Evaluation of oxidizer temperature drop in a combustion chamber

4th International Conference on Launcher Technology "Space Launcher Liquid Propulsion" 3-6 December 2002 – Liege (Belgium)

EVALUATION OF OXIDIZER TEMPERATURE DROP IN A COMBUSTION CHAMBER

R. Schmehl and J. Steelant

ESTEC/ESA Section of Aerothermodynamics Keplerlaan 1 2200 AG Noordwijk The Netherlands Phone: +31 71 565 3868 Fax: +31 71 565 5421 e-mail: [email protected]

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4 International Conference on Launcher Technology "Space Launcher Liquid Propulsion" 3-6 December 2002 - Liege (Belgium)

1

Introduction

The typical gas flow in a Laval nozzle operating at low back pressures and supersonic outflow is characterized by a steep downstream drop in pressure and temperature. In case of a two-phase flow generated exclusively from spray evaporation and excluding additional energy sources like combustion, the initial temperature will be close to the boiling temperature at the corresponding chamber pressure. On the other hand, the downstream drop of pressure and temperature imposed by the nozzle flow leads to partial re-condensation of vapor in conjunction with a local temperature increase. This is the scenario characterizing the oxidizer preflow in the upper stage engine Aestus, which is maintained prior to the injection of Monomethylhydrazine (MMH) and hypergolic ignition. Due to the low boiling temperature (T b ≈ 262 K) of Nitrogen Tetroxide (NTO) in the preflow phase (p ≈ 20 kPa), the injected liquid (T l ≈ 300 K) is metastable resulting in flash atomization/evaporation and intensified mass and energy transfer rates. However, it can be derived from energetic considerations that only a small fraction (α ≈ 14%) of the injected liquid NTO can be vaporized in the combustion chamber. Part of the remaining spray is deposited on the side walls and possibly on the base plate of the chamber. Both effects, the temperature drop and the local accumulation of liquid oxidizer, might have a significant influence on the ignition characteristics of the engine and thus play a role in combustion instabilities.. Locally resolved measurements of the preflow in the chamber are difficult due to the stringent access restrictions. Based on a 3d CFD simulation of the steady two-phase flow field in chamber and nozzle, the present study provides detailed data on the conditions prior to ignition of the engine.

2

General approach

To account for compressible effects in the vapor flow ranging from subsonic up to supersonic flow regimes, a "compressible" pressure correction scheme (Karki and Patankar, 1989) is used in conjunction with a body-fitted Finite-Volume discretization of the Reynolds-averaged transport equations for mass moment and energy. The standard k-ε model is included by considering transport of the turbulent kinetic energy k and its dissipation rate ε. The description of the disperse liquid phase is based on Lagrangian droplet tracking and spatial evaluation of disperse phase properties. The effect of vapor flow turbulence on spray dispersion is accounted for by stochastic sampling of velocity fluctuations along the individual droplet trajectories (Gosman and Ioannides, 1983). Additional submodels are employed for liquid atomization, droplet evaporation and heterogeneous condensation as well as secondary breakup and wall-interaction. Accumulation of liquid on the chamber walls is taken into account. However, film transport and re-evaporation of the film is not considered at present. Due to high rates of mass, momentum and energy transfer between vapor and spray, two-way coupling of the numerical descriptions is of particular importance. It is achieved by means of iterative solution procedure updating spray source term distributions prior to the solution of the vapor flow field. A detailed description of this procedure and additional information on the physical modeling framework is given in Schmehl et al. (1998). Not considered are effects of homogeneous condensation and dissociation of NTO in liquid and vapor phase.

3

Droplet evaporation and condensation model

A fundamental element of the present study is an extension of the well-known D 2 -theory to droplet evaporation and condensation in a pure vapor atmosphere, including effects like flash evaporation of superheated droplets and spontaneous condensation on supercooled droplets. The analytical model is based on the initial assumption of spherical symmetric, quasi-steady transport in the vapor phase around the droplet. In this case, the total energy flux Q˙ and vapor mass flux m ˙ vap leaving the droplet are constant with respect to radial position. Q˙ can be related to any spherical differential element in the vapor as well as to the total change of droplet enthalpy H d (see also figure 1) dT d dHd dT =m ˙ vap hl (T b ) − md c p,l =− , Q˙ = m ˙ vap h(T ) − 4πrd2 λ dr dt dt

rd ≤ r < ∞ .

