Evaporation Condensation and Heat transfer Part 12 pot

to obtain closed systems ofthe non-equilibrium flow equations and to elaborate calculation procedures for transport andrelaxation properties.In the present chapter, on the basis of the ki

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0

Fig 5 Non-isothermal Couette flow of PTT & Newtonian fluids

0 0.1

Fig 6 Non-isothermal Couette flow of PTT & Newtonian fluids (Higher Shear rate)

3.3 Thermal runaway

The long term behavior of the fluid maximum temperature with respect to higher values of

Non-Isothermal Flows 9

and in related works cited therein In Fig 7, the maximum temperatures are recorded atconvergence for each value of the reaction parameter until a threshold value of the reactionparameter is reached at which blow-up of the temperature is observed We notice that

explanations relate to the ability of viscoelastic fluid to store energy due to their elasticcharacter Thus while Newtonian fluids would dissipate all the mechanical energy as heat

in an entropic process, viscoelastic fluids on the other hand will partially dissipate some ofthe energy and store some

Fig 7 Thermal Runaway

4 Thermally decomposable lubricants

In this section, we summarize the work in Chinyoka (2008) for the flow of a thermallydecomposable lubricant described by the Oldroyd-B model In this case, the dissipationfunction takes the form,

exothermic reactions is modeled via Arrhenius kinetics The nonlinear polymer stress functionfor the Oldroyd-B model is identically zero,

431

Effects of Fluid Viscoelasticity in Non-Isothermal Flows

The temperature dependence of the viscosities and relaxation time respectively follow aNahme-type law:

5 Flow in heat exchangers

The lubricant fluid dynamics of the previous section is an important problem as far as physical(industrial and engineering) applications are concerned An equally important problem is that

of coolant fluid dynamics, which is necessarily related to heat exchanger design Three majortypes of heat exchangers are in existence, parallel flow, counter flow Chinyoka (2009a) andcross flow Chinyoka (2009b) heat exchangers The parallel flow heat exchangers are quiteinefficient for industrial scale cooling processes and will not be discussed any further Carradiators employ the cross flow heat exchanger design in which liquid coolant is cooled by

a stream of air flowing tangential to the direction of flow of the liquid coolant Counterflowheat exchanger arrangements are normally employed in industrial settings (say distillationprocesses and food processing) in the form of pipe-in-a-pipe heat exchangers, in which themain fluid to be cooled flows in the inner pipe in the opposite direction to the “colder” fluidflowing in the outer annulus

A choice of the coolant fluid which optimizes performance is undoubtedly of majorimportance as far as physical applications are concerned In particular, the coolant fluidshould be capable of resisting large temperature increases as well as also being able to rapidlylose heat This thus provides the impetus for a comparative study of the thermal loadingproperties of inelastic versus viscoelastic coolants In most industrial settings, the focus mayinstead be on the cooling characteristics and properties of fluids whose elastic properties arepredetermined and not subject to choice, say the fluids extracted from distillation processes.The works referenced in this section can still be used to determine the cooling properties ofsuch fluids whether they are inelastic or viscoelastic Such conclusions can be obtained frominvestigations such as those in Chinyoka (2009a;b) In these two cited works, the Giesekusmodel is employed for the viscoelastic fluids In this case, the dissipation function takes theform,

Non-Isothermal Flows 11

problems are similar to those considered previously Convective temperature boundaryconditions are employed at the interfaces and initial conditions are specified appropriately.Typical results for the fluid temperature are displayed in Fig 8 The figure shows the resultsfor a double pipe (pipe in a pipe) counterflow heat exchanger The inner pipe is referred to

the figure shows, as expected, that the core fluid temperature decreases downstream (since it isbeing cooled by the shell fluid) whereas the shell fluid temperature increases downstream As

in the previous sections, a viscoelastic core fluid leads to lower temperatures than an inelasticfluid Chinyoka (2009a;b)

