Transport Properties

The transport properties (e.g., the viscosity, thermal conductivity and binary diffu-sion coefficients) for the mixture gas were evaluated using the Yos’ formula [55] based on the first Chapman-Enskog approximation in the present study.

2.6.1 Collision Cross Section

In order to determine the first order transport properties, we need to calculate the collision cross sections πΩ^(1,1)_i,j and πΩ^(2,2)_i,j . The collision cross section πΩ^(l,s)_i,j is given by

πΩ^(l,s)_i,j =

∫ _∞

0

∫ _π

0

exp(−γ²)γ^2s+3(1−cos^lχ)4πσ_i,jsinχdχdγ

∫ _∞

0

∫ _π

0

exp(−γ²)γ^2s+3(1−cos^lχ) sinχdχdγ

, (2.91)

whereχrepresents the angle of deflection in center of mass system andσ_i,j =σ_i,j(χ, γ) is the differential scattering cross section for species pairiand j. The reduced velocity γ is defined with the relative velocity g of the collision particles as

γ =

[ m_im_j 2 (m_i+m_j)kT

]¹

2

g. (2.92)

Since calculation of the collision cross sections requires much computational cost, it is effective to prepare those data preliminarily.

Gupta’s curve-fittings

According to Gupta, Yos, Thompson and Lee [47], the collision cross section curve-fit are expressed as a function of temperatures as the following formula:

πΩ^(1,1)_i,j = [

exp (

DπΩ^(1,1)_i,j

)]

T

[AπΩ(1,1) i,j

(lnT)²+B

πΩ(1,1) i,j

lnT+C

πΩ(1,1) i,j

]

, (2.93)

πΩ^(2,2)_i,j = [

exp (

D_πΩ^(2,2)

i,j

)]

T

[AπΩ(2,2) i,j

(lnT)²+B

πΩ(2,2) i,j

lnT+C

πΩ(2,2) i,j

]

, (2.94)

where the constants A

πΩ^(l,s)_i,j , B

πΩ^(l,s)_i,j , C

πΩ^(l,s)_i,j and D

πΩ^(l,s)_i,j are obtained by Gupta’s works [47]. If collision particle pair consists of ions,

lnΛ(p_e) = 1 2ln



2.09×10⁻² (

T_tr 1000p^1/4e

)4

+1.52 (

T_tr 1000p^1/4e

)8/3

, (2.95)

wherep_eis the electron pressure in atmospheres. For the pair of collision which consists of electrons, the collision cross section is corrected by multiplying the following factor:

lnΛ(p_e) = 1 2ln



2.09×10⁻² (

T_e 1000p^1/4e

)4

+1.52 (

T_e 1000p^1/4e

)8/3

. (2.96)

Since it is reported that the collision cross section data for N-e and O-e in the Gupta model are not accurate in low temperature region according to Ref. [56], those data should be particularly replaced by an other model. The collision cross sections for N-e and O-e pairs are given by the model of Fertig et al. [57, 58].

Fertig’s curve-fittings

Fertig, Dohr and Fr¨uhauf [57, 58] have performed new curve-fitting of the collision cross sections using five constants as

πΩ^(1,1)_i,j =πexp [

AπΩ^(1,1)_i,j (lnT)⁴+B

πΩ^(1,1)_i,j (lnT)³ +CπΩ^(1,1)_i,j (lnT)²+D

πΩ^(1,1)_i,j (lnT) +E

πΩ^(1,1)_i,j

]

, (2.97)

πΩ^(2,2)_i,j =πexp [

AπΩ^(2,2)_i,j (lnT)⁴+B

πΩ^(2,2)_i,j (lnT)³ +CπΩ^(2,2)_i,j (lnT)²+D

πΩ^(2,2)_i,j (lnT) +E

πΩ^(2,2)_i,j

]

, (2.98)

where the constantsA

πΩ^(l,s)_i,j ,B

πΩ^(l,s)_i,j ,C

πΩ^(l,s)_i,j ,D

πΩ^(l,s)_i,j andE

πΩ^(l,s)_i,j are obtained by Fertig et al.’s works [57, 58]. If the pair of species consists of ions, the collision cross sections should be corrected by use of the Debye radius λ_D, which is given by

λD = vu utϵ₀k

e² (

n_e T_e +

∑ns i=I

Z_i² T_trni

)₋1

, (2.99)

where Z_i shows the number of charges of the species i. The modified collision cross section is expressed as

πΩ^(l,s)_i,j = (λ_D

T^∗ )2

πΩ^(l,s)_i,j ^∗(T^∗). (2.100)

The dimensionless temperature T^∗ is defined by T^∗ =λ_D

(4πε₀kT_tr e²

)

or T^∗ =λ_D

(4πε₀kT_e e²

) .

