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Computers and Chemical Engineering 32 (2008) 1482–1493

Assessment of various diffusion models for the predictionof heterogeneous combustion in monolith tubes

Ankan Kumar, Sandip Mazumder ∗Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210, USA

Received 1 May 2007; received in revised form 12 June 2007; accepted 29 June 2007Available online 14 July 2007

Abstract

In the case of heterogeneous reactions, diffusion is the only mechanism, locally, of transport of species to and from a surface. Thus, accurateprediction of diffusive transport is a prerequisite for accurate prediction of the operation of devices in which heterogeneous reactions occur. Threedifferent diffusion models are examined from the standpoint of both accuracy and efficiency. Two of these models, namely the dilute approximation(DA) model and the Schmidt number (SN) model, are approximate models, and are compared against a rigorous multi-component diffusion(MCD) model derived from the Stefan–Maxwell equation. Both hydrogen–air and methane–air combustion in a monolith channel are studied. Inletequivalence ratio, Reynolds number (flow rate), and wall temperature are considered as parameters. The results show that both the DA model and theSN model are accurate within 2% irrespective of the equivalence ratio or fuel—the worst accuracy being for hydrogen combustion. The DA modeland the SN model produce almost identical results. In comparison to the MCD model, the DA model is approximately twice as computationallyefficient, while the SN model is 2–16 times more efficient. The accuracy and efficiency of the SN model, in conjunction with its simplicity, makesit an attractive choice for the treatment of diffusion in catalytic combustion calculations.© 2007 Elsevier Ltd. All rights reserved.

Keywords: CFD modeling; Catalytic combustion; Heterogeneous combustion; Diffusion; Stefan–Maxwell; Dilute approximation

1. Introduction

Catalytic combustion has found prolific usage in gas turbineand other application areas as a means to reduce pollution andcombat instabilities associated with homogeneous combustion.Numerical simulations are employed routinely within the indus-try and research communities to better understand the operationof catalytic converters. While there has been significant activ-ity in modeling a single tube of a monolithic converter, effortsat modeling a full-scale catalytic converter have been some-what limited mainly due to the extreme computational resourcesneeded for such simulations (Mazumder, 2007). Prior to thedevelopment of computational fluid dynamics (CFD) basedmodels for the simulation of full-scale catalytic converters, it isnecessary to critically examine the underlying physical models

∗ Corresponding author at: Department of Mechanical Engineering, The OhioState University, Suite E410, Scott Laboratory, 201 West 19th Avenue, Colum-bus, OH 43210, USA. Tel.: +1 614 247 8099; fax: +1 614 292 3163.

E-mail address: [email protected] (S. Mazumder).

to be used at the small (or channel) scale from both accuracy andefficiency standpoints. Such an investigation will likely elucidatethe best (optimum from both accuracy and efficiency standpoint)model that ought to be used for full-scale converter simulations.

With increasing trend towards miniaturization, transport phe-nomena in many engineering devices of the next generation arelikely to be dominated by diffusion. Even in devices of largerscale, such as catalytic converters and fuel cells, in which het-erogeneous (or surface) reactions play a critical role in deviceoperation, diffusive transport is expected to be of immenseimportance. In the case of catalytic combustion, since speciestransport to and from the surface reaction site is driven solely bydiffusion (since no-slip boundary conditions have to be satisfiedat the solid surface and advective transport is absent), accurateprediction of diffusion transport is a prerequisite for accurateprediction of catalytic combustion phenomena.

While examples of modeling catalytic combustion phenom-ena in a single monolith tube are abundant in the literature(Braun et al., 2000; Di Benedetto, Marra, Donsi, & Russo, 2006;Mukadi & Hayes, 2002; Quiceno, Perez-Ramırez, Warnatz,& Deutschmann, 2006; Raja, Kee, Deutschmann, Warnatz, &

0098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.compchemeng.2007.06.024


A. Kumar, S. Mazumder / Computers and Chemical Engineering 32 (2008) 1482–1493 1483

Nomenclature

B body force vector (m s−2)cp,k specific heat capacity of species k (J kg−1 K−1)Dkn binary diffusion coefficient of species k into n

(m2/s)Dkn ordinary multi-component diffusion coefficient

(m2/s)Dkn, Dk effective diffusivity of species k into the mixture

(m2/s)h enthalpy of mixture (J/kg)hk enthalpy of the kth species (J/kg)h0

f,k enthalpy of formation of species k at standard state(J/kg)

Jk diffusion mass flux of the kth species (kg m−2 s−1)kc thermal conductivity of mixture (W m−1 K−1)M molecular weight of the mixture (kg/kmol)Mk molecular weight of the kth species (kg/kmol)n unit surface normal vectorN total number of gas-phase speciesp pressure (Pa)q heat flux (W m−2)qC heat flux due to conduction (W m−2)qD heat flux due to inter-diffusion of species (W m−2)qR heat flux due to radiation (W m−2)Rk production rate of species due to heterogeneous

reactions (kmol m−2 s−1)Sk production rate of kth species due to homoge-

neous reactions (kg m−3 s−1)Sh volumetric enthalpy production rate (W m−3)T temperature (K)U mass averaged velocity (m/s)Xk mole fraction of kth speciesYk mass fraction of kth species

Greek lettersΓ kn transformed diffusion coefficient tensor (m2/s)δkn Kronecker deltaΛk molar concentration of species k at fluid–solid

interface (kmol m−3)ρ mixture density (kg m−3)τ shear stress (N m−2)

