Isolated fluid oxygen drop behavior in fluid hydrogen at rocket chamber pressures

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Abstract

A model has been developed for the behavior of an isolated fluid drop of a single compound immersed into another compound in finite, quiescent surroundings at supercritical conditions. The model is based upon fluctuation theory which accounts for both Soret and Dufour effects in the calculation of the transport matrix relating molar and heat fluxes to the transport properties and the thermodynamic variables. The transport properties have been modeled over a wide range of pressure and temperature variation applicable to LOx–H2 conditions in rocket chambers, and the form of the chemical potentials is valid for a general fluid. The equations of state have been calculated using a previously-derived, computationally-efficient and accurate protocol. Results obtained for the LOx–H2 system show that the supercritical behavior is essentially one of diffusion. The temperature profile relaxes fastest followed by the density and lastly by the mass fraction profile. An effective Lewis number calculated using theory derived elsewhere shows that it is larger by approximately a factor of 40 than the traditional Lewis number. The parametric variations show that gradients increasingly persist with increasing fluid drop size or pressure, and with decreasing temperature. The implication of these results upon accurate measurements of fluid drop size under supercritical conditions is discussed.

Introduction

Liquid rocket engine design is not a mature technology in that the issues of reliability and efficiency are unresolved. Current designs are still based upon empirical knowledge and theory that does not portray the complexities of the physical processes and of the environment in the combustion chambers. The extensive review on liquid propellant rocket instabilities compiled by Harrje and Reardon [1] more than twenty years ago remains the base of rocket design despite the increased understanding that many of the approximations made in performing the calculations compromise the validity of the results.

One of the foundations of liquid rocket instabilities is the theory of isolated drop evaporation and combustion in an infinite medium 1, 2 . The early version of that theory was based on the assumption of quasi-steady gas behavior with respect to the liquid phase, an assumption strictly valid only at low pressures where the liquid density is three orders of magnitude larger than that of the gas. Recognizing that at the elevated pressures of liquid rocket chambers the liquid density approaches that of the gas, the quasi-steady assumption was relaxed in other investigations 3, 4, 5, 6, 7 . However, it is only recently that the description of the full complexity of combustion chambers processes was undertaken ; this includes not only the complete unsteady treatment of the conservation equations but also appropriate equations of state with consistent mixing rules and transport properties valid over transcritical\supercritical conditions.

The discussion below emphasizes the LOx system because its behavior sometimes contrasts with that observed for hydrocarbons as noted by Chesnau et al. [8] . Therefore, the results of many interesting studies in the context of Diesel engines or high-pressure gas turbine engines will not be discussed. Such a discussion appears in the review of Givler and Abraham [9] .

Yang et al. [10] described the vaporization of a liquid oxygen drop (LOx) in hydrogen over a wide range of pressures. The model includes thermodynamic nonidealities, variable properties and the assumption of liquid–vapor equilibria. The Soret term accounting for species transport due to thermal gradients and the Dufour term accounting for heat transfer resulting from concentration gradients are neglected. In the side of the drop boundary containing hydrogen, the Soave–Redlich–Kwong (SRK) equation of state (EOS) is used to calculate the compressibility factor while the density of fluid in the LOx side of the boundary is given by curve-fitted experimental data. The thermal conductivities and heat capacities are correlated as functions of density and temperature, the liquid diffusivity is estimated following Scheibel’s [11] procedure, and the binary diffusion coefficients are calculated following the corresponding states method of Takahashi [12]. Calculations are initiated with a drop at subcritical temperature in surroundings where the pressure and temperature are both above the critical point ; once the drop surface temperature reaches the critical point, the assumption is made that it is this surface that continues to define a gas\liquid interfacial boundary and thus, it is its motion that defines the drop regression rate. This surface tracks thus the progression of a liquid entity composed of both LOx and hydrogen in contrast to the pure LOx drop. Plots of the temperature, mole fraction and density all show steep gradients around what is presumably the interface (the interfacial boundary is not indicated on the curves) . Because the radial coordinate is logarithmic, it is impossible to compare the detailed evolution of the heat vs. the mass transfer ; however, plots of the Lewis number show that it reaches values as large as 1.9 on the hydrogen side and 1.3 in the LOx side. The results show the increasing importance of hydrogen diffusion into LOx and the increase in vaporization rate with increasing pressure. Surprisingly, the results show that the classical, quasi-steady-derived D2-law [2] remains valid for all pressures (5–250 atm) and drop diameters (5×10−6– 300×10−6 m) . Hsiao et al. [13] extended the study of [10] to include convective effects. In this new version, the Benedict–Webb–Rubin (BWR) EOS in conjunction with an extended corresponding states principle, replaces the SRK EOS for the purpose of achieving higher-accuracy density predictions and the corresponding states principle is applied to calculate transport properties. Results from the calculation show that the interface defined by the critical temperature deforms and stretches throughout the drop lifetime presenting a convex surface to the incoming flow. The motion of the drop is defined by the motion of its center of gravity and a drag coefficient is calculated.

