Fundamentals of High Pressure CombustionChapter in High Pressure Processes in Chemical Engineering, Edited by Maximillian Lackner,

ProcessEng Engineering GmbH, pp. 53-75, 2010

Justin Foster† and Richard S. Miller††

Department of Mechanical EngineeringClemson University

Clemson, South Carolina 29634-0921

† Doctoral student: jwfoste@clemson.edu†† Associate Professor: rm@clemson.edu

1 Introduction

The objective of the following chapter is to present an overview of the fundamentals of combustionprocesses in high pressure environments. Developing such an understanding is important to thefurther development of modern combustion devices. Many such devices operate under supercriticalpressure conditions. The critical pressure (or “critical locus” for a mixture) of the majority of hydrocar-bon fuels is in the range of approximately ∼ 15 < PC < 30atm [1]. In comparison, aircraft gas turbineengines operate at quasi-steady pressures ∼ 30atm [2]. Furthermore, gas turbine combustion pres-sures have been increasing at a near linear rate for more than 50 years and are expected to continueto do so [2]. Diesel engines obtain pressures as large as ∼ 60atm after ignition [3, 4, 5, 6, 7]. Figure1 presents a schematic of diesel and gas turbine engine combustion in thermodynamic state space.The mixture saturation curves of both diesel fuel and JET-A fuel are both illustrated. For both sys-tems, the fuel is injected at very large pressures into the combustion chamber. For diesel engines thechamber is at approximately 25atm at injection which then rapidly increases with ignition. Rocket en-gines utilizing hydrogen and oxygen represent another example. The critical pressures for hydrogenand oxygen are approximately 13atm and 50atm, respectively [1]. In comparison, rocket combustionchambers achieve pressures well in excess of 100atm [8]. Sutton states that the largest combustionchamber pressure yet achieved in the U.S. is approximately 225atm measured in the space shuttle.The fundamentals of such supercritical pressure combustion processes are the focus of this chapter.

At supercritical conditions the state of the species is often referred to simply as “fluid.” This is be-cause changes in temperature at fixed supercritical pressure do not result in phase change. The fluidsimply becomes less dense with increasing temperature. Figure 2 illustrates this effect by presentingthe variation of density (and compressibility) for both nitrogen (as a model for air) and dodecane whichis a primary component species in most hydrocarbon fuel blends. The pressure is fixed at 50atm andthe properties are predicted by the cubic Peng Robinson equation of state (see below). The hydrocar-bon in particular experiences large variations in the fluid density from “liquid-like” at low temperaturesto ∼ 100kg/m3 at larger temperatures; however, no discontinuity exists. Note in addition, that for thepurposes of the following discussion, a fluid (mixture) is defined as being in the supercritical state ifeither the pressure or temperature are above the critical values. In either case no phase change ispossible.

Related experimental studies typically address fuel jets or atomizers issuing into high pressurechambers, and are primarily based on qualitative visualizations or relatively simple quantitative mea-surements [9, 10, 11, 12, 13, 14, 15, 16, 17]. This is because it is difficult to make intrusive measure-ments within pressurized combustion chambers [18]. Most of these studies do not include combustiondue to further difficulties associated with heat release and associated pressure increases. Exceptionsinclude the work of Candel et al. [13] and Mayer et al. [17]; both of whom conducted experimentsof oxygen-hydrogen flames at pressures as large as 1MPa and 6MPa, respectively. The results ofthese experimental studies show that the behavior of supercritical jets is dramatically altered from

1

Temperature [K]

Pre

ssur

e[a

tm]

500 600 700 800 900 10000

10

20

30

40

50

60

70

LiquidVapor

JET-A Sat. curve

Diesel Sat. curve

Critical Points

Fuel InjectionConditions

Figure 1: Schematic of thermodynamic diesel combustion regimes: Pressure-Temperature diagramindicating typical diesel and JET-A fuel saturation curves [19, 20] (low and high temperature sides ofthe saturation curves correspond to measured values of the bubble point and dew point, respectively).

traditional low pressure liquid jets. As the critical pressure is exceeded the breakup of the liquid coreinto ligaments and droplets ceases abruptly, and further increases in pressure yield behaviors char-acteristic of low pressure gas-gas jets due to the absence of surface tension and latent heat [16]. Asa result, atomizers for “liquid” fuels become increasingly less effective with increases in pressure overthe critical value. In this sense supercritical combustion is somewhat simplified by the absence ofphase change, surface tension, and latent heat. However, other fundamental changes occur at largepressures that must be accounted for in properly describing the physics of high pressure combustion.

2 Theory

In order to more formally investigate the alterations of combustion phenomena due to high pres-sures, we begin with the governing equations for a compressible reacting mixture:

∂ρ

∂t+

∂

∂xj[ρuj ] = 0 , (1)

∂

∂t(ρui) +

∂

∂xj[ρuiuj + Pδij − τij − ρgi] = 0, (2)

∂

∂t(ρet) +

∂

∂xj

[(ρet + P )uj − uiτij +Qj +

N∑α=1

h,αJ j,α − ρgiui

]= Se, (3)

∂

∂t(ρYα) +

∂

∂xj[ρYα uj + Jj,α] = SYα . (4)

The above system of equations represents a generalized formulation of fluid flow incorporating thepotential for combustion through the right hand side source terms. The compressible form of theNavier-Stokes equations describes the variations of the fluid density (ρ), velocity (ui), specific total

2

Temperature [K]

Den

sity

[kg/

m3 ]

Com

pres

sibi

lity,

Z

400 600 800 1000 1200 1400 16000

100

200

300

400

500

600

700

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Density: NitrogenDensity: DodecaneZ: NitrogenZ: Dodecane

Figure 2: Variation of density and compressibility with temperature at P = 50atm for nitrogen anddodecane (C12H26) predicted by the Peng Robinson equation of state.

energy (et), along with the mass fractions of each individual species present in the system (Yα forspecies α). In the above, t is time, xi is the position vector, gi is the gravitational acceleration vector,P is the pressure, and N is the total number of species in the system. In its above form, the system ofequations remains unclosed. Models or additional constitutive relations are required for the viscousstress tensor (τij), the heat flux vector (Qi), the mass flux vector of species α (Ji,α in mass basedform, or J i,α in molar based form), the reaction source terms (Se and SYα), the partial molar enthalpyof species α (h,α = ∂h/∂Xα where Xα is the mole fraction of species α), the internal energy of themixture (implicit in the specific total energy), as well as all related species properties. In addition, anequation of state is also required to close the system.