6 Convection reaction flows

The one dimensional natural convection flow of Newtonian fluids between heated plateshas received considerable attention, see for example the detailed work in Christov & Homsy(2001) and the references therein In fact the steady state case easily yields to analyticaltreatment, White (2005) In physical applications lubricants, coolants and other importantindustrial fluids are usually exposed to shear flow between parallel plates Differentialheating of the plates thus indeed lead to natural or forced convection flow as illustrated

in Christov & Homsy (2001) The previous sections have highlighted the need to employviscoelastic fluids in such industrial applications involving lubricant and coolant fluiddynamics especially if thermal blow up due, say, to exothermic reactions is a possibility Inthis section we revisit the shear flow of reactive viscoelastic fluids between parallel heatedplates and in light of the observations just noted, we investigate the added effects of natural

or forced convection, in essence summarizing the results of Chinyoka (2011)

As before, we use the Giesekus model for the viscoelastic fluid The model problem consists

of a viscoelastic fluid enclosed between two parallel and vertical plates For simplicity, weconsider the case in which the left hand side plate moves downwards at constant speed andthe right hand side plate moves upwards at a similar speed This creates a shear flow withinthe enclosed fluid Additionally, the differential heating of the plates leads to convectioncurrents developing in the flow field Relevant body forces that account for the convectionflow are added to the momentum equation These body forces are of the form:

where i is the unit vector directed vertically downwards, Gr is the Grashoff number and T is

the fluid temperature Typical results are displayed in Figs 9 - 12

As is expected from the results of the preceding sections and as also shown in Chinyoka (2011)the maximum temperatures attained are lower for the viscoelastic Giesekus fluids than forcorresponding inelastic fluids

7 Current and future work

In this section we summarize at a couple of current investigations that may in the future have

an impact on the conclusions drawn thus far

7.1 Shear rate dependent viscosity

The viscoelastic fluids chronicled in the preceding sections were all of the Boger type andhence all had non shear-rate dependent viscosities The reduction of these fluids to inelastic

433

Effects of Fluid Viscoelasticity in Non-Isothermal Flows

0 0.2 0.4 0.6 0.8 1 1.1

1.15 1.2

0.2 0.4 0.6 0.8 1

0 0.05 0.1

0

1.1 1.2

Non-Isothermal Flows 13

thus lead directly to Newtonian fluids! All the comparisons made were thus for viscoelasticfluids against Newtonian fluids We note that the viscoelastic fluids are part of the broaderclass of non-Newtonian fluids It may be important to compare the performance of viscoelasticfluids against other (albeit inelastic) non-Newtonian fluids, i.e the Generalized Newtonianfluids, which are characterized by shear-rate dependent viscosities The current work inChinyoka et al (Submitted 2011b) for example uses Generalized Oldroyd-B fluids, whichcontain both shear-rate dependent viscosity (described by the Carreau model) as well as elasticproperties

7.2 Non-monotonic stress-strain relationships

The viscoelastic fluids used in the preceding sections are also all described by a monotonicstress versus strain relationship No jump discontinuities are thus expected in the shearrates for any of these viscoelastic models and hence they all lead to smooth (velocity,temperature and stress) profiles in simple flows The viscoelastic Johnson-Segalman modelhowever allows for non-monotonic stress-strain relationships in simple flow under certainconditions Chinyoka Submitted (2011a) Under such conditions, jump discontinuities mayappear in the shear-rates and hence no smooth solutions would exist, say, for the velocityChinyoka Submitted (2011a) In particular only shear-banded velocity profiles would beobtainable If the flow is non-isothermal, as in Chinyoka Submitted (2011a), the large shearrates obtaining in the flow would lead to drastic increases in the fluid temperature evenbeyond the values attained for corresponding inelastic fluids This would thus be an example

of a viscoelastic fluid which does not conform to the conclusions of the preceding sections

in which viscoelastic fluids always resisted large temperature increases as compared tocorresponding inelastic fluids

0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1

−0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

y

T(x,y,t)

xFig 9 Temperature distribution in absence of convection flow

435

Effects of Fluid Viscoelasticity in Non-Isothermal Flows

Fig 11 Pressure contours in absence of convection flow.

