Eur. Phys. J. D (2016) 70: 159 DOI: 10.1140/epjd/e2016-70195-4

On the thermodynamic properties of thermal plasmain the flame kernel of hydrocarbon/air premixed gases

Omid Askari, Gian Paolo Beretta, Kian Eisazadeh-Far and Hameed Metghalchi

Eur. Phys. J. D (2016) 70: 159DOI: 10.1140/epjd/e2016-70195-4

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

On the thermodynamic properties of thermal plasmain the flame kernel of hydrocarbon/air premixed gases

Omid Askari1,a, Gian Paolo Beretta2, Kian Eisazadeh-Far3, and Hameed Metghalchi1

1 Northeastern University, 334 Snell Engineering Center, 360 Huntington Ave, Boston, MA 02115-5000, USA2 Universita di Brescia, Dipartimento di Ingegneria Meccanica e Industriale, via Branze 38, 25123 Brescia, Italy3 Tula Technology, 2460 Zanker Rd, San Jose, CA, 95131, USA

Received 15 March 2016 / Received in final form 21 May 2016Published online 2 August 2016 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2016

Abstract. Thermodynamic properties of hydrocarbon/air plasma mixtures at ultra-high temperaturesmust be precisely calculated due to important influence on the flame kernel formation and propagationin combusting flows and spark discharge applications. A new algorithm based on the complete chemicalequilibrium assumption is developed to calculate the ultra-high temperature plasma composition and ther-modynamic properties, including enthalpy, entropy, Gibbs free energy, specific heat at constant pressure,specific heat ratio, speed of sound, mean molar mass, and degree of ionization. The method is appliedto compute the thermodynamic properties of H2/air and CH4/air plasma mixtures for different tempera-tures (1000–100 000 K), different pressures (10−6–100 atm), and different fuel/air equivalence ratios withinflammability limit. In calculating the individual thermodynamic properties of the atomic species neededto compute the complete equilibrium composition, the Debye-Huckel cutoff criterion has been used forterminating the series expression of the electronic partition function so as to capture the reduction of theionization potential due to pressure and the intense connection between the electronic partition functionand the thermodynamic properties of the atomic species and the number of energy levels taken into ac-count. Partition functions have been calculated using tabulated data for available atomic energy levels.The Rydberg and Ritz extrapolation and interpolation laws have been used for energy levels which are notobserved. The calculated plasma properties are then presented as functions of temperature, pressure andequivalence ratio, in terms of a new set of thermodynamically self-consistent correlations that are shownto provide very accurate fits suitable for efficient use in CFD simulations. Comparisons with existing datafor air plasma show excellent agreement.

1 Introduction

To improve efficiency and reduce pollutant formation ininternal combustion engines [1], knowledge of flame kerneldevelopment and flame propagation play important role.A plasma, at very high temperature, will be generated atthe onset of spark discharge. Accurate modeling of thethermodynamic properties of plasma mixtures is essentialto understand the evolution of the plasma channel and itsevolution into the formation of the flame kernel. Duringthe spark discharge in a fuel-air mixture, the electrical en-ergy is injected in a constant volume process followed bya sudden expansion which leads to the formation of fullyionized high temperature plasma through the generationof a shock wave and the consequent dissociation and ion-ization of the mixture. The plasma thermodynamic prop-erties and its degree of ionization have important effectson flame ignition, structure, and propagation.

a e-mail: omidmech@gmail.com

During the past decades a significant progress inplasma applications such as cutting, spraying, arc heat-ing, re-entry of space-vehicles, nuclear rockets, CFD sim-ulation of high-temperature flow fields and spark ignitionhas happened. To model and control the plasma flow inthese applications, energy, mass, and momentum trans-fer are very important. As a result, the thermodynamicproperties of plasma mixtures at high temperatures mustbe estimated by means of sophisticated models to ensureaccurate simulations of the plasma flow field. In many ap-plications it is possible to model plasma mixtures behaviorby the equation of state of an ideal gas in local thermo-dynamic equilibrium (LTE).

In the last six decades, many papers have been pub-lished on the thermodynamic properties of plasma mix-tures with particular attention to air species becauseof their importance in the aero-thermodynamic analy-sis of hypersonic flows surrounding a space vehicle dur-ing its reentry into the Earth’s atmosphere. Gilmore [2,3]reported results for composition and thermodynamicproperties of air for temperatures between 1000 K

Page 2 of 22 Eur. Phys. J. D (2016) 70: 159

and 24 000 K. Calculations assume a perfect gas mixture inlocal thermodynamic equilibrium, including dissociationand ionization. Hansen and Heims [4], and Hansem [5,6]studied the space-vehicle traveling at high speed which ex-cite the air to high temperature, resulting in dissociationand ionization of air. His method was valid to predict thethermodynamic properties of air up to 15 000 K and con-sidered dissociation and only first ionization of nitrogenand oxygen. Rosenbaum and Levitt [7] derived expres-sions for the composition, specific volume, enthalpy, andentropy of hydrogen plasmas up to 100 000 K.

Lick and Emmons [8] calculated the thermody-namic properties of helium plasmas at temperatures upto 50 000 K. They considered a mixture composed ofneutral helium atoms, singly and doubly ionized he-lium atoms, and electrons. Brown calculated equilibriumhigh-temperature thermodynamic properties of the atmo-spheres of Earth [9], Mars [10] and Venus [11] for studyingflights in planetary atmospheres in the shock and bound-ary layers. Drellishak et al. [12] calculated equilibriumcomposition and the thermodynamic properties of argonplasma for five pressures of 0.1, 0.5, 1.0, 2.0 and 5.0 atm,and for the temperature range from 5000 K to 35 000 K.Kubin and Presley [13] calculated the thermodynamicproperties of hydrogen plasmas for a temperature up to20 000 K. Hydrogen atoms were assumed to have only sixenergy levels. They neglected the reduction in the ioniza-tion potential and arbitrarily cut off the electronic parti-tion function at six terms.

Patch and McBride [14] obtained thermodynamicproperties for H+

3 and H+2 at temperatures between 298 K

and 10 000 K. Patch [15] calculated the equilibriumcomposition of a hydrogen plasma for pressures rang-ing from 1 to 100 atm at temperatures of 400 Kto 40 000 K. Nelson [16] calculated and tabulated thermo-dynamic properties of an atomic hydrogen-helium plasmafor temperatures from 10000 K to 100 000 K. Pateyronet al. [17,18] calculated thermodynamic properties suchas specific enthalpy and specific heat of the Ar-H2 andAr-He plasmas up to 30 000 K using partition functionsof species. Sher et al. [19] calculated the specific heat ca-pacity and mole fractions of the air at high temperaturesusing a simplified thermodynamic model to study the for-mation of spark channels. Capitelli et al. [20–24] calcu-lated thermodynamic and transport properties of air athigh temperatures assuming that the plasma is in localthermodynamic equilibrium (LTE) for temperatures upto 100 000 K. D’Angola et al. [25] calculated thermody-namic and transport properties of high temperature equi-librium air plasmas in a wide range of pressure (0.01 atmto 100 atm) and temperature range (50 to 60 000 K).Rat et al. [26] performed an alternate derivation of trans-port properties in a nonreactive two-temperature plasmawithout Bonnefoi’s assumptions. They applied their modelfor a two-temperature argon plasma and figured that thetheory of transport coefficients of Devoto and Bonnefoiunderestimates the electron thermal conductivity. Mur-phy calculated the transport coefficients such as vis-cosity, thermal conductivity and electrical conductivityof different species including hydrogen and mixtures of

argon-hydrogen [27] and helium and its mixture with ar-gon [28] using local thermodynamic equiblirium at at-mospheric pressure in the temperature range from 300to 3000 K.

The above literature shows that most of the worksdealt only with air and some fundamental gases like, hy-drogen, helium, argon, and their mixtures. It also revealsa shortage of exact curve-fit correlations of thermody-namic properties of fuel/air plasma mixtures such as hy-drogen/air and methane/air which have many importantapplications in automotive and aviation industries [29,30],as well as in other studies dealing with plasma mixtures.The combustion community suffers from lack of these kindof correlations which in CFD simulations of plasma mix-ture flows can be easily implemented and effectively reducenumerical complications and computation time. Curve-fitcorrelations have only been reported in few works withhigh emphasis on air plasma.

The purpose of this paper is to provide the thermo-dynamic properties and equilibrium composition of hy-drogen/air and methane/air plasma mixtures. The corre-lations we propose provide the equilibrium compositionand the thermodynamic properties such as enthalpy, en-tropy, Gibbs free energy, specific heat at constant pres-sure, specific heat ratio, speed of sound, mean molar mass,and degree of ionization for a wide range of tempera-tures (1000–100000 K), pressures (10−6–102 atm) anddifferent equivalence ratios within flammability limit (formethane/air mixture flammability limit is 0.6 < φ < 1.4and for hydrogen/air mixture flammability limit is 0.5 <φ < 5.0). To calculate the thermodynamic properties ofa plasma mixture, it is necessary to evaluate the individ-ual properties of the pure components and calculate theequilibrium composition. For polyatomic molecules andmolecular ions, which are the dominant components dur-ing the dissociation phase, the individual thermodynamicproperties are extracted from the NASA database [31].For monoatomic molecules, atomic ions, and electrons,the individual thermodynamic properties are computedby statistical thermodynamic methods using a rigorouscalculation of the electronic partition functions. Griem’sself-consistent model [32] is used for the reduction of theionization potential based on Debye-Huckel length. Thismodel uses a cut-off of the electronic partition functionexpansion series in order to prevent its divergence prob-lem. The effect of the number of energy levels on ther-modynamic properties is illustrated by comparing theresults with the ground state method. In order to deter-mine the equilibrium composition, the complete equilib-rium method based on Gibbs free energy minimizationassuming ideal gas equation of state, mass conservation,and electrical neutrality [33,34] has been applied.

The paper is structured as follows. Section 2 discussesthe importance of plasma study in flame kernel model-ing in the spark ignition process. Section 3 illustrates themodel and the various assumptions in detail. Section 4presents and discusses the results. Section 5 gives someconclusions. Appendix gives the coefficients of the pro-posed correlations.

Eur. Phys. J. D (2016) 70: 159 Page 3 of 22

Fig. 1. Schematic of flame propagation model [35].

2 Plasma application in spark ignition process

It is a well-known fact that in the spark and laser igni-tion, high temperature ionized gases is the source of flamekernel formation and propagation. In conventional sparkignition, electrical energy is supplied through an externalsource (e.g. a spark plug). A part of electrical energy isconverted to thermal energy by ionizing the gases. Thisconversion process involves the formation of a plasma ker-nel which can potentially form a flame kernel. The forma-tion of spark kernel consists of two parts [35]. The first, theshorter stage involves a pressure wave emission; the secondstage which is longer is a constant pressure process. In thesecond stage the diffusion of reactants and ions concludethe initial flame kernel [35]. In both stages the high densityelectrical energy creates ions in an extremely high temper-ature environment. In our previous work, we calculatedthe properties of the initial spark kernel by employing athermodynamic model validated by experiments [35]. Foran initial plasma kernel the energy balance for the controlvolume shown in Figure 1, is given by:

∂E

∂t= mb

(huf

+ cpuTu

)+

dSE

dt− Qcond − Qrad − P V ,

(1)

E = m(ubf

+ cvbTb

), (2)

where E is the energy of the burned gas region, m themass of the gas, T the temperature, h the specific en-thalpy, u the specific internal energy, cp the specific heatat constant pressure, cv the specific heat at constant vol-ume, SE the supplied spark energy, t the time, Qcond theconductive energy losses, Qrad the radiated energy losses,P the pressure and V the volume of the kernel, u, b and fsubscripts refer to unburned, burned and formation.

By solving equations (1) and (2), the kernel tempera-ture at several conditions was calculated. Figure 2 showsthe temperature of kernel at different initial radii. It canbe seen that the temperature is high enough for the for-mation of plasma ions. We have shown that at temper-atures higher than 6000 K the thermodynamic proper-ties such as cv and cp in equations (1) and (2) are strong

Temperature(K)

0

10000

20000

30000

40000

50000

60000

70000

0 0.5

Tim

1 1

e (ms)

.5 2 2.5

ri = 0.45 mmri = 0.5 mmri = 0.55 mm

3

Fig. 2. The effect of initial radius on air kernel temperature,Ti = 7000 K, discharge energy = 24 mJ [35].

function of ionization processes [36]. In previous work [35]due to low concentration of methane molecule the thermo-dynamic properties of methane/air plasma were approxi-mated by considering only air and neglecting methane inthe mixture. This rough assumption can be resolved bykeeping all the components available in the real plasmamixture which will enhance the accuracy of thermody-namic properties by including hydrogen, helium, carbon,argon and neon ions in the computations. Laminar burn-ing speed [37–40], an important thermo-physical proper-ties of combustible mixtures, can be calculated using themethod introduced in previous publication [35] in conjunc-tion with exact plasma simulation of hydrocarbon/air pre-mixed gases in this study at ultra-high pressures at whichthe flame is always cellular and unstable.