(1)

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Tg > Tb 1a : T d < T b , 1b : T d > T b , 1∞: T d = T b ,

superheated vapor heat-up, flash evaporation, evaporation,

dT d /dt > 0, T s < T b , m ˙ vap = 0 dT d /dt < 0, T s = T b , m ˙ vap > 0 dT d /dt = 0, T s = T b , m ˙ vap > 0

T g < T b supersaturated vapor 2a : T d < T b , heat-up + condensation, 2b : T d > T b , 1: evaporation, 2: no mass transfer, 3: condensation, 2∞: T d = T b , condensation,

dT d /dt > 0, dT d /dt < 0, dT d /dt < 0, dT d /dt < 0, dT d /dt = 0,

T s = Tb , T s = Tb , T s = Tb , T s = Tb , T s = Tb ,

m ˙ vap < 0 m ˙ vap > 0 m ˙ vap = 0 m ˙ vap < 0 m ˙ vap < 0

Table 1: Interphase transfer regimes based on instantaneous temperature conditions Solution of the energy balance requires a further closure assumption. The classical D 2 -theory - cases 1a and 1∞ in table 1 - is retrieved by dividing the droplet lifetime into a heat-up and a evaporation phase and integrating equation (1) from r = rd to r → ∞. All other cases require a modeling of the energy flux Q˙ i from the interior of the droplet to its surface, which for m ˙ vap , 0 is at the boiling temperature T b . Using dT d Q˙ i = 4πrd2 α s (T d − T b ) = −md c p,l dt

and

m ˙ vap, f =

Q˙ i ∆hv (T b )

(2)

in order to define the vaporization rate at pure flash evaporation, equation (1) can be integrated resulting in an implicit expression for the total vaporization rate     c p,re f (T g − T b ) λ B re f   T 2  with BT = . (3) ln 1 + m ˙ vap = 4πrd  , m ˙ vap, f  c p,re f ∆hv (T b ) 1− m ˙ vap

The reference values λre f and c p,re f are evaluated according to the 1/3-rule at the temperature T re f = 2/3T b + 1/3T g (Sparrow and Gregg, 1958). 60000

50000

m ˙ vap

Tg PSfrag replacements m ˙ vap h(T )

Q˙ i Td Ts

4πrd2 λ

dT dr

40000

αs

replacements ∆T = T d − T b α α s,min s: Dd = 5 µm Dd = 10 µm Dd = 100 µm

m ˙ vap m ˙ vap h(T ) dT 4πrd2 λ dr Td Ts Tg Q˙ i

30000

20000

α s,min : Dd = 5 µm

10000

0

Dd = 10 µm Dd = 100 µm 0

10

20

30

40

50

∆T = T d − T b

Figure 1: Mass and energy transfer model (left) and empirical heat transfer coefficient (right) Using an empirical heat transfer coefficient α s in equation (2) it is possible to cover a wide range of physical transport processes within the droplet for different levels of superheat ∆T : Pure heat conduction in the liquid for 2

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small ∆T up to intensive mixing at large ∆T due to internal cavitation. Zuo et al. (2001) suggest a correlation which is based on the IC engine fuel spray data by Adachi et al. (1997)   760 ∆T 0.26 , 0 ≤ ∆T ≤ 5 ,    2.33 , αs =  27 ∆T 5 < ∆T ≤ 25 , (4)    13800 ∆T 0.39 , 25 < ∆T

and illustrated in figure 1 (right). In the present study, α s is limited by a size-dependent minimum value α s,min = λl /rd , which is an order of magnitude estimate for the effect of internal heat conduction except for the initial flashing phase. As a first approximation, this estimate is also used to cover negative values of ∆T when Tg < Tb . In a final step, the effect of forced convection due to relative movement between droplet and vapor are accounted for by means of an empirical correction of the vaporization rate m ˙ ∗vap = cfh m ˙ vap ,

with

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1

cfh = 1 + 0.276 Re 2 Pr 3 .