Non-Isothermal Flows 15

0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1

−0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

y

T(x,y,t)

xFig 12 Temperature distribution in presence of convection flow

8 Conclusion

We conclude that non-Newtonian fluids play a significant role in non-isothermal flows ofindustrial importance In particular, viscoelastic fluids are important in industrial applicationswhich require the design of fluids with increased resistance to temperature build up Forimproved thermal loading properties, energetic and entropic effects of the (viscoelastic)fluids however need to be carefully balanced, say by varying the elastic character of thefluids Viscoelastic fluids, say of the Johnson-Segalman type, that exhibit shear banding inexperiment however may not be suitable for the aforementioned applications as they can lead

to rapid blow up phenomena, faster than even the corresponding inelastic fluids All thequantitative (numerical) and qualitative (graphical) results displayed were computed usingsemi-implicit finite difference schemes

9 References

R.B Bird, C.F Curtiss, R.C Armstrong, O Hassager (1987), Dynamics of polymeric liquids

Vol 1 Fluid mechanics, Second edition, Wiley, New York

T Chinyoka, Y.Y Renardy, M Renardy and D.B Khismatullin, Two-dimensional study of drop

deformation under simple shear for Oldroyb-B liquids, J Non-Newt Fluid Mech 31

(2005) 45-56.

T Chinyoka, Computational dynamics of a thermally decomposable viscoelastic lubricant

under shear, Transactions of ASME, J Fluids Engineering, December 2008, Volume 130,

Issue 12, 121201 (7 pages)

T Chinyoka, Viscoelastic effects in Double-Pipe Single-Pass Counterflow Heat Exchangers,

Int J Numer Meth Fluids, 59 (2009) 677-690.

437

Effects of Fluid Viscoelasticity in Non-Isothermal Flows

T Chinyoka, Modelling of cross-flow heat exchangers with viscoelastic fluids, Nonlinear

Analysis: Real World Applications 10 (2009) 3353-3359

T Chinyoka, Poiseuille flow of reactive Phan-Thien-Tanner liquids in 1D channel flow,

Transactions of ASME, J Heat Transfer, November 2010, Volume 132, Issue 11, 111701 (7 pages) doi:10.1115/1.4002094

T Chinyoka, Two-dimensional Flow of Chemically Reactive Viscoelastic Fluids With or

Without the Influence of Thermal Convection, Communications in Nonlinear Science

and Numerical Simulation, Volume 16, Issue 3, March 2011, Pages 1387-1395.

T Chinyoka, Suction-injection control of shear banding in non-isothermal and exothermic

channel flow of Johnson-Segalman liquids, submitted

T Chinyoka, S Goqo, B.I Olajuwon, Computational analysis of gravity driven flow of a

variable viscosity viscoelastic fluid down an inclined plane, submitted

C.I Christov and G.H Homsy, Nonlinear Dynamics of Two Dimensional Convection in a

Vertically Stratified Slot with and without Gravity Modulation, J Fluid Mech 430(2001) 335-360

M Dressler, B.J Edwards, H.C Öttinger (1999) “Macroscopic thermodynamics of flowing

J.D Ferry (1981), Viscoelastic properties of polymers, Third edition, Wiley, New York

D.A Frank-Kamenetskii (1969), Diffusion and Heat Transfer in Chemical Kinetics, Second

edition, Plenum Press, New York

M Hütter, C Luap, H.C Öttinger (2009) “Energy elastic effects and the concept of temperature

G.W.M Peters, F.P.T Baaijens (1997) “Modelling of non-isothermal viscoelastic flows”, J

