. [6,7]. For the ®rst time, self-ignition of the polymethyl methacrylate (PMMA) solid fuel by the hot air¯ow as well as sustained combustion under supersonic cross¯ow conditions without external aids were demonstrated. Flameholding limits were presented as well. The research revealed that the eect of increasing the air¯ow rate is to reduce the combustion eciency and to increase fuel regression rate. Recent test results by Cohen and Natan [8] indicated an increase in regression rate and a decrease in combustion eciency for higher inlet air stagnation temperatures. Computationally, Jarymowycz et al. [9] conducted a comparative numerical study of the combustion of a hydroxyl-terminated polybutadiene (HTPB) solid fuel under supersonic cross ¯ow. They concluded that both inlet temperature and pressure have strong in¯uences on the burning rate of the fuel sample. However, the selected con®guration 155
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R. Ben-Arosh et al.
Fig. 1. Solid fuel scramjet combustor geometry.
was a hollow cylinder of a solid fuel without an inlet step that is essential for ¯ame holding in actual combustors. Ben-Arosh et al. [10] conducted an orderof-magnitude analysis showing that the fuel-air mixing process in a supersonic combustion solid fuel ramjet combustor does not limit combustion more than it does in a conventional subsonic solid fuel ramjet (SFRJ). Furthermore, in a detailed numerical study in the same work [10] the results indicate that high mixing eciencies may be obtained. The main objective of the present research is to investigate the feasibility and to predict the characteristics of solid-fuel dump-type scramjet combustors. 2. THEORETICAL FORMULATION
2.1. General The geometry considered in the analysis is shown in Fig. 1. A uniform supersonic air¯ow enters a two-dimensional axisymmetric chamber through a sudden expansion inlet characterized by a step height H. The chamber is 20H long, while solid polyethylene fuel in the form of hollow cylinder occupies the initial 12H-long section along the chamber wall. Ethylene gas is added (injected) into the combustor from the side-wall as a result of the decomposition of the solid fuel. The conservation equations{ can be written in the following generalized form [11]: divrVf ÿ Gf gradf Sf
1
where the ®rst and second left-hand side terms represent convection and diusion, respectively, and the right-hand side term is a source term. f stands for the conserved property and Gf is the diusion coecient. The terms of the relevant dierential equations [the conservation of mass, two velocity {See Appendix.
components, enthalpy, six species (C2H4, O2, CO2, H2O, N2, CO), turbulence kinetic energy, and turbulent dissipation] are presented in Table 1, where se=1.3, sk=1.0. The speci®c enthalpy h is expressed by: h
N X
1 Yi hi W 2 V 2 2 i1
2
where the individual species enthalpy is de®ned by: hi
T Tref
Cpi dT h0f i
3
accounting for the variation of speci®c heat with temperature given from CHEMKIN chemical kinetic code combined with the PHOENICS computational ¯uid dynamics (CFD) code. Turbulence kinetic energy generation rate Pk is given by: ( " 2 2 # @V @W Pk mt 2 @r @z 4 2 ) @V @W : @z @r The turbulent Prandtl number was set to 0.9, and the Eucken formula [12] was used for the laminar Prandtl number. Table 1. Terms of dierential equations f
Gf
1 V W h Yi k e
0 me me mlam mt Prlam Prt me/sm me/sk me/se
Sf 0
ÿ@ P 1 @ @V @ r r @ r rm @ r ÿ@ P 1 @V @ @W @ z r @ z @ z m @ z
0
oi Pkÿre e k Cl Pk ÿ C2 re
Solid Fuel Scramjet
2.2. Boundary conditions Inlet stagnation temperature of 1300 K and static pressure of 3.5 atm were used to simulate conditions resulting from ¯ight Mach number of 5 at 15 km altitude and inlet air¯ow Mach number of 1.5. At the combustor symmetry line, radial velocity component and radial derivatives of all other dependent variables were set to zero. At the wall, no slip condition was employed. Computation of the nearwall region used the generalized wall functions of Ref. [13], taking into account the fuel gas velocity. The local mass ¯ow rate and the fuel gas velocity at the wall were determined by mass conservation between gas/solid phases at the fuel surface: Vg
r_ rf : rg
5
Basically, at steady state combustion the fuel regression rate can be calculated from r_
q_ w rf Hv, eff
6
where qÇ w is the heat ¯ux to the wall, rf is the condensed fuel density, and Hv,e is the eective heat of gasi®cation (or vaporization) of the fuel. It was assumed that the driving force in the heat transfer mechanism is the dierence between the ¯ame and wall temperatures: q_ w h Tfl ÿ Tw :
7
The solid fuel surface temperature was assumed to be approximately constant and was set to 800 K according to Refs. [14] and [15]. Flame temperature was determined as the maximum temperature at each axial location. The heat transfer coecient was determined using Stanton number: h St r W Cp
8
where Stanton number, St, was computed from skin friction coecient calculated by the wall functions. For the exit boundary, the conditions of constant static pressure (the recommended code outlet boundary condition) and zero gradient for all other dependent variables were used. The selection of the outlet pressure is described in Ref. [10]. 2.3. Chemical kinetics The simpli®ed 2-equation ethylene combustion model of Westbrook and Dryer [16] was applied: C2 H4 2O2 4 2CO 2H2 O
9
1 CO O2 $ CO2 : 2
10
Rate constants were adopted from Refs. [16] and [17]. The chemical reactions and the species production rates were calculated by the CHEMKIN
157
chemical kinetics package combined with the PHOENICS code. 2.4. Numerical method For the solution of the described problem the PHOENICS Computational Fluid Dynamics (CFD) code was employed. The code is based on the SIMPLEST algorithm developed by Spalding and Patankar [18]. The validation of the PHOENICS code for supersonic ¯ow over a backward facing step is presented in Refs. [10] and [19]. The computational grid was non-uniform, with smaller and uniform cells from the wall and up to a distance of half step above the inlet step, and growing cells from there up to the symmetry line. The in¯uence of grid density was examined by the authors in their previous work [10] that investigated the same geometric con®guration as in the present case, and a 70 105 grid was found as most adequate for the purpose of the present investigation.
