Chemical Non-equilibrium Model Research Articles

Numerical investigation of thermochemical non-equilibrium effects in Mach 10 scramjet nozzle

Abstract High-temperature non-equilibrium effects are prominent in scramjet nozzle flows at high Mach numbers. Hence, the thermochemical non-equilibrium gas model incorporating the vibrational relaxation process of molecules in the hydrocarbon-air reaction is developed to numerically simulate the flow of a hydrocarbon fuel scramjet nozzle at Mach 10. Besides, the results computed by the models of the thermally perfect gas, chemically non-equilibrium gas, and thermally non-equilibrium chemically frozen gas are applied for comparative studies. Results indicate that chemical non-equilibrium effects are more significant for the flow-field structure and parameters compared to thermal non-equilibrium effects. Meanwhile, vibrational relaxation and chemical reactions interact in the flow-field. The heat released from the chemical reactions in the flow-field of the thermochemical non-equilibrium gas model makes the thermal non-equilibrium effects weaker compared to the thermally non-equilibrium chemically frozen gas model; the chemical reactions in the thermochemical non-equilibrium gas model are more intense than in the chemically non-equilibrium gas model. Due to the slow relaxation of vibrational energy, the thermal non-equilibrium models predicted nozzle thrust lower than the thermal equilibrium models by approximately 1.11% to 1.33%; when considering the chemical reactions, the chemical non-equilibrium models predicted nozzle thrust higher than the chemical frozen models by approximately 7.30% to 7.54%. Hence, the structural design and performance study of the high Mach numbers scramjet nozzle must consider thermochemical non-equilibrium effects.

Transport and retention of ciprofloxacin with presence of multi-walled carbon nanotubes in the saturated porous media: impacts of ionic strength and cation types.

The environmental fate and risks of ciprofloxacin (CIP) in the subsurface have raised intensive concerns. Herein, the transport behaviors of CIP in both saturated quartz sand and sand/multi-walled carbon nanotubes (MWCNTs) mixtures under different solution ionic strength of the solution and coexisting cation types were investigated. Batch adsorption experiments highlighted growing adsorptive capacity for CIP with the increasing content of MWCNTs in the MWCNTs-quartz sand mixtures (from 0.5% to 1.5%, w/w). Breakthrough curves (BTCs) of CIP in the MWCNTs-quartz sand mixtures were well fitted by the two-site chemical nonequilibrium model (R2 > 0.833). The estimated retardation factors for CIP increased from 9.68 to 282 with growing content of MWCNTs in the sand column, suggesting the presence of MWCNTs significantly inhibited the transport of CIP in saturated porous media. Moreover, the values of retardation factors are negatively correlated with the ionic strength and higher ionic strength could facilitate the transport of CIP in the saturated porous media. Compared with monovalent cations (Na+), the presence of divalent cations (Ca2+) significantly facilitated the transport of CIP in the columns due to the complexation between CIP and Ca2+ as well as deposition of MWCNTs aggregates on the sand surface. Results regarding CIP retention in columns indicated that MWCNTs could enhance the accumulation of CIP in the layers close to the influent of sand columns, while they could hinder upward transport of CIP to the effluent. This study improves our understanding for transport behaviors and environmental risk assessments of CIP in the saturated porous media with MWCNTs.

High-Temperature Nonequilibrium Air Chemistry from First Principles

We present first-principles calculations for chemically reacting five-species air (N2, N, O2, O, NO) over a range of gas temperatures (T=5000–30,000 K), relying exclusively on ab initio potential energy surfaces (PESs) from the University of Minnesota Computational Chemistry group to describe the forces between atoms. We use these PESs within direct molecular simulations (DMSs) and quasi-classical trajectory (QCT) calculations to determine the coupling of internal energy relaxation to chemical reactions. From DMS we extract the internal energy populations of diatomic species during the quasi-steady-state (QSS) dissociation phase and, for all diatomic species, observe depleted high-energy tails relative to corresponding Boltzmann distributions. A comparison of thermochemical equilibrium rate coefficients (from QCT) with those during QSS (from DMS) helps quantify the macroscopic effects of vibrationally depleted distributions on dissociation. In contrast, Zeldovich exchange reactions are almost unaffected by these vibrationally depleted distributions. Unlike dissociation, they do not exhibit significant vibrational bias and take place at near-thermal rates at all temperatures studied. Furthermore, we quantify the amount of vibrational and rotational energy removed and/or gained in exchange and dissociation reactions. Such macroscopic quantities are of interest for enhancing the fidelity of multitemperature nonequilibrium chemistry models used in computational fluid dynamics codes.