Non-Newtonian Fluid Mech., Vol 68, pp 205-224

B Straughan (1998), Explosive Instabilities in Mechanics, Springer

F Sugend, N Phan-Thien, R.I Tanner (1987) “A study of non-isothermal non-Newtonian

extrudate swell by a mixed boundary element and finite element method”, J Rheol.,Vol 31(1), pp 37-58

P Wapperom, M.A Hulsen (1998) “Thermodynamics of viscoelastic fluids: the temperature

F.M White, Viscous fluid flow, 3rd edition, McGraw-Hill ISE, 2005

Different Approaches for Modelling

of Heat Transfer in Non-Equilibrium

Reacting Gas Flows

E.V Kustova and E.A Nagnibeda

Saint Petersburg State University

Russia

1 Introduction

Modelling of heat transfer in non-equilibrium reacting gas flows is very important andpromising for many up-to-date practical applications Thus, calculation of heat fluxes isneeded to solve the problem of heat protection for the surfaces of space vehicles enteringinto planet atmospheres

In high-temperature and hypersonic flows of gas mixtures, the energy exchange betweentranslational and internal degrees of freedom, chemical reactions, ionization and radiationresult in violation of thermodynamic equilibrium Therefore the non-equilibrium effectsbecome of importance for a correct prediction of gas flow parameters and transport properties.The first attempt to take into account the excitation of internal degrees of freedom incalculations for the transport coefficients was made in 1913 by E Eucken Eucken (1913),who introduced a phenomenological correction into the formula for the thermal conductivitycoefficient Later on, stricter analysis for the influence of the excitation of internal degrees

of freedom of molecules on heat and mass transfer was based on the kinetic theory of gases.Originally, in the papers concerning kinetic theory models for transport properties, mainlyminor deviations from the local thermal equilibrium were considered for non-reacting gasesFerziger & Kaper (1972); Wang Chang & Uhlenbeck (1951) and for mixtures with chemicalreactions Ern & Giovangigli (1994) In this approach non-equilibrium effects were takeninto account in transport equations by introducing supplementary kinetic coefficients: thecoefficient of volume viscosity in the expression for the pressure tensor and corrections to thethermal conductivity coefficient in the equation for the total energy flux Such a description

of the real gas effects becomes insufficient under the conditions of finite (not weak) deviationsfrom the equilibrium, in which the energy exchange between some degrees of freedom andsome part of chemical reactions proceed simultaneously with the variation of gas-dynamicparameters In this case characteristic times for gas-dynamic and relaxation processes becomecomparable, and therefore the equations for macroscopic parameters of the flow should

be coupled to the equations of physical-chemical kinetics The transport coefficients, heatfluxes, diffusion velocities directly depend on non-equilibrium distributions, which may differsubstantially from the Boltzmann thermal equilibrium distribution In this situation, theestimate for the impact of non-equilibrium kinetics on gas-dynamic parameters of a flow

21

and its dissipative properties becomes especially important In recent years, these problemsreceive much attention, and new results have been obtained in this field on the basis of thegeneralized Chapman-Enskog method Nagnibeda & Kustova (2009), see also references inNagnibeda & Kustova (2009) The kinetic theory makes it possible to develop mathematicalmodels of a flow under different non-equilibrium conditions, i.e to obtain closed systems ofthe non-equilibrium flow equations and to elaborate calculation procedures for transport andrelaxation properties.