3. RESULTS AND DISCUSSION
3.1. Velocity ®eld Figures 2 and 3 present the ¯ow Mach number distribution of a non-reacting case and that of the present reacting case, respectively. Comparison reveals that the velocity ®eld changes substantially. The inlet fan at the step corner, which served to elevate the Mach number from 1.5 to a maximum value of 3.6 in the non-reacting case, almost disappears in the reacting case, resulting in a maximum Mach number of 2.1. Broad areas of supersonic ¯ow in the non-reacting case become subsonic due to the heat release from the chemical reactions. A mixed supersonic (at the center)/subsonic (at the circumference) jet is formed at the combustor exit when combustion takes place, replacing the almost whole supersonic non-reacting stream. 3.2. Temperature Temperature distribution of the reacting ¯ow®eld is presented in Fig. 4. The computation results reveal that sustained combustion can exist within the combustor for the inlet and geometry conditions under investigation. A diusion ¯ame with a maximum temperature of 2846 K is formed supplying a substantial heat addition to the ¯ow. The static and stagnation temperatures increase from 900 K and 1300 K at the inlet to mass-averaged values of 2189 K and 2494 K, respectively, at a distance of 18H from the inlet. The region between the wall and ¯ame center is heated too due to the chemical reactions and the lateral heat transfer from the ¯ame. At the combustor axis, the temperature remains almost unchanged.
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Fig. 2. Mach number map for supersonic ¯ow over a backward facing step.
Fig. 3. Reacting ¯ow®eld Mach number map (nominal case).
Fig. 4. Reacting ¯ow®eld temperature map (nominal case).
3.3. Flame characteristics Flame temperature and radial position change with axial distance. The change in radial position of the ¯ame is demonstrated in Fig. 5 for several crosssections along the combustor. Flame center was de®ned by the local maximum temperature. From Fig. 5 one can see that the further the axial distance
along the combustor the further the radial position of the ¯ame from the wall. This shift is explained by the diusive nature of the ¯ame, where the ¯ow dominates the ¯ame development and characteristics. The ¯ame temperature also varies along the combustor (Fig. 6), increasing continuously with the axial distance to a maximum ¯ame temperature of 2846 K close to the reattachment cross-section. In
Solid Fuel Scramjet
Fig. 5. Flame temperature distribution at various axial locations.
Fig. 6. Maximum ¯ame temperature vs axial location.
159
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R. Ben-Arosh et al.
the recirculation region the average temperature (at a 3-step-heights distance) is 2530 K. Downstream the reattachment point, the temperature is slightly decreasing with axial location, but remains high. 3.4. Chemical species Radial pro®les of the chemical species at axial locations of 3H, 10H and 18H are presented in Figs 7, 8 and 9, respectively. Oxygen mass fraction decreases from its maximum value at the centerline to almost zero beyond the ¯ame. This decrease is very steep at the recirculation region and becomes moderate further downstream. Similarly, fuel vapor fraction decreases gradually from the wall towards the ¯ame position. Beyond the ¯ame, no fuel vapors are apparent. From Fig. 9 one can see that although the fuel grain length is only 12H, there is still a substantial amount of unburned fuel vapors close to the exit (at 18H distance). The main combustion products are CO2 and H2O. Very low values of CO mass fraction are found (maximum value of 0.009), resulting from the high conversion rate of CO into CO2 in the operating conditions of near stochiometric fuel/air ratio. Higher CO fractions are expected for fuel-rich conditions or at higher temperatures.
The combustion eciency is de®ned here by the ratio between the reacted fuel mass ¯ow rate and the maximum value of fuel mass ¯ow rate that can react. The maximum value is the stoichiometric ¯ow rate for fuel-rich conditions, and the total entering fuel ¯ow rate for fuel-lean mixture conditions. For the nominal conditions, an equivalence ratio of 1.14 yielded combustion eciency of 70% at the combustor exit. Calculations reveal stagnation pressure losses of 60% resulting from the heat and mass addition to the ¯ow within the combustor. 3.6. Fuel regression rate Figure 10 presents the axial variation of the fuel regression rate. Along the recirculation region there is a gradual increase of the fuel regression rate from 0.5 mm/s near the inlet step up to 1.8 mm/s close to the reattachment point. A similar behavior is also typical to subsonic solid fuel ramjet combustors [20]. This may be expected since the recirculation region is subsonic for both subsonic and supersonic ¯ow combustors. Downstream of the reattachment, a higher and almost constant regression rate of 3 mm/s was calculated. In comparison, in subsonic ¯ow combustors the downstream section is characterized by slightly decreasing regression rates.
3.5. Combustor performance
3.7. Recirculation region
One of the ways to evaluate combustor performance is by computation of the combustion eciency.
Figure 11 illustrates the axial velocity variation at the ®rst near-wall grid point (0.1 mm). The reattach-
Fig. 7. Chemical species and temperature radial distribution at 3H distance from inlet.
