RIGOROUS MODELING OF THE FULL CHEMICAL EQUILIBRIUM FLOWS WITH

REFERENCE TO HYPERSONIC AERODYNAMIC AND HEAT TRANSFER

PROBLEMS.

Grigoriy A. Tirskiy

Institute of Mechanics Moscow State University and

Moscow Institute of Physics Technology Russia

1. Introduction

2. Mass- and heat transfer transport equations solved with respect to the fluxes (fluxes via “forses”, Hirshfelder, Curtis, Bird).

3. Mass and heat transfer equations solved with respect to the gradients of the hydrodynamic variables ("forces" via fluxes, Kolesnikov, Ttirskiy)

4. Different forms of the energy and heat influx equations for a mixture

5. The Navier-Stokes equations for flows of mixtures of gases and plasma in thermochemical and ionization equilibrium. The effective transport coefficients.

6. Conclusions

1. Introduction

Large amount of heat may be carried as chemical enthalpy

of molecules which diffusive because of concentration gradients

and as a result, heat conductivity in reacting gases may be

considerably higher then in nonreacting mixtures.

Heat transfer in dissociating gases was first treated

theoretically by:

1. W. Nerst (Festschr. Ludwig Boltsmann Gewidment,

p.904, 1904)

2. P.A. Dirac (Poc. Cambridge Phil. Soc. 1924, vol. 22, p.

132)

3. J. von Meixner (Z. Naturforsch. 1952, vol. 7a, pp 553-

559)

4. I. Prigogine and R. Buess (Acad. Roy. Belg. 1952, vol

38, ser. 5, pp. 711-851)

5. R. von Hasse (Z. Naturforsch. 1953, vol. 8a, #11, pp

729-740)

6. F. Waelbrock, S. Lafleur and I. Prigogine (Physica,

1955, vol. 21, pp 667)

Available papers (Hirschfelder (1957), Butler and Brokaw

(1957), Brokaw (1960), Krinberg (1965), Luchina (1875) and all)

do not completely and exactly solve the problem of the

hydrodynamic description of chemically equilibrium flows of

multycomponent gas mixtures and a plasma when the

components have different diffusion properties.

First, in a number of papers, the energy equation for

chemically equilibrium flows is described as for a homogeneous

gas but with the conventional thermal conductivity replaced by

an effective thermal conductivity.

Second these papers are limited solely to the calculation of the

effective thermal conductivity and only in isobaric flows (∇p =

0).

Third, the derivation of the expression for the effective thermal

conductivity λr is based on the use of the mass transfer

equations of the components in the Stefan-Maxwell form

without making allowance for the effect of thermo- and

barodiffusion.

Fourth, only molecular heat transfer with a specified element

composition was considered, that is, no allowance was made for

molecular mass transfer in the form of the diffusion of elements

which, of necessity, manifests itself when there is a

temperature gradient in a multicomponent mixture where the

components have different diffusion properties even when the

effects of thermo diffusion and baro diffusion are neglected.

Finally, and this is the most important remark, the initial

formulae for the transport coefficients were based on the use of

the lowest approximations, that is, the first non-zero

approximations when they were found in the form of a series of

Sonin polynomials using the Chapman-Enskog method (CEM)

for the solution of linearized Boltzmann's equation.

It is necessary to take account of higher approximations when

calculating the transport coefficients of a plasma in order to

obtain quantitatively correct results. It has been repeatedly

pointed out in the literature (Kaneko (1960) ), that the formulae

which are conventionally used for the transport coefficients,

obtained in the lower approximations of the Chapman-Enskog

method (the first approximation for the diffusion coefficients

and the second approximation for the thermal conductivity and

thermo diffusion coefficient), do not reduce to the exact

formulae for the transport coefficients, obtained in (Sprinter

(1953), (1962) ) by the method of statistical mechanics of

charged particles, in the case of a completely ionized plasma.

Calculations (Devoto (1966), (1967) ) for certain single-element

ionized gases such as argon and hydrogen, for example, show

that, for the correct determination of the transport coefficients,

it is necessary to take account of a minimum of three terms in

the expansion of the perturbed distribution functions in series

over Sonin polynomials (the third approximation).