In the present chapter, on the basis of the kinetic theory developed in Nagnibeda & Kustova(2009), the mathematical models for calculation of heat transfer in strong non-equilibriumreacting mixtures are proposed for different conditions in a flow

2 Theoretical models

The theoretical models adequately describing physical-chemical kinetics and transportproperties in a flow depend on relations between relaxation times of various kinetic processes.Experimental data show the significant difference in relaxation times of various processes

At the high temperature conditions which are typical just behind the shock front, theequilibrium over the translational and rotational degrees of freedom is established for asubstantially shorter time than that of vibrational relaxation and chemical reactions, andtherefore the following relation takes place Phys-Chem (2002); Stupochenko et al (1967):

τ tr < τ rot  τ vibr < τ react ∼ θ. (1)

the mean time of the variation of gas-dynamics parameters In this case for the description

of the non-equilibrium flow it is necessary to consider the equations of the state-to-statevibrational and chemical kinetics coupled to the gas dynamic equations It is the most detaileddescription of the non-equilibrium flow Transport properties in the flow depend not only

on gas temperature and mixture composition but also on all vibrational level populations ofdifferent species Kustova & Nagnibeda (1998)

More simple models of the flow are based on quasi-stationary multi-temperature or

temperatures, the near-resonant vibrational energy exchanges between molecules of the samechemical species occur much more frequently compared to the non-resonant transitionsbetween different molecules as well as transfers of vibrational energy to the translational androtational ones and chemical reactions:

τ tr < τ rot < τ VV1  τ VV2 < τ TRV < τ react ∼ θ. (2)

molecules of different species and TRV transitions of the vibrational energy into other modes.Under this condition quasi-stationary (multi-temperature) distributions over the vibrationallevels establish due to rapid energy exchanges, and equations for vibrational level populationsare reduced to the equations for vibrational temperatures for different chemical species

Different Approaches for Modelling of Heat Transfer in Non-Equilibrium Reacting Gas Flows 3

Heat and mass transfer are specified by the gas temperature, molar fractions of species andvibrational temperatures of molecular components Chikhaoui et al (1997)

For tempered reaction regime, with the chemical reaction rate considerably lower than thatfor the internal energy relaxation, the following characteristic time relation takes place:

τ tr < τ int  τ react ∼ θ, (3)

non-equilibrium chemical kinetics can be described on the basis of the maintaining thermalequilibrium one-temperature Boltzmann distributions over internal energies of molecularspecies while transport properties are defined by the gas temperature and molar fractions ofmixture components Ern & Giovangigli (1994); Nagnibeda & Kustova (2009) The influence

of electronic excitation of atoms and molecules on the transport properties under the lastcondition is also considered in Kustova & Puzyreva (2009)

Finally, if all relaxation processes and chemical reactions proceed faster than gas-dynamicparameters vary, the relaxation times satisfy the relation:

τ tr < τ int < τ react  θ. (4)

chemically equilibrium or weakly non-equilibrium (Brokaw (1960); Butler & Brokaw (1957);Vallander et al (1977)) In this case the closed system of governing equations of a flowcontains only conservation equations, and non-equilibrium effects in a viscous flow manifestthemselves mainly in transport coefficients

In the present contribution, for all these approaches, on the basis of the rigorous kinetictheory methods, we propose the closed sets of governing equations of a flow, expressions fortransport and relaxation terms in these equations and formulas for the calculation of transportcoefficients The results of applications of proposed models for particular flows are brieflydiscussed The comparison of the results obtained for heat transfer in different approachesbehind shock waves, in nozzle flows, in the non-equilibrium boundary layer and in a shocklayer near the re-entering body is discussed

2.1 State-to-state model

2.1.1 Distribution functions and governing equations

The mathematical models of transport properties in non-equilibrium flows of reacting viscousgas mixtures are based on the first-order solutions of the kinetic equations for distribution

particle velocities u, coordinates r and time t In the case of strong deviations from thermal

and chemical equilibrium in a flow, the kinetic processes may be divided for rapid and slow

(2009)

∂ f cij

∂t +uc · ∇ f cij= 1ε Jrapcij +Jslcij, (5)

where Jrapcij , Jsl

441

Different Approaches for Modelling of Heat Transfer in Non-Equilibrium Reacting Gas Flows

J cijrapdescribes elastic collisions and rotational energy exchange whereas the operator for slow