Solid Fuel Scramjet
Fig. 8. Chemical species and temperature radial distribution at 10H distance from inlet.
Fig. 9. Chemical species and temperature radial distribution at 18H distance from inlet.
161
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Fig. 10. Axial variation at the fuel regression rate.
Fig. 11. Axial velocity variation at the ®rst near-wall grid point (0.1 mm).
Solid Fuel Scramjet
ment point at the end of the recirculation zone is obtained at an axial distance of 5.9 step heights. This is a much larger distance compared to the nonreacting case (1.4H ) [10]. A similar behavior was reported by Bussing and Merman [21] for a Mach 3 combustor. Combustion seems to have the opposite eect in subsonic ¯ow combustors, where heat addition causes shortening of the reattachment distance [22].
163
4.2. Increased combustor inlet Mach number (1.75) For the same ¯ight conditions, increasing the combustor inlet Mach number from the nominal value of 1.5 to 1.75, implies higher velocity and lower static temperature at the combustor inlet. The computed combustion eciency (70%) and equivalence ratio (1.11) are similar to those of the nominal case, while local and exit Mach numbers are higher. 4.3. Increased ¯ight Mach number
4. PARAMETRIC STUDY
4.1. Shorter fuel grain Following the results of the nominal case, which indicate a substantial amount of unburned fuel gas, a shorter fuel grain (9H instead of 12H long) has been examined. The overall combustor length is kept constant, hence the average residence time of the fuel gas is increased. The total equivalence ratio reduces to a lean value of 0.7, while the combustion eciency reaches a higher value of 90%. Figure 12 presents the species radial pro®les for the shorter fuel grain case. For this case, lower values of fuel mass fraction are found near the exit. However, this con®guration has two disadvantages: ®rst, the total energy added to the ¯ow is reduced (exit calculated stagnation temperature reduces to 2389 K), and second, the volumetric loading fraction of the fuel decreases.
For the nominal combustor inlet Mach number of 1.5, increasing the ¯ight Mach number from the nominal value of 5 to 6 elevates the stagnation and static temperatures at the combustor inlet from 900 K and 1300 K to 1225 K and 1777 K, respectively. As a result, the whole temperature pro®le is raised. Figure 13 presents the radial temperature pro®les in the middle of the chamber (10H ) for three dierent cases. For the nominal case the radial location of the ¯ame center is shifted towards the fuel wall. The temperature pro®les of ¯ight Mach 5 cases (with Min=1.5 and 1.75) dier mainly around the combustor center, and almost unite close to the wall. Figure 14 shows the in¯uence of inlet conditions on the axial regression rate pro®les. The highest regression rate values were calculated for the ¯ight Mach 6 case, where both the inlet temperature and air mass ¯ux are higher than in the nominal case. The regression rate for the combustor inlet Mach 1.75 case exceeds the nominal case values as
Fig. 12. Chemical species and temperature radial distribution at z = 18H for a short (9H ) fuel grain.
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Fig. 13. Temperature radial distribution at z = 10H for various inlet conditions.
Fig. 14. Regression rate distribution for various inlet conditions.
Solid Fuel Scramjet
165
Fig. 15. Mach number map for a 0.5-cm-step-height combustor.
well due to the increased mass ¯ux. This behavior agrees with the known general dependence of regression rate on inlet mass ¯ux and temperature [6,9,15]. 4.4. Enlarged combustor A double-size combustor yields drastic changes in combustor behavior. The two-fold larger combustor could not sustain combustion, possibly due to insuf®cient mixing. Mixing in solid fuel combustors, where the added fuel velocity at the wall is very low, is diusion controlled, while distances relevant to mixing are proportional to the combustor diameter. Hence, larger combustors may have a reduced mixing capability than smaller ones. 4.5. Smaller step Reducing step height while keeping other dimensions (e.g. combustor diameter and length) unchanged, might have several qualitative in¯uences on the ¯ow pattern. First, the inlet/exit area ratio is reduced, hence the acceleration eect attributed to enlargement of the cross-section in supersonic ¯ow is reduced. This might cause choking of the ¯ow in the case that the cross-section enlargement is insucient to compensate for ¯ow deceleration due to heat and mass addition. Another important consideration is the ¯ame-holding aspect. For a particular geometry, fuel and inlet conditions, ¯ameholding capability is mainly dependent on step size. Reducing the step height below a critical value would result in combustor extinction. In the present case, step height was reduced to one-half of the nominal one. Figure 15 presents the ¯ow Mach number distribution of this case. Mach number drops from 1.5 at the inlet to 0.7 downstream of step as a result of thermal choking due to the heat addition in chemical reactions. In order to avoid choking situations special care should be taken during the design of supersonic combustors.