Abbreviations

Finally, to simplify the equations developed hereafter, the abbreviations are intro-duced as

∆⁽¹⁾_i,j(T) = 8 3

[ 2m_im_j πkT(m_i+m_j)

]1/2

πΩ^(1,1)_i,j , (2.101)

∆⁽²⁾_i,j(T) = 16 5

[ 2m_im_j πkT(m_i +m_j)

]1/2

πΩ^(2,2)_i,j , (2.102)

A^∗_i,j = πΩ^(2,2)_i,j πΩ^(1,1)_i,j

, (2.103)

B_i,j^∗ = 5πΩ^(1,2)_i,j −4πΩ^(1,3)_i,j πΩ^(1,1)_i,j

. (2.104)

The mole fraction is defined by

X_i = ρ_i M_i

∑ns j

ρ_j Mj

. (2.105)

2.6.2 Viscosity

According to Hirschfelder, Curtiss and Bird [52], the viscosity for mixture gas is given with rigorous first order kinetic theory by,

µ=−

H_1,1 H_1,2 · · · H_1,ns X₁ H_1,2 H_2,2 · · · H_2,ns X₂ ... ... . .. ... ... H_1,ns H_2,ns· · · H_ns,ns X_ns

X₁ X₂ · · · X_ns X₀ H1,1 H1,2 · · · H1,ns

H_1,2 H_2,2 · · · H_2,ns ... ... . .. ... H_1,nsH_2,ns· · · H_ns,ns

, (2.106)

where

H_i,i = X_j² µi

+

∑ns j̸=i

2X_iX_j mi +mj

RT pDi,j

[ 1 + 3

5 m_j mi

A^∗_i,j ]

, (2.107)

Hi,j =− 2X_iX_j m_i+m_j

RT pD_i,j

[ 1− 3

5A^∗_i,j ]

, i̸=j. (2.108)

The viscosity of the pure speciesµi is expressed in the self diffusion coefficient Di,i as µ_i = 5

6 ρD_i,i

A^∗_i,i . (2.109)

Compared with the diagonal elements H_i,i, the off-diagonal elements H_i,j are small.

Primary contributions of the viscosity are born by H_i,i. Assuming that A^∗_i,j = 5/3, the off-diagonal elementsH_i,j are exactly vanished. The viscosityµfor the multicomponent gas can be expressed by only H_i,i. The approximate expression is given by Yos [55] as

µ=

∑ns i









m_iX_i

∑ns j

Xj∆⁽²⁾_i,j(T)







. (2.110)

2.6.3 Thermal Conductivity

Translational Degree of Freedom

The translational component of the thermal conductivity is given by

λ_tr=−

L1,1 L1,2 · · · L1,ns X1

L_1,2 L_2,2 · · · L_2,ns X₂ ... ... . .. ... ... L_1,ns L_2,ns · · · L_ns,nsX_ns

X₁ X₂ · · · X_ns X₀ L_1,1 L_1,2 · · · L_1,ns L_1,2 L_2,2 · · · L_2,ns

... ... . .. ... L_1,ns L_2,ns· · · L_ns,ns

, (2.111)

where

L_i,i =− 4X_i²

λ_i,tr,mono − 16 25

T p

∑ns j̸=i





 X_iX_j

(15

2 m²_i + 25

4 m²_j −3m²_iB_i,j^∗ + 4m_im_jA^∗_i,j ) (m_i+m_j)²D_i,j





,

(2.112) L_i,j = 16

25 T

p

X_iX_jm_im_j (mi+mj)²Di,j

[55

4 −3B_i,j^∗ −4A^∗_i,j ]

i̸=j. (2.113) The translational contribution of the thermal conductivity of the pure speciesλi,tr,mono

is expressed as

λ_i,tr,mono = 15

4 Rµ_i. (2.114)

The translational component of the thermal conductivity can be simplified in the same fashion with the viscosity for multicomponent gas mixture. The approximate

expres-sion is given by Yos [55] as

λ_tr = 15 4 k

∑ns i̸=e









Xi

∑ns j

α_{i j}X_j∆⁽²⁾_i,j(T)