In this paper, we introduce equations for the mass transfer of

the elements and heat for thermochemicalry equilibrium flows

and for all the effective transport coefficients in them in any

approximation, which are free from any of the constraints in the

papers cited above. These equations are based on simpler and

more exact equations for the mass transfer of the components

and energy obtained previously in a form which is suitable for

the numerical and analytic solution of problems and are solved

with respect to the "forces" (the gradients of the hydrodynamic

variables) via the fluxes.

ADDITIONAL NOTES. 1. Quasineutrality is assumed in the case of a plasma

(quasineutrality) breaks down electrically conducting walls-

electrodes) and it is assumed that there is no external force

fields.

2. In addition to the defects, which has been noted above,

substitutions of the expressions for the fluxes obtained by

Hirschfelder, Curtiss, Bird into the conservation equations leads

to a system of equations which cannot be solved with respect to

higher derivatives of the required function. At present, there are

no general methods for the effective solution of such systems of

equations even in the approximation of the various

asymptotically simplified versions of the Navier-Stockes

equations.

2. MASS- AND HEAT-TRANSFER TRANSPORT

EQUATIONS SOLVED WITH RESPECT TO THE FLUXES

(FLUXES VIA “FORSES”, HIRSHFELDER, CURTIS, BIRD).

In order to close the system of conservation equations for

mixtures of gas and plasma, it is necessary to have explicit

expressions for the mass diffusion fluxes and total heat flux, the

so-called transport equations.

We will initially give the mass and heat transfer equations in a

multycomponent, partially ionized mixture of gases in the well

known classical form (fluxes via “forces”)

NiTDDm

mm Tijij

N

j

jii ,,1,ln

12 K=∇−ρ= ∑

=dJ

(1)

k

N

j k

Tk

k

N

jkq

DnkThT dJJ ∑∑

==

−+∇′−=11 ρ

λ (2)

( ) ⎟⎠

⎞⎜⎝

⎛−+∇−+∇= ∑

=

N

kikk

iiiii p

cpcxx

1ln FFd ρρ

(3)

Where di are the vectors of the diffusion “forces”. λ′ is not

genuine heat transfer coefficient.

In order to obtain an expression for the true thermal

conductivity λ. in any approximation and, correspondingly, the

heat flux, it is necessary to solve equation (1) with respect to the

vectors di (i =1,..., N). The formal solution can be written in the

form

Tm

DEnm

En

N

k k

Tkik

k

N

k k

iki ln

12

12 ∇+= ∑∑

==

ρρ Jd (4)

where Eik are the elements of the matrix which is the inverse of

the matrix with the elements Dkjmj. Substituting solution (4) into

(2) we obtain the required expression

∑ ∑= =

⎟⎟⎠

⎞⎜⎜⎝

⎛−+∇−=

N

jj

N

k k

Tk

jjq

Dnm

kThT1 1

JJρ

ρλ (5)

with the true thermal conductivity, which is equal to

∑∑= =

+′=N

i

N

j ji

Tj

Tiij

mDDE

nk

1 1 ρρλλ

(6)

Hence, the classical Chapman-Enskog approach in6 give

extremely complex expressions for the true thermal conductivity

and the thermal diffusion coefficient which are not very suitable

for solving problems in gas dynamics. The transport equations

in form (1-3) are not used when solving actual hydrodynamic

problems because of their complexity in particular case of

partially or fully ionized mixtures when it is necessary to take

into account the higer approximations ( ξ≥ 2).

The double summation in (5) is the diffusion thermal

effect and the coefficients in front of Ji are complex

expressions in which is the inverse of a matrix, since the

coefficients Eik are the elements of a matrix which is the

inverse of a matrix in which, in turn, the elements are

ratios of determinants of orders (Nξ + 1) and Nξ

3. MASS AND HEAT TRANSFER EQUATIONS SOLVED WITH RESPECT TO THE GRADIENTS OF THE HYDRODYNAMIC VARIABLES ("FORCES" VIA FLUXES, KOLESNIKOV, TIRSKIY)

We will now derive simple and accurate transport equations in

any approximation which are equivalent to (1) and (2). The idea

behind the derivation of these transport equations was put

forward in (Tirskiy (1974), Gens and Tirskiy (1972) ) and finally

implemented in (Kolesnikov and Tirskiy (1982) [1]).