The above system of equations applied to low pressure combustion and appropriate closures havebeen well documented in the literature and are beyond the scope of the present work. Here we focusonly on alterations to the typical forms due to large pressure. However, recent review articles addresscombustion kinetics [21, 22], transport model effects [22, 23], direct numerical simulations of turbu-lent non-premixed flames [24], turbulent combustion modeling [25], turbulent flame structure [26],experimental measurements of turbulent flames [27], large eddy simulation of turbulent combustion[28, 29], and future directions in turbulent combustion research [30]. In addition, high pressure fluidbehavior under non-reacting conditions is reviewed in Refs. [31, 32].

The following contains a step by step evaluation of the above system of equations and unclosedterms. The focus is on bringing attention to the individual physical phenomena that may be alteredat high pressures from their traditional low pressure treatment with emphasis on the author’s con-tributions which are primarily in the subject of molecular transport phenomena [33, 34, 35, 36, 37,38, 39, 40, 41, 42]. This body of research has focussed on direct numerical simulations (DNS) ofhigh pressure mixing and combustion. DNS refers to a computational simulation in which all lengthand time scales of the flow are solved using very high order accurate methods, and without resortto any turbulence or subgrid modeling [43, 24]. Although limited to relatively low Reynolds numbers,DNS is a powerful tool as it provides a reference “exact” flow solution from which any variable orstatistical quantity can be calculated. The author’s research group has most recently focussed on therole of Soret and Dufour cross diffusion in high pressure heptane, methane, and hydrogen-oxygen

3

laminar diffusion flames [40, 41, 44, 45], and on subgrid analyses of transitional three dimensionalhydrogen-oxygen flames [42, 46] relevant to future predictive large eddy engineering simulations.

The particular objectives of the chapter are to examine the impact of high pressure on the choiceof the: (1) equation of state (EOS), (2) chemical kinetics, (3) constitutive relations, and (4) propertymodels. For each topic a general review is first provided, followed by details of the formulationscurrently being employed by the author. Note, however, that radiation is neglected. Though oftenneglected in low pressure gas flames, radiation is known to be important in sooting flames [47, 48]and in some cases for liquid fuel systems [49]. In fact, radiation may be important even in non-sootingflames at large pressure as the fluid densities are much larger and radiative absorption may occurin a manner similar to liquid fuels at low pressure. However, the author is not aware of any work onsuch effects and they are, therefore, not included in the following.

2.1 Equation of State

While ideal gas behavior is nearly universally assumed for typical atmospheric pressure mixing andcombustion research, such an assumption can lead to very large errors in high pressure combustionsystems. For example, Fig. 2 illustrates that the compressibility factor, Z = P/(ρRT ) (where T isthe mixture temperature and R is the mixture specific gas constant), can be as small as ≈ 0.35 fordodecane under practical operating conditions. A variety of EOS are employed in the high pressureliterature; including the ideal EOS [31]. The author’s own work incorporates real gas effects throughthe cubic Peng-Robinson equation of state [1]:

P =RT

v −Bm− Am

v2 + 2vBm −B2m

, (5)

where R is the universal gas constant, v is the specific molar volume, and Am and Bm are appropri-ately defined mixture parameters. This form was chosen due to its relative simplicity, as well as theavailability of a simple correction which can be used to substantially increase its accuracy [33].

The mixing rules adopted by the author’s research group [36] are those recommended by Harstadet.al. [33]:

Am =∑α

∑β

XαXβAαβ, (6)

Bm =∑α

XαBα. (7)

Aαβ = 0.457236(RT cαβ)

2[1 + Cαβ

(1−

√T/T c

αβ

)]2/P c

αβ , (8)

Bα = 0.077796 RT cαα/P

cαα , (9)

Cαβ = 0.37464 + 1.54226Ωαβ − 0.26992Ω2αβ , (10)

where the superscript c indicates critical properties. The diagonal elements of the critical matricesare equal to their pure substance counterparts; i.e. T c

αα = T cα, P

cαα = P c

α, and Ωαα = Ωα where Ωα isthe acentric factor of species α. The off diagonal elements are evaluated as follows:

T cαβ =

√T cααT

cββ(1− kαβ), P c

αβ = Zcαβ(RT c

αβ/vcαβ), (11)

vcαβ =1

8

[(vcαα)

1/3 +(vcββ

)1/3]3, Zc

αβ =1

2

(Zcαα + Zc

ββ

), Ωαβ =

1

2(Ωαα +Ωββ) , (12)

where the diagonal elements of each of the above symmetric matrices are also equal to the puresubstance values. The binary interaction parameter, kαβ, is a function of the species being consideredand is taken to be kαβ = 0.1 for α = β and kαα = 0.

4

In order to quantify real gas effects on turbulent flows both linear stability analyses and DNS wereconducted for non-reacting 2D temporally developing binary mixing layers comprised of nitrogen-heptane, oxygen-hydrogen, and 3,methylhexane-heptane streams [44]. Ambient pressures in therange 1atm → 100atm were considered. The formulation included models for the temperature, pres-sure, and concentration dependence of all species properties which were validated with experimentaldata (described below). For each case considered, mean initial flow profiles were first obtained assolutions of the similarity equations. Linear stability analysis was then used to calculate the most un-stable wavelengths which were used to generate forcing functions for the DNS. All simulations wererepeated with both the real and ideal gas models. Large differences were observed in the behaviorsof the mixing layers based on the choice of state model, including; variations in the growth rates, andmarked differences in the predicted densities, heat capacities, and sound speeds. Alterations to thedensity stratification were concluded to be the primary source of errors in the predicted mixing layerdevelopment by the ideal gas law. These results are consistent with a previous study of hydrogen,oxygen, heptane, and nitrogen binary mixing layers reported in [50].