J cijrap=Jtrcij+Jrotcij,

The integral operators (6) are given in Ern & Giovangigli (1994); Nagnibeda & Kustova (2009).For the solution of the kinetic equations (5), (6) modification of the Chapman-Enskog methodfor rapid and slow processes Chikhaoui et al (1997); Kustova & Nagnibeda (1998) is used.This method makes it possible to derive governing equations of the flow, expressions forthe dissipative and relaxation terms in these equations and algorithms for the calculation oftransport and reaction rate coefficients The solution of the equations (5), (6) is sought as the

The solution of the kinetic equations in the zero-order approximation

is specified by the independent collision invariants of the most frequent collisions whichdefine rapid processes These invariants include the momentum and particle total energywhich are conserved at any collision, and additional invariants for the most probable collisions

which are given by any value independent of the velocity and rotational level j and depending arbitrarily on the vibrational level i and chemical species c The additional invariants appear

due to the fact that vibrational energy exchange and chemical reactions are supposed to befrozen in rapid processes Based on the above set of the collision invariants, the zero-orderdistribution function takes the form

j =ε c

j , Z rot ci =Z rot c = 8π2I c kT

the symmetry factor

invariants of rapid processes

The closed set of equations for the macroscopic parameters n ci(r, t), T(r, t), and v(r, t)followsfrom the kinetic equations and includes the conservation equations of momentum and totalenergy coupled to the relaxation equations of detailed state-to-state vibrational and chemical

Different Approaches for Modelling of Heat Transfer in Non-Equilibrium Reacting Gas Flows 5

kinetics Nagnibeda & Kustova (2009):

at different vibrational states, U is the total energy per unit mass:

c at the i-th vibrational level, ε c is the energy of formation of the particle of species c.

The source terms in the equations (9) are expressed via the integral operators of slowprocesses:

R ci=∑

j



and characterize the variation of the vibrational level populations and atomic numberdensities caused by different vibrational energy exchanges and chemical reactions

For this approach, the vibrational level populations are included to the set of mainmacroscopic parameters, and the equations for their calculation are coupled to the equations

of gas dynamics Particles of various chemical species in different vibrational states represent

of molecules at different vibrational states Thus the diffusion of vibrational energies is thepeculiar feature of diffusion processes in the state-to-state approximation

2.1.2 Transport properties

In the zero-order approximation of Chapman-Enskog method

P(0)=nkTI, q(0)=0, V(0)

The set of governing equations in this case describes the detailed state-to-state vibrational andchemical kinetics in an inviscid non-conductive gas mixture flow in the Euler approximationNagnibeda & Kustova (2009) Taking into account the first-order approximation makes itpossible to consider dissipative properties in a non-equilibrium viscous gas

The first-order distribution functions can be written in the following structural formKustova & Nagnibeda (1998):

The functions Acij, Ddk

The viscous stress tensor is described by the expression:

relaxation pressure appear in the diagonal terms of the stress tensor in this case due to rapidinelastic TR exchange between the translational and rotational energies The existence of therelaxation pressure is caused also by slow processes of vibrational and chemical relaxation If

The diffusion velocity of molecular components c at the vibrational level i is specified in the

state-to-state approach by the expression Kustova & Nagnibeda (1998); Nagnibeda & Kustova(2009):

dk

each chemical and vibrational species

The total energy flux in the first-order approximation has the form:

2kT +  ε ci rot+ε c

i+ε c



n ciVci, (18)

probable processes which, in the present case, are the elastic collisions and inelastic TR- and

RR rotational energy exchanges In the state-to-state approach, the transport of the vibrationalenergy is described by the diffusion of vibrationally excited molecules rather than the thermalconductivity In particular, the diffusion of the vibrational energy is simulated by introducingthe independent diffusion coefficients for each vibrational state It should be noted thatall transport coefficients are specified by the cross sections of rapid processes excepting therelaxation pressure depending also on the cross sections of slow processes of vibrationalrelaxation and chemical reactions