5. CONCLUSIONS
The feasibility of solid fuel combustion under supersonic cross¯ow conditions in SFRJ con®guration has been demonstrated numerically. A diusion ¯ame with a maximum temperature of 2846 K is established supplying signi®cant heat addition to the ¯ow. The resulting regression rate increases gradually from the inlet cross-section to the reattachment zone. Further downstream, a higher and approximately constant regression rate is demonstrated. The heat release from the chemical reaction reduces the local Mach number, changes the pressure ®eld and increases the ¯ow recirculation zone. The parametric investigation reveals that the operation of the solid fuel scramjet combustor is sensitive to the design parameters. Inappropriate con®guration might cause extinction of the ¯ame or choking of the ¯ow. Fuel regression rate, ¯ame characteristics and combustion eciency are all aected by geometrical and operating conditions. APPENDIXAPPENDIX
Nomenclature Cp D H Hv,e h h 0f k ` M qÇ w P Pk Pr r Çr S St T t V V
speci®c heat diusion coecient step height eective heat of vaporization speci®c enthalpy; heat transfer coecient speci®c heat of formation turbulence kinetic energy Prandtl mixing length Mach number heat ¯ux to the wall pressure turbulence kinetic energy generation rate Prandtl number radial coordinate regression rate source term Stanton number temperature time radial velocity velocity vector
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W axial velocity mass fraction of species i Yi z axial coordinate Greek e turbulence dissipation m viscosity f conserved property r density s Prandtl or Schmidt number o chemical source Subscripts e eective f fuel ¯ ¯ame g gas phase i ith chemical species in combustor inlet lam laminar t turbulent w wall REFERENCES
1. Billig, F. S., Tactical missile design concepts. Johns Hopkins APL Technical Digest, 1983, 4, 139±154. 2. Cheng, Sin-I, Hypersonic propulsion. Prog. Energy Combust. Sci, 1989, 15, 183±202. 3. Witt, M. A., Investigation into the feasibility of using solid fuel ramjets for high supersonic/low hypersonic tactical missiles. M.Sc. thesis, Department of Aeronautics and Astronautics, Naval Postgraduate School, Monterey, CA, 1989. 4. Angus, W. J., An investigation into the performance characteristics of a solid fuel scramjet propulsion device. M.Sc. thesis, Department of Aeronautics and Astronautics, Naval Postgraduate School, Monterey, CA, 1991. 5. Angus, W. J., Witt, M. A., Laredo, D. and Netzer, D. W., Solid fuel supersonic combustion. La Recherche Aerospatiale 1993, 1±8. 6. Ben-Yakar, A. and Gany, A., Experimental study of a solid fuel scramjet. In 30th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Indianapolis, IN, 1994 (AIAA 94-2815). 7. Ben-Yakar, A., Natan, B. and Gany, A., Investigation of a solid fuel scramjet combustor. Journal of Propulsion and Power, 1998, 14, 447±455. 8. Cohen, A. and Natan, B., Experimental investigation of a supersonic combustion solid fuel ramjet. Journal of Propulsion and Power, 1998, 14, 880±889.
9. Jarymowycz, T., Yang, V. and Kuo, K., A numerical study of solid fuel combustion under supersonic cross¯ows. In 20th Joint Propulsion Conference, Orlando, FL, 1990 (AIAA-90-2076). 10. Ben-Arosh, R., Natan, B., Spiegler, E. and Gany, A., Fuel-air mixing in solid fuel scramjet combustors. International Journal of Turbo and Jet Engines, 1998, 15, 223±234. 11. Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, 1980. 12. Bird, R. B., Stewart, E. W. and Lightfoot, E. N., Transport Phenomena. Wiley, New York, 1960. 13. Launder, B. E. and Spalding, D. B., The numerical computation of turbulent ¯ows. Computer Methods in Applied Mechanics and Engineering, 1974, 3, 269± 289. 14. Hadar, I. and Gany, A., Fuel regression mechanism in a solid fuel ramjet. Propellants, Explosives, Pyrotechnics, 1992, 17, 70±76. 15. Schulte, G., Pein, R. and Hogl, A., Temperature and concentration measurements in a solid fuel ramjet combustion chamber. Journal of Propulsion and Power, 1987, 3, 114±120. 16. Westbrook, C. K. and Dryer, F. L., Simpli®ed reaction mechanisms for the oxidation of hydrocarbon fuels in ¯ames. Combustion Science and Technology, 1981, 27, 31±43. 17. Dryer, F. L. and Glassman, I., High-temperature oxidation of CO and CH4. In Proceedings of the 14th Symposium (International) on Combustion, The Combustion Institute, 1972, pp. 987±1003. 18. Patankar, S. V. and Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic ¯ows. International Journal of Heat and Mass Transfer, 1972, 15, 1787. 19. Halupovich, Y., Natan, B. and Rom, J., Numerical solution of the turbulent supersonic ¯ow over a backward facing step. Fluid Dynamics Research, 1999, 24, 251±273. 20. Zvuloni, R., Gany, A. and Levy, Y., Geometric eects on the combustion in solid fuel ramjets. Journal of Propulsion and Power, 1989, 5, 32±37. 21. Bussing, T. R. A. and Murman, E. M., Numerical investigation of 2-dimensional H2-Air ¯ameholding over ramps and rearward facing steps. In AIAA/ASME/ SAE/ASEE 21st Joint Propulsion Conference, Monterey, CA, 1985 (AIAA 85-1250). 22. Pits, W. and Daily, J. W., Combustion in a turbulent mixing layer formed at a rearward-facing step. AIAA Journal, 1983, 21, 1565±1570.