, (2.115)

where

α_{i j} = 1 + [1−(m_i/m_j)] [0.45−2.54 (m_i/m_j)]

[1 + (m_i/m_j)]² . (2.116) On the other hand, the thermal conductivity of electrons is given by

λ_e = 15 4 k









X_e

∑ns j

αejXj∆⁽²⁾_e,j(Te)







, (2.117)

Internal Degree of Freedom

The contribution of the internal degree to the thermal conductivity λ_int, such as the rotational and vibrational, is given by

λ_int =

∑nm i

Xi(λi,int−λi,tr,mono) D_i,i

∑ns j

X_j D_i,j

. (2.118)

Hirschfelder et al. expressed λ_i,int as the following form:

λ_i,int= 1 4

[(

15−6ρ_iD_i,i µi

)

C_p,int,i^′ − (

15−10ρ_iD_i,i µi

) C_v,int,i^′

]

µ_i, (2.119) where

C_p,int,i^′ =C_p,tr+C_p,int,i, (2.120)

C_v,int,i^′ =C_v,tr+C_v,int,i. (2.121)

With Eqs. (2.114) and (2.119), we obtain the following expression as

λ_i,int−λ_i,tr,mono =ρ_iD_i,iC_v,int,i. (2.122)

Substituting Eq. (2.122) with Eq. (2.109) into Eq. (2.118), one obtains

λ_int =k

∑nm i









(Cv,int,i

R_i )

X_i

∑ns j

X_j∆⁽¹⁾_i,j(T)







. (2.123)

2.6.4 Electrical Conductivity

The electrical conductivity is introduced by the first rigorous kinetic theory ac-cording to Hirschfelder, Curtiss and Bird [52]. The following expression is given by Yos [55]:

σ = e² kT_e

X_e

∑ns j̸=e

X_j∆⁽¹⁾_e,j(T_e)

. (2.124)

2.6.5 Diffusion

Multicomponent Diffusion Model

The diffusion velocity V_s^j is defined with the species-averaged velocity v_s^j as

V_s^j =v_s^j−v^j, (2.125)

where the mass-averaged velocity v is given by v^j =

∑ns s=1

ρ_s

ρv_s^j. (2.126)

With Eqs. (2.125) and (2.126), one obtains the following relation:

∑ns s=1

ρV_s^j = 0. (2.127)

Following Hirschfelder, Curtiss and Bird [52], the diffusion velocity is given by V_s^j = n²

n_s

∑ns k=1

m_kD_s,kd^j_k− D^T_s ρ

∂lnT

∂x^j , (2.128)

d^j_s ≡ ∂

∂x^j (n_s

n )

+ (n_s

n − ρ_s ρ

)∂lnp

∂x^j − ρ_s p

( F_s^j−

∑ns k=1

ρ_k ρF_k^j

)

, (2.129) whereD_s,k andD_s^T are the multicomponent mass diffusion coefficient and the thermal diffusion coefficient, respectively. In addition, F_s^j in d^j_s represents the external force acting on the species s. Equation (2.128) involves four components of the diffusion velocity caused by different ways: (1) the gradient in the concentration (ordinary diffusion), (2) the pressure gradient (pressure diffusion), (3) the external force gradient

(forced diffusion) (4) the temperature gradient (thermal diffusion). In the present study, the effects of the pressure, the forced and thermal gradients are neglected because those terms are small compared with that of the ordinary diffusion. Hence, Eq. (2.128) can be expressed by

V_s^j = n² n_s

∑ns k

m_kDs,k

∂

∂x^j (n_s

n )

. (2.130)

According to Hirschfelder, Curtiss and Bird [52], the binary diffusion coefficientDs,k is given by

Ds,k = kT

p∆⁽¹⁾_s,k(T). (2.131)

Following Curtiss and Hirschfelder [59], it is possible to simplify the Eq. (2.130) by a rearrangement as

∑ns k=1

n_sn_k Ds,k

(V_s^j −V_k^j)

