paul,p. 98 a re-evaluation of the means used to calculate transport properties of reacting flows

495

Twenty-Seventh Symposium (International) on Combustion/The Combustion Institute, 1998/pp. 495–504

A RE-EVALUATION OF THE MEANS USED TO CALCULATE TRANSPORTPROPERTIES OF REACTING FLOWS

PHILLIP PAUL1 and JURGEN WARNATZ2

1Combustion Research Facility

Sandia National Laboratories Livermore

Livermore, CA 94550, USA2Interdisciplinary Center of Scientific Computing (IWR)

Heidelberg University

D-69120 Heidelberg, Germany

Simulations of laminar combustion and other reactive flow processes (like chemical vapor deposition,plasma etching, etc.) are presently carried out in most cases using the transport code TRANFIT attachedto the CHEMKIN package. The approach used is based on experimental data from 1975 and is nowoutdated, especially in view of recent work presented in the literature.

The new approach described here seeks to remove the deficiencies of former transport models by usingthe following features: (1) representation of transport data of light species at high temperature by switchingto an exponential repulsive potential, (2) use of effective potential parameters to handle the intermolecularforces in an easy and elegant way, if polar molecules are considered, and (3) use of a simplified formulafor binary thermal diffusion factors, based on an expansion for large values of the mass ratio of the speciesincluded.

This paper presents the new transport model in terms of a complete set of equations. The molecularparameters provided allow a complete treatment of the oxidation of H2 and H2/CO mixtures (data forspecies taking place in the oxidation of hydrocarbons and in other reaction systems are not yet available).To demonstrate the consequences of the new transport model for combustion processes, results have beengenerated by implementing the model in a code for the simulation of premixed laminar flames.

Introduction

Simulations of laminar combustion and other re-active flow processes (like chemical vapor deposi-tion, plasma etching, etc.) are presently carried outin most cases using the transport code TRANFIT[1,2] attached to the CHEMKIN package [3]. Theapproach taken [4] is based on the Chapman–En-skog theory (see, e.g., Ref. [5]). It uses a Stockmayer12-6-3 potential (equivalent to the well-known Len-nard–Jones 12-6 potential for the interaction of non-polar molecules) and was tested extensively in thesimulation of laminar premixed flames (with theMIXFLA package [6–8]). The underlying database[4] was developed on the basis of the experimentaldata existing in 1975 and is now outdated, especiallyin view of recent work done by Mason and cowork-ers [9–12].

The need to use a new approach stems from thefact that transport properties of light particles cannotbe appropriately described by the 1/r12 repulsivepart of the Stockmayer or Lennard–Jones potential,which seems to be too stiff at high characteristictemperatures [5]. This is quantitatively demon-strated by Paul [13] on the basis of experimental re-sults and ab initio calculations on H2 viscosity and

on H-atom diffusivity given in the literature. Thus,the new approach described here seeks to removethis and some other deficiencies of former transportmodels by using the following features [13]:

1. representation of transport data of light speciesat high temperature T (i.e., at high characteristictemperatures T* 4 kBT/e . 10; e 4 potentialwell depth, kB 4 Boltzmann constant) by switch-ing to an exponential repulsive potential;

2. use of effective potential parameters to handlethe intermolecular forces in an easy and elegantway if polar molecules are considered; and

3. use of a simplified—but sufficiently accurate—formula for binary thermal diffusion factors,based on an expansion for large values of the massratio of the species included.

In light of the serious problems with species or pairtransport properties, the discussion on whether fullmulticomponent formulations for mixture transportproperties need to be used turns out to be minor.

This paper presents the new transport model interms of a complete set of equations (reduced col-lision integrals are given in the literature [9,13]). Themolecular parameters presented allow a completetreatment of the oxidation of H2 and H2/CO

496 LAMINAR PREMIXED FLAMES

mixtures (data for species involved in the oxidationof hydrocarbons and other reaction systems are notyet available). To demonstrate the consequences ofthe new transport model for combustion processes,results have been generated by implementing themodel in a code for the simulation of premixed lam-inar flames.