3 Method of calculation

The calculations described here, and the resulting correla-tions, are based on the following three assumptions, whichare valid for many plasma mixtures: (1) the plasma is inlocal thermodynamic equilibrium; (2) the plasma mixtureis quasi-neutral; and (3) the plasma mixture and its indi-vidual components obey the ideal gas equation of state.Under such conditions, Boltzmann statistics is applica-ble. Calculations have been carried out in two differenttemperature ranges, called dissociation and ionization. Toobtain realistic calculations, all minor species have beenconsidered, for an overall of 133 species as listed in Table 1for both methane/air and hydrogen/air plasma mixtures.

3.1 Dissociation temperature range

Increasing the temperature of a gas mixture causes themolecules not only to vibrate but also to dissociate into el-emental atoms. Depending on the initial gas mixture com-position the dissociation temperature range is typically

Page 4 of 22 Eur. Phys. J. D (2016) 70: 159

Table 1. List of the 133 species considered in the present calculations for H2/Air and CH4/Air plasma mixtures.

Plasma mixture Available species

CH, CH+, CH2, CH2OH, CH2OH+, CH3CH3O, CH3OHCH4,

C2H, C2H2, CH2CO, C2H3, CH3CN, CH3CO, C2H4, C2H4O, CH3CHO,

C2H5, C2H6, C3H7, C3H8, CN, CN+CN−, CNN, CO, CO+, CO2, CO+2

NCO, HCO, HCO+, HCN, HCCN, HCCO, HNC, HNCO, HNO, OH, OH−

H2/Air OH+, HO2, HO−2 , H2O, H2O

+, H2O2, NH, NH+, NH2, NH3, NO, NO+

and NO2, NO−2 , N2ON2O

+H2, H−2 , H+

2 , H, H−, H+He, He+, He+2, C, C−,

CH4/Air1 [41] C2, C−2 C+

2 , C+, C+2, C+3, C+4, C+5, C+6N, N−, N2, N−2 , N+

2 , N+, N+2,

N+3, N+4, N+5, N+6, N+7, O, O−, O2, O−2 , O+

2 , O+O+2, O+3, O+4

O+5, O+6, O+7, O+8, Ne, Ne+, Ne+2, Ne+3, Ne+4, Ne+5, Ne+6, Ne+7

Ne+8, Ne+9, N+10Ar, Ar+, Ar+2, Ar+3, Ar+4, Ar+5Ar+6, Ar+7, Ar+8,

Ar+9, Ar+10, Ar+11, Ar+12, Ar+13, Ar+14, Ar+15Ar+16, Ar+17, Ar+18, e−

1 N2: 78.084%, O2: 20.946, Ar: 0.9335%, CO2: 0.03398%, Ne: 0.001818% and He: 0.000702%.

characterized by lots of chemical reactions all actively con-tributing to determine the equilibrium composition. Un-der such conditions, the accurate approach to compute theequilibrium composition for given temperature and pres-sure is a complete equilibrium calculation [42], namely,Gibbs free energy minimization constrained by elementalconservation and electrical neutrality. The minimizationalgorithm used here is based on Lagrange’s multipliers asimplemented in the RAND method [33,34]. The thermo-dynamic properties for the molecules and molecular ionsare taken from NASA database [31] up to 20 000 K.

3.2 Ionization temperature range

At even higher temperatures the gas mixture is character-ized by the formation of positively charged ions and un-bound electrons created by the ionization reactions. Ionsand electrons have a significant effect on plasma propertiesat high temperatures and because of that a rigorous sta-tistical model which is described in the following sectionshas been developed.

3.2.1 Thermodynamic properties of individual monoatomicspecies

The important step in calculating the high temperatureproperties of plasma mixtures is to calculate the pure-substance thermodynamic properties of the individualmonoatomic species participating in the ionization reac-tions, i.e., neutral atoms, positive atomic ions, and elec-trons. For gaseous species the thermodynamic functionsmay be calculated from spectroscopic constants using thepartition function concept [43]. Using the statistical ther-modynamics formulation, the partition functions and theirfirst and second derivatives discussed in detail in Sec-tion 3.2.2, we can compute all the pure-substance ther-modynamic properties. The specific enthalpy (kJ/kg) andspecific entropy (kJ/kg K) of the pure ith species are given

respectively by

hii (T, P ) = 2.5RT

Mi+

RT 2

Mi

1Q∗

ei

∂Q∗ei

∂T, (3)

sii (T, P ) = 2.5R

Mi+

RT

Mi

1Qei

∂Qei

∂T

+R

Miln

[Qei

(2πMi

NA

)3/2

(kT )5/2 h−3p P−1

]

(4)

where, following reference [42] the double subscript iiin hii and sii is used to denote pure-substance specificproperties, and to distinguish them from the partial prop-erties hi and si of the same component in the mixture;moreover,

Q∗ei

= Qeie−ε∗ikT (5)

∂Q∗ei

∂T=[∂Qei

∂T+

Qei

kT 2

(ε∗i − T

∂ε∗i∂T

)]e

−ε∗ikT (6)

where ε∗i is the energy of formation of the ith species, Qei isthe electronic partition function of the ith species, Mi is itsmolar mass, R the universal gas constant, NA Avogadro’snumber, k the Boltzmann constant, hp the Planck con-stant, P the pressure and T the gas temperature. Thepure-substance specific Gibbs free energy (kJ/kg) is cal-culated from enthalpy and entropy, gii = hii − Tsii, i.e.,

gii (T, P ) = μii (T, P ) =R

kMi

(ε∗i − T

∂ε∗i∂T

)

− RT

Miln

[Qei

(2πMi

NA

)3/2

(kT )5/2 h−3p P−1

]

(7)

where the first equality is a reminder that for the puresubstance the specific Gibbs free energy is equal to thechemical potential. The pure-substance specific heat atconstant pressure is evaluated as the partial derivative

Eur. Phys. J. D (2016) 70: 159 Page 5 of 22

of the specific enthalpy with respect to temperature,

cpii (T, P ) = 2.5R

Mi+ 2

RT

Mi

1Q∗

ei

∂Q∗ei

∂T

− RT 2

Mi

(1

Q∗ei

)2(∂Q∗ei

∂T

)2

+RT 2

Mi

1Q∗

ei

∂2Q∗ei

∂T 2,

(8)

where∂2Q∗

ei

∂T 2=[∂2Qei

∂T 2+

2kT 2

(ε∗i − T

∂ε∗i∂T

)∂Qei

∂T

+1

k2T 4

(ε∗i − T

∂ε∗i∂T

)2

Qei

− 2kT 3

(ε∗i − T

∂ε∗i∂T

)Qei −

1kT

∂2ε∗i∂T 2

Qei

]e

−ε∗ikT

(9)

and we note that, differently from the standard idealgas model, the enthalpy and the specific heat here de-pend, though slightly, on pressure through the pressure-dependent cut-off criterion that we adopt for the electronicpartition function, as explained in the next section.

3.2.2 Partition function

To calculate the pure-substance thermodynamic proper-ties and the equilibrium mass fractions of all the speciespresent in the plasma, the knowledge of partition functionsand their derivatives are prerequisites. In general, the par-tition function of a molecular system can be expressed bytranslational and internal contributions

Q = QtrQint. (10)

The internal contribution may be due to the rotational,vibrational, and electronic motions within the particle(Qint = QrQvQe). For an atomic system (atoms, atomicions and electrons) the rotational and vibrational parti-tion functions take the value of unity, so translational andelectronic partition functions for such species are:

Qtr =(

2πmkT

h2p

)3/2RT

P(11)

Qe =∑

n

gne−εnkT =

∑n

(2Jn + 1) e−εnkT (12)

where m is the mass of the molecule, εn is the electronicenergy of the nth level of the species under consideration,gn its statistical weight, and Jn the corresponding angularmomentum quantum number.

However, the summation in equation (12) diverges be-cause the statistical weight increases rapidly with thenumber of energy levels (e.g., for hydrogen atoms, gn ∝n2). This behavior is correct just for a hypothetical iso-lated atomic species, while interactions with other speciesin the plasma mixture limit the number of energy levels.So an appropriate cut-off criterion is required for the ter-mination of electronic partition function of atomic speciesto define an upper limit for the aforementioned series.

3.2.2.1 Cut-off criteria

The only problem to calculate the pure-substance ther-modynamic properties of plasma mixtures is that the se-ries for the electronic partition function of atomic speciesdoes not have an upper bound. The exponential terms inequation (12) approach a finite limit corresponding to theionization potential, which is the upper bound to the en-ergy but the statistical weight increases as the square ofthe number of energy levels and consequently the seriesdiverges. On the other hand, it has been observed experi-mentally that as the temperature increases, due to a polar-izing effect of neighboring charged particles, the ionizationpotential of particles in a plasma is lowered [44]. In orderto prevent numerical divergence and match the empiricallyobserved lowering of the ionization potential, a criterionis needed for terminating the series of the electronic par-tition function. A review of various cut-off criteria can befound in reference [45,46]. These cut-off criteria may besummarized as the following types:

1. no dependence on temperature or pressure [46,47];2. dependence on temperature only [46,48];3. dependence on temperature and pressure (or number

density) [12,32].

The ionization potential of an atomic species in the pres-ence of other ions and electrons is decreased due to theaction of the Coulomb fields. This reduction depends onthe number densities of the charged particles or the gaspressure. This means that the pure-substance thermody-namic properties of single atomic species in the ioniza-tion range depend on both temperature and pressure. Thewell-known Griem model [32,49] adopts the cut-off crite-rion to include only energy levels below reduced ionizationpotential,

εn � IP − ΔIP. (13)

In current work the observed energy levels reported byMoore [50] and NIST1 have been completed with theRydberg-Ritz formulas using the isoelectronic sequencemethod [51]. For the atomic species including only oneelectron which are called hydrogenic species (H+, He+,C+6, N+7, O+8, . . . ) the statistical weight (gn) and energy(εn) of nth level have been evaluated using the followingrelations,

gn = 2n2, (14)

εn = IP

(1 − 1

n2

). (15)

3.2.2.2 Reduced ionization potential

The reduced ionization potential, ΔIP adopted by theGriem model [32,49] depends on the plasma compositionvia the Debye-Huckel length lD and by making a self-consistent solution for the problem. The reduction of theionization potential of an atomic specie of charge z is

ΔIP i =(zi + 1) e2

lD, (16)

1 http://www.nist.gov/pml/data/asd.cfm (2015).

Page 6 of 22 Eur. Phys. J. D (2016) 70: 159

where the Debye length is defined as

lD =

⎡⎢⎢⎣ kT

4πe2

(NS∑i=1

Niz2i

)⎤⎥⎥⎦

1/2

, (17)

where e is the charge of an electron, NS is the number ofall species present in the plasma including electrons andNi is the number density of ith species. Equation (16)applies for densities and temperatures for which the De-bye theory is valid. Griem [32] and Cooper [52] derive thecriterion for the validity of the Debye theory as

NS∑i=1

Ni � 18πl3D

. (18)

3.2.3 Complete equilibrium solution based on Gibbsfree energy minimization

This method is based on Gibbs free energy minimizationunder the assumption of Gibbs-Dalton mixture of idealgases [42] and subject to the constraints of element con-servation and electrical neutrality [33,34]. To determinecomplete equilibrium composition at a given temperatureand pressure, the Gibbs free energy of the mixture mustbe minimized, subject only to the constraints of elementconservation, electrical neutrality, and non-negativity ofthe mole numbers, i.e.,

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Ns∑i=1

aijni = qjnCkHlj = 1, . . . , m

Ns∑i=1

ai0ni = 0

ni � 0 ∀i

(19)

where nCkHlis the mole number of hydrocarbon CkHl

in the initial fuel-air mixture, ni is the mole number ofspecies i in the equilibrium plasma mixture, Ns the num-ber of species present in the plasma mixture, m the num-ber of different elements, aij represents the number of el-ements of type j that compose the atom or molecule ofspecies i so that qj (dimensionless) represents the (fixed)number of elements of type j in the mixture per unit molenumber of hydrocarbon in the initial fuel-air mixture, andai0 represents the electrical charge of species i. In whatfollows we will denote by q the vector q1, . . . , qm of thegiven amounts of the elements in the mixture per unitamount of initial fuel. For an initial CkHl-air mixture withequivalence ratio φ and molar composition of air given byxa

N2N2+xa

O2O2+xa

H2OH2O+xa

ArAr+xaCO2

CO2+xaNeNe+

xaHeHe the values of the qj ’s are listed in Table A.1 in

Appendix.Under the assumption of Gibbs-Dalton mixture of

ideal gases, the Gibbs free energy of the reacting mix-ture at a generic non-equilibrium composition can be

written as:

G (T, P, {ni}) =Ns∑i=1

niμi,off (T, P, {ni})

=Ns∑i=1

ni

⎡⎢⎢⎢⎣μ0

ii (T ) + RT ln

⎛⎜⎜⎜⎝ ni

Ns∑j=1

nj

⎞⎟⎟⎟⎠

+ RT ln(

P

P 0

)⎤⎥⎥⎥⎦ , (20)

where, following [42], μi,off(T, P, {ni}) denotes the chem-ical potential of species i in the so-called “surrogate sys-tem”, namely, the frozen-composition non-reacting mix-ture (hence the “off” subscript) at stable equilibrium withthe same temperature T , pressure P , and composition{ni} as the actual non-equilibrium state of the reactingmixture, and μ0

ii is the chemical potential of pure species iat temperature T and the standard pressure P 0 = 1 atm,which we have defined in terms of partition functions inthe preceding section.