2.2 Chemical Kinetics

Combustion kinetics can be included in reacting flow simulations with varying levels of complexityfrom simple single step reactions to complete detailed mechanisms incorporating hundreds of reac-tions and species [21, 22]. The computational requirements of turbulent flame DNS typically precludethe use of fully detailed mechanisms with the possible exceptions of very simple hydrogen-oxygenor hydrogen-air systems. These simulations therefore often require the use of reduced chemistrymodels in which a relatively small number of reactions and species are retained. Low pressure com-bustion kinetics also typically only include temperature dependence via the Arrhenius rate law andneglect pressure dependence of the combustion pathways. However, it is well known that thesepathways are affected by pressure and high pressure combustion simulations must make use of ap-propriate mechanisms. A review of the physics of chemical kinetics is beyond the scope of the currentwork and the reader is referred to Refs. [21, 22] for recent reviews. Suffice it to say that the choicesof appropriate mechanisms which contain pressure dependence and have been validated at highpressures is substantially limited in comparison to low pressure mechanisms.

Three different chemical kinetics mechanisms have been incorporated in the author’s DNS code.Each mechanism was chosen from the literature based on having pressure dependent reaction ratesand prior validations with high pressure experimental data. A detailed 24 − step, 12 species mecha-nism applicable to both H2/O2 and H2/Air combustion proposed by Sohn et al. [51] has been incor-porated and includes NOx chemistry [52]. Reduced mechanisms for two different hydrocarbon fuels(methane and heptane) are also incorporated. For the CH4 oxidation, a 5-step, 8 species reducedmechanism developed by Hewson and Bollig [53] has been employed. For the C7H16 oxidation, a7-step, 10 species reduced mechanism developed by Bollig et al. [54] is employed. Furthermore, a6-step, 7 species NOx chemistry developed by Hewson and Bollig [53] is added to both the methaneand heptane mechanisms. Results of the author’s research described below are based on thesemechanisms.

2.3 Constitutive Relations

Constitutive relations for the shear stress tensor, the heat flux vector, and the mass flux vector arerequired to close Eqs. (1) - (4). To the author’s knowledge, the shear stress tensor is nearly universallymodeled with the same Newtonian fluid model typically employed in low pressure combustion:

τij = µ

[∂ui∂xj

+∂uj∂xi

− 2

3

∂uk∂xk

δij

], (13)

where µ is the mixture viscosity and δij is the Kronecker delta function. This model works well for

5

low pressure gases and liquids of interest in hydrocarbon combustion. The author is not aware ofany evidence that it is any less applicable at larger pressures. In contrast, the heat and mass fluxvectors have varying levels of complexity even at low pressures. Such molecular transport effectscan be important for both non-premixed and premixed flames as both involve substantial temperatureand concentration gradients within the local flame zone. In their simplest Fourier and Fickian formsthese are modeled as: Qi = −κ∂T/∂xi and Ji,α = −ρD⋆αβ

m ∂Yα/∂xi, respectively, where κ is themixture thermal conductivity and D⋆αβ

m is the mass diffusion coefficient characterizing the rate ofdiffusion of species α into species β (within the mixture). However, these forms are well known tobe overly simplified for real reacting systems as they inherently neglect several potentially importantphenomena.

Before reviewing the literature related to molecular transport effects in combustion processes, it isfirst useful to define three different transport phenomena not typically included in the above simplevector formulations. For the purposes of the present work we define “multicomponent diffusion” tobe the diffusion of one species due to concentration gradients of the other species (ie. not only itsown). “Differential diffusion” is defined as a variation in the mass diffusion coefficients of individualspecies with respect to one another. Finally, “cross diffusion” is defined as the diffusion of thermalenergy (mass) in the presence of concentration (temperature) or pressure gradients. Cross diffusionof thermal energy is referred to as the Dufour effect, and cross diffusion of mass if referred to asthe Soret effect. Another transport issue highly important in combustion research involves the Lewisnumber. The Lewis number can be defined for each binary species interaction as Leαβ = ρCp/D

⋆αβm

(where Cp is the constant pressure heat capacity of the mixture) and describes the ratio of the ratesof thermal diffusion to mass diffusion. The Lewis number is often assumed to be unity in many engi-neering combustion models [25]. This implies that all Dαβ

m are equal to each other and is, therefore,a subset of neglecting differential diffusion.

2.3.1 Molecular Transport in Low Pressure Combustion

A recent review of (low pressure) combustion simulations reveals that the impact of the chosenmolecular diffusion model can be as large as the observed differences between available chemicalkinetics models [22]. Under low pressure conditions, the generalized Stefan-Maxwell relations providethe most comprehensive means of predicting multicomponent diffusion incorporating cross diffusion(eg. Refs. [55, 56, 57]) and provide expressions for all of the associated transport properties. Theformulation requires the solution of a coupled set of N ×N matrix equations for an N species systemrequiring substantial computational time [58]. Other simplified models have also been addressed,including various forms of Fick’s law [59, 60, 61]. Of particular interest to this study, DNS of turbulentflames have historically incorporated either only very simple chemistry models (eg. single step), verysimplified diffusion models (discussed above), or both [24, 62, 63]. Some recent exceptions arereviewed below.