(5)

Except for the heat-up of a droplet at T g > T b , the extended flash-evaporation and condensation model is described by the following Lagrangian derivatives dmd = −m ˙ ∗vap dt

2

and

α s 4πrd dT d =− (T d − T b ) . dt md c p,l

(6)

For each time step of the numerical integration scheme, equation (3) has to be solved iteratively. Better suited in this respect is a formulation in terms of a relative rate presented by Zuo et al. (2001) " ! ! # m ˙ vap, f m ˙ vap, f −1 2 λre f ln 1 + 1 + 1+ BT . (7) m ˙ vap,c = m ˙ vap − m ˙ vap, f = 4πrd c p,re f m ˙ vap,c m ˙ vap,c Here, m ˙ vap,c represents the part of the vaporization rate attributed to heat conduction within the vapor phase.

4

Single droplet in quiescent vapor atmosphere

To demonstrate the versatility of the modelling framework, mass and energy transfer of a single droplet in an infinite quiescent vapor atmosphere is computed and assessed. The initial droplet size is 100 µm, temperature and pressure of the gas phase are constant within each computation (p g = 20 kPa except for case 2a where pg = 50 kPa). The results of the numerical integration of equations (6) are summarized in figure 2. Case 1a/1∞ reproduces the characteristic evolution of droplet temperature and diameter in high temperature atmospheres as predicted by classical D 2 -theory. The increase of droplet diameter in the heat-up phase can be attributed to the decrease of liquid density with increasing temperature. A slight overshoot in droplet temperature due to the limited stepsize of the Runge-Kutta-Fehlberg solver is automatically corrected by the model applying a tiny amount of flash evaporation. The other extreme, flash-evaporation from ∆T = 38 K is depicted in case 1b/1∞. Due to the high mass transfer rate, droplet size spontaneously reduces to 95 µm, which corresponds to a mass loss of ∆md = 15% within a very short time frame ∆t = 2 ms. However, after this initial phase of large superheat, the temperature and diameter trajectories are rapidly converging towards the steady-state solution of D2 -theory. Case 2a/2∞ illustrates the condensation of a metastable vapor phase on a cool droplet. Initially, spontaneous condensation on the cool droplet surface will increase the temperature of a very thin surface layer up to the boiling temperature. In the absence of convective mixing processes due to cavitation, the energy transport into the droplet is controlled by heat conduction and thus much slower compared to the flash-evaporation regime. It should however be noted, that the description of Q˙ i in this case is solely based on the admittedly crude estimate for α s,min . The other extreme, case 2b/2∞, presents the initial flash-evaporation of a superheated droplet followed by condensation of the metastable vapor phase on the droplet surface. The point of transition from evaporation to condensation takes place when the energy flux Q˙ i provided by the droplet interior equals the conductive energy loss to the low temperature vapor phase (T s = T b > T g ). 3

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300

replacements

260

85

80 0.2

250

D [µm]

85

0

270

Case 1a/1∞: heat-up → evaporation Case 2601b/1∞: flash-evaporation → evaporation Case 2a/2∞: heat-up 250 & condensation → condensation 0

0.05

0.1

0.15

t

0.05

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80 0.2

t 105

95

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T

Td Tb Tg Dd m ˙ vap

280

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PSfrag replacements 100

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Case 2b/2∞: flash-evaporation → condensation

105

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250 0

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D [µm]

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Case 2b/2∞: 0 0.05 0.1 0.15 t flash-evaporation → condensation Case 1b/1∞: flash-evaporation → evaporation

290

T

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Case 1a/1∞: heat-up → evaporation Case 2601b/1∞: flash-evaporation → evaporation

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Td Tb Tg Dd ˙ vap 90m 95

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T


PSfrag replacements 100

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D [µm]

replacements

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D [µm]

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Case 2a/2∞: heat-up & condensation → condensation

T

Case 1a/1∞: heat-up → evaporation

80 0.2

t

Figure 2: Computed droplet behavior in different interphase transfer regimes (see also table 1) Both cases of superheated droplets, 1b and 2b, are encountered in the course of iteratively coupled two-phase flow simulations at saturated vapor conditions. Even if the converged steady-state solution of the flow field will locally be characterized by T g ≈ T b , source term underrelaxation techniques will introduce larger oscillations of vapor temperature.