=−n² ∂

∂x^j (n_s

n )

. (2.132)

Based on Eq. (2.132), assuming that V_s^j = const for s ̸= k, the diffusion velocity V_s^j for multicomponent mixtures is given by Yos [55] as the following form:

V_s^j =− 1−C_s

∑ns k̸=s

X_k/Ds,k

1 X_s

∂X_s

∂x^j, (2.133)

where the effective diffusion coefficient is defined as D_s ≡ 1−C_s

∑ns k̸=s

X_k/Dsk

C_s

X_s. (2.134)

Thus, the mass flux is expressed by

ρ_sV_s^j =−ρD_s∂X_s

∂x^j . (2.135)

Ambipolar diffusion

Since electron’s mass is very small compared with the heavy particle, the electrons tend to diffuse faster than the charged species. An ambipolar electric field is formed in the plasma, when the electrical potential drop between the diffused electrons (negative charge) and the remaining charged species (positive) is caused. The positive charged species are forced and accelerated by the electric field, while the electrons are decel-erated. Thus, the ambipolar diffusion should be considered in the partially or fully multicomponent plasma. In the present study, the effect of the ambipolar diffusion for charged species is approximately evaluated by

D^a_s = (

1 + T_e T_tr

)

D_s. (2.136)

where Ds is the effective diffusion coefficient of the ionic species in the absence of the ambipolar electric field.

Electron diffusion coefficient

The effective diffusion coefficient of the electron is given by

D_e= m_e

∑ns s=I

D_s^aX_s

∑ns s=I

m_sX_s

. (2.137)

2.6.6 Summary of Transport Properties

When the electron temperature is largely different from the translational tempera-ture of the heavy species, presence of the electrons should be considered in the transport properties calculation. The transport properties for gas mixture in thermochemical nonequilibrium are summarized here.

Viscosity for mixture gas

µ=

∑ns i̸=e









m_iX_i

∑ns j̸=e

X_j∆⁽²⁾_i,j(T_tr) +X_e∆⁽²⁾_i,e(T_e)







+ m_eX_e

∑ns j

X_j∆⁽²⁾_e,j(T_e)

. (2.138)

Thermal conductivity

Translational contribution of thermal conductivity in a multicomponent gas mixture except for electrons:

λ_tr = 15 4 k

∑ns i̸=e









X_i

∑ns j̸=e

α_{i j}X_j∆⁽²⁾_i,j(T_tr) + 3.54X_e∆⁽²⁾_i,e(T_e)







. (2.139)

Rotational contribution of thermal conductivity in a multicomponent gas mixture:

λrot =k

∑nm i=M









X_i

∑ns j̸=e

X_j∆⁽¹⁾_i,j(T_tr) +X_e∆⁽¹⁾_i,e(T_e)







. (2.140)

Vibrational contribution of thermal conductivity in a multicomponent gas mixture:

λ_vib =k

∑nm i=M









(Cv,vib,i

R_i )

X_i

∑ns j̸=e

X_j∆⁽¹⁾_i,j(T_tr) +X_e∆⁽¹⁾_i,e(T_e)







. (2.141)

Thermal conductivity of electrons:

λe= 15

4 k X_e

∑ns j̸=e

1.45X_j∆⁽²⁾_e,j(T_e) +X_e∆⁽²⁾_e,e(T_e)

. (2.142)

Electric conductivity

σ = e² kTe

X_e

∑ns j̸=e

X_j∆⁽¹⁾_e,j(T_e)

. (2.143)

Binary diffusion coefficient

Di j = kT_tr

p∆⁽¹⁾_i,j(T_tr), for i̸= e and j ̸= e, (2.144) Di j = kT_e

p∆⁽¹⁾_i,j(T_e), for i= e or j = e, (2.145) where

∆⁽¹⁾_i,j(T) = 8 3

[ 2m_im_j πkT(m_i+m_j)

]1/2

πΩ^(1,1)_i,j (T), (2.146)

∆⁽²⁾_i,j(T) = 16 5

[ 2m_im_j πkT(m_i+m_j)

]1/2

πΩ^(2,2)_i,j (T), (2.147) αi j = 1 + [1−(mi/mj)] [0.45−2.54 (mi/mj)]

[1 + (m_i/m_j)]² . (2.148)

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