The New Transport Model

Molecular Interaction Potentials

In recent transport program packages [1,2,4], mo-lecular interaction between two molecules i and jwas described by the Stockmayer 12-6-3 potential,which is equivalent to the well-known Lennard–Jones 12-6 potential for the interaction of nonpolarmolecules (fd-id and fd-d 4 0),

12riju (r, x ) 4 4e 1 [1 ` n* f (x )]ij ij ij ij d1id ij51 2r

6 3r rij ij1 d* f (x ) • (1)ij d1d ij1 2 1 2 6r r

where uij is the interaction potential (dependent onintermolecular distance r and angle xij between theinduced and permanent dipoles), eij is the potentialwell depth, rij is a mean molecule diameter, fd-id andfd-d are angle-dependent functions for dipole–in-duced dipole and dipole–dipole interaction, and n*ij(not used in the following) and /2eij are2 3d* 4 l rij ij ij

mean reduced polarizability and mean reduced di-pole moment (l 4 dipole moment). Usually this po-tential is integrated with respect to the angle-depen-dent part (assuming constant fd1id and fd1d) to yieldan angle-independent Krieger potential [5], forwhich reduced collision integrals X(m,n)* are avail-able [14]. It is equally plausible to perform this in-tegration with respect to the angle-dependent po-tential parts using a thermally orientation-averagedpotential of the form [15]

u(r, x) exp[1u(r, x)/(k T)]dxB##x

û (r)& 4ij

exp[1u(r, x)/(k T)]dxB##x

(2)

The huge advantage of this central (or spherical) po-tential is that it can be expanded in a series that givesat low order

12 6r 2c e d* rij ij ij ijû (r)& 4 4e 1 1 ` n* ìj ij ij31 2 1 2 1 2 4r 3 T r

(eff.) (eff.)12 6r rij ij(eff.)4 4e 1ij 31 2 1 2 4r r

(3)

which is an effective Lennard–Jones 12-6 potentialand allows easy incorporation of dipole–dipole anddipole–induced dipole interaction terms into the re-duced collision integrals by defining effective Len-nard–Jones and exponential repulsive parameters.Formally, the expansion delivers a value k 4 1; how-ever, a value k 4 1/4 works much better [13,16].

For the reasons discussed in the Introduction, athigh characteristic temperatures T* 4 kBT/e . 10,it is advisable to switch to an exponential repulsivepotential uij(r) 4 Vij exp(1r/qij) (see, e.g., Ref. [9]).An analogous treatment leads to an effective poten-tial

(eff.) (eff.)û (r)& 4 V exp(1r/q ) (4)ij ij ij

For the determination of effective potential pa-rameters (and then of reduced collision integrals),first the quantities (ai 4 polarizability)

2 2 2 2a l ` a l l li j j i i jv 4 and D 4 (5)ij ij6 64e r 24e rij ij ij ij

are defined. vij is dimensionless and is zero if bothspecies i and j are nonpolar; Dij is zero unless bothspecies are polar. Then the effective parameters be-come

(eff) (eff)2e 4 e (1 ` v ` D //T) , rij ij ij ij ij

11/64 r (1 ` v ` D /T) (6)ij ij ij

(eff) 12V 4 V*(1 ` v ` D /T) ,ij ij ij ij

(eff) 11/6q 4 q*(1 ` v ` D /T)ij ij ij ij

Thus, the transport property calculation is in termsof 6 reduced parameters for every collision pair: eij,rij, 4 Vij /eij, 4 qij /rij, aij, and lij. A collectionV* q*ij ij

of potential parameters for species involved in theoxidation of H2 and H2/CO mixtures is presented inTable 1; the combination rules for species pairs (in-cluding a seventh parameter, the normalized inter-action dispersion energy 4 C6/er6) are given inC*6a later section.

Pure Species and Species Pair TransportCoefficients

For the computation of pure species viscositiesand binary diffusion coefficients, the usual relationsare used together with higher order corrections (hi

and dij) [5]. The shear viscosity gi is given by (mi 4mass of molecule i; expressions for the reduced col-lision integral and for are given in Refs.X* E*ij ii

[9,13])

5 m •k T 1 ` hi B in 4 with hi i2 (2,2)!16 p r X * (t*)ii ii

34 (8 E* 1 7) (7)ii196

RE-EVALUATED TRANSPORT PROPERTIES OF REACTING FLOWS 497

TABLE 1Molecular data for the determination of intermolecular potentials [13]

Species e/k[K] r [10110 m] l [Debye] a [10130 m3] V* q* C*6

CO 98.40 3.652 0.1098 1.95 5.31 2 104 0.1080 2.630CO2 245.30 3.769 0.0 2.65 2.80 2 106 0.0720 1.860H 5.42 3.288 0.0 0.667 3.70 2 104 0.1010 6.586H2 23.96 3.063 0.0 0.803 1.14 2 105 0.1030 4.245H2O 535.21 2.673 1.847 1.45 3.50 2 107 0.0640 1.612H2O2 368.11 3.499 1.573 2.230 8.23 2 105 0.0830 2.322HO2 365.56 3.433 2.09 1.950 5.30 2 105 0.0860 2.450O 57.91 3.064 0.0 0.802 5.06 2 105 0.0840 2.740O2 121.10 3.407 0.0 1.600 1.32 2 106 0.0745 2.270OH 281.27 3.111 1.655 0.980 7.73 2 104 0.1010 3.226N2 98.40 3.652 0.0 1.750 5.31 2 104 0.1080 2.180