Multicomponent diffusion has primarily been considered for laminar flames. Differences in molec-ular transport models have been shown to result in variations in observed diffusion rates, as well asin alterations to local flame curvature and strain rates [22]. Ignition characteristics in high pressureheptane flames have also been shown to be affected both by the assumption of unity Lewis number,as well as by the inclusion of Soret and Dufour cross diffusion [64]. In contrast to laminar flames,the overall impact of molecular transport is generally considered to be diminished in turbulent flamesdue to effects such as a reduction of the local scalar bounds and dominating effects of turbulent stir-ring. However, recent low pressure investigations have shown that molecular transport can have verysignificant impact on turbulent H2/O2 flames [57] as well as for turbulent CH4 jet flames [65, 66].In the latter case differential diffusion effects were shown to be particularly important in the laminarregion near the jet base, yet persist considerably downstream into the turbulent regions through astrong history effect. A recent review by Lipatnikov and Chomiak [23] focuses specifically on pre-mixed flames and shows that these are also substantially impacted by molecular transport effects,

6

even at moderate and high turbulence Reynolds numbers. Recent experiments further support theconclusion that differential diffusion and molecular transport effects are important (albeit diminished)for high Reynolds number turbulent flames [66, 67, 68, 69].

DNS of turbulent or transitional flames that include detailed multicomponent molecular diffusionmodels and realistic detailed or reduced kinetic mechanisms have only recently appeared in theliterature. DNS results have clearly shown molecular transport to have significant impact on a varietyof flames. For example, maximum flame temperatures in both premixed and non-premixed hydrogenflames are altered by differential diffusion in 2D simulations using detailed chemistry models [70, 71].Ethylene flames have been simulated in 2D and 3D using a 19 species reduced kinetics mechanismcoupled with a 4 step soot formation model [47, 48]. The formulation included a comprehensivemulticomponent diffusion model and realistic property values. The 3D jet simulation required ∼ 228×106 numerical grid points to fully resolve the flame. Among other findings, the location of the sootformation and its proximity to the flame were significantly affected by differential diffusion.

Relatively little research has been done regarding combustion simulations incorporating compre-hensive multicomponent diffusion that includes Soret and Dufour cross diffusion. However, studiesthat do include these effects find them to be of substantial importance to proper flame modeling evenat low pressure (for both “light” and “heavy” species, and for soot) [72, 73, 74, 75, 76, 77, 78]. Rosneret al. [75] criticize the seemingly widely held view in the combustion community that Soret diffusion isnot important for proper (low pressure) combustion modeling, and offer evidence that such effects areindeed important. Several studies have addressed these effects under atmospheric pressure and lowMach number conditions (although under kinetic theory derivations [79] which are inapplicable to thehigh pressure dense fluids discussed below). Greenberg [72] simulated an atmospheric hydrogen-airflame and found that thermal diffusion fluxes are comparable in magnitude to the Fickian contributionsand recommended that they be incorporated into flame codes. Garcia-Ybarra et al. [73] performeda theoretical estimation of the influence of Soret and Dufour diffusion of wrinkled premixed flamesby assuming a weak cross-diffusion coupling. Under conditions of large activation energy they con-clude that Soret mass diffusion cannot be neglected when analyzing flame front dynamics (Dufoureffects were found to be negligible). Ern and Giovangigli [74] numerically simulated 2D steady lam-inar methane-air and hydrogen-air diffusion flames using a more complete formulation of the Soretand Dufour fluxes together with complex chemistry and accurate models for transport coefficients. Inagreement with the above theoretical study it was concluded that Soret effects must be consideredfor accurate flame calculations when the molecular weights of the constituent species substantiallyvary. For example, in the hydrogen-air flame the Soret contribution to the mass flux vector comprisedas much as 50% of the total flux vector magnitude. Important alterations to species concentrationswere observed, including; H, H2, H2O2 and HCO in the methane flame.

2.3.2 Molecular Transport in High Pressure Mixing and Combustion

Given the difficulties associated with experimental measurements in high pressure environments[18], numerical simulations are ideally suited to the task. While additional high pressure mixing andcombustion studies exist in the literature (eg. Refs. [51, 64, 80, 81, 82, 83, 84, 85]), relatively few at-tempts have been made to incorporate all of the associated transport phenomena described above.For high pressure flows outside of the range of validity of kinetic theory, the proper theory for formu-lating the molecular fluxes is non-equilibrium thermodynamics (NEQT) [86]. However, in contrast tothe Stefan-Maxwell relations, NEQT does not provide expressions for the transport properties. Themixture viscosity, thermal conductivity, mass diffusion coefficients, and thermal diffusion coefficientsmust therefore be provided through additional theory or models. Bellan’s group at Jet PropulsionLaboratory have conducted extensive research into high pressure binary species mixing with heatand mass diffusion fluxes derived from NEQT and Keizer’s fluctuation theory [87, 88, 89, 90, 91, 92](the author’s interests in this area started as a Postdoc with this group). Their derivation includesthe potential for both heat and mass diffusion in the presence of temperature, pressure, and concen-

7

Pr

α BK21

0 1 2 3 4 510-4

10-3

10-2

10-1

100

101

102

CH4/C3H8: Tr=1.12CH4/C4H10: Tr=1.12CH4/C4H10: Tr=1.28CH4/C4H10: Tr=1.04Ar/CO2: Tr=1.57N2/CO2: Tr=1.66H2/N2: Tr=4.48H2/CO2: Tr=2.11CH4/CO2: Tr=1.44

Figure 3: Compilation of experimental data from Ref. [97] for thermal diffusion coefficients in binarymixtures as a function of reduced pressure and reduced temperature.

tration gradients. The “thermal diffusion factors” were identified as the species properties related toSoret and Dufour cross diffusion. Real gas effects were accounted for with the cubic Peng-Robinsonequation of state due to its relative computational efficiency, and the availability of a simple correctionwhich can be used to substantially increase its accuracy [33].