The minimization is done using an equilibrium compo-sition solver based on the well-known algorithm developedin reference [34] and convergence is considered satisfiedbased on the following condition:

Δni

ni< 10−15 ∀i. (21)

As shown in reference [42], the solution can be formallyexpressed in terms of a set of m + 1 Lagrange multipliersλj(T, P, q), with j = 0, 1, . . . , m, so that the complete-equilibrium mole fractions are given by

xi(T, P, q)=P 0

Pexp

⎡⎣ 1

R

m∑j=0

λj(T, P, q)aij− 1RT

μ0ii(T )

⎤⎦.

(22)Once the mole fractions xi are obtained, the mass fractionsare readily found from yi(T, P, q) = xiMi/

∑k xkMk.

3.2.3.1. Iterative solution

The iterative solution to find the mole fractions of allspecies at a given temperature, pressure, and elementalcomposition proceeds as follows. Using the last calculatedmole fractions, we estimate the number densities Ni usingthe ideal gas law

Ni(T, P, q) =ni

V=

ni

n

P

RT= xi (T, P, q)

P

RT(23)

to evaluate (i) the Debye length; (ii) the lowered ioniza-tion potential; and (iii) the maximum number of energylevels based on the cut-off criterion. Then the partitionfunction is calculated by summing over all such energy

Eur. Phys. J. D (2016) 70: 159 Page 7 of 22

levels. The species mole fractions are then re-calculatedusing these partition functions and the equilibrium com-position solver. The new mole fractions are then used anda new Debye length is determined. In general, the previ-ous value of the Debye length and the new calculated valuedo not agree. Therefore, the new value is used to computeagain an improved partition function, and the process isrepeated until convergence is reached on the value of theDebye length.

3.3 Mixture thermodynamic properties

Once the complete-chemical-equilibrium mole fractions ofall the species are obtained, the (mass) specific prop-erties of the plasma mixture are determined using theGibbs-Dalton mixture model. Showing explicitly all de-pendences, we have

h (T, P, q) =NS∑i=1

yi(T, P, q)hii (T, P ) (24)

s (T, P, q) =NS∑i=1

yi (T, P, q)

×(

sii (T, P ) − R

Miln (xi (T, P, q))

)(25)

g (T, P, q) = h (T, P, q) − Ts (T, P, q) . (26)

The mean molar mass is calculated as

Mt (T, P, q) =

[NS∑i=1

yi (T, P, q)Mi

]−1

(27)

The mass specific gas constant is

R (T, P, q) =R

Mt (T, P, q). (28)

The density and the mass specific volume v(T, P, q) =1/ρ(T, P, q) can be obtained from the relation

ρ (T, P, q) ∼= Mt (T, P, q)P

RT(29)

and therefore the (mass) specific internal energy isgiven by:

u (T, P, q) =NS∑i=1

yi(T, P, q) uii (T, P )

∼= h (T, P, q) − RT

Mt (T, P, q). (30)

In the last two relations we use the ’approximately equalto’ symbol because the electronic partition functions de-pend (slightly) on pressure through the cut-off criterion,and therefore neither the individual species nor the mix-ture strictly obey the ideal gas equations of state. How-ever, we have verified that the departure is essentially neg-ligible for all species. The specific heat for oxygen atom

and its ions versus temperature for three different pres-sures of 10−6, 1 and 100 atm are plotted in Figure 3. Asit is obvious, the effect of pressure is very negligible es-pecially for the ions, which are the main species at hightemperatures.

To compute specific heats at constant pressure andvolume, cp,off and cv,off , the specific heat ratio γoff , theisoentropic exponent γs,off , and the speed of sound χoff ,we must recall that according to the standard model ofchemical kinetics (again, for a fully explicit discussionsee [42]) these properties are generally defined at arbi-trary compositions in terms of the so-called “surrogatesystem”. Here, the plasma composition is that of completechemical equilibrium, hence the surrogate system is thefrozen-composition, non-reacting mixture at stable equi-librium with the given temperature T and pressure P ,and the (fixed) composition (T, P, q). Therefore, the par-tial derivatives with respect to temperature must be eval-uated keeping the γi’s fixed, so that we have

cp,off (T, P, q) =(

∂h

∂T

)P,y

=NS∑i=1

yi (T, P, q) cpii (T, P ),

(31)

cv,off (T, P, q) =(

∂u

∂T

)v,y

=NS∑i=1

yi (T, P, q) cvii (T, P )

= cp,off (T, P, q) − R

Mt (T, P, q), (32)

γoff (T, P, q) =cp,off (T, P, q)cv,off (T, P, q)

=(

1 − R

Mt (T, P, q) cp,off (T, P, q)

)−1

,

(33)

γs,off (T, P, q) =ρ

P

(∂P

∂ρ

)s,y

= γoffρ

P

(∂P

∂ρ

)T,y

= γoff (T, P, q) , (34)

χoff (T, P, q) =

√(∂P

∂ρ

)s,y

=

√γoff (T, P, q)RT

Mt (T, P, q), (35)

with y the mass fraction, and where of course, we used theMeyer relations, cpii = cvii + R/Mi. Equilibrium specificheats at constant pressure will be calculated as:

cp,eq =(

∂h

∂T

)P,q

=NS∑i=1

yicpii (T, P )

+NS∑i=1

hii (T, P )(

∂yi

∂T

)P,q

. (36)

All of the above properties can be easily computed ifthe following three properties are known as functions oftemperature T and pressure P , for the given elemental

Page 8 of 22 Eur. Phys. J. D (2016) 70: 159

Fig. 3. Specific heat at constant pressure for oxygen atom and its ions versus temperature for three different pressures, 10−6,1 and 100 atm.

composition q

g = g (T, P, q) (37)Mt = Mt (T, P, q) (38)

cp,off = cp,off (T, P, q) . (39)

Indeed, for example, from the latter we can find

s = −∂g (T, P, q)∂T

,

cp,eq

T=(

∂s

∂T

)P,q

= −∂2g (T, P, q)∂T 2

,

(∂s

∂P

)T,q

= −∂2g (T, P, q)∂T∂P

(40)

and similarly from h = g + Ts and u = h − RT/Mt

we may find all the other partial derivatives. It is forthis reason that in Section 3.5 we propose correlationsfor the relations (37), (38) and (39) (see Eqs. (47), (48)and (43) respectively). The coefficients for these correla-tions have been obtained by running the complete chem-ical equilibrium composition calculations for various tem-peratures, pressures, and initial compositions, so as tocompute the mass fractions yi(T, P, q) needed in the rela-tions (26), (27) and (31), which lead to relations (37), (38)and (39). In Section 3.5 we discuss the functional formschosen to correlate efficiently these relations. The coeffi-cients for hydrogen-air and methane-air plasmas are listedin Appendix.

3.4 Ideal gas model validation

As part of the preliminary computations, we checked thevalidity of the ideal gas model assumption. In order to con-sider an ionized gas as an ideal one, it is necessary that theenergy of the Coulomb interaction between neighboringparticles be small in comparison with the thermal energyof the particles [53], meaning

Nt �(

kT

Λ2e2

)3

= 2.2 × 1014

(T

Λ2

)3 [m−3

], (41)

where Nt is the number density of the plasma mixture, Λis the degree of ionization defined as

Λ =NI

NI + Nn, (42)

where NI is the number density of ions and Nn isthe number density of neutral atoms. The virial correc-tions to thermodynamic properties are negligible espe-cially for temperatures higher than 2000 K and pressuresup to 1000 atm [54,55]. Virial corrections have very sig-nificant effects for very low temperatures and very highpressures [56] meaning far from plasma conditions. In-equality (41) is comfortably satisfied for all plasmas atall pressure and temperatures considered in this work.

3.5 Fitting of thermodynamic properties

It is desired to have analytical expressions to evaluate thethermodynamic properties of a plasma mixture without

Eur. Phys. J. D (2016) 70: 159 Page 9 of 22

iteration. Such expression would be valuable for exam-ple as a subroutine in a computer program or a CFDcode. Fitting thermodynamic functions for plasma mix-ture properties in a wide range of temperatures (1000–100 000 K) and pressures (10−6–102 atm) is a complicatedproblem because of the non-monotonic behavior of someproperties as functions of temperature, in particular, theequilibrium specific heat cp,eq. In order to correlate thecomputed values of cp,eq, we have chosen a modified Hillequation in conjunction with a modified Log-normal dis-tribution function. These functions provide the capabilityto capture the peaks and valleys of the equilibrium spe-cific heat. The number of terms for first (modified Hill)part of the cp,eq correlation depends on how well we canfit cp off to the exact data points and the number of termsfor the second (modified Log-normal) part depends on thenumber of peaks in cp,eq.

The least squares fitting procedure is as follows. Firstwe find the coefficients for cp,eq, then we find the coef-ficients for the integration constants of the specific en-thalpy and the specific entropy obtained by integration ofthe functional form of the correlation for cp,eq accordingto h =

∫cp,eqdT + const and s =

∫(cp,eq/T )dT + const

respectively. The integration constants are obtained byleast square fitting the calculated data for enthalpy andentropy to the correlations developed by integration. Theadvantage of this approach is that the main set of coef-ficients for the equilibrium specific heat, the specific en-thalpy and the specific entropy are the same and are con-sistent with the thermodynamic relations provided in theprevious sections.

Below, we present the analytical expressions that wepropose for the correlations of the frozen and equilibriumspecific heat at constant pressure, the specific enthalpy,the specific entropy, the specific Gibbs free energy, and themean molar mass. The results of the least square fittingsof our calculated data using these correlations are given inAppendix. The number of terms in the summations havebeen chosen so that the relative errors of the correlationsare always less than 2%.

Frozen specific heat at constant pressure (kJ/kg K)

cp,off =8∑

i=1

aoffi

1 +(boffi /T

)coffi

. (43)

Equilibrium specific heat at constant pressure (kJ/kg K)

cp,eq =8∑

i=1

aoffi

1 +(boffi /T

)coffi

+16∑

i=9

aeqi exp

[−(

ln (T/beqi )

ceqi

)2]. (44)

Specific enthalpy (kJ/kg)

h = λ1+T

8∑i=1

aoffi

(1−2F1

(1,

1coffi

, 1+1

coffi

,−(

T

boffi

)coffi

))

−√

π

2

16∑i=9

aeqi beq

i ceqi exp

((ceq

i )2

4

)

× erf(

ceqi

2− ln (T/beq

i )ceqi

). (45)

Specific entropy (kJ/kg K)

s = λ2 +8∑

i=1

aoffi

coffi

ln((

boffi

)coffi + (T )coff

i

)

+√

π

2

16∑i=9

aeqi ceq

i erf(

ln (T/beqi )

ceqi

), (46)

where erf and 2F1 (a, b, c, x) are the error function andthe hypergeometric functions2, respectively. As a result ofequations (45) and (46), the specific Gibbs free energy is

g = λ1 − Tλ2

+ T8∑

i=1

aoffi

(1−2F1

(1,

1coffi

, 1+1

coffi

,−(

T

boffi

)coffi

))

− T

8∑i=1

aoffi

coffi

ln((

boffi

)coffi + (T )c

offi

)

−√

π

2

16∑i=9

aeqi beq

i ceqi exp

((ceq

i )2

4

)

× erf(

ceqi

2− ln (T/beq

i )ceqi

)

− T√

π

2

16∑i=9

aeqi ceq

i erf(

ln (T/beqi )

ceqi

). (47)

Mean molar mass (kg/kmol)

Mt = λ3 −8∑

i=1

aMi

1 +(bMi /T

)cMi

, (48)

where the various coefficients are function of pressureand equivalence ratio using polynomial surface (PSmn)

2 http://www.mathworks.com/matlabcentral/

fileexchange/43865-gauss-hypergeometric-function/

content/hyp2f1mex/hyp2f1.m, p. 43865.