TABLE 2Coefficients of Pi(T) for the determination of ki [13] (Pi (T) 4 1 for the monatomics)

Species P1 P2 P3 P4 P5 P6

CO 0.9133026 10.3384788 10.26450910 0.03584491 0.01961665 10.00108565CO2 1.5518010 10.2911856 10.54380010 0.02452900 0.05427209 0.0H2 1.1004753 10.3268305 10.32212841 0.03198276 0.02408867 10.00075892H2O 0.5931511 10.1095666 10.03359944 0.0 0.0 0.0H2O2 0.0243916 10.2087738 0.07258789 0.01300192 0.0 0.0HO2 10.422988 10.1763043 0.16050169 0.01131380 0.0 0.0O2 0.7759023 10.3787674 10.23802530 0.04579047 0.01837284 10.00170918OH 0.7319990 10.3669068 10.21580729 0.04311154 0.01598934 10.00157644N2 1.0367960 10.3182594 10.29313580 0.03122558 0.02160071 10.00079964

In an analogous way, the coefficients for binarydiffusion of a species i into species j are given by theexpression (q 4 total mass density)

3 2k Tm 1 ` dB ij ijD 4ij 2 (1,1)!8 p qr X *(T*)ij ij

m mi jwith m 4 (8)ijm ` mi j

again, with a higher order correction (an expressionfor is given in Refs. [9,13])C*ij

c ai2d 4 1.3(6C* 1 5) (9)ij ij 1 1 c ` c abi i

with (xi 4 mole fraction of species i; the indices areordered such that mj # mi)

2 (1,1)!x X *ic 4 , a 4i 2 (2,2)x ` x 8(1 ` 1.8m /m ) X *i j j i ii

2and b 4 10[1 ` 1.8m /m ` 3(m /m ) ]j i j i

The thermal conductivities actually are not com-puted in a similar way due to the difficulties con-nected with transport of internal degrees of freedom(Eucken correction [1,2,4,5]). Instead, polynomialsPj are used to describe the deviation of the heat con-ductivity ki from its monatomic part ,(mon)ki

(mon)k 4 P (T)k with P (T)i i ii

2(P ` P )(y ` P )yi,1 i,3 i,54 (10)2 3(1 ` P )(y ` P )(y ` P )yi,2 i,4 i,6

where y 4 ln(T/K). Values of the Pi,1 . . . Pi,6 aregiven in Table 2. The monatomic part is de-(mon)kitermined in the usual way from the viscosity of spe-cies i (equation 7) by the relation

15 k •gB i(mon)k 4 (11)i 4 mi

It remains to derive an expression for the binarythermal diffusion factors , which are second-Taij

order transport properties with a complicated de-pendence on composition, molecular mass, and col-lision integrals. Bzowski et al. [9], for example, find


that calculated values of may involve uncertain-Taij

ties that are an order of magnitude larger than theuncertainties in calculated values for viscosity or bi-nary diffusion coefficients. They also note that errorsin the experimental data for the thermal diffusionfactor involve uncertainties of similar magnitude. Toavoid complicated computations with doubtful re-sults, one can use the fact that thermal diffusion isimportant only for light species [4,5], that is, for Hand H2 in combustion processes. Therefore, thethermal diffusion factor (as given in Refs. [5,13])Taij

may be expanded for large values of the ratio mi/mj

to give the approximation (n 4 particle numberdensity, given in Refs. [9,13])B*ij

n [1 1 F(n )](6C* 1 5)ij ij ijTa ùij (5 1 12B*)c ` (4n /3)(m nD /g )cij i ij j ij j j

1/2x mi iwith c 4 , n 4 (12)i ij 1 2x ` x m ` mi j i j

The function F(n) is an empirical correction thatmakes the approximation reasonably accurate forvalues of mj/mi up to a value of 1/2 and is given as

2 4F(n) 4 7.99027(1 1 n) ` 76.0603(1 1 n)

In these equations, the indices, again, are orderedsuch that mj # mi, which is consistent with the nor-mal sign convention for the thermal diffusion factor(i.e., above the inversion temperature is a positiveTaij

number; thus, the thermal diffusion of heavy specieswill be toward a cooler region).

