Multicomponent Thermodynamic and Transport Properties for Ionized

AIAA 2014-2966 AIAA Aviation 16-20 June 2014, Atlanta, GA 11th AIAA/ASME Joint Thermophysics and Heat Transfer Conference

Development of Mutation++: MUlticomponent Thermodynamics And Transport properties for IONized gases library in C++ James B. Scoggins∗ Thierry E. Magin†

Downloaded by Alessandro Turchi on July 30, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2014-2966

von Karman Institute for Fluid Dynamics, Chauss´ee de Waterloo, 72, B-1640 Rhode-St-Gen`ese, Belgium The development of the Mutation++ library is described which has been designed to compute thermodynamic and transport properties of ionized gases including equilibrium compositions and species production rates due to finite-rate elementary reactions. Thermodynamic properties are obtained from three separate databases: the NASA 7- and 9-coefficient polynomials and a custom database which describes each atom and molecule as a rigid-rotator and harmonic oscillator. Mixture thermodynamic properties are suitably formulated from species quantities. Transport properties are derived from kinetic theory which provides relationships for macroscopic transport coefficients based on microscopic collision integrals. Finally, the chemical production rates for species based on elementary chemical reactions including third body reactions is detailed.

I.

Introduction

Space exploration is one of the boldest and most exciting endeavors that humanity has undertaken, and it holds enormous promise for the future. After the successful manned missions to the Moon and many probe entries in the atmospheres of outer planets, our next challenges include bringing back samples to Earth by means of robotic missions and continuing manned exploration, which aims at sending human beings to Mars and returning them safely home. Of the many design challenges associated with these goals is the difficult task of accurately predicting the heat-flux to the surface of the spacecraft thermal protection system (TPS) during planetary entry. An inaccurate prediction can be fatal for the crew or the success of robotic missions. Large safety factors are often added to vehicle TPS thicknesses to avoid such catastrophes at the expense of additional cost, weight, and reduced payload margins. The following complex phenomena that affect the surface heat-flux are considered as potential “mission killers:” 1) Radiation of the plasma in the shock layer, and 2) Complex surface chemistry on the thermal protection system material. However, current entry-vehicle design paradigms largely decouple many of these physico-chemical processes due to numerical and modeling constraints, making it difficult to infer from current design tools the complex, coupled phenomena occurring for any given entry problem. Thus, our poor understanding of the coupled mechanisms of flow, radiation, and ablation leads to the difficulties in heat flux prediction. This research focuses on addressing these issues through the development of new multiphysics, numerical methods aimed at strongly coupling the flow, radiation, and material response fields surrounding an entryvehicle. In the long-term, a hypersonics flow solver will be developed which makes use of an existing CFD framework such as SU21 and this solver will be implicitly coupled with a material response solver for carbonphenolic ablators. This coupling will allow the flux of pyrolysis and ablation gases generated by the thermal degradation of the ablator to be injected into the boundary layer computed by the CFD solver. In addition, the divergence of the radiative heat flux will be computed and coupled with the flow solver in order to complete the multiphysics coupling. As a first step however, a new library called MUlticomponent Thermodynamic And Transport properties for IONized plasmas in C++ (Mutation++ ) has been developed in order to provide accurate modeling of ∗ Ph.D.

Candidate, Aeronautics and Aerospace Department, Student Member AIAA Professor, Aeronautics and Aerospace Department, Member AIAA

† Associate

1 of 15 American Institute of Aeronautics and Astronautics Copyright © 2014 by James B. Scoggins and Thierry E. Magin. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

the various gases which will be necessary to close the governing equations that detail the flow and material fields. The development of this library and its validation is presented in the proceeding sections.

II.

Overall Design

The Mutation++ library is largely based on its Fortran 77 predecessor Mutation developed by Magin.2 It has been designed with the following constraints in mind: 1) high fidelity of the physical models, ensuring the laws of thermodynamics are satisfied and results are validated against existing experimental data, 2) low computational cost, 3) modern, object oriented, extensible framework, and 4) detailed in-source and user’s guide documentation in order to facilitate model improvement and collaboration. This document will focus on the physical modeling that is employed in Mutation++ . The properties provided by the library are necessitated by the closure of the governing equations for the systems of interest (flow field and material field).


A.

Governing Equations

The Navier-Stokes equations describing the flow field can be readily derived from the appropriate solution of the Boltzmann equation. This derivation is performed in many standard texts (see for instance3 ). In this section, we assume thermal equilibrium of the internal energy modes and chemical non-equilibrium of the gas mixture. It is further assumed that the gas is a weakly magnetized, quasi-neutral plasma. The species continuity equations which result from conservation of the species mass is ∂ ~i = ω˙ i ρi + ∇ · (ρi~v ) + ∇ · ρi V ∂t

∀i ∈ S,

(1)