Page 10 of 22 Eur. Phys. J. D (2016) 70: 159

Fig. 4. Comparison between calculated data (solid line) andfitted data (symbols) for the equilibrium specific heat at con-stant pressure cp,eq of a stoichiometric H2/air plasma mixturefor P = 10−6, 1 and 102 atm.

of degree m in φ and degree n in ln(P ) as follows,

PSmn = ξ00 +m∑

i=1

ξi0φi +

n∑j=1

ξ0j (ln (P ))j

+

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

m∑k=1

n−k∑l=1

ξklφk (ln (P ))l for m < n

n−1∑k=1

n−l∑l=1

ξklφk (ln (P ))l for m = n

n∑l=1

m−l∑k=1

ξklφk (ln (P ))l for m > n

(49)

a, b, c = exp (PSmn) (50)λ = PSmn. (51)

The degree of m and n for methane/air mixture are con-sidered 2 and 5, respectively. For hydrogen/air mixturewe have two sets of degree, first one is m = 4 and n = 3and the second one is m = 5 and n = 2 to provide thebest fit. In this way all the thermodynamic properties areexpressed as a function temperature, pressure and equiv-alence ratio. The values of the ξkl coefficients that resultfrom our least square fitting procedure are reported inAppendix for H2/air and CH4/air plasma mixtures in Ta-bles A.2 and A.3, respectively. Figure 4 shows the compar-ison of the equilibrium specific heat at constant pressurefor H2/air plasma mixtures between calculated and fitteddata which indicates excellent agreement.

4 Results and discussion

In this section we present and discuss some of our cal-culated chemical equilibrium compositions and thermo-dynamic properties for H2/air and CH4/air plasmas inthe temperature range 1000–100000 K, pressure range

Fig. 5. Comparison of values of the equilibrium specific heatat constant pressure cp,eq computed using our self-consistentmethod (solid line) and the so-called ground state method(dashed line), for a stoichiometric H2/air plasma at three dif-ferent pressures, 10−6, 1 and 100 atm.

10−6–102 atm, and for different equivalence ratios withinflammability limits (for methane/air mixture flammabil-ity limit is 0.6 < φ < 1.4 and for hydrogen/air mixtureflammability limit is 0.5 < φ < 5.0). Calculated valueshave been fitted using the analytical correlations proposedin the previous section and the fitting coefficients are tab-ulated in Appendix.

Figure 5 shows an important effect that must be care-fully taken into account in the interest of accuracy, namely,the effect of the excited energy levels on the specific heatat constant pressure for stoichiometric H2/air plasma mix-ture. Figure 5 compares the results obtained using our self-consistent method considering excited energy levels withthose of the so-called ground state method. As it can beseen, for temperatures below 15 000 K the effect of excitedenergy levels on the values of the specific heat and henceall the properties is negligible, because at relatively lowtemperatures the high order terms in the electronic par-tition function expansion are indeed negligible. But fortemperatures above 15 000 the effect is significant and,importantly, it exhibits a strong pressure dependence. Thereason is that, differently from the ground state method,our self-consistent method takes into account the experi-mental observation that the number of excited energy lev-els is a function of both temperature and pressure. In otherwords, Figure 5 shows that considering just the groundstate or fixing the number of excited energy levels inde-pendent of pressure, may introduce very large errors inthe estimated thermodynamic properties at high plasmatemperatures, especially for derivative properties like thespecific heats.

In order to validate our calculations with data existingin the literature for high temperature plasmas, due to lackof data for H2/air and CH4/air plasma mixtures at hightemperatures, we make a comparison with air plasma. Thereasons to choose air are that first it forms a large por-tion of typical fuel/air mixtures and also many researches

Eur. Phys. J. D (2016) 70: 159 Page 11 of 22

Fig. 6. Comparisons of the values of the equilibrium specificheats at constant pressure cp,eq of air plasma mixture obtainedwith present study (solid line) and those obtained by D’Angolaet al. [25] (◦), Hansen [6] (×), Sher et al. [19] (�), Cressaultet al. [57] (�) and Bottin [58] (+) for three different pressures:(a) = 10−2, (b) = 1, and (c) = 100 atm.

have already been done on air plasma. Figure 6 comparesthe equilibrium specific heat at constant pressure of airplasma calculated in this study with those of D’Angolaet al. [25] (up to 60 000 K), Hansen [6] (up to 15 000 K),Sher et al. [19] (up to 50 000 K), Cressault et al. [57](up to 30 000 K), and Bottin [58] (up to 15 000 K) forthree different pressures, 10−2, 1, and 100 atm. The datafor Sher et al. [19] and Cressault et al. [57] are availableonly for atmospheric pressure. The underestimate in Sheret al.’s results is due to the very simple method he usedto find individual properties. The results of this studyare shown an excellent agreement in low and high pres-sures with D’Angola et al. [25] (up to 60 000 K). Figure 7compares the equilibrium compositions of air plasma com-

Fig. 7. Comparisons of the selected species mole fraction ofair plasma mixture obtained with present study (dashed line)and those obtained by Gilmore [2] (�) and Hilsenrath andKlein [54] (◦) at pressure of 1 atm.

puted in this study with tabulated data by Hilsenrathand Klein [54] (up to 15 000 K) and Gilmore [2] (up to24 000 K) at atmospheric pressure for temperature rangeof 300 to 25 000 K. As it is obvious from Figures 6 and 7,the results of our calculations are in very good agreementwith existing data, in spite of the slight differences in aircomposition assumed in the different studies. We are nowin the position to examine our results to illustrate the im-portant effects of temperature and pressure on all prop-erties. Figure 8 shows the mole fractions of the neutraland ionized single species (only those with mole fractionsgreater than 2×10−6) for a stoichiometric CH4/air plasmamixture at atmospheric pressure.

Figure 9 shows the mean molar mass Mt and the de-gree of ionization Λ, over the temperature range 1000–100 000 K for different pressures (10−6, 1, and 100 atm)for a stoichiometric H2/air plasma. As shown in Figure 9a,increasing the temperature at a given pressure leads toa decrease in mean molar mass due to the increase inmole numbers resulting from dissociation. The stepwisedecrease in the molar mass is connected to the successive

Page 12 of 22 Eur. Phys. J. D (2016) 70: 159

Fig. 8. Mole fractions for selected species in a stoichiometricCH4/air plasma at atmospheric pressure.

ionization of atomic species, while the slope becomes zeroin transitions from one ionization to another. This be-havior is very manifest at low pressures and becomes lessevident at high pressures. The dissociation reactions aresignificantly favored at low pressures and consequently thetransition from partially ionized gas (Λ < 1) to fully ion-ized gas (Λ = 1) takes place at lower temperature for lowpressures than for high pressure. This is seen in Figure 9b,where the fully ionized condition for P = 10−6 atm ob-tains around 10 000 K whereas for P = 100 atm it occursaround 50 000 K.

Figure 10a shows the equilibrium specific heat atconstant pressure for a stoichiometric CH4/air plasmamixture. It shows some distinct peaks with increasing tem-perature where the temperature dependence of the com-plete chemical equilibrium composition is very high. Thesepeaks are connected first to dissociation of the molecules

Fig. 9. (a) Mean molar mass Mt and (b) degree of ioniza-tion Λ at three different pressures (10−6, 1 and 100 atm) for astoichiometric H2/air plasma mixture.

Fig. 10. (a) Equilibrium and frozen specific heat at constantpressure cp,eq and (b) specific enthalpy for a stoichiometricCH4/air plasma mixture at three different pressures, 10−6, 1and 100 atm.

and then to successive ionization of the atomic species.When pressure decreases the peaks become sharper andshift to lower temperature. This effect, again, is due tothe favorable effect of low pressure on dissociation andionization reactions. As a result, for each given temper-ature, more chemical energy is stored at low pressure,resulting in the specific enthalpy of the plasma mixturebeing a decreasing function of pressure, as shown in Fig-ure 10b. Figure 10a compares the frozen and equilibriumspecific heats at constant pressure, cp,off and cp,eq, definedby equations (31) and (36), respectively, for a stoichio-metric CH4/air plasma at different pressures. As it can be

Eur. Phys. J. D (2016) 70: 159 Page 13 of 22

Fig. 11. (a) Gibbs free energy and (b) equilibrium specificheat at constant pressure for a H2/air plasma mixture at at-mospheric pressure for three different equivalence ratio of 1, 3and 5.

seen, the effect of chemistry is clearly visible and when-ever dissociation and ionization start to play a role andbecome important, cp,eq presents a peak. When maximumionization is reached in the mixture, no further reactionstake place, so the rates of change of mole fractions withtemperature, (∂yi/∂T )P,q, become small and values ofequilibrium cp,eq decrease and reach the frozen cp,off val-ues. This behavior is shown in Figure 10a for the pressureof 10−6 atm around 75 000 K. It is important to em-phasize again that the pressure dependence of the mix-ture enthalpy and equilibrium specific heat has two sig-nificant reasons. One is the pressure dependence of thecomplete chemical equilibrium composition and its rate ofchange with temperature. In fact, chemical composition,computed with the complete equilibrium method whichis a function of pressure. The second reason is related tothe electronic partition functions of the individual species,which depend on the cut-off criteria that determine themaximum number of energy levels that need to be consid-ered. As discussed in Section 2.2.2.1, our cut-off criteriaare functions of the Debye length, which in turn is relatedto number densities and, hence, to pressure.

Figure 11 shows the effect of equivalence ratio on thespecific Gibbs free energy and the equilibrium specific heatat constant pressure for H2/air plasma mixture. At eachgiven temperature, the Gibbs free energy is a decreasingfunction of equivalence ratio whereas the equilibrium spe-cific heat at constant pressure increases with equivalenceratio. The effect on the specific heat is most significant at

Fig. 12. Speed of sound in a stoichiometric H2/air plasmamixture at three different pressures, 10−6, 1 and 100 atm.

Fig. 13. Specific heat ratio γoff and isentropic exponent γs,off

for a stoichiometric CH4/air plasma mixture at three differentpressures of 10−6, 1 and 100 atm.

the first and third peaks, around 3800 K and 15 000 K, re-spectively. As already seen in Figure 8, the 3800 K peak isdue to the dissociation reaction to create atomic hydrogen,while the 15 000 K peak is due to the ionization reactionto form ionic hydrogen. Increasing the equivalence ratiois equivalent to increasing the hydrogen percentage in themixture and therefore the more hydrogen in the mixturethe more hydrogen dissociation and ionization reactionscontribute to the mixture properties.

Figure 12 shows speed of sound χoff defined by equa-tion (35). Speed of sound is higher at low pressures, lead-ing to higher Mach numbers at low pressures for each giventemperature. The differences between frozen specific heatratio γoff defined by equation (33) and isentropic expo-nent γs,off defined by equation (34) are shown in terms ofpressure and temperature in Figure 13. As it can be seenin Figure 13, the frozen specific heat ratio is higher atlow pressures and the isentropic exponent is always lower

Page 14 of 22 Eur. Phys. J. D (2016) 70: 159

than its corresponding specific heat ratio. Under the con-dition of non-reacting flows or when maximum ionizationis reached in the mixture, the rates of change of massfractions with respect to temperature become negligibleand the ratio of specific heats and the isentropic exponentbecome identical as it is shown in Figure 13 for the pres-sure of 10−6 atm. For higher pressures this convergencehappens in higher temperatures.

5 Conclusion

A comprehensive model has been developed to cal-culate thermodynamic properties of hydrogen/air andmethane/air plasma up to 100 000 K for a wide range ofpressures and fuel/air equivalence ratios. The model isbased on statistical thermodynamics and complete chem-ical equilibrium of all species. For both hydrogen/airand methane/air plasma mixtures the model considers133 species. Properties such as enthalpy, entropy, Gibbsfree energy, specific heat at constant pressure, specific heatratio, speed of sound, mean molar mass, and degree of ion-ization have been calculated. The results have been com-pared to available experimental data and the agreement isexcellent. For each mixture and fuel/air equivalence ratioconsidered, the properties have been summarized in theform of curve-fitted correlations as suitably defined func-tions of temperature, pressure and equivalence ratio. Inaddition, the results have also allowed the following con-clusions relevant for other existing state-of-the-art modelsof thermodynamic properties:1. considering just the ground state or fixing the number

of energy levels independently of the temperature andthe pressure produces very large errors in the estimates

Table A.1. Values of the qj ’s as a function of the equivalence ratio φ.