Cross diffusion effects have been shown to be substantially enhanced under high pressure condi-tions for pertinent species; particularly when large variations in the species’ molecular weights arepresent (although the Dufour effect is generally found to be negligible) [35, 36, 37, 38, 39, 40, 55, 75,74, 88, 89, 93, 94, 95, 96, 97, 90, 98, 99, 91, 100, 101, 102, 76, 103, 104, 92, 77, 105, 106, 107, 108,44, 41]. This occurs because the “thermal diffusion coefficient” (the fluid property related to Soret dif-fusion) increases by up to two orders of magnitude at high pressure [97]. The enhancement of Soretdiffusion under high pressure conditions has received relatively little attention in the combustion com-munity; however, it has been well documented in the petroleum reservoir research community (albeitfor low temperatures). Experimental data from one such study Ref. [97] have been compiled in Fig.3 in reduced variable form (ie. normalized by the mixture critical properties; defined as mole fractionweighted averages of the individual species critical values). The data in the figure are typically for50/50 mole fraction binary mixtures; however, substantial variations with mixture fraction are alsopresent (see also [40]). In the limit of low pressure the dimensionless thermal diffusion coefficientsapproach kinetic theory values and are typically ∼ 0.01 → 0.1. However, as the pressure is increasedthese values show a maximum at approximately twice the mixture critical pressure as defined. Underthese conditions the thermal diffusion coefficients are ∼ 1 → 10 for most species. The trends alsoreveal that the coefficients increase with increasing molecular weight ratio of the species pairs, anddecrease with increasing temperatures above the critical mixture values. One model recently devel-oped by the author makes use of these trends by curve fitting the data in the sense of the principle ofcorresponding states (see below).

Bellan’s group applied their formulation to the DNS of non-homogeneous turbulent binary speciesmixing in both 2D and 3D temporally developing mixing layers formed by the merging of supercritical

8

pressure nitrogen and heptane [34, 35, 100, 109, 110], and hydrogen and oxygen [101, 110] streams.Results of these studies illustrate the relative effects of the thermal diffusion factors on the first andsecond order flow statistics, and showed qualitative agreement with experimental flow visualizationsof supercritical jet mixing evolutions [111]. A later study [50] illustrated the significant impact Soretdiffusion has on the stability and the similarity flow profiles for temporally developing binary mixinglayers. They then conducted a priori analyses of the DNS simulations examining the behavior ofvarious subgrid terms models with a primary focus on terms related to the real gas state equation.Neither real property models, combustion, nor mixtures comprised of greater than two species haveyet been considered by this group.

Extremely little research exists on cross diffusion effects in combustion at large pressures (otherthan the author’s work). Briones et al. [78] simulated partially premixed laminar counterflow H2 −Air flames at high pressures incorporating Soret diffusion. Global flame structure changes due tothe Soret effect were found to be negligible to this reaction. Nevertheless, the Soret effect causedthe transition between high and low pressure reaction limits to occur at lower pressures than forsimulations not including this effect. Ignition characteristics have also been observed to be altered bythe Soret effect in high pressure heptane flames [64]. Oefelein [112, 113] conducted both DNS andLES of LOx −H2 jet flames at large pressure and incorporated detailed diffusion, property models,and detailed combustion kinetics. Their results show substantial real gas and transport effects onthe flame. Near critical injection conditions resulted in large spatial variations of the mixture density,compressibility, and molecular transport properties.

The author’s research group has incorporated the general multicomponent forms of the heat andmass flux vectors derived by Bellan’s group from NEQT [86] and fluctuation theory [87]:

Qj = −

κ+N−1∑α=1

N∑β>α

XαXβααβBKααβ

BK

Rρ

MmDαβ

m

∂T

∂xj(14)

−N∑

α=1

Xα

N∑β =α

[Mβ

M2m

XβααβBKρDαβ

m

]v,α

∂P

∂xj

−N−1∑γ=1

N∑α=1

RT

N∑β =α

[Mβ

M2m

XβααβBKρDαβ

m

]ααγD

∂Xγ

∂xj,

J j,α = −N∑

β =α

nDαβm

XαXβ

T

Mβ

MmααβBK

∂T

∂xj(15)

−N∑

β =α

nDαβm

RT

−MαMβ

M2m

XαXβv,β +MαMα

M2m

XαXβv,α

∂P

∂xj

−N−1∑γ=1

N∑

β =α

[−MαMβ

M2m

XαnDαβm αβγ

D +MβMβ

M2m

XβnDβαm ααγ

D

] ∂Xγ

∂xj.

In the above, n is the molar density (n = ρ/Mm), the mixture molecular weight is Mm =∑N

α=1XαMα,the universal gas constant is R, and the partial molar volume of species α is v,α. Several propertiesappear in the above: the “mass diffusivity” within the mixture is Dαβ

m for species pair α, β, and the“mass diffusion factor” pairs are ααβ

D which can, in general, be derived from the chosen equationof state along with the Gibbs Duhem relation. The dimensionless Bearman-Kirkwood form of the“thermal diffusion factor” pairs, ααβ

BK , are related to the Soret and Dufour cross diffusion. In addition

9

to the above, the “mass diffusion coefficient” is defined as the product of the mass diffusivity andthe mass diffusion factor (D⋆αβ

m = Dαβm ααβ

D ) and the “thermal diffusion coefficient” is defined as theratio of the thermal diffusion factor to the mass diffusion factor (αBK/αD). These are the quantitiestypically measured in (binary species) experiments. The “coefficients” therefore implicitly includemass diffusion factor effects (see below). Details can be found in Refs. [40, 44].

The heat and mass flux vectors described by Eqs. (15) and (16) contain the potential for all threeforms of diffusion processes described above. Multicomponent diffusion is associated with the sum-mations over all species in terms proportional to ∂Xα/∂xi appearing in the mass flux vector. Dif-ferential diffusion is incorporated when non-equal values of the species’ mass diffusion coefficientsare applied. Finally, cross diffusion appears in both vectors. The Dufour effect is associated withthe terms proportional to ∂Xα/∂xi and ∂P/∂xi in Eq. (15). The Soret effect is associated with termsproportional to ∂T/∂xi and ∂P/∂xi in Eq. (16). The terms involving the pressure gradient are typi-cally neglected in low pressure and low Mach number cross diffusion studies. However, Miller [36]found these terms to be important for compressible flows. Although typically found to be negligible,the Dufour terms should not be arbitrarily neglected while retaining the Soret terms, as violations ofthe second law may occur [86]. Unlike the Stefan Maxwell relations, the above forms of the heat andmass flux vectors require additional closures for the properties: κ, Dαβ

m , and ααβBK .