Schmidt, 2000; Ramanathan, Balakotaiah, & West, 2004; Stutz& Poulikakos, 2005, and the references cited therein), to thebest of the authors’ knowledge, none of these studies have criti-cally examined the effect of the diffusion model on the predictedresults. The decision with regard to the choice of the diffusionmodel has been largely arbitrary despite it being well-known thatcatalytic combustion, being a high-temperature phenomenon, isdiffusion-limited in many cases (Hayes & Kolaczkowski, 1997;Quiceno et al., 2006). Review of the literature reveals that dif-fusion models that have been used in the simulation of catalyticcombustion belong to either of the following two broad cate-gories: (1) the dilute approximation model and its variations, in

which nitrogen is treated as a diluent and the Fick’s law, in mod-ified form, is used (Andreas & Poulikakos, 2005; Braun et al.,2000; Deutschmann & Schmidt, 1998; Hayes, Liu, Moxom, &Votsmeier, 2004; Mukadi & Hayes, 2002; Quiceno et al., 2006;Raja et al., 2000; Ramanathan et al., 2004; Stutz & Poulikakos,2005) and (2) the multi-component diffusion model, in which theStefan–Maxwell equations, in their primitive form, are solved(Moustafa, 2003; Saracco, Veldsink, Versteeg, & Wim, 1995;Salmi & Waerna, 1991).

In a system comprised of two species, i.e., a binary system,diffusive transport of a certain species is dictated by the gradi-ent of its own concentration, as described by the Fick’s law ofdiffusion (Bird, Stewart, & Lightfoot, 2001). In a system withmultiple species, the diffusion of a species is governed not onlyby its own concentration gradient, but also by the concentra-tion gradient of the other species in the system, and is generallyreferred to as multi-component diffusion (MCD) (Bird et al.,2001). In a multi-component system, use of Fick’s law resultsin violation of overall mass conservation, as will be discussedshortly.

If the mass fraction of a certain species is large everywhere ina mixture and this species is also non-reacting, it can be deemeda “buffer” or “diluent”. If a diluent is present in a mixture,the diffusion flux of the other (active) species is widely mod-eled using the so-called dilute approximation (Bird et al., 2001;Hirschfelder, Curtiss, & Bird, 1954). In the dilute approxima-tion, the Fick’s law is still used although, strictly speaking, itviolates mass conservation (Bird et al., 2001). To accommodateits use, the conservation of the diluent is not enforced and all theerrors due to non-satisfaction of overall mass conservation areabsorbed by the diluent. In other words, the diluent serves therole of an infinitely large reservoir.

The use of the dilute approximation for modeling mass trans-fer has been prolific (references cited in Andreas & Poulikakos,2005; Braun et al., 2000; Deutschmann & Schmidt, 1998; Hayeset al., 2004; Mukadi & Hayes, 2002; Quiceno et al., 2006; Raja etal., 2000; Stutz & Poulikakos, 2005). It is generally believed thatthe use of dilute approximation for multi-component systems isvalid when the mass fraction of the diluent is large—a notionthat stems out of the “reservoir” concept. As to what consti-tutes “large” remains a matter of conjecture. A few studies haveattempted to investigate the validity of the dilute approximationby comparing against approximate models for multi-componentdiffusion (Desilets, Proulx, & Soucy, 1997; Ern & Giovangigli,1999; Sutton & Gnoffo, 1998), although not for catalytic con-version applications. Nevertheless, these studies and recentcalculations performed by the authors (Kumar & Mazumder,2007) indicate that the dilute approximation may not always beaccurate, and its validity depends on the specific application inquestion and/or the operating conditions being used.

The first objective of the present study is to quantify theerrors incurred by using the dilute approximation specifi-cally for catalytic combustion applications. Another simplifiedmodel, referred to earlier as the Schmidt number model, andto be described later, is also examined. The accuracy of thesetwo approximate models is assessed by comparing the resultsobtained using these models with those obtained using a rigorous


1484 A. Kumar, S. Mazumder / Computers and Chemical Engineering 32 (2008) 1482–1493

MCD formulation derived from the Stefan–Maxwell equation(Bird et al., 2001). The second objective is to compare thecomputational efficiency of the three models in question. Tomeet these two objectives, steady state catalytic combustion cal-culations are performed within a single monolith tube undera wide range of operating conditions. Both methane–air andhydrogen–air combustion are studied. Single-step chemistry isused for both fuels. The inlet equivalence ratio, the flow rate,and the temperature of the tube wall are treated as parameters.The governing conservation equations of mass (both overall andindividual species), momentum, and energy are solved usinga conservative finite-volume formulation, and the conversionpercentages, mass fraction distributions, and temperature distri-butions in each case are carefully analyzed.

2. Governing equations

The governing equations are the equations of conserva-tion of mass (both overall and individual species), momentumand energy, and are written as (Bird et al., 2001; Kuo, 1986;Mazumder, 2007):

Overall mass :∂

∂t(ρ) + ∇ · (ρU) = 0 (1)

Momentum :∂

∂t(ρU) + ∇ · (ρUU) = −∇p + ∇ · � + ρB

(2)

Energy :∂

∂t(ρh) + ∇ · (ρUh) = −∇ · q + Sh (3)

Species mass :∂

∂t(ρYk) + ∇ · (ρUYk)