ElementalCkHl/air mixture

H2/air mixture CH4/air mixture

atom numbers (k = 0, l = 2) (k = 1, l = 4)

qN1

φ

(k +

l

4

)2xa

N2

xaO2

0.78084

0.20946φ

4 × 0.78084

0.20946φ

qO1

φ

(k +

l

4

)(2 +

xaH2O

xaO2

+2xa

CO2

xaO2

)1

φ+

0.0003398

0.20946φ

4

φ+

4 × 0.0003398

0.20946φ

qC k +1

φ

(k +

l

4

)xa

CO2

xaO2

0.5 × 0.0003398

0.20946φ1 +

2 × 0.0003398

0.20946φ

qH l +1

φ

(k +

l

4

)2xa

H2O

xaO2

2 4

qAr1

φ

(k +

l

4

)xa

Ar

xaO2

0.5 × 0.009335

0.20946φ

2 × 0.009335

0.20946φ

qNe1

φ

(k +

l

4

)xa

Ne

xaO2

0.5 × 0.00001818

0.20946φ

2 × 0.00001818

0.20946φ

qHe1

φ

(k +

l

4

)xa

He

xaO2

0.5 × 0.00000702

0.20946φ

2 × 0.00000702

0.20946φ

especially for second order properties at high temper-ature conditions. In particular, the accuracy of theground state method accuracy decreases with increas-ing the pressure and temperature;

2. transition from partially ionized gas (Λ < 1) to fullyionized gas (Λ = 1) takes place at relatively low tem-peratures;

3. the speed of sound is higher at low pressures, leading tohigher Mach numbers at low pressures for each giventemperature.

This paper was made possible by NPRP award [NPRP 7-1449-2-523] from Qatar National Research Fund (a member of theQatar Foundation). This statement is solely the responsibilityof the authors.

Appendix: Fitting coefficients for hydrogen/airand methane/air plasma mixtures

In this supplementary material we present the coeffi-cients to calculate frozen and equilibrium specific heat atconstant pressure, specific enthalpy, specific entropy, andmean molar mass, according to equations (43), (44), (45),(46) and (48), respectively, for hydrogen-air and methane-air plasma mixtures. For our calculations and correlations,we assumed the following molar composition of air (nowater vapor): xa

N2= 0.78084, xa

O2= 0.20946, xa

H2O = 0,xa

Ar = 0.009335, xaCO2

= 0.0003398, xaNe = 0.00001818,

xaHe = 0.00000702. Table A.1 gives the corresponding val-

ues of the qj ’s that we considered, as functions of theequivalence ratio φ. Tables A.2 and A.3 show the fittingcoefficients of hydrogen/air and methane/air plasma mix-ture, respectively.

Eur. Phys. J. D (2016) 70: 159 Page 15 of 22

Table

A.2

.H

2/air

pla

sma.

c p,o

ff(k

J/kg

K),

c p,e

q(k

J/kg

K),

h(k

J/kg),

s(k

J/kg

K)

ξ 00

ξ 10

ξ 01

ξ 20

ξ 11

ξ 02

ξ 30

ξ 21

ξ 12

ξ 03

ξ 40

ξ 31

ξ 22

ξ 13

aoff

10.3

25766

0.2

95999

−0.0

10454

0.1

7606

0.0

16251

0.0

00125

−7.5

1E

-02−0

.004551

−0.0

00141

2.0

2E

-06

0.0

07738

3.9

2E

-04

1.5

5E

-06

6.1

1E

-07

aoff

2−0

.356532

0.2

0411

−0.0

14564−0

.321786

0.0

53117

−0.0

09868

0.0

87119

−0.0

21719

0.0

04184

−0.0

0035

−0.0

06685

2.4

2E

-03

−1.6

3E

-04

1.0

5E

-04

aoff

30.1

49062

0.4

82639

0.0

1784

−0.1

41731−0

.005376

0.0

04729

0.0

31576

0.0

02638

−0.0

01093

0.0

00193

−2.8

4E

-03−2

.99E

-04

3.6

8E

-05

−4.4

2E

-05

aoff

40.4

92105

−0.0

27053

0.0

09439

−0.0

7043

−0.0

16436

3.6

6E

-04

0.0

2252

0.0

01266

−0.0

01508

4.4

4E

-05

−2.1

3E

-03−6

.75E

-05

3.6

6E

-05

−5.5

4E

-05

aoff

50.4

95306

0.0

46521

−0.0

27897−0

.065072

0.0

05746

−1.3

7E

-04

0.0

17576

−0.0

00864

9.9

6E

-04

2.3

8E

-05

−0.0

01632

−4.3

1E

-05−7

.25E

-05

3.7

4E

-05

aoff

6−0

.246024

0.2

60661

−0.1

11618−0

.240513

0.0

32134

−0.0

19551

0.0

72196

0.0

00282

5.7

0E

-03

−0.0

00876−6

.88E

-03−2

.39E

-04−1

.77E

-04

2.0

8E

-04

aoff

7−0

.052569

0.5

00379

0.0

08356

−0.2

95294

0.0

39514

−0.0

00679

0.0

58712

−0.0

15687

0.0

00776

−6.9

4E

-05−4

.01E

-03

1.5

7E

-03

−0.0

00109

2.4

9E

-05

aoff

8−1

.316139

−0.6

3594

0.0

02881

0.2

08148

−0.0

47142−0

.005715

−0.0

21114

0.0

235

0.0

03315

−0.0

00247

−0.0

00488

−2.8

7E

-03−4

.53E

-05

1.6

9E

-04

boff

15.7

23293

1.0

2107

−0.0

02698−0

.511197

0.0

05373

−4.8

5E

-05

0.1

09718

−0.0

0247

0.0

00164

6.6

0E

-06

−0.0

08195

3.1

2E

-04

−5.7

6E

-05−4

.07E

-06

boff

28.9

52488

−0.0

65389

0.0

38513

0.0

44519

0.0

07632

−0.0

01154

−0.0

13942−0

.005278

0.0

02049

−7.4

6E

-05

1.7

4E

-03

7.0

8E

-04

−1.6

6E

-04

7.6

0E

-05

boff

39.5

91297

0.0

05342

0.0

60889

−0.0

04212−0

.000505−8

.48E

-05

0.0

0092

−0.0

00135

1.7

9E

-05

−5.0

8E

-05−5

.58E

-05

2.6

3E

-05

4.0

0E

-06

4.0

5E

-06

boff

410.3

1002

−0.0

03176

0.0

61256

0.0

0335

0.0

00614

0.0

00559

−0.0

00694

1.2

3E

-04

6.0

0E

-05

−1.5

2E

-05

5.3

9E

-05

−7.5

2E

-06

2.5

3E

-06

1.0

5E

-06

boff

510.7

5484

−0.0

04941

0.0

66266

−0.0

0014

−0.0

00329

0.0

01644

0.0

00297

1.0

7E

-04

3.3

3E

-05

2.8

7E

-05

−3.9

9E

-05−1

.43E

-05−1

.69E

-06

1.8

1E

-06

boff

611.0

9656

0.0

05438

0.0

54597

−0.0

0577

0.0

02364

0.0

00248

0.0

01791

−0.0

0011

3.9

6E

-04

−2.9

0E

-05−1

.80E

-04−1

.59E

-05−1

.85E

-05

1.4

6E

-05

boff

711.3

3268

0.0

59837

0.0

43133

−0.0

35995

0.0

0807

−0.0

01844

0.0

08301

−0.0

01454

8.3

3E

-04

−0.0

00124−6

.66E

-04

1.1

0E

-04

−3.5

4E

-05

3.1

2E

-05

boff

811.0

3422

1.0

89476

−0.0

09898−0

.513685

0.1

09938

0.0

01625

0.0

83932

−0.0

41693

1.0

3E

-03

0.0

00145

−0.0

04116

4.2

0E

-03

−3.6

0E

-04−3

.94E

-05

coff

10.1

24054

0.8

40514

0.0

40843

−1.2

10174−0

.072285−5

.72E

-05

0.3

85634

0.0

19032

0.0

00658

5.7

8E

-06

−3.6

7E

-02−1

.57E

-03−3

.99E

-05

1.0

1E

-06

coff

22.2

71393

−0.1

72845−0

.019733

0.0

74026

0.0

32179

−0.0

03477

−0.0

21718−0

.009678

0.0

0277

−0.0

0016

0.0

01756

4.7

1E

-04

−1.3

5E

-04

8.9

3E

-05

coff

32.4

8469

−0.0

32591−0

.045116

0.0

46243

−0.0

0277

−0.0

06517

−0.0

1555

−0.0

00701

0.0

00415

−0.0

00372

1.6

3E

-03

1.4

4E

-04

−1.9

0E

-05

2.0

7E

-05

coff

42.6

01802

0.0

23603

−0.0

43885

0.0

47296

0.0

10316

−0.0

03402

−0.0

16312−0

.001063

7.9

6E

-04

−1.5

3E

-04

1.5

7E

-03

8.1

4E

-05

−9.9

7E

-06

3.3

6E

-05

coff

52.5

93675

−0.0

43896−0

.053577

0.0

29797

−0.0

08094−0

.000805

−0.0

06784

0.0

01325

−0.0

01015

5.5

2E

-05

0.0

00612

1.1

3E

-05

7.9

8E

-05

−3.5

1E

-05

coff

62.8

2907

0.0

23139

−0.0

28702

0.0

07746

−0.0

04316

0.0

03463

−0.0

07304−0

.002063−1

.31E

-03

0.0

00248

0.0

00934

2.5

8E

-04

3.8

8E

-05

−3.7

2E

-05

coff

72.6

17703

0.0

25931

−0.0

32416−0

.096552−0

.031419−0

.001768

0.0

42323

0.0

12062

−0.0

00329−5

.35E

-05−4

.70E

-03−9

.89E

-04

0.0

00225

3.0

5E

-05

coff

82.6

79309

0.1

88426

−0.0

78739−0

.038423

0.0

23823

−0.0

03399

−0.0

07311−0

.009759

0.0

01192

−8.7

7E

-05

0.0

01749

0.0

01131

−5.8

3E

-06

8.2

3E

-05

Page 16 of 22 Eur. Phys. J. D (2016) 70: 159

Table

A.2

.C

onti

nued

.

c p,o

ff(k

J/kg

K),

c p,e

q(k

J/kg

K),

h(k

J/kg),

s(k

J/kg

K)