2.4 Property Modeling

Several mixture properties appear in the above formulation which require modeling. The molar heatcapacity, molar specific enthalpy, partial molar volume, partial molar enthalpy, and the mixture molarinternal energy (u = h − Pv) are all directly related to the chosen EOS through the Gibbs functionand should be derived from it for self consistency of the formulation [1]:

Cp = C0p − T

(∂P/∂T )2v,X(∂P/∂v)T,X

−∑α

XαR− T∂2Am

∂T 2K1, (16)

h = h0+ Pv −

∑α

XαRT +K1

(Am − T

∂Am

∂T

), (17)

v,α =−1

(∂P/∂v)T,X

[RT

v −Bm+

RTBα

(v −Bm)2+

2Am(v −Bm)Bα

(v2 + 2vBm −B2m)2

−2∑

β AαβXβ

v2 + 2vBm −B2m

], (18)

h,α = h0α+Pv,α−RT+(Am−T

∂Am

∂T)

v,α − vBα/Bm

v2 + 2vBm −B2m

+K1

[∂Am

∂Xα− T

∂2Am

∂Xα∂T− (Am − T

∂Am

∂T)Bα

Bm

],

(19)where:

K1 =1

2√2Bm

ln

[v + (1−

√2)Bm

v + (1 +√2)Bm

]. (20)

The above departure functions yield the thermodynamic deviations from the low pressure mixtureheat capacity (C0

p) and enthalpy (h0) which may be found in various reference books (eg. Ref. [1]).Particular forms of the thermodynamic terms and gradients appearing in the above derived from thePeng Robinson EOS and applied in the author’s work may be found in Ref. [36].

A variety of approaches to modeling the mixture viscosity, thermal conductivity, and the mass dif-fusion coefficients are available; including look up tables, purchased software, etc. However, theauthor has chosen only approaches based on the principle of corresponding states in order to sim-plify the procedure and allow the DNS code to be self contained. The principle of correspondingstates assumes that all species exhibit universal behavior when appropriately normalized by theircritical parameters (pressure, temperature, volume, compressibility, and acentric factor - estimationprocedures can be used for species in which these properties are not available [1]). Such models

10

can therefore be applied to general species even for cases in which no experimental data is available.Details of all chosen models may be found in Refs. [1, 38, 44]. In brief, the Lucas Method is chosenfor calculating the mixture viscosity, and the method of Steil and Thodos is chosen to calculate themixture thermal conductivity. Low pressure diffusion coefficients are evaluated using the method ofFuller et al., and high pressure corrections are made based on a correlation proposed by Takahashi.These models actually describe the “binary” diffusion coefficient for the rate of diffusion of species αinto pure species β. In the above formulation; however, the coefficients actually describe the ratesof diffusion of species α into species β within a larger mixture. Relatively little is known about thealteration of binary diffusion coefficients in the presence of complex mixtures and this issue remainsin need of further investigation.

Each model’s accuracy has been assessed for properties in which experimental data is available.If significant errors were found then the difference between the model and the data (the departurefunction) was evaluated and curve fit. Note that the model is based on the mass diffusion coefficients,and not on the mass diffusivity, within the mixture. Therefore, the binary mass diffusion factors aremodeled with the ideal mixing assumption [114] rather than attempting to derive them directly fromthe EOS and the Gibbs-Duhem relation. In this case the mass diffusion factors take values: ααβ

D = 1

for α = β, ααβD = −1 for α = N , and are null otherwise. The procedure is justified in Refs. [38, 40, 44].

The use of these realistic property models also enables investigations of the actual Lewis num-bers found in high pressure combustion. In examining high pressure binary species droplet diffusion,Harstad and Bellan [90] defined an effective Lewis number. They observed that the effective Lewisnumber in supercritical mixtures can be 2-40 times larger than its typical definition (even when crossdiffusion is negligible) and that both forms increase with pressure. However, these simulations werepurely for low temperature binary mixing and utilized somewhat simplified property models. As men-tioned above, many combustion models assume unity Lewis number of all species. As such, non-unityLewis number effects have often been addressed in the literature in the general sense of parametervariation studies. However, very little data is available concerning actual Lewis numbers in realisticflame conditions (at any pressure). Therefore, distributions of all species pair Lewis numbers from our1D laminar flame simulation were reported for each of the hydrogen, methane, and heptane flamesand as functions of pressure in Refs. [41]. Lewis numbers were shown to have strong cross flame de-pendence as well as pressure dependence. Averages over all species at the flame zone ranged fromapproximately 0.5 → 2 with rms values as large as approximately unity. Maximum and minimum Lewisnumbers observed within the flame regions ranged from approximately 0.1 → 5 among the variousspecies combinations. Figure 4 presents statistics of the Lewis number distributions observed in onedimensional laminar flame simulations for H2−Air, H2−O2, CH4−Air (methane), and C7H16−Air(heptane) flames as a function of pressure. In each case the minimum, maximum, mean, and rootmean square of all existing Leαβ in the long time flames were calculated; in this case at the locationof maximum oxidizer mass flux rate. The simulations are essentially the same as those published inRef. [40] and utilize the combustion mechanisms described above. Maximum, mean, and rms Lewisnumbers are somewhat reduced at large pressures. However, substantial departures from the unityLewis number assumption are observed at all pressures, as well as substantial differential diffusion(ie. non-equal Lewis numbers).