Combination Rules for Species Pairs

The usual combination rules for species-pair po-tential parameters (arithmetic mean for diameters,geometric mean otherwise) are extended here bycorrections described in Refs. [9,13,17,18]. The fol-lowing procedure gives the information needed forthe treatment of species pairs:

1. Determine rij 4 (ri ` rj), ai 4 ri[1 1 ( /C*6,i

2.2)1/6] for all , 2.2 and ai 4 0 otherwise;C*6,i

2. determine eij (ri 1 ai)3(rj 1 aj)3/[rij 1 (aie e! i j

` aj)/2]6 with Xij 4 ai/aj (C6,j/2/ X ` 1/X! ij ij

C6,i)1/2;

3. and, finally, determine 4 rj/q* q*r ` q* 2r*ij i i j ij

and 4 qij/eij .q q q /2qj ijV* (V /q ) V /q )ij i i /2 j jt y

Mixture Transport Coefficients

Full multicomponent formulations for mixturetransport properties are avoided here; the relationsgiven later do not vary from the multicomponenttreatment by more than the error limits attributedto the species transport properties.

The problem of internal degrees of freedom(Eucken correction) was already mentioned; it is alsorelevant, if mixture heat conductivities are consid-ered. To avoid this problem here (see, e.g., the well-known Eucken–Hirschfelder approach [19]), formixture heat conductivities, a formulation of Masonand Saxena is taken [20] (N 4 number of speciesinvolved):

N N 11

k 4 x k x ` 1.065 x Umix o i i i o j ij1 2i41 j41, j?i

1/4 1/2 2 11/2(mon)m k mi iiwith U 4 1 ` 8 1 `ij 3 1 2 1 2 4 3 1 24(mon)m k mj jj

(13)

In an analogous way, the mixture shear viscosity gmixis given by the semiempirical expression

N N 11

g 4 x g x ` x Umix o i i i o j ij1 2i41 j41, j?i

1/4 1/2 2 11/2m g mj i iwith U 4 1 ` 8 1 `ij 3 1 2 1 2 4 3 1 24m g mi j j

(14)

For diffusion coefficients of a species i into the mix-ture, the old formalism of Stefan [21] agrees wellwith full multicomponent models [22] (within theuncertainty of the input data and models used tocompute the binary diffusion coefficients):

N

x mo j jj41, j?i

D 4 (15)i,mix N N

x m x /Do j j o j ijj41 j41, j?i

For the mixture thermal diffusion factor, the usualexpression (following Chapman and Cowling [23])

N

T Tk 4 x • x a (16)i,mix i o j ijj41

is taken, where the sum of the over the mixtureTki

is zero. In analogy to the binary expression [5]

2c M M1 2T TD 4 k D with i 4 1,2 (17)i,binary i 12q

the thermal diffusion coefficient is evaluated as (c 4total molar concentration, Mi 4 molar mass, M 4mean molar mass)

2 ¯c M MiT TD 4 k D (18)i,mix i,mix i,mixq

This should be sufficiently accurate as only thermaldiffusion of H and H2 are important in combustionprocesses, which thus the mixture behaves like a bi-nary one, formed by the light species considered andthe rest of the mixture.


Premixed Flames Considered, Choice of theReaction Mechanism

As previously mentioned, transport properties oflight species like H and H2 and of polar species likeH2O experience the largest modifications, if the so-phisticated transport model described earlier is ap-plied to reacting flow systems. Therefore, conse-quences of the new formulation can be tested best,if the combustion of hydrogen (or mixtures of hy-drogen with carbon monoxide) is considered. A fur-ther (preliminary) reason for this choice is that datafor species involved in the oxidation of hydrocarbonsand other reaction systems are not yet available.

The reaction mechanism used is based on a datacompilation by Baulch et al. [27,28], which had beenslightly modified for practical application [24,25].This mechanism attributes rather high rate coeffi-cients to the sensitive reactions H ` O2 → OH `O and OH ` CO → CO2 ` H to be able to simulateexperimental data on H2-air and H2-CO-air flames.This procedure has been harshly criticized [29], butin light of the results of this paper, it is evident thatdeficiencies of the (slow) old transport model had tobe compensated by fast rate-limiting reactions. Thisfitting now can be removed, and the desired slowrate coefficients can be used for the reactions men-tioned earlier [26], which are similar to those usedin the GRI mechanism [30]. The reaction mecha-nism is listed in Table 3.

Results and Discussion

Pure Species and Species Pair TransportCoefficients

As already mentioned in the Introduction, thisnew approach changes (1) the data for light speciesat high temperature by switching to an exponentialrepulsive potential, (2) data for polar molecules byusing effective potentials, and (3) thermal diffusionfactors, using an expansion for large values of themass ratio of the species included. Whereas the ther-mal diffusion coefficients (leading to a rather smallcorrection in any case) do not change considerably,there are noticeable changes with respect to theother points addressed.