ξ 00

ξ 10

ξ 01

ξ 20

ξ 11

ξ 02

ξ 30

ξ 21

ξ 12

ξ 03

ξ 40

ξ 31

ξ 22

ξ 13

aeq

11.2

58564

1.5

15934

−0.1

33356−0

.585117−0

.005996

−0.0

0369

1.1

5E

-01

0.0

01849

4.6

5E

-05

−5.4

8E

-05−8

.53E

-03−1

.75E

-04−2

.57E

-05−6

.61E

-06

aeq

22.4

95274

−0.0

06412

−0.1

01442−0

.001124

0.0

03565

−0.0

01871

−0.0

0025

−0.0

00344

0.0

00359

−2.6

0E

-05

7.3

8E

-05

5.3

1E

-05

9.2

5E

-07

1.4

4E

-05

aeq

32.9

2697

0.3

51267

−0.1

35045−0

.065859

0.0

01769

−0.0

04247

0.0

09786

−0.0

01006

0.0

00225

−1.0

1E

-04−6

.42E

-04

1.1

5E

-04

−5.0

5E

-05

2.7

2E

-06

aeq

43.2

3209

0.0

15997

−0.1

05329−0

.018725

0.0

02379

−0.0

03153

0.0

03897

−0.0

00443

0.0

00244

−7.8

4E

-05−2

.79E

-04

3.8

2E

-05

−2.0

6E

-05

5.1

2E

-06

aeq

53.4

12773

−0.0

11158

−0.0

9399

−0.0

04827

0.0

02356

−2.2

4E

-03

0.0

00617

−0.0

00623

1.4

6E

-04

−3.8

4E

-05−1

.19E

-05

3.6

6E

-05

−3.6

7E

-05−2

.98E

-06

aeq

63.5

06561

0.0

18226

−0.1

02517

−0.0

2345

0.0

06194

−0.0

02934

4.8

6E

-03

−0.0

01432

6.0

5E

-04

−7.9

6E

-05−3

.46E

-04

1.0

8E

-04

−5.7

2E

-05

1.6

5E

-05

aeq

73.5

06037

0.0

00662

−0.0

97295−0

.017872

0.0

0285

−0.0

02919

0.0

04808

−0.0

00183

6.1

7E

-04

−7.5

1E

-05−4

.27E

-04

8.7

8E

-06

−3.0

7E

-05

2.4

7E

-05

aeq

82.1

4331

−0.0

00928

−0.1

19999−0

.005795

0.0

02536

−0.0

05078

1.3

9E

-03

0.0

00209

2.9

3E

-05

−1.3

6E

-04−0

.000108

−2.7

1E

-05−4

.82E

-06−1

.16E

-05

beq

18.1

85678

0.0

02758

0.0

5985

0.0

14419

0.0

01688

0.0

01762

−4.7

0E

-03−7

.83E

-05

5.5

8E

-05

4.3

0E

-05

4.3

4E

-04

−1.5

4E

-05−9

.05E

-06−1

.08E

-06

beq

28.8

69649

−0.0

29608

0.0

58091

0.0

05957

−0.0

01627

0.0

01486

−8.5

9E

-04

0.0

00281

−8.0

9E

-05

3.4

6E

-05

5.1

3E

-05

−3.3

9E

-05−6

.57E

-06−6

.29E

-06

beq

39.6

04007

0.0

09

0.0

72687

−0.0

02042

0.0

0139

0.0

02126

0.0

00147

−0.0

00341

3.9

6E

-05

5.0

6E

-05

4.9

2E

-06

1.7

5E

-05

−1.6

7E

-05−2

.44E

-06

beq

410.3

1207

−0.0

10974

0.0

70462

0.0

05786

−0.0

0021

0.0

01827

−0.0

0126

3.7

2E

-05

−2.4

4E

-05

3.4

4E

-05

9.0

9E

-05

−7.2

8E

-06

1.5

0E

-06

3.1

9E

-08

beq

510.7

6373

−6.6

9E

-03

0.0

6755

0.0

03914

0.0

00246

0.0

01925

−1.2

0E

-03−2

.92E

-04−7

.93E

-05

4.4

5E

-05

1.2

2E

-04

3.8

3E

-05

4.0

7E

-06

−2.2

9E

-06

beq

611.0

8908

0.0

10521

0.0

64075

−0.0

05576

0.0

02253

0.0

01883

0.0

00692

−0.0

00794

7.3

3E

-05

4.3

4E

-05

−2.4

0E

-06

8.6

7E

-05

−4.3

1E

-06

2.2

8E

-06

beq

711.3

5902

−0.0

03179

0.0

643

0.0

00868

0.0

00499

0.0

01791

−1.9

9E

-04−0

.000105

3.2

1E

-05

4.1

1E

-05

1.5

8E

-05

1.0

3E

-05

9.7

1E

-07

1.5

9E

-06

beq

811.7

1941

−0.0

00954

0.1

00886

−0.0

01549

2.5

9E

-04

0.0

0645

7.0

9E

-04

2.1

3E

-05

5.0

1E

-05

2.1

0E

-04

−7.9

1E

-05−6

.90E

-06−7

.80E

-06−6

.52E

-07

ceq

1−1

.39116

−0.1

43516

0.0

94047

0.0

09853

−0.0

10312

0.0

01449

0.0

04833

0.0

00999

−0.0

00667−5

.16E

-05−6

.10E

-04−3

.78E

-05

4.7

9E

-05

−8.8

5E

-06

ceq

2−1

.596411−0

.187092

0.0

54767

0.1

41901

−0.0

07228

−0.0

01475

−0.0

37177

0.0

04669

0.0

00588

−0.0

00139

3.2

3E

-03

−5.9

6E

-04

5.6

0E

-06

4.0

5E

-05

ceq

3−1

.547451

0.0

66859

0.0

63709

−0.0

25208

0.0

07269

0.0

02743

0.0

04076

−0.0

02569

−0.0

00272

9.5

8E

-05

−2.2

8E

-04

2.5

7E

-04

1.9

0E

-05

−1.5

4E

-05

ceq

4−1

.736177

−0.0

4716

0.0

47029

0.0

17783

−0.0

16571

0.0

01021

−0.0

02458

0.0

04972

−0.0

00898−7

.14E

-06

6.9

8E

-05

−4.1

5E

-04

1.6

0E

-04

−1.6

3E

-06

ceq

5−1

.878835

0.0

80386

0.0

44049

−0.0

38666

0.0

05369

0.0

02548

0.0

06026

−0.0

02326

−0.0

00166

9.2

0E

-05

−2.3

4E

-04

2.8

0E

-04

1.9

5E

-05

−5.8

5E

-06

ceq

6−1

.965198−0

.013076

0.0

34078

0.0

06134

0.0

02581

−0.0

00673

−0.0

00811−9

.27E

-05

3.3

9E

-04

−5.0

9E

-05

7.5

9E

-06

1.7

8E

-05

2.9

1E

-05

2.1

9E

-05

ceq

7−2

.131654

0.3

94373

−0.0

01909−0

.264532

0.0

1199

−0.0

03503

0.0

66224

−0.0

05314−4

.40E

-05−0

.000177

−0.0

05564

6.3

0E

-04

7.2

4E

-06

−3.3

3E

-07

ceq

8−1

.65582

−0.0

68852

0.0

82272

0.0

20623

−0.0

01633

4.7

8E

-04

−0.0

01202

0.0

018

7.3

3E

-04

−0.0

00106

−1.0

1E

-04−2

.24E

-04−4

.79E

-05

2.4

3E

-05

λ1

1136593

162278.7

4919.2

63

−110923.4

8700.3

56

−739.1

671

27501.8

4−2

420.6

42

563.6

006

−58.3

6081

−2283.9

62

270.0

947

−15.1

6192

22.3

6501

λ2

−55.0

3018−0

.340312

−1.2

06588−4

.652112−1

.127481

0.0

90793

1.5

362

0.1

72984

−0.0

55993

0.0

03532

−0.1

48127

−0.0

10352

0.0

03772

−0.0

01654

Eur. Phys. J. D (2016) 70: 159 Page 17 of 22

Table

A.2

.C

onti

nued

.

Mt

(kg/km

ol)

ξ 00

ξ 10

ξ 01

ξ 20

ξ 11

ξ 02

ξ 30

ξ 21

ξ 12

ξ 03

ξ 40

ξ 31

ξ 22

ξ 13

aM 2

2.2

66963

−1.0

96096

−0.0

0123

0.2

74652

0.0

00149

−0.0

00236

−0.0

49081

0.0

00282

0.0

00132

−8.4

7E

-06

0.0

03594

−6.9

4E

-05

−1.1

6E

-05

5.0

0E

-06

aM 3

1.9

52006

−0.3

61525

−0.0

00424

0.0

58866

0.0

00428

1.3

7E

-05

−0.0

07083

−0.0

00103

−8.0

2E

-06

4.5

3E

-07

3.9

3E

-04

8.1

7E

-06

−8.1

9E

-07

−8.7

5E

-07

aM 4

0.8

60893

−0.6

65758

−0.0

01402

0.1

00323

−0.0

01688

−4.6

5E

-05

−0.0

12112

0.0

00394

1.1

1E

-05

−6.3

3E

-07

6.8

8E

-04

−1.6

6E

-05

1.5

4E

-05

5.3

4E

-06

aM 5

0.2

23394

−0.4

53833

0.0

04065

0.0

51948

0.0

00174

1.4

6E

-04

−0.0

05189

−5.4

9E

-05

2.4

6E

-06

−4.5

2E

-06

0.0

0026

3.9

2E

-06

−7.2

0E

-07

1.7

4E

-07

aM 6

−0.3

48842

−0.3

56119

0.0

03212

0.0

36125

−0.0

00909

0.0

00266

−0.0

03732

0.0

00157

−5.2

5E

-05

7.3

2E

-06

2.1

1E

-04

−5.9

4E

-06

8.9

6E

-06

2.0

5E

-07

aM 7

−0.6

38468

−0.2

55556

0.0

00653

0.0

07127

0.0

01264

−0.0

00129

0.0

01732

−0.0

0061

−1.0

2E

-05

−8.1

6E

-06

−1.7

5E

-04

8.4

5E

-05

2.3

3E

-06

−4.6

7E

-07

aM 8

−2.5

2645

−0.1

53855

−0.0

05414

−0.0

21514

0.0

07352

−0.0

00151

0.0

04023

−0.0

02828

1.2

6E

-05

−1.3

7E

-05

−6.6

7E

-05

3.3

3E

-04

4.9

5E

-06

2.7

7E

-07

bM 18.1

95886

−0.0

93007

0.0

60415

0.0

73774

2.1

4E

-05

0.0

0176

−0.0

19171

0.0

00454

2.9

7E

-05

3.9

9E

-05

0.0

0163

−6.2

8E

-05

7.7

0E

-07

7.9

3E

-07

bM 28.8

23667

0.0

04928

0.0

57076

−0.0

10969

−0.0

00867

0.0

01424

0.0

0319

3.9

4E

-05

−6.0

4E

-05

2.7

0E

-05

−2.8

5E

-04

3.1

9E

-06

1.8

4E

-06

−1.8

2E

-06

bM 39.5

43688

0.0

07416

0.0

71456

−0.0

01467

0.0

00837

0.0

02073

0.0

00139

−0.0

00143

1.2

6E

-05

4.4

1E

-05

−3.7

6E

-06

9.9

3E

-06

−1.0

8E

-06

3.4

3E

-08

bM 410.2

9095

0.0

06853

0.0

68933

−0.0

01976

5.1

3E

-05

0.0

0178

0.0

00386

1.0

7E

-05

−4.3

5E

-06

3.4

1E

-05

−3.0

5E

-05

−1.1

2E

-06

5.3

1E

-07

−2.1

4E

-07

bM 510.7

4839

−0.0

01804

0.0

64935

0.0

00328

−0.0

00475

0.0

01479

3.6

6E

-05

1.3

2E

-04

−9.4

5E

-06

2.4

4E

-05

−9.8

8E

-06

−8.9

4E

-06

3.7

4E

-06

6.2

7E

-07

bM 611.0

8649

−0.0

01236

0.0

62025

0.0

00507

0.0

00623

0.0

01417

−0.0

00169

−0.0

00108

6.8

6E

-05

2.2

9E

-05

1.4

6E

-05

7.8

1E

-07

−7.1

6E

-06

1.7

0E

-06

bM 711.3

8418

−0.0

42498

0.0

72513

0.0

17401

−0.0

09966

0.0

02303

−0.0

02689

0.0

0227

−6.9

6E

-04

4.4

1E

-05

1.4

2E

-04

−1.3

4E

-04

8.1

3E

-05

−1.2

8E

-05

bM 811.6

7614

−0.1

07179

0.0

67361

0.0

50281

−0.0

12522

0.0

02773

−0.0

09217

0.0

03612

−4.7

4E

-04

0.0

00103

0.0

00578

−3.1

4E

-04

5.9

5E

-05

−5.9

0E

-06

cM 12.4

74094

−0.3

75539

−0.0

73192

0.3

16855

−0.0

03339

−0.0

01935

−0.0

88337

0.0

02056

4.1

4E

-05

−2.6

3E

-05

7.9

0E

-03

−2.4

6E

-04

−5.7

9E

-06

−6.6

5E

-07

cM 22.4

55022

0.0

80699

−0.0

48362

−0.0

68114

0.0

00985

−0.0

00758

0.0

18978

−0.0

00135

4.5

1E

-05

−8.3

0E

-06

−0.0

01707

8.1

7E

-06

−8.6

0E

-06

−1.3

4E

-06

cM 32.3

01124

−0.0

35348

−0.0

66133

0.0

11457

−0.0

00816

−0.0

01908

−0.0

01977

0.0

00224

1.3

9E

-05

−3.9

4E

-05

1.3

3E

-04

−2.0

0E

-05

6.8

7E

-07

1.2

8E

-06

cM 42.5

95419

0.0

32911

−0.0

48502

−0.0

0837

0.0

01987

−0.0

01769

0.0

01646

−0.0

00276

1.1

3E

-04

−4.0

1E

-05

−1.3

1E

-04

1.5

6E

-05

−5.0

2E

-06

4.5

5E

-06

cM 52.6

11683

0.0

0722

−0.0

47775

−0.0

07095

−0.0

02323

−1.1

5E

-03

0.0

02062

0.0

00398

−0.0

00167

−7.7

7E

-06

−0.0

00172

1.4

6E

-05

3.4

9E

-05

6.3

3E

-07

cM 62.7

03058

0.0

27752

−0.0

52293

−0.0

25826

−0.0

07296

−0.0

0201

0.0

07507

0.0

00979

−6.8

1E

-04

−8.6

7E

-05

−0.0

00656

4.4

0E

-05

9.2

7E

-05

−8.6

3E

-06

cM 72.6

46768

−0.0

70787

−0.0

59517

0.0

409

−0.0

0632

−0.0

05176

−0.0

09092

0.0

0191

−0.0

00245

−0.0

00216

6.8

5E

-04

−1.8

9E

-04

2.5

3E

-05

−3.5

2E

-06

cM 82.6

72429

0.1

33919

−0.0

69491

−0.0

19029

0.0

3175

−0.0

01516

−0.0

143

−0.0

14129

0.0

00213

−1.6

9E

-05

0.0

02694

0.0

01664

−3.1

6E

-05

6.8

8E

-06

ξ 00

ξ 10

ξ 01

ξ 20

ξ 11

ξ 02

ξ 30

ξ 21

ξ 12

ξ 40

ξ 31

ξ 22

ξ 50

ξ 41

ξ 32

aM 1

1.3

68591

2.4

93734

−0.0

02591

−2.2

66136

0.0

06045

−3.6

2E

-05

0.8

6761

−0.0

04278

4.1

4E

-05

−0.1

53462

0.0

01101

−2.2

8E

-05

0.0

10227

−9.3

9E

-05

2.9

6E

-06

λ3

24.9

3836

9.3

44645

−0.0

00745

−14.9

7752

0.0

01263

−6.5

1E

-05

6.4

56078

−7.6

7E

-05

0.0

00129

−1.1

97114

−0.0

00104

−4.3

7E

-05

0.0

81749

1.5

5E

-05

4.2

1E

-06

Page 18 of 22 Eur. Phys. J. D (2016) 70: 159

Table

A.3

.C

H4/air

pla

sma.