~i is the species diffusion where ρi is the density of species i, ~v is the mass averaged bulk velocity of the gas, V velocity for species i, and ω˙ i is the mass production rate of species i due to elementary chemical reactions. For equilibrium flows, it is only necessary to enforce global continuity which is determined by summing the species conservation equations ∂ ρ + ∇ · (ρ~v ) = 0, (2) ∂t P provided that elemental remixing is not taken into account. ρ = i∈S ρi is the total mixture density. Conservation of total momentum yields ∂ (ρ~v ) + ∇ · (ρi~v ⊗ ~v ) + ∇ · P = 0, ∂t where the shear stress tensor is defined as i h 2 P = p + η∇ · ~v I − η ∇~v + (∇~v )T . 3

(3)

(4)

p is the mixture pressure and η is the bulk viscosity. Finally, conservation of total energy yields ∂ (ρE) + ∇ · (ρH~v ) + ∇ · ~q + ∇ · (P~v ) = 0, ∂t

(5)

where the heat flux is expressed as q= ~

X i∈S

~i + p ρi h i V

X i∈S

~i − λ∇T. kTh i + kTe i V

(6)

Material response models often use simplified conservation laws and require only a subset of the closure properties that are found in the Navier-Stokes equations.4, 5 It is therefore sufficient to provide the thermodynamic and transport properties and species production rates identified by the NS equations and the material response models will also be closed.

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III.

Thermodynamics

The governing equations described in the above section require the explicit knowledge of the mixture energy and enthalpy. If the flow is in equilibrium, the composition of the gas is explicitly defined by the minimization of the Gibbs free energy of the mixture with suitable mass balance constraints. For reacting flows, the equilibrium constant of each elemental reaction will depend on the Gibbs free energy of the participating species. Species specific heats are also important in many applications such as computing the Euken corrections to the thermal conductivity or numerically solving for the mixture temperature from a given energy density using a Newton-Raphson iterative procedure. A.

Thermodynamics of pure gases

For thermally perfect gases in which the effect of surrounding particles on the internal energy state of each particle is negligible, mixture thermodynamic properties can be treated as a summation of pure species properties. The Gibbs energy of a pure species i is defined as


gi ≡ hi − si T,

(7)

where hi and si are the pure species enthalpies and entropies respectively. The specific heats are defined as ∂hi ∂ei cpi ≡ , cvi ≡ (8) ∂T p ∂T V Therefore, a thermodynamic library must be capable of providing pure species enthalpies, energies, and entropies. The remaining thermodynamic functions are then readily obtained from expressions like Eqns. (7) and (8). Several types of data exist for computing pure species thermodynamic properties. Mutation++ currently includes two possibilities: 1) direct evaluation of species partition functions or 2) explicit evaluation of thermodynamic functions from a given set of polynomials. In the following subsections, both databases will be described including the resulting equations for the evaluation of the species thermodynamic properties. 1.

Partition functions

The thermodynamic properties of a pure gas may be derived from a statistical mechanics approach using the so-called partition function.6 The partition function for species i is defined as X θki gki exp − , (9) Qi ≡ T k

where gki and θki ≡ ǫki /kB are the degeneracy and energy of the kth micro-state of species i, respectively. The specific energy and entropy of a pure species i, normalized by the gas constant Ru , are related to the partition function by ∂ e˜i =T 2 (ln Qi ) ∂T (10) ∂ Qi +1+T (ln Qi ) , s˜i = ln Ni ∂T where Ni is the number of particles in the volume considered. The partition function may be split into separate energy types assuming no coupling exists between the different energy modes. Y Qm (11) Qi = i m

For gases which are considered to have weak interactions amongst individual particles, such as perfect gases, the translational energy may be separated from the internal energy of each particle. Furthermore, above a few Kelvin, the electronic energy of an atom or molecule can be assumed independent of the remaining rovibrational energy states with a negligible loss in the accuracy of the partition function.6 For relatively simple molecules, the rovibrational energy may also be split into separate modes. The most basic description of each of these modes is derived from the so called Rigid-Rotor (RR) and Harmonic-Oscillator (HO) models. 3 of 15 American Institute of Aeronautics and Astronautics

Table 1: Summary of pure species thermodynamic properties related to each energy mode. m

˜ em i [K]

˜ m [K] h i

translation

3 2T

e˜Ti + T

rotation (RR)

Li 2 T

vibration (HO)


electronic

e˜R i

i−1 V P V θik −1 k θik exp T E −1 P θik E E QE i k gik θik exp − T h

e˜Vi e˜E i

˜m ˜ sm gim [ - ] i − hi /T, −˜ 3 5 2πmi 5 1 2 ln T − ln p + 2 ln kB h2P Li T 2 ln θiR − ln σi h V i P −θ − k ln 1 − exp Tik ln QE i