5

Single droplet in nozzle flow

The objective of the preceding section was a systematic assessment of the modeling framework for temperature and pressure levels expected in the oxidizer preflow phase during start-up. In a next step, evaporation and condensation of individual droplets in the 3d vapor flow field will be investigated. The focus now will be on (1) the computational robustness of the model under the extreme conditions imposed by the transsonic vapor flow in combustion chamber and nozzle and (2) the influence of the initial size of the droplets which are injected at the base plate. At present, two-way coupling is not considered.

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The vapor flow field is based on a single-block discretization of a 180 o section of combustion chamber and nozzle by 29580 volume elements. In absence of spray source terms in the vapor phase transport equations, the nozzle flow has to be enforced by an hypothetic inflow of vapor at the base plate. Thus, total temperature and pressure are prescribed in such a way, that a chocked flow of saturated vapor is maintained in the chamber at the measured pressure p = 20 kPa with a total vapor mass flow rate defined as a fraction of the total liquid ˙ NT O,g = 0.14 M˙ NT O . Using the "compressible" pressure correction scheme, a converged flow field is flow rate M obtained after 1500 solver iterations. Three droplet sizes are considered: D d,0 = 25 µm, 100 µm and 400 µm. The initial velocity is assumed to be close to the injection velocity of the liquid oxidizer, v l ≈ 20m/s, a value which can be derived from the internal ˙ NT O . The initial temperature of the droplets is T d,0 = 300 K. The injector geometry and total mass flow M results of the trajectory integrations are illustrated in figures 3, 4 and 5. After scaling of Dd and m ˙ vap the 300

25

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T

260 22 240

Td Tb Tg Dd m ˙ vap pg

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D [µm]

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replacements

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x Figure 3: Trajectory data for Dd,0 = 25 µm

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replacements T

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75 70 65

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x Figure 4: Trajectory data for Dd,0 = 100 µm

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4 International Conference on Launcher Technology "Space Launcher Liquid Propulsion" 3-6 December 2002 - Liege (Belgium) 300

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x Figure 5: Trajectory data for Dd,0 = 400 µm interphase transfer behavior can be divided in three characteristic phases which are similar for all three droplet sizes: (1) Violent flash-evaporation in the atomizer near field, (2) asymptotic approach of thermal equilibrium in the chamber flow and (3) further acceleration and evaporation of the droplets in the nozzle. This last phase will be affected significantly by two-way coupling of spray and vapor and by homogeneous condensation of supersaturated vapor. It is at this point not further considered but to emphasize, that the oscillatory behavior of the vaporization rate m ˙ vap is an amplified reaction to the stepwise decrease of vapor phase temperature and pressure. The latter is a result of an insufficient spatial discretization in the supersonic flow regime in order to resolve the steep gradients in flow properties. The major effect of droplet size is on the dynamic behavior of the droplet, i.e. how quick it reacts to changes in the boundary conditions imposed by the vapor flow: The 25 µm-droplet rapidly evaporates 33% of its original mass within a fraction of the total chamber length (47% up to the nozzle exit). The 100 µm-droplet has lost 45% of its mass at the nozzle throat where it still shows a slight ∆T = 5 K of superheat (71% lost at the exit). Finally, the 400 µm-droplet passes the throat flash-evaporating at ∆T = 10 K with 60% mass evaporated (89% at the nozzle exit). Irrespectively of their size, the droplets further evaporate in the supersonic part of the nozzle flow. Driving force is the steep drop of pressure accompanied by a significant decline in boiling temperature. Conversely, the saturated vapor is reacting to the pressure drop by homogeneous condensation. This discontinous phenomenom is not taken into account in the present computations but has been investigated in detail in a previous study (Mazoué, 1996). Thus, the thermal inertia of droplets entering the supersonic flow results in flash-evaporation, whereas at the same time the supersaturated vapor adapts to the local flow conditions by spontaneous condensation. It should further be noted, that computations with activated deformation and secondary breakup modelling indicated, that only the 25 µm-droplet actually passes the throat. The large droplets are destroyed by the action of aerodynamic forces in the converging nozzle flow. This phenomenum greatly enhances interphase transfer and mixing by drastically reducing the size spectrum of the two-phase flow.