In any approximation, the equations for the transfer of mass

of the components (the Stefan-Maxwell relations) take the form

[1]

( )

( )( ) ( ) ( )ξξ

ξξ

ξρρ

ξ

ijij

ijTiij

N

j ij

ji

Tij

jN

j ij

ijjii

fTk

xx

NiTkiifxx

d

DDD

D

11)(

ln)(

,,1,ln)(1

)(

1

1

=∇−−=

=∇−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∑

∑

=

=

VV

JJK

(7)

where fij=fji=(1-ϕij), the correction coefficient for the higher

approximation thermal diffusion relations kTi, expressed via the

ratio of the N(ξ-1)+1 determinants to N(ξ-1) order determinant,

i.e. the determinants are N orders less without matrix inversion.

The total heat flux in any approximation has the form

( ) ( )∑ ∑∑= ==

+∇−=++∇−=N

j

N

jj

Tjj

j

TjN

jjjq hT

knkThT

1 11

JJJJ ξλρ

ξλ (8)

where jjjTjTj

jTjjjTjjTj

mkRxkTRhmkThh

/,//==

+=+=

α

αα

where the true thermal conductivity in any approximation

could be expressed via the ratio of the N(ξ-1)+1 determinants to

N(ξ-1) order determinant and so more simple than expression

(6). αTj=kTj/xj – thermodiffusion factor.

Hence, the transport equations with the exact and simpler

transport coefficients are obtained without the double inversion

of matrices and serve as a basis for solving hydrodynamic

problems associated with investigations of the flows of a

multicomponent plasma and, in particular, as if they are

specially represented in a form which is convenient for

converting the equations of motion of multicomponent mixtures

of thermochemically equilibrium flows to canonical form with a

complete set of all the effective transport coefficients, which will

be done later.

4.DIFFERENT FORMS OF THE ENERGY AND HEAT

INFLUX EQUATIONS FOR A MIXTURE

The total heat flux (8), expressed in terms of the temperature

and the specific enthalpy of the mixture hi (i=1,…N), will then be

,11

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−∇−=⎥

⎦

⎤⎢⎣

⎡−∇−= ∑∑

==

N

kk

Tkkk

N

kk

Tkpq hchhhTc JJJ

µσ

σµ

σµ

σµ

σµ

(9)

λµ

σ pc=

where σ is the Prandtl number, defined in terms of the

coefficient of viscosity of the mixture µ, the thermal conductivity

of the mixture λ. and the heat capacity of the mixture cp; hTi is

the specific enthalpy of the i-th component written taking

account of thermal diffusion (see (8)).

Substituting expression (8) into energy equation, we obtain the

energy equation, written in terms of the total enthalpy of the

mixture H=h+v2/2

∑=

+−∂∂

=N

kkkkH vF

tdtdH

1div ρρρ J (10)

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−⎟⎟

⎠

⎞⎜⎜⎝

⎛∇−⋅+∇−= ∑

=

N

kk

TkkkH hchvH

1

2

2ˆ JvJ

µστ

µσ

σµ

were τ̂ -tensor of viscosity stress.

We next explicitly introduce the heats of reactions qi. We

transform the energy and heat influx equations, explicitly

separating out of the terms, which contain the heats of chemical

reactions, including ionization reactions.

The total heat flux (9) will then be equal to

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛−∇−∇−= ∑∑

+==

N

Lkk

Tkkk

N

jj

Tjjjq qcqhchh

11

** JJJµσ

µσ

σµ

(11)

Here cj* and Jj

* is the concentrations and diffusion flaxes of

the chemical elements.

∑∑+=+=

+=+=N

Lkk

k

jkjjj

N

Lkk

k

jkjjj m

mc

mm

cc1

*

1

* ,, JJJ νν (12)

NLim

qqTqBAN

jTjij

N

jTi

ii

Tiijiji ,,1,1),(

1 1K+=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−=−= ∑ ∑

= =

αναν

(13)

It follows from expression (9) that, if when there is a thermal

diffusion effect, one introduces the concept of generalized heats

of reaction qTi=qi-βTi/mi, instead of qi and the concept of

generalized specific enthalpies of the basis components

hTj=hj+(αTj/mj)kT, instead of hj, the structure of the expression

for the total heat flux Jq(9) will be identical to the expression for

Jq when there is no thermal diffusion effect.