Derivations of the contributions of each energy mode to the thermodynamic functions of a pure species may be found in standard texts on statistical thermodynamics such as Ref. 6. A summary these expressions is listed in Table 1. Corrections also exist for the RRHO model which take into account anharmonicity effects caused by the coupling between the moment of inertia and rotation of a molecule as well as the centrifugal forces on vibration, however these corrections are currently neglected in Mutation++ . Note that the expressions in Table 1 require knowledge of several constants for each species. L denotes the linearity of a molecule. It is 2 for linear molecules and 3 for nonlinear molecules. The steric (or symmetry) factor, σi , is defined as 1 for unsymmetric molecules (CO, NO, etc.) and 2 for symmetric ones (N2 , O2 , m CO2 , etc.). Finally, the θik values are characteristic temperatures for each energy level k of mode m for E species i and gik is the degeneracy of each electronic energy level k for species i. These basic data are typically determined through quantum mechanics calculations requiring accurate force potentials between each interacting particle in a molecule. Data for many species, including those used in Air and Martian atmospheric modeling, have been collected in the works of Gurvich et al.7–12 Following Eqns. (10) and (11), the thermodynamic properties of each pure species are simply the sum of the properties attributed to each energy mode belonging to the type of particle. Therefore, the specific energy and enthalpies of a species i are given by   i ∈ Satoms  {T, E}, X F m ˜ (12) ∀ m ∈ {T, R, V, E}, i ∈ Smolecules {e or h}i = Ri {˜ e or h} i + hi   m  {T }, i = e−

where hF i is the formation enthalpy of the species i. The formation enthalpy is defined as zero for a set of reference species (typically N2 , O2 , Ar, C(gr), etc.). All other species derive the formation enthalpy based on a formation reaction in which the reactants are the reference species and the product is the species of interest. The entropy of a species i is given by  {T, E}, X i ∈ Satoms , s˜m ∀ m ∈ si = Ri i {T, R, V, E}, i ∈ Smolecules (13) m se− = Re− s˜Te− + ln 2 ,

where the ln 2 term is added for the spin contribution to the entropy of the free electron. It is trivial to derive analytic expressions for the specific heats of each species based on Eq (8) and Table 1 which are not included here for brevity sake but are implemented in Mutation++ . As an example, the contributions of each energy mode to the enthalpy of N2 and CO2 are plotted in Fig. 1. As expected, the translational and vibrational contributions are linear in temperature. The vibrational enthalpy quickly asymptotes to a linear function because the rotational energy mode is fully excited quickly V due to the relatively low characteristic temperatures of vibration for each molecule (θN = 3408.464 K, 2 V θCO2 = {932.109 K, 932.109 K, 1914.081 K, 3373.804 K}). The electronic energy levels are the last to be fully excited, exhibited in Fig. 1 by the fact that the electronic enthalpy remains close to zero until about 10 000 K for N2 and 7500 K for CO2 .


12.5

12.5

hT hR hV hE

7.5

5

2.5

0

hT hR hV hE

10

hm [MJ kg−1 ]

hm [MJ kg−1 ]

10

7.5

5

2.5

0

2500

5000

10000

7500

12500

15000

0 0

2500

5000

T [K]

10000

12500

15000

(b) CO2

(a) N2


7500

T [K]

Figure 1: Enthalpy contributions for the N2 and CO2 molecules from each energy mode using the expressions in Table 1.

2.

NASA polynomial databases

The thermodynamic databases of NASA Glenn Research Center developed by McBride et al.13, 14 provide curve-fits for pure species thermodynamic properties at a standard state pressure of 1 bar in the form of 7- or 9-coefficient polynomials. The coefficients are computed based on least-squares fitting to accurate thermodynamic data either computed by the authors or retrieved from the literature. In general, these fits are more accurate then the simple RRHO model presented above because they take into account the most accurate partition function expressions and/or available experimental data. Beyond the original species found in Refs. 13 and 14, extensions to these databases have been created by other researchers such as the work of Burcat et al.15 Because of this, the NASA polynomial format remains a widely used and valuable database format for thermodynamic properties. In the case of the 7-coefficient database, the nondimensional specific heat at constant pressure is given by 5 X c˜◦pi = aik T (i−1) (14) k=1

The enthalpy and entropy of each species is then related by the standard thermodynamic relations Z ◦ ˜ hi = c˜◦pi dT + ai6 Z ◦ c˜pi ◦ s˜i = dT + ai7 T

(15)

For ideal gases and solids, the pure-substance specific heats and enthalpies are only functions of temperature ˜ ◦ . The species entropy is related to the standard state entropy by such that cpi = Ri c˜◦pi and hi = Ri h i  s˜◦ − ln p, i = gas i (16) si = Ri s˜◦ , otherwise i

The species Gibbs energies are then readily obtained from Eq (7). Note that in practice, the relations for enthalpy, entropy, and Gibbs energy allow an analytical formulation of the expressions which are implemented in Mutation++ . B.

Mixtures of pure gases

Mixture quantities are derived from pure species quantities through mixing rules. For a perfect gas, mixture thermodynamic properties are simply the sum of pure species properties weighted by the composition of the 5 of 15 American Institute of Aeronautics and Astronautics

mixture. For example, the mixture enthalpy per unit mass, h, is given as X ρh = ρi h i .

(17)

i∈S

The mixture entropy is similarly computed from species entropies with the addition of the entropy of mixing. X ρs = ρi (si − Ri ln Xi ) , (18) i∈S

where Xi is the mole fraction of species i. The mixture energy is computed via X X ρe = ρh − p = ρi h i − pi .


i∈S

(19)

i∈S

Mixture specific heats are defined in terms of partial derivatives of the mixture enthalpy and energy like the pure species values from Eq (8). ∂h ∂e cp = , cv = (20) ∂T p ∂T V For flows in chemical nonequilibrium, the mixture enthalpy and energy are functions of the local temperature and composition and the specific heats are called frozen (denoted cp,fr and cv,fr ). From Eqns. (17) and (19), the frozen specific heats are thus X X cp,fr = Yi (∂hi /∂T )p = Yi cpi i∈S

cv,fr =

X

i∈S

Yi (∂ei /∂T )V =

i∈S

X

Yi cvi .