Acta Astronautica Vol. 45, No. 3, pp. 155±166, 1999 # 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765/99 $ - see front matter S0094-5765(99)00113-7
THEORETICAL STUDY OF A SOLID FUEL SCRAMJET COMBUSTOR RACHEL BEN-AROSH, BENVENISTE NATAN{, ELYESER SPIEGLER and ALON GANY Faculty of Aerospace Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel (Received 25 September 1998) AbstractÐThe combustion of a solid fuel under supersonic cross ¯ow conditions was investigated theoretically. A two-dimensional, axisymmetric, turbulent (k±e ), two-reaction, six-species reactive ¯ow model was developed and solved numerically. The nominal case investigated simulated stagnation conditions resulting from ¯ight Mach number of 5 at 15 km for a designed combustor entrance Mach number of 1.5. The results demonstrate the feasibility of solid fuel combustion under supersonic cross ¯ow where the diusion ¯ame supplies a substantial heat addition to the ¯ow. The sensitivity of the combustor performance to ¯ight Mach number, combustor inlet Mach number, combustor size, step size and length of fuel grain was studied. The eects on ¯ow pattern, regression rate pro®les as well as ¯ame temperature and location were evaluated. Typical computed combustion eciencies of the order of 0.7±0.9 support potential use of solid fuel scramjet. # 1999 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The demand for cost reduction and increased dependability of transporting payload to orbit has led to a constantly increasing interest in development of modern air breathing propulsion systems for hypersonic vehicles. Consequently, attention is being focused on the supersonic combustion ramjet (commonly known as scramjet). Ramjet engines operate at supersonic ¯ight Mach numbers. In the conventional ramjet, the air¯ow is slowed down to subsonic ¯ow velocities throughout the combustion chamber in order to achieve better ¯ame stabilization and combustion eciencies. However, for ¯ight Mach numbers above 5, better performance (higher speci®c impulse) can be achieved if the combustor ¯ow Mach number remains supersonic [1,2]. The scramjet engine is usually powered by liquid fuels. For certain applications, however, one can see an advantage in employing solid fuels. The system design is greatly simpli®ed, storage is very convenient, and a feeding system is not required. Hence, low cost propulsion system is enabled. However, unlike the case of liquid fuel combustion, the use of a solid fuel gives no direct control on fuel ¯ow rate and injection velocity. The solid fuel undergoes degradation and gasi®cation because of heat feedback from the hot gas ¯ow, resulting in regression of the solid wall and establishment of diusion ¯ame within the boundary layer above the wall. Flame holding is achieved by the inlet step.
{Corresponding author.
In contrast to the liquid fuel scramjets, only a few publications are available on solid fuel supersonic combustion. Preliminary experimental eorts aimed at operating such combustors were made by Witt [3] and Angus [4], who conducted initial screening tests to determine geometry that could provide ignition and ¯ame stabilization. These tests resulted in selection of a speci®c con®guration for further study performed by Angus et al. [5]. Their experiments yielded combustion eciencies between 48% and 87% and stagnation pressure losses of 50±70% for inlet Mach number of 1.5 and exit Mach number of 1.1±1.4. However, in all of those experiments some injection of hydrogen was required to sustain combustion. Following these works, an experimental investigation was performed by Ben-Yakar et al. [6,7]. For the ®rst time, self-ignition of the polymethyl methacrylate (PMMA) solid fuel by the hot air¯ow as well as sustained combustion under supersonic cross¯ow conditions without external aids were demonstrated. Flameholding limits were presented as well. The research revealed that the eect of increasing the air¯ow rate is to reduce the combustion eciency and to increase fuel regression rate. Recent test results by Cohen and Natan [8] indicated an increase in regression rate and a decrease in combustion eciency for higher inlet air stagnation temperatures. Computationally, Jarymowycz et al. [9] conducted a comparative numerical study of the combustion of a hydroxyl-terminated polybutadiene (HTPB) solid fuel under supersonic cross ¯ow. They concluded that both inlet temperature and pressure have strong in¯uences on the burning rate of the fuel sample. However, the selected con®guration 155
156
R. Ben-Arosh et al.
Fig. 1. Solid fuel scramjet combustor geometry.
was a hollow cylinder of a solid fuel without an inlet step that is essential for ¯ame holding in actual combustors. Ben-Arosh et al. [10] conducted an orderof-magnitude analysis showing that the fuel-air mixing process in a supersonic combustion solid fuel ramjet combustor does not limit combustion more than it does in a conventional subsonic solid fuel ramjet (SFRJ). Furthermore, in a detailed numerical study in the same work [10] the results indicate that high mixing eciencies may be obtained. The main objective of the present research is to investigate the feasibility and to predict the characteristics of solid-fuel dump-type scramjet combustors. 2. THEORETICAL FORMULATION
2.1. General The geometry considered in the analysis is shown in Fig. 1. A uniform supersonic air¯ow enters a two-dimensional axisymmetric chamber through a sudden expansion inlet characterized by a step height H. The chamber is 20H long, while solid polyethylene fuel in the form of hollow cylinder occupies the initial 12H-long section along the chamber wall. Ethylene gas is added (injected) into the combustor from the side-wall as a result of the decomposition of the solid fuel. The conservation equations{ can be written in the following generalized form [11]: divrVf ÿ Gf gradf Sf
1
where the ®rst and second left-hand side terms represent convection and diusion, respectively, and the right-hand side term is a source term. f stands for the conserved property and Gf is the diusion coecient. The terms of the relevant dierential equations [the conservation of mass, two velocity {See Appendix.