From the expressions (17), (18), it is seen that the energy flux and diffusion velocities includealong with the gradients of temperature and atomic number densities also the gradients ofall vibrational level populations with multi-component diffusion coefficients depending onthe vibrational levels of colliding molecules In the state-to-state approach, the transport of

Different Approaches for Modelling of Heat Transfer in Non-Equilibrium Reacting Gas Flows 7

vibrational energy is associated with diffusion of vibrationally excited molecules rather thanwith heat conductivity This constitutes the main feature of the heat transfer and diffusion

diffusion velocities and heat flux obtained on the basis of one-temperature, multi-temperature

or weakly non-equilibrium approaches

Ddk cij , B cij , F cij , and G cij:

They were introduced in Nagnibeda & Kustova (2009) for strongly non-equilibrium reactingmixtures similarly to those defined in Ferziger & Kaper (1972) for a non-reacting gas mixtureunder the conditions for weak deviations from the equilibrium

For the transport coefficients calculation in the state-to-state approximation, the functions

peculiar velocity and those of Waldmann-Trübenbacher in the dimensionless rotationalenergy For the coefficients of these expansions, the linear transport systems are derived inKustova & Nagnibeda (1998), Nagnibeda & Kustova (2009), and the transport coefficients areexpressed in terms of the solutions of these systems

Solving transport linear systems for multi-component mixtures in the state-to-stateapproximation is a very complicated technical problem because a great number of equationsshould be considered A simplified technique for the calculation of the transport coefficientskeeping the main advantages of the state-to-state approach, is suggested in Kustova (2001).The assumptions proposed in this paper made it possible to noticeably reduce the number ofmulti-component diffusion and thermal diffusion coefficients and simplify the expressions forthe diffusion velocity and heat flux:

2kT+ε ci

j

rot+ε c

i+ε c



n ciVci (22)

to the vibrational level Ferziger & Kaper (1972) It is important to emphasize that in these

on the vibrational quantum number

The systems for the calculation of the diffusion, viscosity and thermal conductivity coefficientscan be solved using the efficient numerical algorithms elaborated in Ern & Giovangigli (1994)for the solution of linear algebraic systems, or more traditional techniques used in classicalmonographs on the kinetic theory Chapman & Cowling (1970); Ferziger & Kaper (1972);Hirschfelder et al (1954)

445

Different Approaches for Modelling of Heat Transfer in Non-Equilibrium Reacting Gas Flows

The expression for the total energy flux may be presented as a sum of contributions of differentprocesses:

conductivity of translational and rotational degrees of freedom, mass diffusion, thermaldiffusion, and the transfer of vibrational energy

heat flux variation behind a shock wave and in a nozzle flow along its axis found in thestate-to-state approach

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

HC TD MD DVE

(b)

q

D/q

x/R

of x/R (R is the throat radius).

We can see that the contribution of thermal diffusion to the heat flux is small in both flowswhile mass diffusion of atoms is important in the whole flow region Diffusion of vibrationallyexcited molecules plays more important role behind a shock than in a nozzle Close to theshock front, heat conduction and mass diffusion compensate each other, and the main role in

not negligible only close to the throat (but does not exceed 15%)

The model presented in this section gives a principle possibility to take into accountthe state-to-state transport coefficients in numerical simulations of viscous conducting gas

influence of state-to-state vibrational and chemical kinetics on the dissipative processeswas studied using this model in the flows of binary mixtures of air componentsbehind shock waves Kustova & Nagnibeda (1999) and in the nozzle expansion ofbinary mixtures Kustova, Nagnibeda, Alexandrova & Chikhaoui (2002) and 5-component airmixture Capitelli et al (2002) However, even taking into account proposed simplifications oftransport coefficients mentioned above, the problem of implementation of the state-to-statemodel of transport properties in numerical fluid dynamic codes for the flows ofmulti-component reacting mixtures remains time consuming and numerically expensive forapplications particularly if many test cases should be considered Indeed, the solution ofthe fluid dynamics equations coupled to the equations of the state-to-state vibrational andchemical kinetics in a flow requires numerical simulation of a great number of equations