(21)

i∈S

In chemical equilibrium, the mixture composition is a function of temperature and pressure and this contribution must be taken into account. Thus, X cp,eq = cp,fr + (∂Yi /∂T )p hi i∈S

cv,eq = cv,fr +

X

(∂Yi /∂T )V ei

(22)

i∈S

It is often necessary to know the local sound speed of the gas for CFD calculations. For a general equation of state, the sound speed is defined to be ∂p 2 a ≡ . (23) ∂ρ s For an ideal gas this equates to the following for frozen and equilibrium mixtures respectively: a2fr = γfr

p ρ −1

(24)

a2eq = γeq (∂ρ/∂p)

where γ = cp /cv is the ratio of specific heats per mass. Note that for an ideal gas, ρ = pMw,mix /(Ru T ), and thus at equilibrium X ∂ρ ρ ρ ∂Xi = + . (25) Mw,i ∂p Mw,mix ∂P p i∈S

C.

Linearly constrained multiphase equilibria

For a chemical system composed of any number of ideal phases, the set of indices which denote all species in is S = 1, . . . , nS = ∪m∈P Sm where nS is the total number of species considered, P = the system 1, . . . , nP is the set of phase indices with nP the number of phases, and Sm denotes the set of species 6 of 15 American Institute of Aeronautics and Astronautics

indices belonging to phase m. The equilibrium composition of the mixture is the one which minimizes the mixture Gibbs energy while satisfying elemental mass balance constraints.16 It is convenient to write the molar Gibbs energy of the mixture, normalized by Ru T . Since all phases are ideal, the normalized Gibbs energy is X gj (T, p) G ˜ ¯ G= (26) = + ln Nj − ln Npj Nj , Ru T Rj T j∈S

where Nj is the number of moles of species j and gj is the Gibbs function of pure species j at the system ¯p is the phase moles for the phase to which species temperature and pressure (evaluated from Eq (7)), and N j j belongs. X ¯p = N Nk , ∀ j ∈ S (27) j


k∈Ppj

The subscript pj ∈ P ∀ j ∈ S is used to denote the phase index to which the species index, j, belongs. The total moles of atoms of each element i in the system, cei , can be computed from the simple elemental mass balance relation X e Bji Nj = cei ∀ i ∈ E, (28) j∈S

e Bji

where is equal to the number of i atoms belonging to species j and E = {1, . . . , nE } is simply the set of all element indices. It is often useful to impose other constraints on the system. Therefore, we consider any nG general linear constraints on the number of moles of each species. Using matrix notation, the total constraints imposed on the composition are given as B T N = c,

(29)

and nC = nE + nG are total number of linear constraints. For a given B, c, and a fixed temperature and ˜ while satisfying the linear pressure, the equilibrium composition for the system is the one which minimizes G constraints in Eq (29). Using the well known Lagrange multiplier method for the solution of the above constrained minimization problem, the so called element potential equations are derived (see for instance Ref. 17 for the derivation). ! X g j ¯pj exp − Nj = N + λi Bji , ∀ j ∈ S. (30) Rj T i∈C

¯p , the nC element potentials, λi , Note that when written in terms of species mole fractions, Xj = Nj /N j completely define the equilibrium composition for a fixed temperature and pressure. Physically, the λi values represent an amount of Gibbs energy associated with each element at equilibrium. The substitution of Eq (30) into Eqs. (27) and (29) leads to a nonlinear system of nC + nP equations and as many unknowns ¯m , λi ). (N Well known equilibrium codes such as StanJan17 or CEA18 use specialized forms of the Newton-Raphson iterative method to solve these equations. It has been documented that such procedures can fail when the nonlinear system is numerically stiff.19 Therefore, an alternative procedure known as the Gibbs function continuation (GFC) method developed by Pope20, 21 for single phase mixtures is adopted instead. While the details of this method can be quite cumbersome, it is sufficient to mention that the GFC method employs a robust continuation algorithm which guarantees convergence for all well posed equilibrium problems. In Mutation++ , the GFC method is extended to multiphase in a unique continuation-Krylov iterative scheme called MPGFC (for multiphase GFC). The complete analysis of the MPGFC method will be presented in a future work as it is beyond the scope of the present article. The equilibrium composition of Air and CO2 mixtures at 1 atm as computed by Mutation++ are given in Fig. 2. The results are obtained using the NASA 9-coefficient database.14 The equilibrium Air composition is then used in Fig. 3 to compute several mixture thermodynamic properties based on the formulations from the preceding sections.

IV.