The energy equation, written in terms of the enthalpy will be

]}[{

1

1

**

J

J

∑

∑

+=

=

⎟⎠

⎞⎜⎝

⎛+∇+

+⎟⎠

⎞⎜⎝

⎛+∇−∇+=

N

Lkk

Tkkk

L

jj

Tjjj

qcq

hchhdivdtdp

dtdh

µσ

µσ

σµρ

written in terms of the temperature will be

−+⎟⎠

⎞⎜⎝

⎛ −∇+= ∑∑+==

N

Lkkkkk

N

kkTp wqJTRT

dtdp

dtdTc

11div &

rαλρ

kk

N

kkkk

N

kkk m

RRvFeJh =⋅++⋅∇− ∑∑== 11

ˆ:ˆ rrrρτ

5. THE NAVIER-STOKES EQUATIONS FOR FLOWS OF MIXTURES OF GASES AND PLASMA IN THERMOCHEMICAL AND IONIZATION EQUILIBRIUM. THE EFFECTIVE TRANSPORT COEFFICIENTS.

In the case of local thermodynamic equilibrium in the flow,

the diffusion equations for the reaction products (15) in the

asymptotic limit as tch → ∞ are replaced by the conditions of

chemical equilibrium, which, when the reactions are expressed

in the form (13), will be written as

∏ ∑= =

=−==L

j

L

jiji

pi

j

J Nip

TKxx

i

ij

1 1,,1,1,

)(Kννν

ν

(16)

Using (15), the energy equation, written in terms of total

enthalpy H, for locally thermodynamic equilibrium flows will be

eHt

pdt

dH Jdiv−∂∂

=ρ (17)

Where

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−

∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛∇−⋅+∇−= ∑

=

N

jjj

effjj

eff

eff

eH bcapQavH

1

****2

,2

ˆ JvJµσ

ρντ

µσ

σµ

prpeffpreffeff

effpeff ccc

c+=+== ,, λλλ

λµ

σ

Hence it is now possible, using conditions (16) to eliminate the

concentration gradients and the diffusion fluxes of the reaction

products. (∇ck and Jj, k=L+1,…,N). λr is given on fig. 2, σeff –on

fig.3.

Expressions for a(ν,Q ), aj*, bj

* are :

NNLNN

NLLLL

NL

NNLNN

NLLLL

NL

pr

aav

aavQQ

amkQva

aaQ

aaQQQ

amkc

,1,

,11,11

1

,1,

,11,11

1

0

)(),(

0

)(

L

MMMM

L

L

L

MMMM

L

L

+

++++

+

+

++++

+

∆−=

∆−=

NNLNN

NLLLjL

NL

jj

aav

aavQQ

amkQva

,1,*

,11,1*

,1

1

**

0

)(),(

L

MMMM

L

L

+

++++

+

∆−=

NNLN

NLLL

ik

NNLNjN

NLLLjL

TN

TL

j

Tj

bb

bbbb

bbd

bbdQQ

bmQdb

,1,

,11,1

,1,,

,11,1,1

1

*

det)(

0

)(1),(

L

MM

L

L

MMMM

L

L

+

+++

+

++++

+

==∆

∆−=

kTqm

QqmQ

bbQ

bbQQQ

b

TiiT

iTii

Ti

NNLNT

N

NLLLT

L

TN

TL

r ==∆

−=

+

++++

+

,,

0

)(1

,1,

,11,11

1

L

MMMM

L

L

λ

Unlike the case of a homogeneous noreacting gas, additional

terms, proportional to ∇p, ∇cj* and Jj

* are omitted in the paper

cited above, have appeared in Eq. (17) when there are

equilibrium reactions. The concentrations of the elements cj* and

the diffusion fluxes of the elements Jj* (j=1,…,L) will be found

from the diffusion equations of the elements:

Ljdt

dcj

j ,,10div **

K==+ Jρ (18)

Supplemented by the corresponding transport equations of the

elements, which getting from (7) if we eliminate the diffusion

fluxes of the reaction products Jk (k=L+1,…,N) and thereby

obtain the Stefan-Maxwell relations for the chemically

equilibrium flows

∑=

=+−=L

ll

eij

jjj

jej Lj

SmS

1

*)(*)( ,,1, KJJd αµµ (19)

Where diffusion force vectors dj(e) for chemically equilibrium

flows will be

LjpKTmm

Kc pje

jj

Tjjej ,,1,lnln *)(**)( K=∇+∇⎟⎟

⎠

⎞⎜⎜⎝

⎛−+∇= δd (20)