Different Approaches for Modelling of Heat Transfer in Non-Equilibrium Reacting Gas Flows 9

for the vibrational level populations of all molecular species Moreover, the closed system

of macroscopic equations in the state-to-state approach includes the state-dependent rate

and theoretical data on these rate coefficients and especially on the cross sections ofinelastic processes are rather scanty Due to the above problems, simpler models based onquasi-stationary vibrational distributions are rather attractive for practical applications.Such approaches are considered in the next section

2.2 Quasi-stationary models

In quasi-stationary approaches, the vibrational level populations are expressed in terms of afew macroscopic parameters, consequently, non-equilibrium kinetics can be described by aconsiderably reduced set of governing equations Commonly used models are based on theBoltzmann distribution with the vibrational temperature different from the gas temperature.However, such a distribution appears not to be justified under the conditions of strongvibrational excitation, since it is valid solely for the harmonic oscillator model, whichdescribes adequately only the low vibrational states In the present section, the transportkinetic theory is considered on the basis of the non-Boltzmann vibrational distributions takinginto account anharmonic vibrations

2.2.1 Distribution functions and governing equations

Quasi-stationary models follow from the kinetic equations (5) under the conditions (2) forthe relaxation times In this case, the integral operator of the most frequent collisions in the

transitions between molecules of the same species along with the operators of elastic collisionsand collisions with rotational energy exchanges:

J cijrap=Jtrcij+J cijrot+JVV1

between molecules of different species, the operator describing the transfer of vibrational

reactions Jreact

cij :

J cijsl =JVV1

For the solution of Eqs (5) with the collisional operators (24) and (25), the distributionfunction is expanded into the generalized Chapman-Enskog series in the small parameter

ε ∼ τ VV1/θ In the zero-order approximation, the following equation for the distribution

the vibrational quanta in a molecular species c, and any value independent of the velocity, vibrational i and rotational j quantum numbers and depending arbitrarily on the particle chemical species c Conservation of vibrational quanta presents an important feature of

447

Different Approaches for Modelling of Heat Transfer in Non-Equilibrium Reacting Gas Flows

collisions resulting in the VV1vibrational energy exchange between the molecules of the samespecies The existence of a similar invariant for VV transitions in a single-component gaswas found for the first time in Treanor et al (1968) where a non-equilibrium quasi-stationarysolution of balance equations for the vibrational level populations was found using thisinvariant This solution is now called the Treanor distribution.

species, the number of vibrational quanta in a given species keeps constant The existence ofthe other additional invariants is explained by the fact that under the condition (2), chemicalreactions are supposed to be slow and remain frozen in the most frequent collisions

Taking into account the system of collision invariants we obtain the zero-order distributionfunctions:

from the energy of the zero-th level The functions (27) represent the local equilibriumMaxwell-Boltzmann distribution of molecules over the velocity and rotational energy levelsand the nonequilibrium distribution over the vibrational states and chemical species For thevibrational level populations, from Eq (27) it follows:

translational-rotational and vibrational degrees of freedom are not isolated in the most

The expression (29) yields the non-equilibrium quasi-stationary Treanor distribution

single-component gas, the distribution (29) describes adequately only the populations of

It is explained by the fact that the number of vibrational quanta is conserved only at low

conductivity of translational and rotational degrees of freedom, mass diffusion, thermaldiffusion, and the transfer of vibrational energy

heat flux variation behind a shock wave and in a nozzle flow... vibrational quanta in a molecular species c, and any value independent of the velocity, vibrational i and rotational j quantum numbers and depending arbitrarily on the particle chemical species c Conservation... important role behind a shock than in a nozzle Close to theshock front, heat conduction and mass diffusion compensate each other, and the main role in

not negligible only close to the throat