= −∇ · Jk + Sk ∀ k = 1, 2, . . . , N (4)

where ρ is the mixture density, p the pressure, τ the shear stresstensor, and B is the body force vector. Eqs. (1) and (2) are the wellknown Navier–Stokes equations, and need no further discussion.In Eq. (4), Yk is the mass fraction of the kth species, Jk is themass diffusion flux of the kth species, and Sk is the productionrate of the kth species due to homogeneous chemical reactions.The total number of gas-phase species in the system is denotedby N. In Eq. (3), Sh represents the net source due to viscousdissipation and other work and heat interactions, and q denotesthe net heat flux due to molecular conduction, radiation, andinter-species diffusion, and is written as Bird et al. (2001):

q = qC + qR + qD = −kc∇T + qR +N∑

k=1

Jkhk (5)

where hk is the enthalpy of the kth species, kc the thermal con-ductivity of the mixture, and h is the enthalpy of the mixture(=∑N

k=1hkYk). The heat flux due to inter-diffusion of species,qD, is often neglected in reacting flow formulations withoutany justification. In many practical applications of multi-speciesflows, this term can be comparable or larger than the Fourier

conduction flux qC, and can result in net heat flux that is oppo-site in direction to the imposed temperature gradient (Kumar& Mazumder, 2007). In the above formulation, the enthalpy ofthe kth species, hk, includes the enthalpy of formation and thesensible enthalpy, and is written as

hk(T ) = h0f,k +

∫ T

T0

cp,k(T ) dT (6)

where h0f,k is the enthalpy of formation of species k at the stan-

dard state and cp,k is the specific heat capacity of species k. Thespecies enthalpy is generally computed using standard thermo-dynamic databases, such as the JANNAF database.

Eq. (1) can be derived from Eq. (4) if and only if the followingtwo constraints are satisfied:

At any point in space :N∑

k=1

Yk = 1 (7)

At any arbitrary cutting plane :N∑

k=1

Jk · n = 0 (8)

where n is the surface normal to the cutting plane in question.Eq. (7) is generally enforced either by solving only N − 1 equa-tions from the set in Eq. (4), and then using Eq. (7) directly todetermine the mass fraction of the last species, or by normaliz-ing the calculated mass fractions by their sum, resulting in anindirect correction strategy (Mazumder, 2006; Wangard, Dandy,& Miller, 2001). In diffusion dominated (i.e., low mass transportPeclet number) systems, non-satisfaction of the constraint givenby Eq. (8) results in a serious inconsistency. The inconsistencyis a result of the fact that the sum of the species conservationequations over all species does not result in the continuity equa-tion (Eq. (1)). Rather, it results in a continuity equation with aspurious mass source.

In general, the mass diffusion flux, Jk, includes diffusion dueto concentration gradients, temperature gradients (Soret diffu-sion), and pressure gradients (pressure diffusion) (Bird et al.,2001). Here, only mass transport due to concentration gradientsis considered since it is known to be the dominant mechanismof diffusion in such systems. The most common approach formodeling diffusive transport of species due to concentration gra-dients is to use the Fick’s law of diffusion. In a binary system,consisting of species A and B, the diffusion flux is accuratelydescribed by the Fick’s law of diffusion (Bird et al., 2001):

JA = −ρDAB∇YA, JB = −ρDBA∇YB (9)

where DAB is the binary diffusion coefficient of species A intoB, and is equal to DAB, which is the binary diffusion coeffi-cient of species B into A. Using the mass fraction summationconstraint (Eq. (7)), it can be readily shown that JA = −JB, i.e.,Eq. (8) is automatically obeyed. The same law, when used formulti-component systems with more than two species, leads toviolation of mass conservation due to non-satisfaction of theconstraint given by Eq. (8).



2.1. Multi-component diffusion (MCD) model

In a multi-component system, diffusion is best described bythe Stefan–Maxwell equation (Bird et al., 2001), which implic-itly relates molar fluxes of species to mole fraction gradients:

∇Xi = M

ρ

N∑j = 1

j �= i

(XiJj

MjDij

− XjJi

MiDij

)(10)

where M is the mixture molecular weight, Mk the molecularweight of the kth species, and Xk is the mole fraction of thekth species. The Stefan–Maxwell equation has been formu-lated in such a manner that Eq. (8) is satisfied for an arbitrarymulti-component system. Unfortunately, it can only be usedfor pure diffusion problems, and is not amenable for use ina CFD framework since it is not an equation of the general-ized advection–diffusion form. Upon significant manipulation,Eq. (10) can be re-written as (Mazumder, 2006; Wangard et al.,2001).

Jk = −ρMk

M2

N∑n=1

MnDkn∇Xn (11)

where Dkn is the ordinary multi-component diffusion coefficient(Bird et al., 2001; Mazumder, 2006; Wangard et al., 2001), and isdifferent from the binary diffusion coefficient Dkn. Specifically,while the binary diffusion coefficients are independent of themole fractions, the ordinary multi-component diffusion coeffi-cients are strong nonlinear functions of the mole fractions. Also,it is worth noting that while Dkk = 0, Dkn �= Dnk. The ordinarymulti-component diffusion coefficients can be computed usingwell-known relationships (Bird et al., 2001; Sutton & Gnoffo,1998):

Dij = M

Mj

(Fji − Fii)

|F | ,

Fij = Xi

Dij

+ Mj

M

∑k �=i

Xk

Dik

∀j �= i (12)

where |F| is the determinant of the matrix whose elements are Fij

for j �= i. Note that Fii = 0. Fji are the transposed cofactors of thematrix F. Eq. (11) essentially states that in a multi-componentsystem, diffusion of a certain species is governed not only by itsown concentration gradient, but also the concentration gradientof the other species in the system.