The model of Delplanaque and Sirignano [14] is similar to that of [10] in that Soret and Dufour effects are neglected and phase equilibrium is assumed ; additionally, since the calculations are not pursued beyond the drop surface reaching the critical point, the mixture is approximated by a gas with mixture-averaged properties. The Chueh and Prausnitz [15] version of the Redlich–Kwong (RK) EOS is used and the thermophysical and transport properties are correlated from data with no correction for real gas behavior. The diffusion coefficients are calculated using dilute gas theory and it is uncertain whether the discussed high-pressure effects [16] are included. The liquid density is evaluated using the Hankinson–Brobst–Thomson method [16] and the isobaric heat capacity is obtained through an ideal gas correlation [16] . Since there is a unique relationship between heat capacities and the enthalpy as expressed by the EOS, it is not clear that this thermodynamic relationship is satisfied by this procedure. Results from the calculations show that the LOx drop surface reaches the critical temperature in a time much smaller than the characteristic thermal diffusion time.

Morerecently, Haldenwang et al. [17] constructed a model similar to those in [10] and [14] , but identified the drop surface with the location where thermodynamic equilibrium occurs. The RKS EOS is used in the model and Cp is estimated independently of the EOS, possibly introducing inconsistencies in the thermodynamics. The binary diffusion coefficients were calculated through empirical correlations [16] in liquids and dense gas, and the thermal conductivity was calculated using standard mixing rules and experimental data for pure substances. In this study, the subcritical regime is defined as that for which the mass fraction at the surface remains higher than that corresponding to the value for which the surface temperature is the critical temperature during the entire drop lifetime. With this definition, the evolution of a LOx drop initially at 100 K in surroundings at 1000 K and 8 MPa is classified as subcritical behavior although obviously there is no material drop surface at those conditions. Calculations performed with different thermodynamic properties yield results qualitatively similar but quantitatively different. Just as in [10] , the D2-law remains valid both in the subcritical and supercritical regimes. Comparisons with the microgravity observations of Sato [18] for n-octane show similar trends : the drop lifetime decreases with increasing pressure in the subcritical regime and increases with increasing pressure in the supercritical regime. To evaluate the impact of the assumed interface location (at the critical temperature [10] vs. the saturation temperature [17] ) , results are presented with the model of [17] using each assumption, and they are further compared with those of [10] ; although there is qualitative agreement between the drop lifetime vs. the reduced pressure curves found by the authors using each assumption, their variation is not similar to that of [10] . A further detailed evaluation of the influence of the interface definition reveals that, in contrast to the situation when the interface is located at the critical point, an increase in radius is obtained in the transcritical regime when the surface is assumed to be at thermodynamic equilibrium. This effect is attributed to the increase in the difference between equilibrium mass fraction and critical mass fraction with increasing pressure. The obvious conclusion from this study is that it is not only the conservation equations, EOS, and transport properties that must be accurately modeled, but also the interface processes.

Haldenwanget al. [17] also note that the results of [10] and [14] for identical conditions have more than one order of magnitude discrepancy, and that their own results do not agree with those of either one of these studies ; this indicates that it is only through comparisons with experiments that the cause of the difference could be resolved. However, experiments with LOx–hydrogen combinations are prohibitively expensive because of the associated safety aspect. Additionally, the interpretation of the results is not straightforward as will be discussed below.