6

Conclusions

During start-up of an upper-stage rocket engine, the low pressure in the combustion chamber causes flashatomization and flash-evaporation of the injected liquid oxidizer. Therefore, the numerical description of droplet evaporation has been extended to account for increased interphase transfer rates in this regime. The

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same modelling framework has been employed to describe vapor condensation on droplets. In this way, the physical capabilities of the model have been extended to two-phase flows in saturated as well as supersaturated, metastable vapor flows. Computational results for single droplets in the nozzle flow clearly indicate a rapid decrease in droplet sizes and a correspondingly large sink of local enthalpy values closely downstream of the injection point. As a consequence, iterative coupling between vapor flow and spray proves to be problematic. This holds in particular for the realistic configuration with a closed-off vapor inlet at the base plate because interphase transfer then represents the exclusive source of mass, momentum and energy to maintain a high speed flow in the nozzle. The intrinsic difficulty is the decoupling of spray and vapor phase into separate numerical schemes with limited means of information exchange (Schmehl et al., 1998). Thus, future actions will address advanced methodologies for iterative solution of strongly coupled two-phase flows in order to evaluate the local temperature and pressure distribution in a combustion chamber of an upperstage engine.

Acknowledgements The authors would like to acknowledge the interest and support of ESA for this research activity which is conducted at ESTEC in the frame of an internal research fellowship.

Nomenclature αs BT cp cfh Dd h, hl ∆hv λ µ md m ˙ vap ˙ M Pr = µre f c p,re f /λre f Re = vrel Dd ρg /µre f t ∆T = T d − T b Td Ts Q˙

J/(sm2 K) J/(kgK) m J/kg J/kg W/(mK) kg/(ms) kg kg/s kg/s s K K K J/s

heat transfer coefficient droplet interior to surface Spaldings heat transfer number specific heat capacity correction factor to account for forced convetion droplet diameter specific enthalpy gas, liquid enthalpy of vaporization thermal conductivity dynamic viscosity droplet mass vaporization rate mass flow rate Prandtl number Reynolds number time degree of droplet superheat thermal mean temperature of droplet surface temperature total heat flux

References Adachi, M., V. G. McDonell, D. Tanaka, J. Senda, and H. Fujimoto (1997). Characterization of fuel vapor concentration inside a flash boiling spray. SAE Paper 970871. Gosman, A. D. and E. Ioannides (1983). Aspects of Computer Simulation of Liquid-Fueled Combustors. Journal of Energy 7(6), 482–490. Karki, K. and S. Patankar (1989). Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA-Journal 27, 1167–1174.

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Mazoué, F. (1996, Jan.). Ariane 5 EPS Venting Study: Numerical Study of Condensation During the Venting of Ariane 5 Upper Stage Propellant Tanks. Report, European Space Agency. Schmehl, R., G. Klose, G. Maier, and S. Wittig (1998). Efficient Numerical Calculation of Evaporating Sprays in Combustion Chamber Flows. In RTO-MP-14, pp. 51.1–51.13. Sparrow, E. M. and J. L. Gregg (1958). The variable fluid property problem in free convection. Transactions of the ASME 80, 879–886. Zuo, B., A. Gomes, and C. J. Rutland (2001). Studies of superheated fuel spray structures and vaporization in GDI engines. 11th International Multidimensional Engine Modeling User’s Group Meeting, http://www.erc.wisc.edu.

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