Expressions for coefficients Sj, α jl(e), δj

(e), KTj*, Kpj

* are given

in[1-3]. It is important to note that, if all of the binary

coefficients are equal to one another, then δj(e)≡ 0. at fig 1 one

could see δj(e)

TWO-ELEMENTS MIXTURE

All solar system planet atmospheres consist mostly of two

elements. That are:

Earth O2, N2

Mars CO2

Venus CO2

Jupiter H2, He

Saturn H2, He

For two-elements equilibrium partially ionized mixture

diffusion of elements is described accurately by generalized Fick

law:

0

lnln

**

(e)****

**

=+

⎟⎟⎠

⎞⎜⎜⎝

⎛∇⎟⎟⎠

⎞⎜⎜⎝

⎛−+∇+∇−=−=

βα

ααααα

βα δµ

JJ

JJ

rr

rrT

mm

KPKcS

jTP

here α and β are indexes of the two chemical elements

6. CONCLUSIONS

Finally, we draw a number of fundamental conclusions.

1. When account is taken of thermal diffusion in the energy

equation, this leads to the replacement of the specific

enthalpies of the basis components by the effective enthalpies:

hTj = hj + kTαTj (j = 1,…,L) (8) and the heats of reaction qi by

the "effective heats": qTi = qi -βTi (j (i = L + 1,…,N) (13).

2. If a partially ionized, chemically equilibrium mixture of

gases is formed by the heating of initially two-element mixture

of gases, then the diffusion flux of the jth element will depend

solely on tile single gradient of its own (jth) concentration and

the gradients of the remaining hydrodynamic parameters that

is, the cross-effects of barodiffusion and thermal diffusion.

3. The existence of components in a moving gas mixture with

unequal binary diffusion coefficients or taking account of

thermal and barodiffusion leads to a state of affairs where the

concentrations of the elements ci* (i = L + 1,…,N) do not retain

a constant value in the flow, even when there is no delivery

(blowing) of the substance from the walls around which the

flow occurs. For this reason, the diffusion fluxes of the

elements Jj* ≠ 0.

In other words, Eqs (18), together with relations (19), do not,

in general, admit of the trivial solution Jj* = 0, cj

* = cjw* = const

(i = 1,…,L) due to the existence of the effects of

multicomponent diffusion of components with different binary

diffusion coefficients. This leads to a state of affairs where the

chemically equilibrium composition of the mixture at the point

being considered will not only depend on the pressure and

temperature but also on the concentrations of the chemical

elements, which change in the flow in accordance with (18)

and (19). This has not been taken into account in any of the

papers cited above and other. This is the main difference

between an exact calculation of the equilibrium composition of

the mixture in the flow and the numerous approximate

calculations for being at rest or moving mixture or an inviscid,

non-heat conducting and diffusion-free gas mixture, when the

element chemical composition is specified and is constant in

each flow field and the concentrations of the components and

the thermodynamic properties of the mixture depend solely on

two variables: the pressure and temperature.

The equilibrium transport coefficients were calculated using

the final expressions derived [2, 3]. At fig. 4 higher

approximation account areas are given in the space of pressure-

temperature when calculating effective transfer coefficients for

thermal equilibrium air.

. Figure 1

.

Figure 2

.

Figure 3

Figure 4

References

1. Kolesnikov A.F., Tirskiy G.A. Equation of hydrodynamics for

partially-ionized multycomponent mixtures of gases,

employing higher approximations of transport transfer

coefficients. Fluid Mechanics – Soviet Research, #4, 70-97,

July-August, 1984.

2. Vasil’evskii S.A., Tirskiy G.A. Rigorous modeling of the full

and partially chemical equilibrium flows with reference to

hypersonic aerodynamic and heat transfer problems // Proc.

West East High Speed Flow Field 2002. Aerospace Application

from High Subsonic to Hypersonic Regimes. Marseille, France.

April 22-26, 2002. Ecole Polytechnique Universitaire de

Marseille. Universite de Provence. P. 367-373.

3. Tirskiy G.A. The hydrodynamic equations for chemically

equilibrium flows of a multycomponent plasma with exact

transport coefficient. J. Appl Math& Mechs. Vol 63, #6, pp.

841-861, 1999.

The work received financial support from the RFBI (project 03-

01-00542-a), leading scientific schools program (project

1899.2003.1) and universities of Russia program (project

04.01.167).