c p,o

ff(k

J/kg

K),

c p,e

q(k

J/kg

K),

h(k

J/kg),

s(k

J/kg

K)

ξ 00

ξ 10

ξ 01

ξ 20

ξ 11

ξ 02

ξ 21

ξ 12

ξ 03

ξ 22

ξ 13

ξ 04

ξ 23

ξ 14

ξ 05

aoff

10.4

97235

−0.0

10146

0.0

06146

0.1

32319

−0.0

04705

−0.0

04655

−1.3

2E

-04

0.0

03779

−0.0

00401

−0.0

02773

0.0

00293

−7.4

1E

-06

−0.0

00171

4.8

0E

-06

2.9

7E

-08

aoff

2−0

.492009

0.0

63951

0.0

24955

−0.0

78895

0.0

03734

−0.0

0132

−0.0

02522

0.0

05689

−0.0

00704

−0.0

01277

−0.0

00208

−4.8

4E

-05

1.8

2E

-05

−2.4

0E

-05

1.5

8E

-08

aoff

30.1

36237

0.3

0128

0.0

01404

−0.0

62065

0.0

13145

0.0

04856

−0.0

0231

−0.0

03133

0.0

00619

0.0

01107

−1.7

3E

-04

2.6

6E

-05

4.7

2E

-05

−1.7

9E

-06

9.9

0E

-09

aoff

40.4

7622

0.0

09606

0.0

00554

−0.0

20947

−0.0

08987

7.7

0E

-05

−0.0

01349

−0.0

00732

0.0

00153

−1.3

1E

-04

1.1

7E

-05

8.1

7E

-06

−5.6

6E

-06

1.7

9E

-06

1.4

9E

-08

aoff

50.4

58757

0.0

64372

−0.0

36187

−0.0

58107

0.0

24215

−2.2

6E

-04

−0.0

10297

0.0

03277

−1.7

1E

-05

−0.0

00196

0.0

00329

−4.0

2E

-06

2.2

4E

-05

1.3

7E

-05

4.9

5E

-09

aoff

60.0

42087

−0.4

79118

−0.0

01569

0.3

30713

−0.1

0352

−0.0

0276

0.0

81911

−0.0

08013

4.7

8E

-05

0.0

04647

−7.1

2E

-04

1.6

7E

-05

2.0

7E

-05

−3.7

9E

-05

−3.1

7E

-08

aoff

7−0

.271327

1.3

38115

−0.0

10471

−0.8

84048

0.2

33342

0.0

04274

−0.1

68054

0.0

05653

−0.0

00159

−0.0

03721

4.9

7E

-04

−3.6

9E

-05

0.0

00314

6.1

6E

-05

−3.0

7E

-08

aoff

8−0

.999978

−1.0

1828

0.0

45122

0.4

48092

−0.0

54759

−0.0

07118

0.0

19203

0.0

14534

−0.0

00117

−0.0

08288

0.0

0084

5.8

0E

-05

−0.0

00528

−9.8

1E

-06

2.2

1E

-06

boff

14.8

83268

0.7

68414

−0.0

76613

−0.1

85001

0.0

68064

0.0

18553

−0.0

20469

0.0

04553

0.0

02715

−0.0

08897

−0.0

00384

1.0

2E

-04

−0.0

00601

−4.5

7E

-05

7.9

2E

-09

boff

28.9

11241

−0.0

07505

0.0

48443

0.0

03303

−0.0

01354

0.0

00606

−0.0

00434

0.0

00603

−0.0

002

7.8

5E

-05

−1.6

9E

-04

−1.5

2E

-05

4.2

4E

-05

−7.0

2E

-06

−1.0

9E

-08

boff

39.5

82611

0.0

06211

0.0

61922

−0.0

03823

−0.0

02991

0.0

00358

0.0

00438

−0.0

00227

−4.8

5E

-05

1.2

7E

-04

4.0

7E

-05

−1.5

6E

-06

7.0

2E

-06

3.1

3E

-06

−5.9

4E

-09

boff

410.3

1256

−0.0

05942

0.0

6253

0.0

00768

0.0

01855

0.0

00335

−0.0

00259

8.3

8E

-06

−8.2

9E

-05

−4.3

3E

-05

−3.8

7E

-05

−3.4

2E

-06

−4.9

3E

-07

−2.0

0E

-06

4.9

5E

-09

boff

510.7

5312

−0.0

00119

0.0

67008

−0.0

02887

−0.0

02425

0.0

01812

0.0

00917

−3.5

4E

-04

3.0

2E

-05

2.6

4E

-04

6.8

2E

-06

−5.1

0E

-07

1.3

3E

-05

1.4

1E

-06

−8.9

1E

-09

boff

611.1

1139

−0.0

39429

0.0

65116

0.0

22776

−0.0

13731

0.0

02108

0.0

08991

−0.0

01642

6.9

4E

-05

0.0

01096

−6.1

5E

-05

1.0

0E

-06

3.9

8E

-05

−2.0

2E

-07

−8.9

1E

-09

boff

711.3

6593

0.0

12699

0.0

57597

−0.0

10485

0.0

05624

−0.0

00397

−0.0

02071

0.0

01757

−2.0

0E

-04

−0.0

00238

1.9

2E

-04

−1.0

2E

-05

−1.1

7E

-05

6.2

8E

-06

−1.3

4E

-07

boff

811.7

7548

−0.3

02768

0.0

9083

0.0

85835

−0.0

91709

0.0

02191

0.0

24234

−0.0

05771

−2.1

5E

-05

0.0

01765

0.0

00206

1.6

9E

-05

3.2

3E

-05

1.8

6E

-05

1.1

8E

-06

coff

1−0

.876133

1.0

31048

−0.0

60732

−0.6

20664

0.0

53666

0.0

25566

−0.0

13877

−0.0

05611

0.0

02462

0.0

02339

−5.6

0E

-04

6.3

2E

-05

8.0

8E

-05

−2.3

7E

-05

−2.6

7E

-08

coff

22.4

25062

−0.1

44835

−0.0

26844

0.0

52425

0.0

1215

−0.0

05304

0.0

02111

−0.0

07377

0.0

00853

0.0

02406

−0.0

00305

6.9

0E

-05

6.7

7E

-05

1.0

3E

-05

1.6

8E

-08

coff

32.5

00415

−0.0

49854

−0.0

03852

0.0

24875

−0.0

07965

−0.0

08848

0.0

00644

0.0

01835

−0.0

02085

−0.0

0087

6.6

0E

-05

−9.8

7E

-05

−3.3

4E

-05

−2.2

3E

-07

1.7

8E

-08

coff

42.6

02944

−0.0

1988

−0.0

43329

0.0

16452

0.0

15017

−0.0

02515

−0.0

00861

0.0

00504

−7.7

3E

-06

−1.3

9E

-04

−1.7

9E

-04

4.5

5E

-06

−4.6

8E

-06

−1.1

2E

-05

−2.9

3E

-20

coff

52.6

18781

−0.0

5165

−0.0

54351

0.0

32002

−0.0

06586

−0.0

02731

0.0

01594

−0.0

01001

−0.0

00112

−8.6

4E

-04

−0.0

00215

−2.7

4E

-06

−6.3

3E

-05

−1.2

5E

-05

1.8

8E

-08

coff

62.7

2428

0.2

97453

−0.0

24522

−0.2

25509

−0.0

32982

0.0

04284

0.0

0253

−0.0

1505

−3.8

7E

-04

0.0

09669

−0.0

00112

−4.8

8E

-05

0.0

00627

5.2

5E

-05

−8.9

1E

-09

coff

72.6

80191

−0.3

91015

−0.1

26438

0.2

92187

0.0

7528

−0.0

136

−0.0

2482

0.0

1997

0.0

00791

−0.0

14036

5.8

8E

-05

1.4

5E

-04

−0.0

00799

−6.4

1E

-05

3.8

7E

-06

coff

81.4

07622

3.4

45276

−0.2

66793

−1.8

81354

0.5

70463

−0.0

04332

−0.3

18174

0.0

11754

−0.0

00518

−0.0

04998

−0.0

00233

−0.0

00116

0.0

00618

4.8

0E

-05

−3.5

7E

-06

Eur. Phys. J. D (2016) 70: 159 Page 19 of 22

Table

A.3

.C

onti

nued

.

c p,o

ff(k

J/kg

K),

c p,e

q(k

J/kg

K),

h(k

J/kg),

s(k

J/kg

K)