The first important fact is (see Fig. 1) that the heatconductivities in the new formulation are slightlylarger for stable nonpolar species (and much morefor H and O) and ;5% smaller for stable polar spe-cies (and much more for OH). The second differ-ence is (see Fig. 2) that diffusion coefficients of lightspecies are much larger (up to ;40% for diffusionof H atoms), if the new approach is used. The onlyexceptions are binary diffusion coefficients of theradical OH, which slightly decrease (;10% for

).DOH1N2

The consequence of these results is that thechanges of both heat conductivities and diffusion co-efficients can (partially) compensate. Thus, the pre-diction of a general trend for reactive flows is diffi-cult and dependent on the actual conditions, asshown below for some selected premixed flames.

H2–Air and H2–O2 Flames

To demonstrate the consequences of the newtransport model for combustion processes, the influ-ence on premixed hydrogen–air and hydrogen–oxy-gen flame velocities (at atmospheric pressure androom temperature in the unburned gas) is shown inFigs. 3 and 4. To guarantee a fair comparison, thesame mixture transport model is taken (equation 13for kmix, equation 15 for Di,mix, and equations 16 and18 for ).TDi,mix

There is no essential change in thermal diffusion(,0.5% change of the flame velocity of a stoichio-metric H2–air flame) but up to 4% change in theflame velocity in the H2–air system (up to 8% inmoderately rich H2–O2 mixtures) as a result of thenew heat conductivities and diffusion coefficients.This reflects the well-known dependency vL 4

, where the thermal diffusivity is given by a 4a/s!k/qcp, which is approximately equivalent to a meandiffusion coefficient [38].

There is good agreement of experiment and simu-lation now in the H2–air flames, where precise mea-surements (including extrapolation to strain-freeconditions) are available; older measurements (andall experimental results for H2–O2 flames; see Fig.4) tend to be too high because of strain effects.

Sensitivity analysis for the fastest H2–air and H2–O2 flames shows that the relative sensitivity of theflame velocity with respect to the rate coefficients ofthe reactions H ` O2 → OH ` O and H ` O2 `M → HO2 ` M are 0.25 and 0.18, respectively. Thismeans that the modifications introduced by the newtransport model have to be compensated by a de-crease of the rate coefficient of one of these reac-tions of up to one-third.

H2-O2-CO-N2 Flames

The consequences of the new transport model forpremixed hydrogen–carbon monoxide–air flame ve-locities (at atmospheric pressure and room tempera-ture in the unburned gas) are demonstrated in Fig.5. As expected, the changes are highest when fastflames (i.e., hotter flames) are considered.

Again, comparison with precise measurements in-cluding extrapolation to strain-free conditions areavailable. Sensitivity analysis of the 50% fuel flamegives the result that the largest relative sensitivitiesof the flame velocity with respect to the rate coeffi-cients of the reactions H ` O2 → OH ` O and CO` OH → CO2 ` H are 0.27 and 0.55, respectively.


TABLE 3Reaction mechanism for the oxidation of H2 and CO [24–26]

Reaction A [cm, mol, s] b E/kJ mol11

1. H2-O2 Reactions (HO2, H2O2 not included)O2 ` H 4 OH ` O 8.70 2 1013 0.0 60.3H2 ` O 4 OH ` H 5.06 2 1004 2.67 26.3H2 ` OH 4 H2O ` H 1.00 2 1008 1.6 13.8OH ` OH 4 H2O ` O 1.50 2 1009 1.14 0.42H ` H ` M* 4 H2 ` M* 1.80 2 1018 11.0 0.00O ` O ` M* 4 O2 ` M* 2.90 2 1017 11.0 0.00H ` OH ` M* 4 H2O ` M* 2.20 2 1022 12.0 0.00

2. HO2 Formation/ConsumptionH ` O2 ` M* 4 HO2 ` M* 2.30 2 1018 10.8 0.00HO2 ` H 4 OH ` OH 1.50 2 1014 0.0 4.20HO2 ` H 4 H2 ` O2 2.50 2 1013 0.0 2.90HO2 ` H 4 H2O ` O 3.00 2 1013 0.0 7.20HO2 ` O 4 OH ` O2 1.80 2 1013 0.0 11.70HO2 ` OH 4 H2O ` O2 6.00 2 1013 0.0 0.00

3. H2O2 Formation/ConsumptionHO2 ` HO2 4 H2O2 ` O2 2.50 2 1011 0.0 15.20OH ` OH ` M* 4 H2O2 ` M* 3.25 2 1022 12.0 0.00H2O2 ` H 4 H2 ` HO2 1.70 2 1012 0.0 15.7H2O2 ` H 4 H2O ` OH 1.00 2 1013 0.0 15.0H2O2 ` O 4 OH ` HO2 2.80 2 1013 0.0 26.8H2O2 ` OH 4 H2O ` HO2 5.40 2 1012 0.0 4.20

4. CO Formation/ConsumptionCO ` OH 4 CO2 ` H 4.76 2 1007 1.23 0.29CO ` HO2 4 CO2 ` OH 1.50 2 1014 0.0 98.7CO ` O ` M* 4 CO2 ` M* 7.10 2 1013 0.0 119.0CO ` O2 4 CO2 ` O 2.50 2 1012 0.0 200.