Finally, the non-dimensional thermal diffusion factors are the species properties directly related toSoret and Dufour diffusion. Relatively little work has been done on proper modeling of this propertyunder high pressure combustion conditions. Early work in the author’s research group (as well asBellan’s group) focussed on overly simplified models. In these models either the Bearman-Kirkwoodor the alternative “Irving-Kirkwood” form of the factor were assumed to be constant (the two formsare not independent and are directly related through the EOS). However, the petroleum reservoirresearch community has addressed this issue in substantially more detail (albeit at low pressure).In order to more deeply explore the role of cross diffusion in high pressure combustion, five modelsfor the high pressure thermal diffusion factor have been incorporated into the DNS code and studied

11

P [ atm ]

Le ij

25 50 75 1000

0.5

1

1.5

2MEANMINMAXRMS

(a) P [ atm ]

Le ij

25 50 75 1000

0.5

1

1.5

2MEANMINMAXRMS

(b)

P [ atm ]

Le ij

25 50 75 1000

1

2

3

4

5MINMAXRMSMEAN

(c) P [ atm ]

Leij

25 50 75 100-2

-1

0

1

2

3

4

5

6 MEANMINMAXRMS

(d)

Figure 4: Lewis number statistics as a function of pressure calculated from 1D laminar diffusionflame simulations [41] at long flame times and at the location of maximum oxygen mass flux rate: (a)H2 −Air, (b) H2 −O2, (c) CH4 −Air, and (d) C7H16 −Air.

in 1D laminar flame simulations [40]. These include the “Hasse model” [115], the “Kempers model”[116], the “Firoozabadi model” [97], and two variations of a model proposed by Curtis and Farrel [95].The first three models are based on semi-theoretical derivations as functions of partial molar volume,partial molar enthalpy, and/or partial molar internal energy differences among species pairs. Thelast is a molecular weight ratio based correlation fit to low pressure experimental data. In agreementwith Ref. [97], the results show that several of the models agree qualitatively with available highpressure data, but that no model is in acceptable quantitative agreement with all of the data. Generallyspeaking, those models capable of predicting hydrocarbon/hydrocarbon data do not fare as well withhydrocarbon/non-hydrocarbon data (and vice versa). All of the models were incorporated into theDNS code and simulations were conducted for 1D laminar flames for each of the hydrogen, methane,and heptane reactions in Refs. [44, 40]. Several of the models were shown to predict unphysicalbehavior (diverging to large very values) at the large temperatures despite good agreement withlow temperature experimental data. These models were therefore not recommended for combustionstudies.

One additional and well recognized problem with the high pressure thermal diffusion factor models

12

x1/δ0

x 2/δ 0

0 10 20 30 40-15

-10

-5

0

5

10

Oxygen (φ=0)

Hydrogen (φ=1)

(a) φ

T/T

0

0 0.25 0.5 0.75 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

(b)

Figure 5: Instantaneous center plane results for a 384 × 384 × 232 resolution, 8 species, 19 step,DNS H2 − O2 flame simulation at Re0 = 850 and t⋆ = 90: (a) centerline contours of the x3 vorticitycomponent and (b) scatter plot of temperature as a function of the mixture fraction. The simulationwas run on 512 processing cores on Clemson University’s “Palmetto” cluster.

is the fact that the partial molar differences are derived from the particular state equation chosen.They are known to be considerably sensitive to both the choice of the state equation as well as tothe particular choice of mixing rules [97]. In order to address the above issues a new and improvedmodel for the thermal diffusion factors has been developed and tested for all of the 1D laminar flames[41, 45]. The model is a curve fit to the experimental data presented in Fig. 3 as a function of reducedthermodynamic variables in the sense of the principle of corresponding states. More specifically, itis a curve fit to the departure between the experimental data and a kinetic theory model. The curvefitting is a function of the reduced pressure, the reduced temperature, the molecular weight ratio,and the mixture mole fraction. As such, it has no sensitivity to the state equation and predicts therange of experimental data more accurately than any of the theoretical models. The curve fittingtakes advantage of the trends observed in the data of Fig. 3 discussed above. Simulations basedon the model show that relatively moderate alterations to flame temperatures and minor speciesconcentrations are expected due to Soret/Dufour cross diffusion in the non-strained, 1D, laminarflame geometry. This occurs because for high combustion temperatures the model predicts a returntowards kinetic theory values. However, in real combustion processes, Soret effects may still beimportant near the fuel jet core and/or along the outer edges of the flame.

3 Ongoing Research and Outlook

The focus of the author’s current research is on DNS of high pressure turbulent flames with em-phasis on modeling testing, development, and validation. Several simulations have been conductedrecently [42] for the hydrogen oxygen flame. All such simulations were for full 3D flow in the temporallydeveloping shear layer configuration. The temporally developing configuration is chosen because itsreduced domain size makes more efficient use of computational resources compared to the spa-tially developing shear or jet flame geometry [117]. The configuration, illustrated in Fig. 5, simulatescounter flowing streams of non-premixed fuel and oxidizer. Periodic boundary conditions are em-ployed in the x1 and x3 directions, while non-reflecting free stream conditions [118, 119] are used inthe x2 direction. The governing equations are solved using fourth order Runge Kutta time integrationwith eighth order central explicit finite differencing for all spatial derivatives (coupled with tenth order

13

explicit filtering) [120]. Parallelization is based on the MPI message passing routines with domaindecomposition in all three coordinate directions.

The largest resolution DNS performed thus far is for a hydrogen-oxygen flame simulated at aReynolds number of 850 (based on the across stream velocity difference, the initial vorticity thick-ness, and free stream average density and viscosity). All parameters and properties are physical:only the problem length scale is adjusted in specifying the Reynolds number. The convective Machnumber is 0.35 and the ambient pressure is 100atm. Initial conditions correspond to a self-similarbinary hydrogen-oxygen (non-reacting) shear layer with forcing imposed at the most unstable wave-length calculated from a separate stability analysis code [44]. The grid spacing is uniform in thestreamwise and spanwise directions, whereas an analytic grid mapping is used in the cross streamdirection. The simulation was conducted using 384× 384× 232 numerical grid points and run on 512processing cores on Clemson University’s new “Palmetto Cluster;” the number 60 fastest supercom-puter in the world at the time of the simulations (www.top500.org). The cluster currently consists of1,283 dual processor quad core nodes (10,264 processing cores) networked with a 10 gigabit persecond Myrinet network.