Chemical Kinetics

For mixtures out of equilibrium, it is necessary to compute the species production rates due to elementary chemical reactions which is added as a source term in the species continuity equations. We consider 7 of 15 American Institute of Aeronautics and Astronautics

1

1

CO2 N2

0.8

0.8

N

Mole Fraction

Mole Fraction

O 0.6

0.4

O

0.6

CO

0.4

C e–

e–

O2

+

0.2

C+

0.2

O

O2

0

O+

N+

NO 0

2500

5000

10000

7500

12500

0

15000

0

2500

5000

T [K]

10000

7500

12500

15000

T [K]

(b) 8 species CO2 mixture.

(a) 11 species air mixture.

150

25000

2.0

cp,fr cp,eq

1.5 1.0 0.5

100

h [MJ kg−1 ]

cp [J kg−1 K−1 ]

20000

15000

10000

0.0 -0.5

5000

0

0

0 0

2500

5000

10000

7500

12500

0

15000

2500

5000

7500

10000

12500

15000

10000

12500

15000

T [K]

T [K]

(a) Frozen and equilibrium specific heats at 1 atm.

(b) Enthalpies.

1.6

5000

γfr , 0.1 atm γeq , 0.1 atm γfr , 1.0 atm γeq , 1.0 atm γfr , 10 atm γeq , 10 atm

1.5

afr , 0.1 atm aeq , 0.1 atm afr , 1.0 atm aeq , 1.0 atm afr , 10 atm aeq , 10 atm

4000

a [m s−1 ]

1.4

1.3

1.2

1.1

10000

5000

hT hR hV hE hF h (Gurvich) h (NASA 9)

50

γ [-]


Figure 2: Equilibrium composition of the 11 species air and 8 species carbon dioxide mixtures at 1 atm.

3000

2000

1000

0

2500

5000

10000

7500

12500

15000

0

0

2500

5000

T [K]

7500

T [K]

(d) Frozen and equilibrium sound speeds.

(c) Frozen and equilibrium ratios of specific heats.

Figure 3: Selected mixture thermodynamic properties for equilibrium Air at 1 atm.

elementary reactions of the form ns X i=1

′

νij Ai ⇋

ns X

′′

νij Ai

∀ j ∈ {1, . . . , nr } ,

i=1


(31)

′

′′

where Ai represents the name of species i, and νij and νij are the stoichiometric coefficients of the reactants and products respectively. Note that any type of elementary reaction can be expressed in this form. The species production rate for each species s due to these elementary reaction processes is computed via ω˙ s = Mw,s

nr X ′′ ′ νsj − νsj τj Θj

(32)

j=1

where τj = kf,j

ns Y

′

νij

ρ˜i

− kb,j


i=1

ns Y

′′

νij

ρ˜i

(33)

i=1

is the rate of progress for reaction j and  Pns α ρ˜ , reaction j is third-body i=1 ij i Θj = 1, otherwise

(34)

is a third-body multiplier for reaction j. The backward rate coefficient is determined by satisfying the equilibrium relation kf,j kb,j = , (35) kC,j

where the equilibrium constant is defined to be KC,j (T ) =

Patm Ru T

∆νj

∆G◦j . exp − Ru T

(36)

The forward reaction rate coefficient must be specified by the user. If an Arrhenius rate law is used, kf,j takes the following form: θd,j kf,j (T ) = Aj T βj exp − , (37) T where the rate constants Aj , βj , and θd,j are given as input quantities by the user for each reaction considered.

V.

Transport Coefficients

Closure of the transport fluxes is achieved through a multiscale Chapman-Enskog perturbative solution of the Boltzmann equation.3 Graille et al.22 have derived a rigorous kinetic model for multicomponent plasmas accounting for the influence of the electromagnetic field and thermal nonequilibrium between free electrons and heavy particles. They showed through a dimensional analysis that the correct scaling of the Boltzmann equation is obtained by using a scaling parameter equal to the square-root of the ratio between the electron mass and a characteristic heavy-particle mass. Magin et al.23 have provided explicit expressions for the transport coefficients in terms of binary collision integrals based on the approach in Ref. 22 for weakly ionized and unmagnetized plasmas. These expressions have been implemented in Mutation++ and are summarized here. For the complete dependence of the following expressions on the collision integrals, the reader is referred to Ref. 23. The dynamic viscosity, η, and heavy particle translational thermal conductivity, λh , are obtained from the first and second Laguerre-Sonine polynomial approximations respectively of the Chapman-Enskog expansion. The resulting expressions require the solution of the linear transport systems X µ µ Gij αj = Xi ∀i ∈ H j∈H

µ=

X

αiµ Xi ,

(38)

i∈H H

where µ = η or λh and Gµ ∈ Rn binary collision integrals.