5. CHO Formation/ConsumptionCHO ` M* 4 CO ` H ` M* 7.10 2 1014 0.0 70.3CHO ` H 4 CO ` H2 9.00 2 1013 0.0 0.00CHO ` O 4 CO ` OH 3.00 2 1013 0.0 0.00CHO ` O 4 CO2 ` H 3.00 2 1013 0.0 0.00CHO ` OH 4 CO ` H2O 1.00 2 1014 0.0 0.00CHO ` O2 4 CO ` HO2 3.00 2 1012 0.0 0.00CHO ` CHO 4 CH2O ` CO 3.00 2 1013 0.0 0.00

6. CH2O Formation/ConsumptionCH2O ` M* 4 CHO ` H ` M* 5.00 2 1016 0.0 320.CH2O ` H 4 CHO ` H2 2.30 2 1010 1.05 13.7CH2O ` O 4 CHO ` OH 4.15 2 1011 0.57 11.6CH2O ` OH 4 CHO ` H2O 3.40 2 1009 1.2 11.90CH2O ` HO2 4 CHO ` H2O2 3.00 2 1012 0.0 54.7CH2O ` CH3 4 CHO ` CH4 1.00 2 1011 0.0 25.5CH2O ` O2 4 CHO ` HO2 6.00 2 1013 0.0 171.

[M*] 4 [H2] ` 6.5 2 [H2O] ` 0.4 2 [O2] ` O.4 2 [N2] ` 0.75 2 [CO] ` 1.5 2 [CO2]


Fig. 1. Heat conductivities for H2

and H2O as function of the tempera-ture. Gray line: old transport model(see Refs. [4,6–8]); black line: newtransport model; points: experimen-tal results [31–37]. The CHEMKINpackage [1–3] is better for H2 due toa different choice of its LJ potentialparameters, but worse for H2O dueto the neglect of a resonance correc-tion for polar species interaction [5].

Fig. 2. Binary diffusion coeffi-cients (related to p 4 1 bar) formixtures with N2 as function of thetemperature. Gray line: old transportmodel (see Refs [4,6–8]); black line:new transport model (the results forO2–N2 are not distinguishable).

Fig.. 3. Flame velocity (at p 4 1 bar, Tu 4 298 K) forH2–air mixtures as function of the H2 mole fraction. Points:experiments after 1970 (see Refs. [6,30]); gray line: oldtransport model: black line: new transport model.

This prominent dependence on the OH ` CO2 re-action means that the modifications introduced bythe new transport model (8% increase of the flamevelocity at maximum) now leaves place for a de-crease of the (rather high) rate coefficient of one ofthese steps up to 15%.

Conclusions and Outlook

A new transport model in terms of a complete setof equations is presented. The molecular parametersprovided allow a complete treatment of the oxidationof H2 and H2/CO mixtures. To demonstrate the con-sequences of the new transport model for combus-tion processes, results have been generated by im-plementing the model in a code for the simulationof premixed laminar flames.

Data for species taking place in the oxidation ofhydrocarbons and in other reaction systems (e.g., to


Fig. 4. Flame velocity (at p 4 1 bar, Tu 4 298 K) forH2–O2 mixtures as function of the H2 unburned gas molefraction. Points: experiments (see Ref. [6]); gray line: oldtransport model; black line: new transport model.

Fig. 5. Flame velocity (at p 4 1 bar, Tu 4 298 K) forH2–CO–air mixtures as function of the fuel percentage inthe unburned gas. Points: experiments ● by McLean et al.[29], and C by Scholte and Vaags [39] with [CO]u 4

19[H2]u; gray line: old transport model; black line: newtransport model.

describe CVD, plasma etching, chemical reactors,etc.) are not yet available but will be provided—atleast for hydrocarbon oxidation—in future. Conse-quences for hydrocarbon combustion can be drawnfrom preliminary results for methane–air flames(where the methane molecular data have been taken

for all hydrocarbon intermediates), showing an in-crease of the flame velocity of a stoichiometric flamefrom 38.1 to 39.1 cm/s or 2.5%; the reaction mech-anism used is similar to that presented in Ref. [25]*).