components, enthalpy, six species (C2H4, O2, CO2, H2O, N2, CO), turbulence kinetic energy, and turbulent dissipation] are presented in Table 1, where se=1.3, sk=1.0. The speci®c enthalpy h is expressed by: h
N X
1 Yi hi W 2 V 2 2 i1
2
where the individual species enthalpy is de®ned by: hi
T Tref
Cpi dT h0f i
3
accounting for the variation of speci®c heat with temperature given from CHEMKIN chemical kinetic code combined with the PHOENICS computational ¯uid dynamics (CFD) code. Turbulence kinetic energy generation rate Pk is given by: ( " 2 2 # @V @W Pk mt 2 @r @z 4 2 ) @V @W : @z @r The turbulent Prandtl number was set to 0.9, and the Eucken formula [12] was used for the laminar Prandtl number. Table 1. Terms of dierential equations f
Gf
1 V W h Yi k e
0 me me mlam mt Prlam Prt me/sm me/sk me/se
Sf 0
ÿ@ P 1 @ @V @ r r @ r rm @ r ÿ@ P 1 @V @ @W @ z r @ z @ z m @ z
0
oi Pkÿre e k Cl Pk ÿ C2 re
Solid Fuel Scramjet
2.2. Boundary conditions Inlet stagnation temperature of 1300 K and static pressure of 3.5 atm were used to simulate conditions resulting from ¯ight Mach number of 5 at 15 km altitude and inlet air¯ow Mach number of 1.5. At the combustor symmetry line, radial velocity component and radial derivatives of all other dependent variables were set to zero. At the wall, no slip condition was employed. Computation of the nearwall region used the generalized wall functions of Ref. [13], taking into account the fuel gas velocity. The local mass ¯ow rate and the fuel gas velocity at the wall were determined by mass conservation between gas/solid phases at the fuel surface: Vg
r_ rf : rg
5
Basically, at steady state combustion the fuel regression rate can be calculated from r_
q_ w rf Hv, eff
6
where qÇ w is the heat ¯ux to the wall, rf is the condensed fuel density, and Hv,e is the eective heat of gasi®cation (or vaporization) of the fuel. It was assumed that the driving force in the heat transfer mechanism is the dierence between the ¯ame and wall temperatures: q_ w h Tfl ÿ Tw :
7
The solid fuel surface temperature was assumed to be approximately constant and was set to 800 K according to Refs. [14] and [15]. Flame temperature was determined as the maximum temperature at each axial location. The heat transfer coecient was determined using Stanton number: h St r W Cp
8
where Stanton number, St, was computed from skin friction coecient calculated by the wall functions. For the exit boundary, the conditions of constant static pressure (the recommended code outlet boundary condition) and zero gradient for all other dependent variables were used. The selection of the outlet pressure is described in Ref. [10]. 2.3. Chemical kinetics The simpli®ed 2-equation ethylene combustion model of Westbrook and Dryer [16] was applied: C2 H4 2O2 4 2CO 2H2 O
9
1 CO O2 $ CO2 : 2
10
Rate constants were adopted from Refs. [16] and [17]. The chemical reactions and the species production rates were calculated by the CHEMKIN
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chemical kinetics package combined with the PHOENICS code. 2.4. Numerical method For the solution of the described problem the PHOENICS Computational Fluid Dynamics (CFD) code was employed. The code is based on the SIMPLEST algorithm developed by Spalding and Patankar [18]. The validation of the PHOENICS code for supersonic ¯ow over a backward facing step is presented in Refs. [10] and [19]. The computational grid was non-uniform, with smaller and uniform cells from the wall and up to a distance of half step above the inlet step, and growing cells from there up to the symmetry line. The in¯uence of grid density was examined by the authors in their previous work [10] that investigated the same geometric con®guration as in the present case, and a 70 105 grid was found as most adequate for the purpose of the present investigation.
3. RESULTS AND DISCUSSION
3.1. Velocity ®eld Figures 2 and 3 present the ¯ow Mach number distribution of a non-reacting case and that of the present reacting case, respectively. Comparison reveals that the velocity ®eld changes substantially. The inlet fan at the step corner, which served to elevate the Mach number from 1.5 to a maximum value of 3.6 in the non-reacting case, almost disappears in the reacting case, resulting in a maximum Mach number of 2.1. Broad areas of supersonic ¯ow in the non-reacting case become subsonic due to the heat release from the chemical reactions. A mixed supersonic (at the center)/subsonic (at the circumference) jet is formed at the combustor exit when combustion takes place, replacing the almost whole supersonic non-reacting stream. 3.2. Temperature Temperature distribution of the reacting ¯ow®eld is presented in Fig. 4. The computation results reveal that sustained combustion can exist within the combustor for the inlet and geometry conditions under investigation. A diusion ¯ame with a maximum temperature of 2846 K is formed supplying a substantial heat addition to the ¯ow. The static and stagnation temperatures increase from 900 K and 1300 K at the inlet to mass-averaged values of 2189 K and 2494 K, respectively, at a distance of 18H from the inlet. The region between the wall and ¯ame center is heated too due to the chemical reactions and the lateral heat transfer from the ¯ame. At the combustor axis, the temperature remains almost unchanged.
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Fig. 2. Mach number map for supersonic ¯ow over a backward facing step.
Fig. 3. Reacting ¯ow®eld Mach number map (nominal case).
Fig. 4. Reacting ¯ow®eld temperature map (nominal case).
3.3. Flame characteristics Flame temperature and radial position change with axial distance. The change in radial position of the ¯ame is demonstrated in Fig. 5 for several crosssections along the combustor. Flame center was de®ned by the local maximum temperature. From Fig. 5 one can see that the further the axial distance
along the combustor the further the radial position of the ¯ame from the wall. This shift is explained by the diusive nature of the ¯ame, where the ¯ow dominates the ¯ame development and characteristics. The ¯ame temperature also varies along the combustor (Fig. 6), increasing continuously with the axial distance to a maximum ¯ame temperature of 2846 K close to the reattachment cross-section. In
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Fig. 5. Flame temperature distribution at various axial locations.
Fig. 6. Maximum ¯ame temperature vs axial location.