In a reacting flow, mass is always conserved while moles arenot. Thus, it is advantageous to re-write Eq. (11) in terms ofmass fractions rather than mole fractions. Using the conversionrelation Yk = XkMk/M, Eq. (11) may be written as (Wangard etal., 2001; Mazumder, 2006):

Jk = −ρ

N∑n=1

Γkn∇Yn (13)

where Γ kn is a new tensor, written as (Wangard et al., 2001;Mazumder, 2006):

[Γ ] = − 1

M2 [M][D][M][C] (14)

In Eq. (14), [M] = diag[M1, M2, . . ., MN], [D] is the matrixnotation for the ordinary multi-component diffusion tensor Dkn,and [C] the Jacobian of the transformation between mass andmole fraction, and is written as

Ckn =(

δkn − Yk

M

Mk

)M

Mn

(15)

where δkn is the Kronecker delta. Substitution of Eqs. (13) in (4)yields:

∂


= ∇ ·(

ρ

N∑n=1

Γkn∇Yn

)+ Sk ∀k = 1, 2, . . . , N (16)

Eq. (16) represents the governing equation for species transportin a multi-component system. It satisfies the constraint posed byEq. (8) automatically.

2.2. Dilute approximation (DA) model

The solution of the equations resulting from the MCD for-mulation (Eq. (16)) is significantly more complicated than thecorresponding equation for the binary diffusion formulation (Eq.(4) with fluxes expressed by Eq. (9)). This is because the speciesmass fractions become tightly coupled through the diffusionterm in Eq. (16) even if homogeneous reactions are absent (i.e.,Sk = 0). To circumvent this complexity, the diffusion flux in amulti-component system is often modeled using the so-calleddilute approximation. Using the dilute approximation, the dif-fusion flux is written as Bird et al. (2001):

Jk = −ρDkm∇Yk (17)

where Dkm is the effective diffusivity of species k into themixture, and is henceforth denoted by Dk for simplicity. Theeffective diffusivity is given by the relation (Bird et al., 2001):

Dkm = Dk = 1 − Xk∑N

i = 1

i �= k

Xi/Dki

(18)

Substitution of Eq. (17) into Eq. (4) yields the appropriatespecies transport equation for the dilute approximation formu-lation:

∂


= ∇ · (ρDk∇Yk) + Sk ∀k = 1, 2, . . . , N (19)

The species transport equations (Eq. (19)) are only weaklycoupled in this formulation through ρ and Dk in the absenceof chemical reactions, making the dilute approximation more



convenient for numerical solution as compared to the MCDformulation. Since the numerical solution of the governing equa-tions is not the focus of this article, these issues are not discussedhere any further. It should suffice to say that in the dilute formula-tion, since the equations are weakly coupled, they can be solvedsequentially if homogeneous reactions are absent. In contrast,the multi-component equations are inherently unstable whensolved sequentially, and necessitates coupled solution to guaran-tee stability and convergence, as shown by Mazumder (2006).Thus, while the MCD formulation is rigorous and guaranteesmass conservation, the accuracy comes at a price—that beingthe necessity to use advanced algorithms for solving the result-ing equations, which, in some cases, may also require morecomputer time.

2.3. Schmidt number (SN) model

The conservation constraint posed by Eq. (8) can be automat-ically satisfied by assuming that the binary diffusion coefficientsfor each of the species pairs are equal. This assumptionessentially translates to producing equal mixture diffusion coef-ficients, Dk, for each species in the system. Since the Schmidtnumber, Sc, is defined as the ratio of the dynamic viscosity, μ,to the dynamic diffusivity, ρD, the Schmidt number for each ofthe species will be equal under this assumption. Thus, if a con-stant Schmidt number is specified, the diffusivity of each speciescan be easily calculated from the relation ρDk = μ/Sc. The factthat the viscosity is a function of the local mixture compositionand the temperature, makes the local diffusion coefficient also afunction of the same quantities, allowing for a spatial variationin the diffusion coefficient. Thus, in this model, the diffusioncoefficient of each species will be equal, but is still allowedto vary spatially based on the local temperature and mixturecomposition.

One critical issue pertaining to this model is the value thatshould be used for the Schmidt number. Di Benedetto et al.(2006) used an arbitrarily chosen value of 0.7 for propane com-bustion. Rather than arbitrarily prescribing a value, the Schmidtnumber was estimated for the current study based on inlet con-ditions. The effective diffusivity of any species is given by Eq.(18). In the SNM model, the effective diffusivity of the fuel,Dfuel, is calculated based on the inlet conditions using Eq. (18).The Schmidt number of the fuel can then be calculated using theformula Scfuel = μ/ρDfuel and once the Schmidt number hasbeen assigned a value, the diffusion coefficient of all the speciescan be calculated using ρDk = μ/Scfuel. This procedure makessure that the Schmidt number is not prescribed arbitrarily and isnot a free tunable parameter in the model.

The SN model is expected to produce inaccurate results. Theinaccuracy in this model stems from the assumption of usingthe same diffusivity (or Schmidt number) for all species in thesystem. On the other hand, despite being somewhat simplis-tic, the SN model has certain distinct advantages. First, unlikethe DA model, it satisfies Eq. (8) automatically. Secondly, thecalculation of the diffusion coefficient reduces to a simple divi-sion operation, thereby eliminating any computer time neededto calculate either Dkm (for the DA model) or Γ kn (for the MCD

model). It has been shown by past studies (Salmi & Waerna,1991) that the CPU times spent in calculation of the diffusioncoefficients, particularly in the case of the MCD model, are nottrivial, and constitute a large portion of the CPU budget. As tohow much accuracy is compromised by using the same diffusioncoefficient and how much efficiency is gained by using the SNmodel remains to be seen.