The effect of slow adsorption of phosphate on its transport during the infiltration process in saturated agricultural soils

ABSTRACT Assessment of phosphorus (P)-infiltrating croplands is essential for the preservation of the water environment. It has been pointed out that a huge discrepancy lies in the different evaluation methods of P adsorption, such as batch experiments and column experiments, which makes it difficult to demonstrate P mobility under flow conditions. The objective of this study was to evaluate the applicability of the convective-dispersion equation using the parameters of the Langmuir-type isotherm obtained from batch experiments with different reaction times: the adsorption capacity of soil (q max) = 0.112 (g kg−1) for a Gray lowland soil with 24 h reaction time, q max = 0.484 (g kg−1) for an Andosol (volcanic ash soil) with 24 h reaction time, and q max = 1.17 (g kg−1) for an Andosol with 32 d reaction time, for describing P mobility in typical Japanese agricultural soils under fast flow conditions. The breakthrough curves of P infiltrating the soil columns demonstrate nonequilibrium P adsorption by the soil. The chemical nonequilibrium model, with a kinetic adsorption rate of α = 0.40 (Gray lowland soil) and 0.098 (Andosol), succeeded in describing the observations in the column experiments. Compared with Gray lowland soil, which is relatively rich in iron oxide, P mobility was largely affected by kinetic sorption in Andosol, which is relatively rich in allophane. It is suggested that the P adsorption capacity of soils should be evaluated reflecting the soil composition in order to simulate the P mobility under flow conditions. In particular, the slow adsorption (long-lasting adsorption) of P by the soil should be considered in the estimation of the P transport.

Effect of Thermal Nonequilibrium on Captured Mass Flow Rate of Scramjet Inlets

The numerical simulation of a scramjet is often based on the thermal equilibrium model, but the flow inside the inlet and isolator is likely to be in thermal nonequilibrium due to shock-based compression. The objective of this study is to investigate the effect of thermal nonequilibrium on the mass flow rate captured by the scramjet inlet. Numerical simulations based on the thermochemical nonequilibrium model (thermal nonequilibrium) and the chemical nonequilibrium model (thermal equilibrium) were adopted to study the mass flow rate of the HyShot II scramjet with the incoming flow Mach numbers ranging from 6.0 to 8.0. The results show that the maximum difference of the mass flow rate between thermal nonequilibrium and equilibrium reaches 10.7% at with . The primary cause of the mass flow rate difference is the oblique shock wave angle difference between nonequilibrium and equilibrium. The relative magnitudes of oblique shock wave angles and flow parameter ratios across the wave under equilibrium and nonequilibrium are theoretically discussed in an intuitive way. A simplified model for the HyShot II is established to predict the mass flow rate difference between thermal nonequilibrium and equilibrium. The result shows that the expression of captured mass flow rate has a significant amplification effect on the slight difference of wave angles between thermal nonequilibrium and equilibrium, which means that the effect of thermal nonequilibrium cannot be ignored for scramjet in some cases.

High temperature non-equilibrium flow characteristics of impinging shock/flat-plate turbulent boundary layer interaction at Mach 8.42

There are fewer reports on the impinging shock/boundary layer interaction in the high Mach number and high-temperature flow than that in the supersonic flow. High-temperature flow characteristics of the impinging shock/flat-plate turbulent boundary layer interaction (IS/FTBLI) at Mach 8.42 are numerically investigated by solving two-dimensional Reynolds averaged Navier–Stokes equations coupling with the thermal–chemical non-equilibrium model. An impinging shock is formed by the wedge with a 10° deflection angle. The inviscid flow parameters ahead of the cowl of a Mach 12 inlet are selected as the free-stream condition of this study. The primary emphasis of this study lies in understanding the thermal–chemical non-equilibrium effects in the IS/FTBLI. Moreover, the chemical non-equilibrium effects similar to previous reports from others are utilized for the comparative analysis. Our findings reveal that the vibrational or thermal non-equilibrium effects exhibit maximum prominence subsequent to the intersection of the impinging shock with separation shock, as well as in the convergence area of compression waves during the flow reattachment. On the other hand, the chemical non-equilibrium effects predominantly result from oxygen dissociation and atomic nitrogen production within the boundary layer; the chemical reactions are most intense within the separation zone. By comparing with a thermally perfect gas, a reduction in the flow separation is observed in the chemical non-equilibrium effects, but the flow separation is enhanced in the thermal–chemical non-equilibrium effects. The insights gained from our research are expected to contribute to the development of flow control technology in hypersonic IS/FTBLI scenarios and aid in configuring wave structures in the inner compression section of high Mach number scramjet inlets.