Circumventing the difficulty of LOx in hydrogen and that of very high pressures, Chesnau et al. [8] present a set of experiments for LOx evaporation in air, nitrogen and helium at 0.1 MPa and 3 MPa. Since the data was acquired through imaging, the drop surface is a measure of steep density gradients. The data shows that at 0.1 MPa, the D2-law is validated within the range of experimental error. In contrast, at 2 MPa the slope of D2 changes with time, which is documented as an unsteadiness in the evaporation constant ; this unsteadiness increases with increasing pressure. These observations seem to disprove the previous theoretical predictions of the D2-law holding over the entire subcritical range. More recently, Chauveau and Gökalp [19] acquired data for LOx in helium at up to 65 bars under gravity conditions. The drop was suspended from a fiber, indicating that it was not pure LOx which becomes supercritical (and thus, has zero surface tension) at 5.043 MPa. Both sets of data show that the evaporation rate increases with increasing pressure, unlike the data of Sato et al. [18] for n-octane and that of Chauveau et al. 20, 21 for n-heptane and methanol showing a minimum in the evaporation rate at the critical point. Chesnau et al. [8] attempt to explain the difference between the variation of the evaporation rate of LOx and hydrocarbons with pressure on the basis of the reduced temperature in the drops surroundings which in their experiments was supercritical for LOx but subcritical for the hydrocarbons. This explanation is not convincing since the data of Sato et al. [18] was acquired for burning n-octane drops and therefore at supercritical surrounding temperature since the critical temperature of n-octane is 570 K. Therefore, the qualitative agreement between Haldenwang et al.’s drop lifetime variation with pressure for LOx [17] and the observations of Sato et al. [18] is suspicious and may indicate a flaw in the model rather than a validation of the model.

The above discussion shows that there is still a wide gap between the current modeling capability of LOx drops in hydrogen and that necessary for advancing the state of the art in liquid rocket motor design. The model presented below constitutes an improvement over the existing models in that it includes Soret and Dufour effects, an accurate form of the chemical potentials valid for general fluids, and thermodynamic nonequilibrium between phases, none of which were included in 17, 10, 13, 14 . This set of conservation equations and a kinetic law (equivalent to the Langmuir–Knudsen law that is valid for liquid evaporation) derived here for dense gas are coupled with accurate EOSs and transport coefficients over the subcritical\supercritical range for both LOx and hydrogen. The model used for the EOSs has been described in detail elsewhere [22] ; these EOSs are obtained by curve fitting data and further extrapolation using the concept of departure function [16] . The importance of Soret and Dufour effects is the subject of discussions in Harstad and Bellan [23] while the thermodynamic equilibrium assumption is shown by Bellan and Summerfield [24] to be unrealistic in certain situations even under subcritical conditions.

Section snippets

Model

Themodel of the conservation equations is based on the fluctuation theory of Keizer [25] , also described by Peacock-Lopez and Woodhouse [26] . The advantage of this theory is that it inherently accounts for nonequilibrium processes and naturally leads to the most general fluid equations by relating the partial molar fluxes, Ji, and the heat flux, q, to thermodynamic quantities.

Numerical method

Asmentioned above, Ma 1, and thus the pressure is calculated as p (r, t) = p (t) +p′ (r, t) where p (t) is specified and p′ (r, t) is a small perturbation calculated from the momentum equation.

The equations are recast in a convenient form for numerical analysis as follows : the density derivatives in Eq. (2).1) are replaced using the relationshipdlnρ=−αvdT+κTdp+1N(mj\m−vj\v)dXjwhere κT=−(1\v)(∂v\∂p)T,Xj is the isothermal compressibility. Combining Eq. (3).1) , (2.1) , (2.10) and (2.11) yields

Results

In order to better understand the trends predicted by this model, we first analyze a baseline calculation and then present a parametric study. During all of our calculations, we monitor the number of phases at each time step and at all locations, for the conditions of this study, only one phase could be found at all times and locations presumably due to the high pressures used here and the peculiarity of the LOx–H2 mixture critical locus. Thus, Eq. (2).16) happens to play no role, although

Conclusions

A model of an isolated fluid drop in quiescent, finite spatial surroundings has been derived using the formalism of fluctuation theory. The model presented here is derived from first principles and incorporates all physical aspects of high pressure behavior including Soret and Dufour effects, high pressure mixture-thermodynamics and mixture transport properties over a wide range of pressures and temperatures.

Results obtained for the LOx–H2 system show that the supercritical behavior is that of

Acknowledgements

This research was conducted at the Jet Propulsion Laboratory under sponsorship from the National Aeronautics and Space Administration, the George C. Marshall Space Flight Center with Mr Klaus W. Gross as technical contract monitor. His continuing interest and support are greatly appreciated.

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