2.4. Boundary conditions

The boundary conditions for the mass and momentum con-servation equations are the no-slip conditions at walls, andappropriate mass flux or pressure boundary conditions at inflowand outflow boundaries. These boundary conditions and theirnumerical implementation are well known and need no furtherdiscussion. The focus of this sub-section is the boundary con-ditions for species and energy associated with heterogeneouschemical reactions at fluid–solid interfaces.

At a reacting surface, the diffusion flux of species is balancedby the reaction flux since the surface cannot store any mass. Atthe heart of surface reaction processes is adsorption and desorp-tion of species at the surface, the treatment of which requiresinclusion of so-called surface-adsorbed species (Coltrin, Kee,& Rupley, 1991). At steady state, the net production rate of thesurface-adsorbed species is zero. In the absence of etching ordeposition of material from the surface (i.e., zero Stefan flux),the reaction-diffusion balance equation at the surface may bewritten as (Coltrin et al., 1991; Mazumder & Lowry, 2001):

Jk · n = MkRk ∀k ∈ gas-phase species (20a)

Rk={

dΛk/dt for unsteady

0 for steady∀k ∈ surface-adsorbed species

(20b)

where Rk is the molar production rate of species k due to het-erogeneous chemical reactions, Λk is the molar concentrationof species k at the fluid–solid interface, and n is the outwardunit surface normal. Since Rk is an extremely nonlinear func-tion of the molar concentrations (or mass fractions) (Coltrinet al., 1991; Mazumder & Lowry, 2001), Eq. (20) representsa nonlinear set of differential algebraic equations (DAE). Thesolution of this stiff set of nonlinear DAE is generally obtainedusing the Newton method, but requires special pre-conditioningto address stiffness and ill-posedness in the case of steady statesolutions. Details pertaining to these numerical issues may befound elsewhere (Mazumder & Lowry, 2001). The solution of(20) provides the near-wall mass fractions and mass fluxes (rep-resented by the left-hand side of Eq. (20a)) of all gas-phasespecies, which appear as sources/sinks for control volumes adja-cent to the surface in a finite-volume formulation (Mazumder &Lowry, 2001).

The balance of energy at the surface yields the followingequation:[−k∇T + qR +

N∑k=1

Jkhk

]F

· n = [−k∇T + qR]S · n (21)



where the subscript “F” denotes quantities on the fluid side ofthe fluid–solid interface, while the subscript “S” denotes quan-tities on the solid side of the same interface. The solution ofEq. (21), which is also a nonlinear equation, yields the tem-perature at the fluid–solid interface, and subsequently providesthe flux of energy at the interface, which can then be used asa source/sink for the cells adjacent to the interface after appro-priate linearization. In this enthalpy formulation, the heat ofsurface reaction actually manifests itself through the

∑Jkhk

term—another reason why the energy carried by species inter-diffusion should never be neglected for such applications. Inthe case of an isothermal wall (Dirichlet boundary condition fortemperature), and in the absence of radiation, the left-hand sideof Eq. (21) can be calculated directly (i.e., without solving Eq.(21) first to obtain the wall temperature) since the temperatureis known already.

Eqs. (1)–(3) along with Eq. (16) (for MCD) or Eq. (19) (forDA), when solved along with the appropriate boundary condi-tions described in the preceding sub-section, will produce flow,temperature and mass fraction distributions of all species withinthe monolith tube.

3. Results and discussion

The governing equations of mass, momentum and energy,described in the preceding section, were solved using a conserva-tive finite volume technique (Ferziger & Peric, 1999; Patankar,1980). The SIMPLE algorithm (Patankar, 1980) was used toaddress pressure–velocity coupling in the Navier–Stokes equa-tions. It is customary to treat individual channels within amonolithic catalytic converter as channels of circular cross-section (Raja et al., 2000), as schematically shown in Fig. 1.Thus, simulations were performed in a single channel with cir-cular cross-section of diameter D = 2 mm and length L = 20 cm.All simulations were performed on a uniform grid with 100 cellsin the axial direction and 30 cells in the radial direction. A gridindependence study for nominal parameter values showed thatcalculations on a 200 × 60 grid yield results that are within 2%of the values from the 100 × 30 grid for local mass fraction val-

Fig. 1. Picture of a full scale catalytic converter and schematic of the singlechannel model geometry.

ues, and less than 0.5% for conversion fraction values, and thusthe coarser of the two meshes was used for parametric studies.

Fuel–air mixture of specified composition is introduced intothe monolith tube at a prescribed velocity. A plug velocity profilewas imposed at the inlet. Variation of the inlet velocity translatesto a variation in the inlet Reynolds number, Re, which is one ofthe parameters in the simulations. The temperature of the wall,Tw, is also a prescribed parameter. In practice, the temperatureof the wall may vary and is dictated by the interplay of thevarious modes of heat transfer. However, to perform such calcu-lations, it would be necessary to model radiation, since radiationis the dominant mode of heat transfer at such elevated temper-atures (Boehman, 1998; Mazumder & Grimm, 2007), and alsoto have good estimates on the heat transfer rates external to thechannels. Thus, it was decided to study temperature effects ondiffusion by using Dirichlet boundary conditions. Conversionof the fuel is assumed to take place because of heterogeneous(surface) reactions at the walls only. This is justified since therange of temperature considered is such that homogeneous com-bustion may be neglected. The tube wall in the region near theentrance was assumed to be non-reacting up to x/L = 0.1 in orderto allow the incoming plug flow to develop to some extent priorto reactions occurring at the walls.