Furthermore, the implementation in non-pre-mixed flame—where changes in the transport coef-ficients can be much more important—is in prepa-ration.

Acknowledgments

Ph. P. acknowledges the support of the U.S. Departmentof Energy, Basic Sciences, Chemical Sciences Division andsupport from the “Deutsche Forschungsgemeinschaft(DFG)” (LEIBNIZ program) for an extended stay at Hei-delberg University. J. W. is thankful for support of this workby the “DFG” (SFB 359, LEIBNIZ program) and the“Fonds der Chemischen Industrie,” and also for the hos-pitality of the Combustion Research Facility, Sandia Na-tional Laboratories in Livermore.

REFERENCES

1. Kee, R. J., Warnatz, J., and Miller, J. A., “A FORTRANComputer Package for the Evaluation of Gas PhaseViscosities, Heat Conductivities, and Diffusion Coef-ficients,” Sandia report SAND83-8209.

2. Kee, R. J., Dixon-Lewis, G., Warnatz, J., Coltrin,M. E., and Miller, J. A., “A FORTRAN ComputerCode Package for the Evaluation of Gas-Phase Mul-ticomponent Transport Properties,” Sandia reportSAND86-8246.

3. Kee, R. J., Rupley, F. M., and Miller, J. A., “CHE-MKIN-II: A Fortran Chemical Kinetics Package forthe Analysis of Gas-Phase Chemical Kinetics,” SandiaNational Laboratories report SAND89-8009.

4. Warnatz, J., Berechnung der Flammengeschwindig-keit und der Struktur von laminaren Flammen, Habi-litationsschrift, Technische Hochschule Darmstadt,1977.

5. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., Mo-

lecular Theory of Gases and Liquids, Wiley, New York,1964.

6. Warnatz, J., Ber. Bunsen-Ges. Phys. Chem. 82:643–649(1978).

7. Warnatz, J., Ber. Bunsen-Ges. Phys. Chem. 83:950–957(1979).

8. Warnatz, J., in Eighteenth Symposium (International)

on Combustion, The Combustion Institute, Pittsburgh,1981, pp. 369–384.

9. Bzowski, J., Kestin, J., Mason, E. A., Uribe, F. J., J.

Phys. Chem. Ref. Data 19:1179–1232 (1990).10. Uribe, F. J., Mason, E. A., and Kestin, J., Phys. A

156:467–491 (1989).

*An actual version of the mechanism is given under:http://reaflow.iwr.uni-heidelberg.de/(keyword: “ftp server”)


11. Uribe, F. J., Mason, E. A., and Kestin, J., J. Phys.

Chem. Ref. Data 19:1123–1136 (1990).12. Assael, G. J., Mixafend, S., Wakeham, W. A., J. Phys.

Chem. Ref. Data 15:1315–1322 (1986).13. Paul, P., “DRFM—A New Package for the Evaluation

of Gas-Phase Transport Properties,” Sandia reportSAND98-8203, Sandia National Laboratories, Liver-more, 1997.

14. Monchick, L. and Mason, E. A., J. Chem. Phys.

55:1676–1697 (1961).15. Keesom, W. H., Phys. Z. 22:129 (1921).16. Gray, C. G., Gubbins, K. E., The Theory of Molecular

Fluids, Clarendon Press, Oxford, 1984.17. Bzowski, J., Mason, E. A., and Kestin, J., Int. J. Ther-

mophys. 9:131–143 (1988).18. Tang, K. T. and Toennies, J. P., J. Chem. Phys.

80:3726–3741 (1984).19. Hirschfelder, J. O., in Sixth Symposium (International)

on Combustion, The Combustion Institute, Pittsburgh,1957, pp. 351–366.

20. Mason, E. A. and Saxena, S. C., Phys. Fluids 5:361–369 (1958).

21. Stefan, J., Sitzungsberichte Akad. Wiss. Wien II 68:325(1874).

22. Paul, P., Personal communication.23. Chapman, S. and Cowling, T. G., The Mathematical

Theory of Non-Uniform Gases, Cambridge UniversityPress, Cambridge, 1970.

24. Warnatz, J., in Twenty-Fourth Symposium (Interna-

tional) on Combustion, The Combustion Institute,Pittsburgh, 1992, pp. 553–579.

25. Warnatz, J., Maas, U., and Dibble, R. W., Combustion,

Springer, Heidelberg, 1996.26. Baulch, D. L., Bowman, C. T., Cox, R. A., Just, Th.,

Kerr, J. A., Pilling, M. J., Stocker, D. W., Troe, J.,Tsang, W., Walker, R. W., and Warnatz, J., unpublishedwork.