Non-equilibrium modeling on the aerothermodynamic characteristics of hypersonic inflatable reentry vehicle

In this study a 2D two-temperature chemical non-equilibrium model is established to investigate the flow characteristics and chemical reactions processes around hypersonic inflatable reentry vehicle at different flight altitudes and reentry velocities. This chemical kinetic model is based on the Gupta model and modified by comparing the species mole fractions at thermal equilibrium condition with the results obtained by Saha equation. In order to validate the model, the calculations are performed for a blunt cone ELECTRE and a cylinder, the predicted heat flux and pressure are compared with experimentally measured data and the maximum differences on heat flux and wall pressure are less than 20% and 5%, respectively. Then the flow characteristics behind the shock wave and wall heat flux are discussed in detail for Titans vehicle. It is found that the pressure declines with the increase of altitude and reentry velocity, this variation trend is opposite to the wall heat flux, whose maximum value increases from 58 kw/m2 to 107 kw/m2 with the rise of flight altitude from 79 to 88 km due to the large temperature and component gradients. This indicates that the aerodynamic force and heat flux should be taken into account simultaneously for the design of inflatable reentry vehicle. By analyzing the chemical reactions processes behind the strong shock wave, it is seen that not only the dissociation reactions of N2 and O2 molecules play an important role, but also the neutral species exchange reactions contribute to the density change of species. The dominant ionization reactions are N+O→NO++e and NO+M→4NO++e+M4, while the ionization processes are weak, and the maximum ionization degree is smaller than 1%. The analysis of dominant chemical reactions can provide a reference for the numerical modeling study of hypersonic reentry vehicle.

Stochastic calibration of a carbon nitridation model from plasma wind tunnel experiments using a Bayesian formulation

In this work, we calibrate a carbon nitridation model for a broad span of surface temperatures from existing plasma wind tunnel measurements by accounting for experimental and parametric uncertainties. A chemical non-equilibrium stagnation line model is proposed to simulate the experiments and obtain recession rates and CN densities, the measured model outputs. First, we establish the influence of the experimental boundary conditions and nitridation parameters on the simulated observations through a sensitivity analysis. Results show that such quantities are mostly affected by the efficiency of nitridation reactions at the gas-surface interface. We then perform model calibrations for each experimental condition and compare them based on the experimental data used. This allows us to check the consistency of the experimental dataset. Using only the trustworthy experimental data, we perform a calibration of Arrhenius law parameters for nitridation efficiencies considering all available experimental conditions jointly, allowing us to compute nitridation efficiencies even for surface temperatures for which there are no reliable experimental data available. The stochastic Arrhenius law agrees well with most of the data in the literature. This result constitutes the first nitridation model extracted from plasma wind tunnel experiments with accurate uncertainty estimates.

Numerical simulation of hydrogen arcjet thruster with coupled sheath model

In the present work, a complete 2D chemical and thermal non-equilibrium numerical model coupled with a relatively simple sheath model is developed for hydrogen arcjet thruster. Conduction heat transfer in the anode wall is also included in the model. The operating voltages predicted by the model are compared with those in the literature and are found to be in close agreement. Power distributions for the various operating conditions are obtained, anode radiation loss primarily determines the thruster efficiency. Higher thruster efficiency was found to be associated with longer arc length. At cathode ion diffusion contribution dominates except at low input current where thermo-field electron current is dominant.