Simulations were carried out for two fuels, namely CH4 andH2. Since the focus of this study is diffusion, rather than detailedreaction kinetics, single-step reaction mechanisms with Arrhe-nius rate expressions were used to model the heterogeneousreactions occurring at the walls. This decision to use a 1-stepmechanism was further prompted by the fact that our own pastexperience (Mazumder & Sengupta, 2002), and those of oth-ers (Deutschmann & Schmidt, 1998; Raja et al., 2000) withmethane combustion using a detailed 24-step reaction mecha-nism (Deutschmann, Behrendt, & Warnatz, 1994) shows thatthe three trace species, namely CO, OH and H2 are three to sixorders smaller in amount than the major species present in the1-step global reaction. The reactions, rates, and the articles fromwhich the rates were obtained are shown in Table 1. As shownin Table 1, the reaction mechanism for hydrogen combustion isfirst-order in hydrogen only, implying that it is valid only forvery lean mixtures in which oxygen is abundant. The mecha-nism for methane combustion is valid for both lean and richmixtures.

Three different parameters were varied for this particularstudy. Reynolds number values of Re = 10, 100 and 1000 wereused for both fuels. For CH4 combustion, three surface temper-atures Tw = 1100, 1200 and 1300 K and three inlet equivalenceratios (Kuo, 1986) Φ = 0.5, 0.75 and 1.0 were considered. For H2combustion, the wall temperatures considered were Tw = 700,800 and 900 K, and the equivalence ratios considered wereΦ = 0.5, 0.6 and 0.7. The inlet temperature for CH4 combus-tion was 600 K while that for H2 combustion was 300 K. Thetemperature values, both wall and inlet, chosen for methane andhydrogen are different because these two fuels have very dif-ferent ignition temperatures. Hydrogen ignition and catalyticcombustion usually occurs at much lower temperatures thanmethane (Deutschmann et al., 1994). For hydrogen combustion,mixtures richer than Φ = 0.7 could not be considered because



Table 1Surface reaction mechanisms used in the current study: rate expressions are of the form AP exp(−Ea/R)[F]a[O2]b, where [F] is the molar concentration of the fuel

Reaction AP Ea/R (K) a b Reference

CH4 + 2O2 ⇒ CO2 + 2H2O 1.3 × 1011 16,189 1 0.5 Song, Williams, Schmidt, and Aris (1991)2H2 + O2 ⇒ 2H2O 1.4 × 103 1792 1 0 Schefer (1982)

Units are in mol, cm, K, s.

of the limitations posed by the reaction mechanism used, asdiscussed earlier.

All transport properties of the fluid, namely viscosity, thermalconductivity, and binary diffusion coefficients were computedusing the Chapman–Enskog equations of kinetic theory (Birdet al., 2001; Hirschfelder et al., 1954). The Lennard–Jonespotentials, which are needed as inputs, were obtained from theCHEMKIN database. For the MCD model, the calculation of thediffusion tensor Γ kn requires calculation of the ordinary multi-component diffusion tensor, Dkn (cf. Eq. (14)). An algorithmproposed by Sutton and Gnoffo (1998) was used for this pur-pose, and care was exercised to avoid repetitive computations ofco-factor determinants.

The solutions were deemed to be converged when the residu-als of each of the equations decreased by six orders of magnitude.The CPU time required to compute each case was obtained

using in-built system functions. The species transport equationswere not solved sequentially (i.e., one species at a time), butrather, using a block-implicit coupled solver, details of whichmay be obtained from Mazumder (2006). Even though the cou-pled solver is not necessary for the DA or the SN model, itwas still used for all three models. This was done to ensure thatobserved differences in CPU times for the three diffusion mod-els are strictly due to the nature of the equations being solvedand not due to differences in the core linear algebraic equationsolver.

3.1. Predictions of conversion fraction

Performance of a catalytic converter is measured by the frac-tion of the fuel that undergoes combustion as it exits the device.Therefore, the prediction of the percentage of fuel that undergoes

Fig. 2. Comparison of conversion fractions predicted using the MCD, the DA and the SN diffusion models: (a) conversion fraction and absolute error (comparedto MCD) in conversion fraction of H2 for Tw = 900 K and Φ = 0.7; (b) conversion fraction and absolute error (compared to MCD) in conversion fraction of CH4 forTw = 1300 K and Φ = 1.0.



combustion as it traverses the length of the monolith is of pri-mary interest in characterizing catalytic converter performance.A large difference in the prediction of conversion percentagedue to the use of different diffusion models would imply thatthe choice of the correct diffusion model would be critical foraccurate prediction of the performance of the catalytic converter.

The local conversion fraction at any axial location x is definedas the ratio of the difference of the mass flux of fuel at the Inlet(x = 0) and the mass flux of fuel at location x to the mass fluxof fuel at the inlet (x = 0). Fig. 2(a) depicts the local conversionpercentage for three different Re with Tw = 900 K and Φ = 0.7for H2–air combustion. Fig. 2(b) depicts the local conversionpercentage for three different Re with Tw = 1300 K and Φ = 1.0for CH4–air combustion. As noted in these figures, the conver-sion fraction values, as predicted using the MCD model, the DAmodel, and the SN model, are almost identical for both CH4–airas well as for H2–air combustion. For the H2–air mixture, the