27. Baulch, D. L., Cobos, C. J., Cox, R. A., Esser, C., Just,Th., Kerr, J. A., Pilling, M. J., Troe, J., Walker, R. W.,and Warnatz, J., J. Phys. Chem. Ref. Data 21:411–734(1992).

28. Baulch, D. L., Cobos, C. J., Cox, R. A., Frank, P., Hay-man, G., Just, Th., Kerr, J. A., Murrels, T. Pilling, M. J.,Troe, J., Walker, R. W., and Warnatz, J., Combust.

Flame 98:59–79 (1994).29. McLean, I. C., Smith, D. B., and Taylor, S. B., in

Twenty-Fourth Symposium (International) on Com-

bustion, The Combustion Institute, Pittsburgh, 1994,pp., 749–757.

30. Frenklach, M., Wang, H., Goldenberg, M., Smith, G.,Golden, D., Bowman, C. T., Hanson, R., Gardiner,W. C., and Lissianski, V., Topical report GRI-95/0058,The Gas Research Institute, Chicago, 1995.

31. Kesselman, P. M., and Litvinov, A. S., Teplo.-Masso-

perenos 7:121–137 (1968).32. Matsunaga, N. and Nagashima, A., J. Phys. Chem.

87:5268–5279 (1983).33. Beaton, C. F., and Hewitt, G. F., (eds.), Physical Prop-

erties for the Design Engineer, Hemisphere, New York,1992.

34. Aleksandrov, A. A., and Matveev, A. B., Them. Eng.

25:58–63 (1978).35. Y. S. Toulokian and C. Y. Ho (eds.), IFI, Plenum, New

York, 1970.36. Assael, G. J., Mixafendi, S., and Wakeham, W. A., J.

Phys. Chem. Ref. Data 15:1315–1322 (1986).37. Y. S. Toulokian and C. Y. Ho (eds.), Properties of Non-

metallic Fluid Elements, McGraw-Hill, New York,1981.

38. Zeldovich, Y. B., Frank-Kamenetskii, D. A., Zh. Fiz.

Khim. 12:100 (1938).39. Scholte, T. G., Vaags, D. B., Combust. Flame 3:511–

524 (1959).

COMMENTS

Daniel E. Rosner, Yale University, USA. One of the mostsensitive tests of the validity of a particular intermolecularpotential is its ability to predict the magnitude and tem-perature dependence of the Ludwig–Soret coefficient aT

(thermal diffusion factor), a dimensionless quantity neededto predict mass transfer rates in disparate molecularweight, highly nonisothermal systems [1,2]. Have such ex-perimental data been used to guide present generalizationsof the more familiar 12:6 Lennard–Jones potential and se-lection of the associated species-specific parameters?

REFERENCES

1. Rosner, D. E., et al., Combust. Sci. Technol. 20:87–106(1979).

2. Rosner, D. E., et al., Electrochem. Soc. 88-4:11–138(1998).

Author’s Reply. It is well known that the thermal diffu-sion factor aT is quite sensitive to the form of the potential.Bzowski et al. [1] find that calculated values of the thermaldiffusion factor may involve uncertainties an order of mag-nitude greater than the uncertainties in calculations of first-order transport properties. They also note that the errorsin the experimental data for the thermal diffusion factorinvolve uncertainties of a similar magnitude. This group hasrecently published an extensive review and revaluation oftransport property experimental data. As a result, they haveprescribed the use of a more advanced set of potential pa-rameter combination rules as well as the use of a hybridpotential (i.e., modified Lennard–Jones at temperaturesless than 10 times the well-depth energy of the collisionpair and an exponential repulsive potential at higher tem-peratures). For our transport model, we have adopted theirfull set of recommendations that include a new set ofpotential parameters, new combination rules, a new poten-tial, the inclusion of higher order corrections to first-order


properties (i.e., Kihara corrections), and the use of a Kiharaformulation for the thermal diffusion factor. In compari-sons with experimental data, they found and we confirmthat the representations for pure species, binary, mixture,and multicomponent properties, as currently used in re-acting flow simulations, perform poorly especially as re-gards mixtures containing disparate mass ratios and thethermal diffusion factor. Within all of the available data,

these advanced models appear to correct the deficienciesof prior models and substantially improve the predictabilityof transport properties.

REFERENCE

1. Bzowski, J., Kestin, J., Mason, E. A., and Uribe, F. J., J.

Phys. Chem. Ref. Data 19:1179–1232 (1990).

paul,p. 98 a re-evaluation of the means used to calculate transport properties of reacting flows

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