An Application of Safety Assessment for Radioactive Waste Repository: Non-Equilibrium Transport of Tritium, Selenium, and Cesium in Crushed Granite with Different Path Lengths

Advection-dispersion experiments (ADE) were effectively designed for inadequate transport models through a calibration/validation process. HTO, selenium (Se), and cesium (Cs) transport in crushed granite were studied using a highly reliable, dynamic column device in order to obtain the retardation factors (R) and the dispersion coefficients (D) by fitting experimental breakthrough curves (BTCs) for various path lengths. In order to conduct a safety assessment (SA) of a deep geological repository for high-level radioactive waste, radionuclide transport in rock systems is necessary to clarify and establish a suitable model. A dynamic column with a radiotracer (HTO, Se(IV), and Cs) was applied to 2, 4, and 8 cm path lengths using a STANMOD simulation. The results showed similar results between the BTCs of Se and Cs by fitting a non-equilibrium sorption model due to the retardation effect. In fact, there was a relatively obvious sorption of Se and Cs in the BTCs obtained by fitting a retardation factor (R) value higher than 1. In addition, a two-region (physical) and a two-site (chemical) non-equilibrium model with either the lowest sum of squared residuals (SSQ) or the root mean square error (RMSE) were applied to determine the Se and Cs sorption mechanisms on granite.

Numerical simulation of the effects of protrusion on DC arc anode attachment

The attachment of the DC arc on the anode is usually affected by surface morphology such as protrusions due to ablation or melting deformation. A three-dimensional thermodynamic and chemical non-equilibrium model is used to numerically simulate the effect of artificially assumed surface protrusions on the arc anode attachment. The numerical simulation results show that the arc deflects toward the protrusions on the anode and attaches to them in a constricted mode, resulting in an increase in the temperature of the arc attachment region. The analysis shows that the presence of protrusion on the anode surface changes the electric field distribution, intensifies the degree of thermodynamic and chemical non-equilibrium in its vicinity, further influences the chemical kinetic process of the plasma around it, which is the main reason for the deflection of the arc toward the protrusions and the arc anode attachment in a constricted mode. In order to verify the numerical simulation results, verification experiments are also performed using similar size scale anode protrusion, and the results showed that the presence of protrusion can indeed cause the deflection of the arc and even cause the ablation of the protrusion.

Non-equilibrium modeling on the plasma–electrode interaction in an argon DC plasma torch

A 2D two-temperature chemical non-equilibrium model considering both cathode and anode space-charge sheath is applied to investigate the plasma–electrode interaction in a laminar argon DC plasma torch working at atmospheric pressure. In order to validate the model, the predicted arc voltage is compared with experimental values under different working conditions, and a reasonable agreement is obtained. The distributions of arc characteristics in the plasma column and electrode regions of a DC arc plasma torch are analyzed in detail. Moreover, the effects of thoriated tungsten, pure tungsten and non-uniform thoriated tungsten cathodes on the plasma characteristics inside the DC plasma torch are investigated. It is found that a constricted arc attachment with the highest current density is obtained at the non-uniform cathode, leading to the largest velocity and electric potential drop in the plasma column. The pure tungsten cathode produces the highest temperatures and heat flux along the cathode surface, which is caused by the largest arc voltage and intense space-charge sheath heating. The temperature and heat flux density on the thoriated tungsten cathode are the lowest, leading to a diffuse distribution of current density. The results indicate that the different cathodes can exert an important influence upon the plasma behavior near the cathode. Furthermore, the torch exit parameters are also slightly different due to their different arc voltages and overall input power.

SU2-NEMO: An Open-Source Framework for High-Mach Nonequilibrium Multi-Species Flows

SU2-NEMO, a recent extension of the open-source SU2 multiphysics suite’s set of physical models and code architecture, is presented with the aim of introducing its enhanced capabilities in addressing high-enthalpy and high-Mach number flows. This paper discusses the thermal nonequilibrium and finite-rate chemistry models adopted, including a link to the Mutation++ physio-chemical library. Further, the paper discusses how the software architecture has been designed to ensure modularity, incorporating the ability to introduce additional models in an efficient manner. A review of the numerical formulation and the discretization schemes utilized for the convective fluxes is also presented. Several test cases in two- and three-dimensions are examined for validation purposes and to illustrate the performance of the solver in addressing complex nonequilibrium flows.

Numerical simulation of anode heat transfer of nitrogen arc utilizing two-temperature chemical non-equilibrium model* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11735004 and 12005010).