Fig. 3. Comparison of mass fraction distributions of H2 for the case of Re = 1000,Tw = 900 K and Φ = 0.7. The radial direction has been scaled by a factor of 20for better visualization. (a) Comparison of local mass fraction between MCDand DA; (b) normalized error in local mass fraction between MCD and DA; (c)comparison of local mass fraction between MCD and SN; (d) normalized errorin local mass fraction between MCD and SN.

differences in predictions of the conversion fractions, for all Re,are numerically larger as compared to the same numbers forCH4–air mixture, as shown in the figures, as well. The largestdifference in the values in the case of H2–air mixture is around1.5% which is larger as compared to the largest difference incase of CH4–air mixture which is as low as 0.05%. This is tobe expected since hydrogen has much higher diffusion coeffi-cient than any other gas (Hirschfelder et al., 1954). The errors inpredictions by both approximate models are not altered signifi-cantly by the Reynolds number, except that they occur spatiallyat different locations, emphasizing the fact that heterogeneouscombustion phenomenon is strongly dependent on the local dif-fusion phenomena at the surface, and is thus, weakly correlatedwith Re based on inlet flow rates.

The results presented in Fig. 2 are for worst-case scenarios,i.e., the highest equivalence ratio for each fuel, in which case,the deviation from the dilute approximation is expected to be

Fig. 4. Comparison of mass fraction distributions of CH4 for the case ofRe = 1000, Tw = 1300 K and Φ = 1.0. The radial direction has been scaled bya factor of 20 for better visualization. (a) Comparison of local mass fractionbetween MCD and DA; (b) normalized error in local mass fraction betweenMCD and DA; (c) comparison of local mass fraction between MCD and SN; (d)normalized error in local mass fraction between MCD and SN.



the most. Both the DA model and the SN model are expectedto produce more accurate results as the equivalence ratio Φ isdecreased. Although not shown here, our calculations confirmedthis expected behavior. Calculations also revealed that the errorsbetween the various models are almost independent of tempera-ture. The errors are marginally higher for the highest temperaturecases, which is consistent with the fact that diffusion coeffi-cients of gases increase with temperature (Hirschfelder et al.,1954).

3.2. Predictions of mass fraction distributions

The mass fraction distributions of the fuels, namely H2and CH4 for two sets of operating conditions are shown inFigs. 3 and 4, respectively. For CH4, the operating conditions areTw = 1300 K, Re = 1000 and Φ = 1.0, while for H2, the operating

Fig. 5. Comparison of temperature distributions for H2 combustion for the caseof Re = 1000, Tw = 900 K and Φ = 0.7. The radial direction has been scaled by afactor of 20 for better visualization. (a) Comparison of local temperature betweenMCD and DA; (b) normalized error in local temperature between MCD and DA;(c) comparison of local temperature between MCD and SN; (d) normalized errorin local temperature between MCD and SN.

conditions are Tw = 900 K, Re = 1000 and Φ = 0.7. Normalizederrors, defined as |mass fraction predicted by MCD − massfraction predicted by the approximate model|/mass fraction pre-dicted by MCD, are also shown in the same figures.

For CH4 (Fig. 4), the mass fractions distribution as pre-dicted by the MCD, the DA model and the SN model are onlymarginally (less than 1% on an average) different from eachother. On the other hand, the mass fraction distribution of H2as predicted from MCD and the DA model show significantdeviation (between 1% and 5% on an average) from each other(Fig. 4). Interestingly, although different diffusion models yielddiscernibly different distributions of mass fractions of H2, thedifference in predicted conversion fractions are insignificant, as

Fig. 6. Comparison of temperature distributions for CH4 combustion for the caseof Re = 1000, Tw = 1300 K and Φ = 1.0. The radial direction has been scaled by afactor of 20 for better visualization. (a) Comparison of local temperature betweenMCD and DA; (b) normalized error in local temperature between MCD and DA;(c) comparison of local temperature between MCD and SN; (d) normalized errorin local temperature between MCD and SN.



shown and discussed earlier. It appears that averaging mass fluxover the cross-section negates local effects.

The temperature distributions in the geometry for the twofuels under consideration, namely H2 and CH4, are shown inFigs. 5 and 6, respectively. As mentioned in Section 2.1, thetemperature field is coupled directly to the species distributionsthrough the species diffusion term (i.e.,

∑Jkhk). Consequently,

the temperature distribution predicted by the three diffusionmodels and their errors relative to each other show the same qual-itative behavior as the mass fraction distributions themselves.The temperature distribution predicted using the DA model andthe SN model show remarkable similarity for both fuels, withless than 0.5% differences in the predicted local temperaturevalues.

In summary, there is a discernable difference between theprediction of species mass fraction distributions between theDA model and the MCD model for H2 combustion, while thedifference in case of CH4 combustion is insignificant. However,the discernable difference in the case of H2 combustion doesnot manifest itself into a significant difference when averagedconversion fractions are calculated. The SN model is comparablein accuracy to the DA model in predicting detailed species massfraction distributions, temperature distributions, and conversionfraction values. Since the accuracy of all the three diffusionmodels is comparable, it is important to consider the amount oftime taken in obtaining the solution for each case. For large-scalecalculations, the model that is computationally most efficient isthe most attractive one.

3.3. Computational efficiency

When the MCD formulation is used, calculation of bothspecies diffusion coefficients as well as solution of the speciestransport equations becomes mathematically complex, andhence, computationally expensive. In the MCD model, thecalculation of the multi-component diffusion coefficients, Γ ij,involves calculation of a N × N tensor, where N is the numberof species in the system. This tensor has to be computed at eachiteration for each control volume. In addition, as discussed ear-

lier, all the species transport equations are tightly coupled andhence have to be solved simultaneously, thereby involving theuse of advanced computational algorithms. On account of thisadded complexity, CFD calculations using the MCD model areexpected to take more CPU time than those where the DA modelor the SN is used.