×nH

are transport matrices depending on the species mole fractions and


Heavy particle thermal diffusion ratios may then be computed via kTh i =

5 X 01 λh Λij αj 2

∀i ∈ H (39)

j∈H

kTh e

= 0,

where Λ01 is another transport matrix based on binary collision integrals and αλh is obtained from the solution of the linear system in Eq (38). The heavy particle thermal diffusion ratios satisfy the expression P h i∈H kT i = 0. Note that free electrons do not contribute to the mixture viscosity, heavy particle thermal conductivity, or the heavy particle thermal diffusion ratios. Expressions for the electron thermal conductivity, λe , and thermal diffusion ratios, kTe i , may be obtained from second or third Laguerre-Sonine approximations (denoted by (2) or (3)) and are given by


λe (2) = λe (3) = and

Xe2 , Λ11 ee Xe2 Λ22 ee

(40) 2,

12 22 Λ11 ee Λee − (Λee )

5 Te Λ01 ∀ i ∈ S, Xe ie 2 Th Λ11 ee 12 Λ01 Λ22 − Λ02 5 Te ie Λee Xe ie ee kTe i (3) = 22 12 2 2 Th Λ11 ee Λee − (Λee )

kTe i (2) =

(41) ∀ i ∈ S,

where Th and Te are the heavy particle and free electron translational temperatures respectively. The Λlk ie and Λlk matrices are complex functions of the binary collision integrals for heavy-electron and electron-electron ee P interactions. The electron thermal diffusion ratios satisfy the relation i∈H kTe i + kTe e Te /Th . Magin et al.24 have studied the convergence of the λe and kTe i due to the Laguerre-Sonine order and found that differences can exist in the levels of approximation even in plasmas with relatively low degrees of ionization. Therefore, the third order expressions are retained in the implementation of Mutation++ . The thermal conductivity due to internal energy transport, λint , is given by the so called Euken corrections, where X X X ρi c m pi P λm = λint = , (42) X j∈H j /Dij m∈{R,V,E}

m∈{R,V,E} i∈Hp

and Dij is the binary diffusion coefficient for the heavy-heavy species pair (i, j). It can be shown that the Euken corrections are in fact exact when the internal energy modes can be split according to rotational, vibrational, and electronic degrees of freedom as in Eq (42).2 The total mixture thermal conductivity is expressed as the sum of all the above thermal conductivities. λ = λh + λe + λint

(43)

Electron electrical conductivity is assumed to be the only significant contributor to the total electrical conductivity and is given by the first and second Laguerre-Sonine approximations which read σe (1) =

4 (Xe qe )2 1 , 2T 00 25 kB e Λee

4 (Xe qe )2 1 σe (2) = . 2T 2 11 00 25 kB e Λee − (Λ01 ee ) /Λee

(44)

Only the second order approximation is retained in Mutation++ . Species diffusion velocities can be obtained from the multicomponent diffusion coefficient matrix Dij by X ~i = − V Dij d~j + kTh j ∇ ln Th + kTe j ∇ ln Te , (45) j∈S


where d~j are the species specific driving forces defined as ∇pj yj p ~ d~j = − ∇ ln p − κj E (46) nkB Th nkB Th P ~ is the electric field. with κj ≡ Xj qj /(kB Th ) − yj q/(kB Th ), q = i∈S Xi qi is the mixture charge, and E P ~ The species diffusion fluxes, Ji ≡ ρi Vi , satisfy the mass conservation constraint such that i∈S Ji = 0. P P Furthermore, the driving forces and κi values are linearly dependent, namely i∈S d~i = 0 and i∈S κi = 0. An equivalent formulation of the species diffusion velocities are found from the solution of the generalized Stefan-Maxwell equations, X ~j = −d~′ i + κi E, ~ ∀ i ∈ H, GVij V j∈S

X

j∈S

(47)

~j = − d~′ e + κe E ~ Th , GVej V Te

yj p ∇pj − ∇ ln p + kTh j ∇ ln Th + kTe j ∇ ln Te . d~j = nkB Th nkB Th

(48)

The diffusion transport system matrix GV is a complex function of binary collision integrals. Selected transport properties as computed by Mutation++ for equilibrium air are given in Fig. 4. 7

4

10

0.1 atm 1.0 atm 10.0 atm

λ λreac λint λh λe

6

3

5

λ [W m−1 K−1 ]

σe [S m−1 ]

10

2

10

4

3

2 1

10

1

0

10 0

0 2500

5000

7500

10000

12500

15000

0

2500

5000

7500

10000

12500

15000

T [K]

T [K]

(b) Individual components of thermal conductivity at 1 atm.

(a) Electric conductivity at 0.1, 1, and 10 atm. 3.0

1e-7

N2

0.1 atm 1.0 atm 10.0 atm

2.5

5e-8

e–

Ji [kg m−2 s−1 ]

2.0

η [mP]


where the modified driving forces are

1.5

1.0

NO

O2

0

O+ N+

O -5e-8

0.5

N 0.0

0

2500

5000

7500

10000

12500

15000

-1e-7

0

2500

5000

T [K]

7500

10000

12500

T [K]

(c) Dynamic viscosity at 0.1, 1, and 10 atm.

(d) Species diffusion fluxes at 1 atm.

Figure 4: Selected transport properties for equilibrium air versus temperature.


15000

A.