A detailed understanding of anode heat transfer is important for the optimization of arc processing technology. In this paper, a two-temperature chemical non-equilibrium model considering the collisionless space charge sheath is developed to investigate the anode heat transfer of nitrogen free-burning arc. The temperature, total heat flux and different heat flux components are analyzed in detail under different arc currents and anode materials. It is found that the arc current can affect the parameter distributions of anode region by changing plasma characteristics in arc column. As the arc current increases from 100 A to 200 A, the total anode heat flux increases, however, the maximum electron condensation heat flux decreases due to the arc expansion. The anode materials have a significant effect on the temperature and heat flux distributions in the anode region. The total heat flux on thoriated tungsten anode is lower than that on copper anode, while the maximum temperature is higher. The power transferred to thoriated tungsten anode, ranked in descending order, is heat flux from heavy-species, electron condensation heat, heat flux from electrons and ion recombination heat. However, the electron condensation heat makes the largest contribution for power transferred to copper anode.

Development and Verification of Nonequilibrium Reacting Airflow Modeling in ANSYS Fluent

High-speed flow problems of practical interest require a solution of nonequilibrium aerothermochemistry to accurately predict important flow phenomena, including surface heat transfer and stresses. The complex coupling of nonequilibrium effects on the aerothermal load remains a challenging aspect of hypersonic flow modeling. As a majority of these flow problems are in the continuum regime, computational fluid dynamics (CFD) is a useful tool for modeling nonequilibrium flows. However, there are a limited number of CFD solvers with nonequilibrium flow modeling capabilities that are highly scalable on parallel architectures and can handle complex geometries. This work demonstrates that a commercial general purpose CFD solver can be effectively modified to include high-fidelity nonequilibrium aerothermochemistry and transport models. The paper presents modifications of a general purpose CFD solver, ANSYS Fluent, using the user defined functions to model salient aspects of nonequilibrium flow in air with a five-species reacting mixture: , , NO, N, and O. The modifications to the solver include Gupta–Yos high temperature mixing laws and Park’s two-temperature nonequilibrium chemistry models. The developed solver was verified for several benchmark nonequilibrium flow problems and compared with the available experimental data and other nonequilibrium flow simulations. The work establishes future possibilities for coupled aerothermal and electromagnetic simulations in thermochemical nonequilibrium.

Thermochemical non-equilibrium effects on aerothermodynamic prediction of laminar double-cone flow

Hypersonic non-equilibrium air flows over a laminar double-cone configuration are numerically investigated. In this paper, the effects of thermochemical non-equilibrium gas, chemical non-equilibrium gas, and thermally perfect gas models on the prediction of the wall-surface heat transfer and pressure are investigated. Comparing to the high-enthalpy experimental data, the results of simulations indicate that all models have a similar tendency, and underpredict the length of the separation region, while the heat transfer peaks are overpredicted. The experimental data for surface pressure and heat transfer is insensitive to the choice of the thermally perfect gas model and non-equilibrium thermochemistry models. Quantitatively, the thermochemical non-equilibrium gas and thermally perfect gas models predict the same separation and reattachment locations within the uncertainty in the simulation, but the chemical non-equilibrium gas model predicts a smaller separation region.

Two-Temperature Chemical Non-equilibrium Modeling of Argon DC Arc Plasma Torch

A 2D two-temperature chemical non-equilibrium model has been developed to investigate the plasma characteristics inside a DC arc plasma torch of argon. In this study the chemical kinetic model is carefully examined in order to properly capture the spatial variations of plasma characteristics, especially near the electrodes and arc fringe, where large gradients of plasma parameters exist. Two different sets of chemical kinetic processes are adopted in the calculation. The first chemical kinetic model [model (a)] includes the ground-state argon atoms, excited argon 4s state, atomic ions, molecular ions, and electrons, while the second model [model (b)] only considers the ground-state argon atoms, excited argon 4s state, atomic ions, and electrons. The predicted exit temperature and arc voltage by model (a) are in better agreement with experimental measurements. A delayed anode attachment with a higher arc voltage occurs in model (a) due to the low electron density in the upstream region of arc-anode attachment. Compared with the wide arc-anode attachment zone of model (b), model (a) exhibits a more constricted anode arc root with higher current density, which leads to a higher heat flux and temperature on the anode. This phenomenon may be explained by the rapid loss of electrons outside the arc-anode attachment zone due to the dissociative recombination of argon molecular ions. Moreover, model (a) produces a longer arc column with slightly higher maximum temperature and velocity on the axis, which is caused by its larger input energy.