aeq

11.1

70969

1.1

20215

−0.1

27371

−0.2

90187

−0.0

0863

−0.0

04768

1.9

7E

-03

0.0

01474

−1.0

2E

-04

−0.0

00784

1.4

4E

-04

8.1

1E

-06

−7.3

7E

-05

−2.3

6E

-07

3.7

7E

-07

aeq

22.4

97633

−0.0

4266

−0.1

02196

0.0

01199

0.0

05102

−0.0

03285

−0.0

01957

0.0

02736

−0.0

00177

−0.0

01204

0.0

00256

1.1

8E

-06

−9.8

1E

-05

2.6

1E

-06

2.4

1E

-07

aeq

32.9

24353

0.1

73386

−0.1

31045

−0.0

22048

−0.0

0109

−0.0

04527

0.0

00571

0.0

00517

−0.0

00182

−3.3

4E

-04

9.1

4E

-05

4.7

6E

-06

−5.0

3E

-05

−5.8

3E

-08

4.2

5E

-07

aeq

43.2

3299

0.0

11512

−0.1

01949

−0.0

13172

0.0

00824

−0.0

03726

0.0

00295

0.0

00233

−0.0

00272

−3.8

2E

-05

9.7

5E

-05

−8.3

8E

-06

−2.7

5E

-05

2.1

6E

-06

−2.6

9E

-08

aeq

53.4

0678

0.0

07096

−0.0

90306

−0.0

09632

0.0

00805

−1.8

0E

-03

0.0

00367

−0.0

00274

−5.1

8E

-05

4.7

4E

-05

−5.8

0E

-07

−6.5

0E

-08

−2.1

2E

-05

−2.0

4E

-06

−4.1

0E

-09

aeq

63.5

286

−0.0

49264

−0.0

82688

0.0

08316

−0.0

30457

−0.0

00632

1.4

0E

-02

−0.0

04309

−1.8

0E

-04

0.0

0196

−8.9

1E

-05

−1.7

1E

-05

5.3

8E

-05

2.0

3E

-06

−4.5

0E

-07

aeq

73.4

93682

−0.0

19162

−0.0

89302

−0.0

13282

0.0

02382

−0.0

0152

0.0

00374

0.0

00427

−3.2

5E

-04

0.0

00234

1.1

6E

-04

−4.3

5E

-05

−7.3

9E

-06

3.6

0E

-06

−1.6

8E

-06

aeq

82.2

21276

−0.0

51149

−0.1

10099

0.0

23218

0.0

04878

−0.0

08491

8.8

2E

-05

0.0

01525

−5.0

1E

-04

−1.0

5E

-03

0.0

00105

1.3

0E

-05

−8.8

5E

-05

−1.6

7E

-06

1.2

9E

-06

beq

18.1

60764

0.0

09346

0.0

53899

0.0

04648

0.0

01257

0.0

0155

3.2

3E

-04

−2.5

8E

-04

1.1

6E

-04

−0.0

00153

−8.7

4E

-05

4.7

0E

-06

−1.7

8E

-05

−5.4

1E

-06

−2.2

7E

-08

beq

28.8

64583

−0.0

11165

0.0

58248

−0.0

00784

−0.0

01456

0.0

00767

6.0

2E

-04

0.0

00983

−5.8

2E

-05

−0.0

00331

1.2

1E

-04

−1.7

5E

-06

−3.5

0E

-05

1.9

7E

-06

1.1

0E

-08

beq

39.6

03162

0.0

03438

0.0

72201

0.0

00246

0.0

00946

0.0

01924

−0.0

00184

0.0

00369

6.6

6E

-05

−2.4

0E

-04

2.8

7E

-05

4.7

9E

-06

−2.5

7E

-05

−9.7

9E

-07

1.4

4E

-07

beq

410.3

1104

−0.0

04258

0.0

6884

−0.0

01763

0.0

02344

0.0

01749

−0.0

01341

8.2

3E

-05

2.5

7E

-05

3.7

0E

-05

−1.2

6E

-05

−3.9

5E

-06

1.8

6E

-05

1.0

0E

-06

−1.6

9E

-07

beq

510.7

6042

−6.3

2E

-06

0.0

64195

−0.0

00807

0.0

03713

0.0

01876

−1.4

2E

-03

6.1

3E

-05

1.2

6E

-04

−2.2

1E

-05

−4.3

8E

-05

5.9

3E

-06

1.1

7E

-05

−1.3

1E

-06

1.1

4E

-07

beq

611.0

9144

0.0

04937

0.0

59122

−0.0

04803

0.0

05487

0.0

01355

−0.0

0263

0.0

00751

1.8

7E

-04

−2.3

8E

-04

4.5

5E

-05

2.2

1E

-05

−3.3

5E

-06

1.5

1E

-06

8.0

9E

-07

beq

711.3

6394

−0.0

1719

0.0

62897

0.0

07962

0.0

00936

0.0

02702

−5.4

2E

-04

0.0

00573

4.0

3E

-04

−0.0

00279

3.6

6E

-05

3.5

5E

-05

−1.2

8E

-05

6.5

6E

-07

1.0

8E

-06

beq

811.7

4009

0.0

43901

0.1

56427

−0.0

19666

−1.0

6E

-03

0.0

13143

1.1

3E

-03

−1.6

2E

-03

−1.6

1E

-03

6.7

9E

-04

−1.0

7E

-04

−3.1

7E

-04

2.7

8E

-05

−2.1

3E

-06

−1.2

4E

-05

ceq

1−1

.783352

0.8

05418

0.0

68536

−0.4

66204

0.0

76076

0.0

10183

−0.0

40024

−0.0

13165

0.0

00133

0.0

0822

−8.6

6E

-04

−6.4

4E

-05

0.0

00685

1.8

7E

-05

−2.4

2E

-06

ceq

2−1

.557631

−0.2

14399

0.0

6253

0.1

38516

−0.0

0792

−0.0

01643

−0.0

00699

−0.0

0013

−0.0

00346

−0.0

01717

−5.5

0E

-05

4.2

5E

-06

−3.6

6E

-05

5.2

3E

-06

1.4

6E

-06

ceq

3−1

.470337

−0.0

66531

0.0

56063

0.0

38827

0.0

1101

0.0

01449

−0.0

04332

0.0

01059

0.0

00399

−0.0

00154

−1.5

5E

-04

2.7

6E

-05

9.4

6E

-05

−1.3

6E

-06

6.2

7E

-07

ceq

4−1

.706674

−0.0

59008

0.0

54224

0.0

28781

−0.0

24559

0.0

03532

0.0

11096

−0.0

07019

−0.0

00268

3.3

6E

-03

−4.5

5E

-04

−6.4

2E

-05

2.0

6E

-04

−1.2

6E

-06

−2.3

9E

-06

ceq

5−1

.911221

0.1

69039

0.0

18469

−0.0

67678

0.0

30464

0.0

01369

−0.0

14073

0.0

0137

0.0

00797

−5.9

8E

-04

−9.6

6E

-05

9.0

2E

-05

5.3

5E

-05

1.0

6E

-06

3.3

0E

-06

ceq

6−1

.885694

−0.1

50879

0.0

18103

0.0

6549

0.0

1824

−0.0

05356

−0.0

12393

0.0

079

−7.6

8E

-06

−2.9

6E

-03

0.0

00568

4.6

2E

-05

−1.0

4E

-04

1.4

6E

-05

2.0

8E

-06

ceq

7−1

.94994

−0.0

21065

0.0

48714

0.0

19304

0.0

15672

0.0

0562

−0.0

04887

0.0

05064

−1.4

8E

-05

−0.0

0335

0.0

00149

−6.8

1E

-05

−0.0

0024

−1.3

9E

-05

−3.6

1E

-06

ceq

8−1

.68478

0.4

40727

0.1

26274

−0.2

30738

−0.0

22052

3.4

9E

-03

0.0

08164

−0.0

19807

−1.6

0E

-04

0.0

1056

−1.3

1E

-03

5.8

0E

-05

7.2

4E

-04

2.3

7E

-06

4.8

6E

-06

λ1

1239204

60992.7

647304.8

3−3

9281

9894.6

82

4912.5

07

−4368.5

46

−643.2

869

−347.3

295

492.5

372

−47.3

9907

−75.0

6363

46.3

7552

1.8

62041

−2.7

74747

λ2

−54.8

455

−6.4

50537

−1.4

53109

3.3

33452

−2.8

85249

0.0

39988

1.8

90681

−0.4

23165

−0.0

12156

0.2

81657

−0.0

17091

−0.0

01429

0.0

1251

0.0

00145

−3.6

4E

-05

Page 20 of 22 Eur. Phys. J. D (2016) 70: 159

Table

A.3

.C

onti

nued

.

Mt

(kg/km

ol)

ξ 00

ξ 10

ξ 01

ξ 20

ξ 11

ξ 02

ξ 30

ξ 21

ξ 12

ξ 03

ξ 40

ξ 31

ξ 22

ξ 13

aM 1

6.4

30346

−20.5

9475

−0.0

17419

35.1

1529

0.0

44612

−0.0

00621

−2.4

9E

+01

−0.0

3549

0.0

01003

−7.6

5E

-06

6.3

14178

8.5

0E

-03

−0.0

00481

1.2

1E

-06

aM 2

0.7

79075

6.1

90502

0.0

26094

−11.1

7136

−0.0

67616

0.0

00817

7.9

19848

0.0

47522

−0.0

01904

−5.4

5E

-06

−2.0

06626

−8.6

9E

-03

1.0

4E

-03

8.1

3E

-06

aM 3

2.0

88849

−0.8

34086

−0.0

04443

1.0

99449

0.0

13696

−1.8

1E

-05

−0.7

72545

−0.0

10785

0.0

00183

4.2

9E

-06

1.9

9E

-01

2.7

4E

-03

−1.1

9E

-04

−3.0

0E

-06

aM 4

1.1

16831

−1.1

49791

0.0

34983

0.7

86855

−0.1

2332

1.1

0E

-05

−0.1

67662

0.1

19168

−0.0

01175

−2.3

9E

-05

−3.7

3E

-02

−3.6

3E

-02

7.0

7E

-04

1.1

8E

-05

aM 5

−3.8

63098

16.3

4086

−0.1

64651

−23.8

4476

0.5

60625

−1.1

9E

-03

14.6

2394

−0.5

85652

1.1

0E

-03

−9.4

8E

-05

−3.2

36748

1.9

6E

-01

1.3

7E

-04

7.4

6E

-05

aM 6

−0.4

16964

−0.0

106

−0.0

09535

−0.2

598

−0.0

55112

−0.0

05661

0.1

15142

0.0

00311

−6.4

8E

-03

−0.0

00509

7.7

3E

-03

2.5

4E

-02

5.6

6E

-03

2.3

0E

-04

aM 7

5.7

76032

−27.2

272

0.1

16906

41.3

3618

−0.3

5928

0.0

00245

−27.2

2584

0.3

82244

0.0

02239

6.9

1E

-05

6.5

4E

+00

−1.3

4E

-01

−0.0

01828

−4.6

0E

-05

aM 8

13.8

5832

−58.6

7581

1.1

88396

78.0

8012

−2.8

25995

0.0

37822

−45.9

7701

2.1

48978

−0.0

68526

−6.6

5E

-06

10.1

2679

−5.1

7E

-01

0.0

29907

−3.8

7E

-05

bM 17.4

44697

3.3

01052

0.0

61611

−5.5

26276

−0.0

08931

0.0

01712

3.8

98451

0.0

05058

−0.0

00182

3.8

0E

-05

−0.9

83131

−4.8

0E

-04

4.0

3E

-05

−4.0

5E

-06

bM 29.0

25076

−0.9

16142

0.0

53798

1.5

11813

0.0

07276

0.0

01327

−1.0

63918

−0.0

05827

0.0

00139

2.7

1E

-05

2.6

9E

-01

1.1

0E

-03

−1.0

4E

-04

−2.3

7E

-06

bM 39.5

11017

0.1

44884

0.0

71576

−0.2

34187

0.0

0052

0.0

02108

0.1

60985

−0.0

00379

−9.2

0E

-06

4.5

6E

-05

−3.9

9E

-02

1.9

4E

-04

6.0

9E

-06

−4.6

6E

-07

bM 410.3

085

−0.0

57311

0.0

71718

0.0

49994

−0.0

09018

0.0

0182

−0.0

0862

8.9

7E

-03

−1.0

7E

-04

3.4

6E

-05

−3.7

9E

-03

−2.7

8E

-03

4.7

1E

-05

−1.1

7E

-06

bM 510.2

7879

1.9

61269

0.0

53431

−2.9

51855

0.0

37831

0.0

01387

1.8

96233

−4.2

2E

-02

−2.1

6E

-04

1.0

1E

-05

−4.4

2E

-01

1.5

1E

-02

2.1

0E

-04

1.0

3E

-05

bM 69.0

69924

8.4

64317

0.0

13068

−12.7

5324

0.1

54179

0.0

00207

8.2

13707

−0.1

73727

−9.8

2E

-04

−9.6

9E

-05

−1.9

2E

+00

6.3

7E

-02

1.2

2E

-03

7.3

6E

-05

bM 79.3

71168

7.8

21829

−0.0

43865

−10.8

5418

0.3

32449

−0.0

00782

6.3

05907

−0.3

7113

−2.1

4E

-03

−0.0

00184

−1.3

0E

+00

1.3

3E

-01

2.0

7E

-03

9.9

9E

-05

bM 814.2

7568

−9.9

73113

0.2

0604

13.8

0429

−0.4

33682

0.0

01587

−8.4

47706

0.3

34434

−9.7

7E

-03

−0.0

00242

1.9

29861

−7.8

5E

-02

5.4

6E

-03

1.1

4E

-04

cM 1−5

.088415

33.7

1181

−0.0

55369

−55.3

3903

−0.0

46527

−0.0

01838

38.7

386

0.0

35283

−0.0

00213

−1.9

1E

-05

−9.7

7E

+00

−8.6

4E

-03

6.7

5E

-06

−2.3

6E

-06

cM 24.0

49066

−7.2

09922

−0.0

63165

11.8

7222

0.0

41731

−0.0

00969

−8.3

33286

−0.0

30271

0.0

00294

−9.0

4E

-06

2.1

05179

6.8

1E

-03

−2.3

7E

-04

−5.6

6E

-06

cM 32.1

06554

0.8

75651

−0.0

61822

−1.5

15111

−0.0

11044

−0.0

01861

1.0

78719

0.0

09621

−0.0

00308

−4.5

3E

-05

−2.7

6E

-01

−2.3

4E

-03

1.3

7E

-04

−1.2

3E

-06

cM 42.5

49737

−0.0

2525

−0.0

75129

0.4

30601

0.0

94872

−0.0

01649

−0.5

50828

−0.0

88378

8.7

0E

-04

−9.9

3E

-06

1.9

5E

-01

2.6

3E

-02

−6.3

3E

-04

−1.7

6E

-05

cM 54.5

38237

−7.5

41132

0.0

6812

10.2

5639

−0.3

86149

−9.5

4E

-05

−5.8

70282

0.3

93446

−0.0

02188

3.0

9E

-05

1.1

93258

−1.2

8E

-01

8.2

3E

-04

−3.5

3E

-05

cM 6−1

.122953

15.0

4155

−0.2

61769

−20.7

1645

0.7

15889

−0.0

01494

11.9

4269

−0.7

54682

−6.1

2E

-04

−6.7

0E

-05

−2.4

41927

2.5

3E

-01

2.6

2E

-05

−1.8

2E

-05

cM 74.5

3999

−7.5

76703

0.0

63945

10.1

5983

−0.3

93544

0.0

00465

−5.5

0541

0.4

55627

0.0

02179

0.0

00287

1.0

0E

+00

−1.7

3E

-01

−0.0

04106

−3.1

4E

-04

cM 83.0

54484

−1.3

66852

−0.1

32466

2.4

81974

0.2

71218

−0.0

0386

−2.3

15441

−0.3

07614

0.0

05328

−8.1

3E

-05

0.7

53935

0.0

99294

−0.0

0235

1.3

8E

-04

λ3

70.2

3199

−189.2

254

−0.0

05767

309.2

387

0.0

2949

0.0

00516

−217.1

366

−0.0

45316

−0.0

01592

−1.8

3E

-05

54.6

2963

0.0

21279

0.0

01129

2.3

5E

-05

Eur. Phys. J. D (2016) 70: 159 Page 21 of 22

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