Collision integrals

The collision integrals referred to in the previous section are a direct consequence of the solution of the Boltzmann equation. They represent integrated cross-sections for binary species pairs which interact with one another. The deflection angle χ for a binary interaction is given from classical mechanics as Z ∞ dr/r 2 r χij = π − 2b (49) , rm 1 mi mj 2 2 2 1 − b /r − φij (r)/ 2 mi +mj g

where b is the impact parameter, r is the distance between particles, rm the distance of closest approach, g the relative velocity of the colliding particles, and φij (r) is the interaction force potential. Collision cross-sections are defined in terms of the deflection angle by Z ∞ (l) 1 − cosl χij b db. (50) Qij = 2π


0

Collision integrals for heavy-heavy particle interactions are computed based on integrating over all thermal speeds of heavy particles. s Z kB Te mi + mj ∞ (l,s) (l) Ωij = (51) exp −g2 g2s+3 Qij dg, ∀ i, j ∈ H, 2π mi mj 0 where g = {mi mj /[(mi + mj )2kB Th ]}1/2 g. For heavy-electron and electron-electron interactions, collision integrals are based on the thermal speed of free electrons. r Z kB Te ∞ (l) (l,s) exp −g2 g2s+3 Qie dg, ∀ i ∈ H, (52) Ωie = 2πme 0

where g = {me /(2kB Te )}1/2 g and

Ω(l,s) ee

=

r

kB Te πme

Z

∞

0

exp −g2 g2s+3 Q(l) ee dg,

(53)

where g = {me /(4kB Te )}1/2 g. As previously mentioned, the transport systems Λ and Gµ are complex functions of collision integrals which are detailed in the work of Magin et al.23 so they are not repeated here. A summary of which collision integrals are required to evaluate each transport coefficient is however given in Table 2 along with the number of individual collision integral pairs for each property. From the table, it is clear that in order to compute the full set of transport properties for a given mixture, several collision integrals must be provided for all the possible collision pairs in the mixture. In general, the collision pairs can be divided into different types depending on the collision partners which include neutral-neutral, ion-neutral, electron-neutral, and ion-ion (charge-charge) pairs. Except for charge-charge collisions, collision integral data for species collision pairs must be taken from the literature. Recent reviews of all possible collision integrals required for Air species as well as Mars and Venus atmospheres have been completed by Wright et al.25, 26 Charged collision pairs are governed by Coulomb potentials shielded by the Debye length, known as the Debye-H¨ uckel potential. The Debye length is defined as 1 ǫ0 kB Te 2 λD = (54) 2ne qe2

Devoto27 and Mason and Munn28 have computed the necessary reduced collision integrals using the DebyeH¨ uckel potential and provided tables for these integrals as a function of the reduced temperature T∗ =

λD 2 qe /kB T

(55)

For this work, these tables have been curve-fitted to 4th -order exponential polynomials of the form (l,s)∗

Ωij

(T ∗ ) = exp[A + B ln(T ∗ ) + C ln(T ∗ )2 + D ln(T ∗ )3 + E ln(T ∗ )4 ]

(56)

The computed coefficients for each integral are given in Table 3 along with the corresponding residual error for each curve-fit. 12 of 15 American Institute of Aeronautics and Astronautics


Table 2: Summary of required collision integral data for each transport coefficient including number of unique collision integrals which much be evaluated. Property

Collision Integrals

η

Ωij

λh

Ωij

λint

Ωij

λe (2)

Ωie

λe (3)

Ωie

kTh i (2)

Ωij

kTe i (2)

Ωie

kTe i (3)

Ωie

~i V

Ωij

σe (1)

Ωie

σe (2)

Ωie

all

Ωij

(1,1)

, Ωij

(1,1)

, Ωij

Number∗

(2,2)

(1,2)

nH (nH + 1) (1,3)

, Ωij

(2,2)

2nH (nH + 1)

, Ωij

(1,1)

nH (nH + 1)/2 , Ωee

(1,3)

, Ωie

, Ωie

(1,2)

, Ωie

, Ωie

(1,1)

, Ωie

(1,1)

, Ωij

(1,1)

, Ωie

(1,1)

, Ωie

(1,1)

, Ωie

(2,2)

(1,3)

(1,2)

(1,1)

(1,4)

ae (3nH + 1) (1,5)

, Ωie

(2,2)

(2,4)

(2,3)

ae (5nH + 3)

, Ωee , Ωee , Ωee

(1,2)

nH (nH + 1)

(1,2)

, Ωie

(1,3)

, Ωee

(1,2)

, Ωie

(2,2)

(1,3)

, Ωie

(1,4)

ae (3nH + 1) (1,5)

, Ωie

(2,2)

(2,3)

(2,4)

ae (5nH + 3)

, Ωee , Ωee , Ωee

(1,1)

nH (nH + 1)/2 + ae nH

(1,1)

ae nH

(1,1)

, Ωie

(1,1)

, Ωij

(2,2)

(1,2)

, Ωie

(1,2)

, Ωij

(2,3)

(1,3)

, Ωee

(2,2)

(1,3)

, Ωij

(2,2)

ae (3nH + 1) (1,1)

, Ωie

(1,2)

, Ωie

(1,3)

, Ωie

(1,4)

, Ωie

(1,5)

, Ωie

,

2nH (nH + 1) + ae (5nH + 3)

(2,4)

Ωee , Ωee , Ωee ∗

ae is 1 if free electrons are present in the mixture, 0 if not.

Table 3: Fits to collision integral functions for charge-charge collisions.