Transport of uranium(VI) in red soil in South China: influence of initial pH and carbonate concentration.

Uranium-contaminated wastewater associated with uranium (U) mining and processing inevitably releases into soil environment. In order to assess the risk of U wastewater contamination to groundwater through percolation, U adsorption and transport behavior in a typical red soil in South China was investigated through batch adsorption and column experiments, and initial pH and carbonate concentration were considered of the high-sulfate background electrolyte solution. Results demonstrated that U adsorption isotherms followed the Freundlich model. The adsorption of U to red soil significantly decreased with the decrease of the initial pH from 7 to 3 in the absence of carbonate, protonation-deprotonation reactions controlled the adsorption capacity, and lnCs had a linear relationship with the equilibrium pH (pHeq). In the presence of carbonate, the adsorption was much greater than that in the absence of carbonate owing to the pHeq values buffered by carbonate, but the adsorption decreased with the increase of the carbonate concentration from 3.5 to 6.5 mM. Additionally, the breakthrough curves (BTCs) obtained by column experiments showed that large numbers of H+ and CO32- competed with the U species for adsorption sites, which resulted in BTC overshoot (C/C0 > 1). Numerical simulation results indicated that the BTCs at initial pH 4 and 5 could be well simulated by two-site chemical non-equilibrium model (CNEM), whereas the BTCs of varying initial carbonate concentrations were suitable for one-site CNEM. The fractions of equilibrium adsorption sites (f) seemed to correlate with the fractions of positively charged complexes of U species in solution. The values of partition coefficients (kd') were lower than those measured in batch adsorption experiments, but they had the same variation trend. The values of first-order rate coefficient (ω) for all BTCs were low, representing a relatively slow equilibrium between U in the liquid and solid phases. In conclusion, the mobility of U in the red soil increased with the decrease of the initial pH and with the increase of the initial carbonate concentrations.

Chemical Non-equilibrium Simulation of Anode Attachment of an Argon Transferred Arc

The diffuse and constricted arc attachment modes to the anode of an argon wall-stabilized transferred arc are investigated using a 2D chemical non-equilibrium model. Detailed argon chemical kinetic processes and self-consistent effective binary diffusion coefficient approximation treatment of diffusion are coupled with a single-fluid, two-temperature model to investigate the plasma characteristics of the near-anode region. The diffuse and constricted arc attachment modes to the anode are obtained self-consistently by changing the gas flow rates. The results show that the high current density and associated self-induced magnetic field of the constricted arc attachment give rise to a large Lorentz force near the anode, resulting in an anode jet. The convection process dominated by the cathode jet is conducive to the radial transport of electrons in the diffuse arc attachment mode, while in the constricted arc attachment mode the direction of electron transport due to the convection caused by the anode jet is opposite to that in a diffuse arc attachment. The effect of diffusion on electron transport is relatively small in the arc central region, while it becomes important in the arc fringe. The recombination processes, especially the dissociative recombination reactions, restrict the electron radial diffusion in the arc fringe. The diffusion current density is the main component of current density in front of the anode due to the steep gradient of electron pressure. It is found that when the diffusion current is not considered, the electric field and Joule heating in front of anode promote a more constricted arc attachment.

Numerical study on the aerothermodynamics of different heatshield configurations for Mars entry capsules

Aeroheating predictions play an important role in the heatshield design of Mars entry capsule. This paper numerically investigates the effects of sphere-cone angle on the aerothermodynamic performances of heatshield configurations. Three-dimensional Navier-Stokes equations with chemical non-equilibrium models are employed to simulate the flowfield around the capsule. The laminar and turbulent heating rates of different heatshield configurations are compared and analyzed in detail. A novel correlation for turbulent heating augmentation in terms of laminar momentum thickness Reynolds number is developed to provide a guidance for engineering design and application. The proposed correlation can be more accurate and applicable due to its consideration of the sphere-cone angle effects. Finally, the laminar and turbulent maximum heat flux and total heat load along the flight trajectory are investigated for all the configurations. The numerical study is expected to illustrate the aeroheating characteristics of different heatshield configurations and provide an insight into the rational configuration design for future Mars entry capsules.