The total CPU time and CPU time per iteration for the CFDcalculation with CH4 fuel is tabulated in Table 2 for Φ = 1.0 andat different wall temperatures and different Re. Other operatingconditions depicted similar behavior, and CPU times for theseare not shown here for the sake of brevity. A few importantobservations can be made from the table. First, the use of theMCD model always takes more time than both the DA and theSN models. Over the whole range of parameters that have beenconsidered in this study (not all of which are shown in Table 2),the use of SN model leads to speed-up factors between 2 and16, where the speed-up factor is defined as the ratio of CPUtimes between MCD and the approximate model in question.The DA model shows a maximum speed-up factor of 1.5. Thesecond important observation is that the speed-up factors are afunction of Re. Highest computational gains from the use of theSN model are manifested at low Re where a speed-up factor of16 has been achieved. The speed-up factor for the DA modelshows increase with increase in Re while for the SN model itdecreases with increase in Re. Since the DA model inherentlyviolates mass conservation, its overall convergence propertiesbecome better as the problem moves from diffusion-dominated(low Re or Peclet number) to advection-dominated (high Re)scenarios. Overall, the SN model leads to the best computationalgains in all cases. As discussed earlier, since the accuracy of theDA model and the SN model are remarkably close, significantcomputational gains can be achieved by using the SN modelwithout sacrificing on accuracy.

It is informative to analyze the CPU times further to under-stand how much of the CPU is actually spent on calculation of thediffusion coefficients and how much is a result of added numer-ical stiffness due to coupling of the species equations. The CPUtime per iteration elucidates the computational effort associatedwith the use of the diffusion model per se. The computational

Table 2CPU time, CPU time per iteration, and associated speed-up factors for CFD calculations with stoichiometric CH4–air inlet mixture

Re = 10 Re = 100 Re = 1000

Time (s) Speed-up Time (s) Speed-up Time (s) Speed-up

CPU timeMCD 11513.1900 1.0000 1155.8750 1.0000 301.0469 1.0000DA 10599.2900 1.0862 845.6250 1.3669 198.4219 1.5172SN 2853.8800 4.0342 454.8910 2.5410 165.2812 1.8214

Re = 10 Re = 100 Re = 1000

Time/Iter (s) Speed-up Time/Iter (s) Speed-up Time/Iter (s) Speed-up

CPU time per iterationMCD 0.6636 1.0000 0.6795 1.0000 0.7001 1.0000DA 0.4144 1.6013 0.4546 1.4947 0.4541 1.5419SN 0.3439 1.9296 0.3852 1.7642 0.3880 1.8045

Results are shown for wall temperature Tw = 1300 K and for different Re. Computations were performed on a 3 GHz Intel Pentium IV processor.



effort required for other operations, such as calculation of flowfield and other properties is nominally the same no matter whichdiffusion model is used. Since all the computations have beenperformed with the same solver, the difference in CPU time periteration is primarily because of the difference in the amount oftime required for calculation of diffusion coefficients. The CPUtime per iteration decreases as the level of complexity of the dif-fusion model decreases: the CPU time per iteration is largest forMCD while it is smallest for the SN model. Since the time takenfor calculation of diffusion coefficient is insignificant for theSN model, the difference between the time taken by the MCDand the SN model is approximately the time taken for diffusioncalculations in MCD. Based on the speed-up factors for the SNmodel from Table 2, the highest value being 1.93 and lowestbeing 1.76, it can be concluded that around 44–49% of totalCPU time is consumed in calculation of the diffusion coeffi-cient tensor when the MCD model is used. Similar observationshave been reported earlier by other authors (Salmi & Waerna,1991) for packed bed reactor modeling. From the same table,it can also be concluded that about 15% of CPU time is spentin calculation of the mixture diffusion coefficients for the DAmodel.

4. Summary and conclusions

Accurate prediction of diffusive transport is a prerequisite foraccurate prediction of the operation of devices in which hetero-geneous reactions occur. In this study, three different diffusionmodels were examined from the standpoint of both accuracyand efficiency. Two of these models, namely the dilute approx-imation (DA) model and the Schmidt number (SN) model,are approximate models, and were compared against a rigor-ous multi-component diffusion (MCD) model derived from theStefan–Maxwell equation. Both hydrogen–air and methane–aircombustion on platinum were studied. Based on the results, thefollowing conclusions can be drawn:

• The DA model and the SN model are accurate within 2% of therigorous MCD model for all the cases that were studied. Boththese diffusion models can, therefore, be used with confidencein catalytic combustion applications.

• There is discernible difference in the predicted species dis-tributions obtained using the MCD and the DA model whenH2 combustion is considered. For CH4 combustion, this dif-ference is not discernable. Therefore, the accuracy of the DAmodel has some dependency on the fuel in question, althoughthe differences in predicted results for H2 are not large enoughto be of concern.

• The predicted conversion fractions using the two approxi-mate models are much more accurate than predictions ofthe detailed species distributions. Thus, from an engineer-ing standpoint, both the DA model and the SN model areacceptable in terms of accuracy.

• In comparison to the MCD model, the DA model is approxi-mately twice as computationally efficient, while the SN modelis 2–16 times more efficient. Since the DA model and the SNmodel produce almost identical results, this implies that the

SN model is the most attractive choice for modeling diffusionin catalytic combustion applications.

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