(1,1)∗ (T ∗ )2 Ωatt (1,1)∗ (T ∗ )2 Ωrep (2,2)∗ (T ∗ )2 Ωatt (2,2)∗ (T ∗ )2 Ωrep A∗att A∗rep ∗ Batt ∗ Brep ∗ Catt ∗ Crep ∗ Eatt ∗ Erep ∗ Fatt ∗ Frep G∗att G∗rep (1,4)∗ (T ∗ )2 Ωatt ∗ 2 (1,4)∗ (T ) Ωrep (1,5)∗ (T ∗ )2 Ωatt ∗ 2 (1,5)∗ (T ) Ωrep (2,4)∗ (T ∗ )2 Ωatt ∗ 2 (2,4)∗ (T ) Ωrep

A

B

C

D

E

R

-7.9270465e-01 -1.3980752e+00 -8.1145738e-01 -1.1089170e+00 -1.8907314e-02 2.8906871e-01 2.8616335e-01 3.2606973e-01 -6.4672764e-01 -5.0077127e-01 -3.8922959e-01 3.6035379e-01 -5.1269011e-01 -1.6610568e-01 -4.2962645e-01 -3.3907302e-01 -2.3064366e+00 -2.6061028e+00 -2.6207402e+00 -2.8756074e+00 -1.5271549e+00 -1.7664116e+00

5.8867723e-01 8.0482070e-01 7.0419264e-01 7.7460857e-01 1.1539718e-01 -3.0170249e-02 -4.8858107e-02 -2.3742462e-02 -1.0737242e-01 -1.0639836e-01 -8.5334423e-02 -7.3192205e-02 -1.6184890e-04 -8.4658420e-02 -8.2092404e-02 -7.8583290e-02 3.9187956e-01 5.7602188e-01 3.6889858e-01 5.3793608e-01 5.5923597e-01 6.4675076e-01

-8.7607125e-02 -9.4801647e-02 -1.2219724e-01 -1.0168153e-01 -3.4459736e-02 -6.9701558e-03 1.3372770e-03 -9.6987749e-03 1.7962157e-02 -1.1478149e-03 8.7430000e-03 1.3517297e-03 8.4938291e-03 -5.1885768e-03 1.5834420e-02 8.4227764e-04 -5.0541034e-02 -8.9813364e-02 -4.5983762e-02 -8.6195947e-02 -1.0268876e-01 -9.8071543e-02

9.3018130e-03 5.2812176e-03 1.3234707e-02 6.7600878e-03 3.9051788e-03 1.5032074e-03 5.6686310e-04 1.7799200e-03 -1.8653043e-03 1.8355926e-03 1.3038899e-04 1.1249000e-03 -2.8416333e-03 2.1711071e-03 -1.5811638e-03 1.3395752e-03 4.9996677e-03 8.4176238e-03 4.4531740e-03 8.5216824e-03 1.2409253e-02 8.6280547e-03

-4.1315208e-04 -8.2652059e-05 -5.6994085e-04 -1.5622212e-04 -1.5513314e-04 -7.5200734e-05 -4.4964175e-05 -8.3938754e-05 8.0134495e-05 -1.1830337e-04 -4.3583770e-05 -8.3717249e-05 1.9700964e-04 -1.2737940e-04 6.0509719e-05 -9.7107273e-05 -2.1443795e-04 -3.1611961e-04 -1.8883719e-04 -3.3719465e-04 -5.8149121e-04 -3.0256691e-04

0.062 0.051 0.045 0.057 0.052 0.008 0.025 0.013 0.011 0.013 0.047 0.016 0.093 0.013 0.036 0.019 0.029 0.023 0.024 0.019 0.081 0.035


ref 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 27 27 27 27 27

B.

Transport Algorithms

In the case of the the heavy particle thermal conductivity and shear viscosity, a linear system must be solved where the linear coefficients are functions of the collision integrals. Classical approaches29 to solve these systems either use a determinate method whereby µ = {η, λh } is computed via " # Gµ αµ µ=− /|Gµ |, (57) (αµ )T 0


or simplified mixture rules such as those developed by Gupta and Yos30 or Wilke.31 Recently, state-of-theart methods were developed by Ern and Giovangigli32 which formulate symmetric positive definite transport systems. This allows the linear systems to be solved quickly using a direct method such as the LDLT decomposition or with iterative methods such as the Conjugate-Gradient method. Solution via LDLT requires a computational cost which scales with O(n3s /6) while the CG method is O(mn2s ), where m is the number of iterations performed. It is therefore obvious that iterative methods are competitive when the number of iterations does not exceed n/6. In practice, it was found that only a single CG iteration is required to compute the shear viscosity of the equilibrium Air-11 mixture with an error of less than 1%.

VI.

Concluding Remarks

Mutation++ is currently being developed to model high enthalpy gases which may or may not be ionized. Currently, the properties required to model equilibrium or single temperature multicomponent flows have been implemented and validated. In future work, a multi-temperature framework will be created which will handle energy transfer between the various energy modes available in the system. This may include a simple two temperature model or could be extended to collisional radiative processes.

Acknowledgments This research is funded through a Fellowship provided by the European Research Council Starting Grant #259354: ”Multiphysics models and simulations for reacting and plasma flows applied to the space exploration program.”

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