Figure 5 presents instantaneous contours of the x3 component of vorticity at the center plane, anda scatter plot of the flame temperature as a function of the mixture fraction [121] (ϕ = 0 indicatespure O2 and ϕ = 1 indicates pure H2). The simulation time corresponds to a point after which thefour initially forced vortices have rolled up and paired. The vorticity contours illustrate the transitionalnature of the flame and the range of turbulence scales attainable at this Reynolds number. Reynoldsnumber requirements for obtaining turbulence transition in non-reacting, high pressure, temporallydeveloping mixing layers have been discussed in Ref. [109]. Note from Fig. 5(b) that the limitedextent of scatter indicates that the hydrogen flame is near equilibrium at this time; however, data atearlier times and for other flames exhibit substantially more scatter and associated non-equilibriumeffects.

This DNS simulation was also further examined to determine subgrid information relevant to largeeddy simulation (LES) [28, 29]. In LES, the governing equations are “filtered” to remove small scalefeatures (the subgrid). As in the Reynolds Averaged Navier Stokes (RANS) approach, filtering resultsin unclosed terms in the equations for the resolved scales that require modeling. DNS provides apowerful tool for examining such models in an a priori manner (as well as the need for modeling).The “exact” solution can be filtered to a courser (fictitious) LES mesh. In this case, the missing sub-grid information can be calculated directly from the DNS results and analyzed. As one example ofthis study, the impact of the subgrid mass flux vector fluctuations were analyzed to determine thesubgrid modeling necessities for reacting flow. Recently, Selle et al. suggested that for a supercrit-ical, non-reacting, hydrogen-oxygen mixing layer, the overall contribution of the subgrid mass fluxvector fluctuations were negligible, and therefore, do not require modeling [122]. Their conclusionswere based on the globally averaged vector magnitude of the gradients of the subgrid mass flux vec-tor being at least one order of magnitude smaller than other terms in the LES filtered mass fractiontransport equation. However, due to the sensitivity of flame dynamics to local phenomena, the au-thor’s current research seeks to examine the subgrid mass flux vector on locally defined regions aswell as globally [42].

One characteristic example is provided in Fig. 6 for the H2 − O2 flame DNS described above. Theassumption of negligible subgrid mass fluxes is investigated by calculating both the filtered massflux vector magnitude, | < Jj,α(Ψ) > |, and the the mass flux vector magnitude calculated usingthe filtered primitive variables, |Jj,α(< Ψ >)| (the brackets, <>, indicate the filter operation, andΨ indicates the thermodynamic state variables from which the mass flux vector is calculated, Eq.(16)]. The former term is the exact (unclosed) form appearing in the filtered LES equations, whereasthe latter term is that available for calculation in the context of an actual LES. If the assumption ofnegligible subgrid mass fluxes is correct, then the two forms should be (approximately) equal. Bothforms can be calculated “exactly” using the DNS data. In this example, the filtering is based on a

14

|JH2(<Ψ>)| / (ρ0U0)*103

|<JH

2 (Ψ)>

|/(ρ

0U0)

*103

0.0 0.5 1.0 1.5 2.0 2.5

0.5

1.0

1.5

2.0

2.5

Figure 6: Scatter plot of the filtered mass flux vector magnitude and the mass flux vector magnitudecalculated as a function of the filtered primitive variables. The data are calculated for the H2 massflux vector from the 3D H2 − O2 flame simulation of Fig. 5. The data are conditioned on regions ofthe hydrogen scalar dissipation exceeding twice its mean value within the flame.

spherical top hat filter with diameters as large as 25 times the streamwise DNS mesh spacing. Whenanalyzed globally the correlation coefficient of the two forms of the mass flux vector ranges from zeroat zero filter width (ie. DNS), to approximately 0.8 → 0.95 for the various species at the largest filterwidth. However, as flame dynamics (eg. extinction and reignition) are known to be sensitive to localphenomena the correlation coefficients have also been calculated by further conditioning on regionsof large (relative to the mean): temperature, reaction rates, filtered scalar dissipation, subgrid kineticenergy, filtered mixture fraction variance, and filtered temperature variance. When analyzed in thismanner a very different picture emerges. Correlation coefficients are dramatically reduced in severalof these regions to as small as approximately 0.2. The scatter plot of Fig. 6 illustrates this point byshowing the two forms of the mass flux vector for the hydrogen species when conditioned on regionsof scalar dissipation being larger than twice its mean within the flame zone. The correlation coefficientfor the data shown is approximately 0.7 (for a perfect correlation all of the data would fall on a straightline along the diagonal). The results of the study clearly illustrate that subgrid mass fluxes cannotbe neglected in LES of high pressure combustion processes without substantially altering the massfluxes of the various species within the flame region. Additional details may be found in Ref. [42], anda comprehensive publication is in preparation [46].

Future research directions include constructing a DNS database for each of the H2−Air, H2−O2,CH4 − Air, and C7H16 − Air flames as functions of the Reynolds number and ambient pressure.Filtered forms of all of the governing equations are being derived and all terms are being analyzed foreach of the flames. Particular emphasis will be on subgrid terms associated with the non-linear EOSas well as on the ultimate role of subgrid mass fluxes on high Reynolds number LES. Other areasof interest include better understanding of the behavior and modeling of mass and thermal diffusioncoefficients in complex mixtures, and the role of radiation in dense fluid high pressure combustion.Finally, the issue of transcritical combustion phenomena has received essentially no treatment in the

15

literature. In some situations a fuel may be injected from very large supercritical pressures into asubcritical, but high pressure, environment. This may be the situation in some diesel applications(although the post ignition pressure rapidly rises to supercritical conditions again). Such an occur-rence would be highly complex to model as phase fronts could be rapidly formed and/or vanish locallywithin the mixture. Large spatial variations in property values near both phase fronts and the criticallocus further complicate the situation. Nevertheless, and despite the complexity of the processes,substantial fundamental understanding of both transcritical and supercritical combustion processesis required in order to aid in the development of future robust and predictive engineering combustionmodels for high pressure environments.

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