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the recirculation region the average temperature (at a 3-step-heights distance) is 2530 K. Downstream the reattachment point, the temperature is slightly decreasing with axial location, but remains high. 3.4. Chemical species Radial pro®les of the chemical species at axial locations of 3H, 10H and 18H are presented in Figs 7, 8 and 9, respectively. Oxygen mass fraction decreases from its maximum value at the centerline to almost zero beyond the ¯ame. This decrease is very steep at the recirculation region and becomes moderate further downstream. Similarly, fuel vapor fraction decreases gradually from the wall towards the ¯ame position. Beyond the ¯ame, no fuel vapors are apparent. From Fig. 9 one can see that although the fuel grain length is only 12H, there is still a substantial amount of unburned fuel vapors close to the exit (at 18H distance). The main combustion products are CO2 and H2O. Very low values of CO mass fraction are found (maximum value of 0.009), resulting from the high conversion rate of CO into CO2 in the operating conditions of near stochiometric fuel/air ratio. Higher CO fractions are expected for fuel-rich conditions or at higher temperatures.
The combustion eciency is de®ned here by the ratio between the reacted fuel mass ¯ow rate and the maximum value of fuel mass ¯ow rate that can react. The maximum value is the stoichiometric ¯ow rate for fuel-rich conditions, and the total entering fuel ¯ow rate for fuel-lean mixture conditions. For the nominal conditions, an equivalence ratio of 1.14 yielded combustion eciency of 70% at the combustor exit. Calculations reveal stagnation pressure losses of 60% resulting from the heat and mass addition to the ¯ow within the combustor. 3.6. Fuel regression rate Figure 10 presents the axial variation of the fuel regression rate. Along the recirculation region there is a gradual increase of the fuel regression rate from 0.5 mm/s near the inlet step up to 1.8 mm/s close to the reattachment point. A similar behavior is also typical to subsonic solid fuel ramjet combustors [20]. This may be expected since the recirculation region is subsonic for both subsonic and supersonic ¯ow combustors. Downstream of the reattachment, a higher and almost constant regression rate of 3 mm/s was calculated. In comparison, in subsonic ¯ow combustors the downstream section is characterized by slightly decreasing regression rates.
3.5. Combustor performance
3.7. Recirculation region
One of the ways to evaluate combustor performance is by computation of the combustion eciency.
Figure 11 illustrates the axial velocity variation at the ®rst near-wall grid point (0.1 mm). The reattach-
Fig. 7. Chemical species and temperature radial distribution at 3H distance from inlet.
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Fig. 8. Chemical species and temperature radial distribution at 10H distance from inlet.
Fig. 9. Chemical species and temperature radial distribution at 18H distance from inlet.
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Fig. 10. Axial variation at the fuel regression rate.
Fig. 11. Axial velocity variation at the ®rst near-wall grid point (0.1 mm).
Solid Fuel Scramjet
ment point at the end of the recirculation zone is obtained at an axial distance of 5.9 step heights. This is a much larger distance compared to the nonreacting case (1.4H ) [10]. A similar behavior was reported by Bussing and Merman [21] for a Mach 3 combustor. Combustion seems to have the opposite eect in subsonic ¯ow combustors, where heat addition causes shortening of the reattachment distance [22].
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4.2. Increased combustor inlet Mach number (1.75) For the same ¯ight conditions, increasing the combustor inlet Mach number from the nominal value of 1.5 to 1.75, implies higher velocity and lower static temperature at the combustor inlet. The computed combustion eciency (70%) and equivalence ratio (1.11) are similar to those of the nominal case, while local and exit Mach numbers are higher. 4.3. Increased ¯ight Mach number
4. PARAMETRIC STUDY
4.1. Shorter fuel grain Following the results of the nominal case, which indicate a substantial amount of unburned fuel gas, a shorter fuel grain (9H instead of 12H long) has been examined. The overall combustor length is kept constant, hence the average residence time of the fuel gas is increased. The total equivalence ratio reduces to a lean value of 0.7, while the combustion eciency reaches a higher value of 90%. Figure 12 presents the species radial pro®les for the shorter fuel grain case. For this case, lower values of fuel mass fraction are found near the exit. However, this con®guration has two disadvantages: ®rst, the total energy added to the ¯ow is reduced (exit calculated stagnation temperature reduces to 2389 K), and second, the volumetric loading fraction of the fuel decreases.
For the nominal combustor inlet Mach number of 1.5, increasing the ¯ight Mach number from the nominal value of 5 to 6 elevates the stagnation and static temperatures at the combustor inlet from 900 K and 1300 K to 1225 K and 1777 K, respectively. As a result, the whole temperature pro®le is raised. Figure 13 presents the radial temperature pro®les in the middle of the chamber (10H ) for three dierent cases. For the nominal case the radial location of the ¯ame center is shifted towards the fuel wall. The temperature pro®les of ¯ight Mach 5 cases (with Min=1.5 and 1.75) dier mainly around the combustor center, and almost unite close to the wall. Figure 14 shows the in¯uence of inlet conditions on the axial regression rate pro®les. The highest regression rate values were calculated for the ¯ight Mach 6 case, where both the inlet temperature and air mass ¯ux are higher than in the nominal case. The regression rate for the combustor inlet Mach 1.75 case exceeds the nominal case values as
Fig. 12. Chemical species and temperature radial distribution at z = 18H for a short (9H ) fuel grain.
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Fig. 13. Temperature radial distribution at z = 10H for various inlet conditions.
Fig. 14. Regression rate distribution for various inlet conditions.
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Fig. 15. Mach number map for a 0.5-cm-step-height combustor.
well due to the increased mass ¯ux. This behavior agrees with the known general dependence of regression rate on inlet mass ¯ux and temperature [6,9,15]. 4.4. Enlarged combustor A double-size combustor yields drastic changes in combustor behavior. The two-fold larger combustor could not sustain combustion, possibly due to insuf®cient mixing. Mixing in solid fuel combustors, where the added fuel velocity at the wall is very low, is diusion controlled, while distances relevant to mixing are proportional to the combustor diameter. Hence, larger combustors may have a reduced mixing capability than smaller ones. 4.5. Smaller step Reducing step height while keeping other dimensions (e.g. combustor diameter and length) unchanged, might have several qualitative in¯uences on the ¯ow pattern. First, the inlet/exit area ratio is reduced, hence the acceleration eect attributed to enlargement of the cross-section in supersonic ¯ow is reduced. This might cause choking of the ¯ow in the case that the cross-section enlargement is insucient to compensate for ¯ow deceleration due to heat and mass addition. Another important consideration is the ¯ame-holding aspect. For a particular geometry, fuel and inlet conditions, ¯ameholding capability is mainly dependent on step size. Reducing the step height below a critical value would result in combustor extinction. In the present case, step height was reduced to one-half of the nominal one. Figure 15 presents the ¯ow Mach number distribution of this case. Mach number drops from 1.5 at the inlet to 0.7 downstream of step as a result of thermal choking due to the heat addition in chemical reactions. In order to avoid choking situations special care should be taken during the design of supersonic combustors.
5. CONCLUSIONS
The feasibility of solid fuel combustion under supersonic cross¯ow conditions in SFRJ con®guration has been demonstrated numerically. A diusion ¯ame with a maximum temperature of 2846 K is established supplying signi®cant heat addition to the ¯ow. The resulting regression rate increases gradually from the inlet cross-section to the reattachment zone. Further downstream, a higher and approximately constant regression rate is demonstrated. The heat release from the chemical reaction reduces the local Mach number, changes the pressure ®eld and increases the ¯ow recirculation zone. The parametric investigation reveals that the operation of the solid fuel scramjet combustor is sensitive to the design parameters. Inappropriate con®guration might cause extinction of the ¯ame or choking of the ¯ow. Fuel regression rate, ¯ame characteristics and combustion eciency are all aected by geometrical and operating conditions. APPENDIXAPPENDIX
Nomenclature Cp D H Hv,e h h 0f k ` M qÇ w P Pk Pr r Çr S St T t V V
speci®c heat diusion coecient step height eective heat of vaporization speci®c enthalpy; heat transfer coecient speci®c heat of formation turbulence kinetic energy Prandtl mixing length Mach number heat ¯ux to the wall pressure turbulence kinetic energy generation rate Prandtl number radial coordinate regression rate source term Stanton number temperature time radial velocity velocity vector
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W axial velocity mass fraction of species i Yi z axial coordinate Greek e turbulence dissipation m viscosity f conserved property r density s Prandtl or Schmidt number o chemical source Subscripts e eective f fuel ¯ ¯ame g gas phase i ith chemical species in combustor inlet lam laminar t turbulent w wall REFERENCES
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9. Jarymowycz, T., Yang, V. and Kuo, K., A numerical study of solid fuel combustion under supersonic cross¯ows. In 20th Joint Propulsion Conference, Orlando, FL, 1990 (AIAA-90-2076). 10. Ben-Arosh, R., Natan, B., Spiegler, E. and Gany, A., Fuel-air mixing in solid fuel scramjet combustors. International Journal of Turbo and Jet Engines, 1998, 15, 223±234. 11. Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, 1980. 12. Bird, R. B., Stewart, E. W. and Lightfoot, E. N., Transport Phenomena. Wiley, New York, 1960. 13. Launder, B. E. and Spalding, D. B., The numerical computation of turbulent ¯ows. Computer Methods in Applied Mechanics and Engineering, 1974, 3, 269± 289. 14. Hadar, I. and Gany, A., Fuel regression mechanism in a solid fuel ramjet. Propellants, Explosives, Pyrotechnics, 1992, 17, 70±76. 15. Schulte, G., Pein, R. and Hogl, A., Temperature and concentration measurements in a solid fuel ramjet combustion chamber. Journal of Propulsion and Power, 1987, 3, 114±120. 16. Westbrook, C. K. and Dryer, F. L., Simpli®ed reaction mechanisms for the oxidation of hydrocarbon fuels in ¯ames. Combustion Science and Technology, 1981, 27, 31±43. 17. Dryer, F. L. and Glassman, I., High-temperature oxidation of CO and CH4. In Proceedings of the 14th Symposium (International) on Combustion, The Combustion Institute, 1972, pp. 987±1003. 18. Patankar, S. V. and Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic ¯ows. International Journal of Heat and Mass Transfer, 1972, 15, 1787. 19. Halupovich, Y., Natan, B. and Rom, J., Numerical solution of the turbulent supersonic ¯ow over a backward facing step. Fluid Dynamics Research, 1999, 24, 251±273. 20. Zvuloni, R., Gany, A. and Levy, Y., Geometric eects on the combustion in solid fuel ramjets. Journal of Propulsion and Power, 1989, 5, 32±37. 21. Bussing, T. R. A. and Murman, E. M., Numerical investigation of 2-dimensional H2-Air ¯ameholding over ramps and rearward facing steps. In AIAA/ASME/ SAE/ASEE 21st Joint Propulsion Conference, Monterey, CA, 1985 (AIAA 85-1250). 22. Pits, W. and Daily, J. W., Combustion in a turbulent mixing layer formed at a rearward-facing step. AIAA Journal, 1983, 21, 1565±1570.