Comparative analysis of mixture approach andEulerian approach using Loschmidt tube

Chethan Mohan Kumar

4. April 2013

Zusammenfassung

The accuracy of using Maxwell-Stefan relations as intercomponentdrag term with the Eulerian approach has been previously established.The drawback of the whole approach, though with its commendableaccuracy is the time consumed in solving for individual velocities ofcomponents. The solver reactingFoam in OpenFOAM which basesitself on the mixture approach gives us possibilities of simulating thewhole flow scenario with lesser computation cost. The solver modi-fiedReactingFoam compiled by Valerio Novaresio is used whichbases itself on the mixture approach. The multicomponentEuler-Foam previously developed is used for the simulation of Eulerian ap-proach. A comparison of accuracy of both the methods are presentedhere.

1 IntroductionThe reactingFoam in OpenFOAM 2.1.1 has a multitude of librarieswhich solves for chemistry and is quite useful in solving the flow scenarioof exhaust gas treatment. The consideration of concentration gradient as adriving force and the modelling of the flux due to this concentration gradientputs the whole simulation closer to the actual flow process. The equationfor the conservation of species is:

∂(ρya)∂t

+∇ · (ρya~υ) +∇ · ~ja = Sya , (1)

where ya is the mass fraction of species, ρ is the fluid density, ~υ is themass averaged velocity of the fluid, ~ja is the mass diffusion-flux of speciesa relative to the mass average velocity ~υ. Sya is the sources or sinks due toreaction of a particular species.

The consideration of the term ~ja can be modelled according to the userrequirements. The solver reactingFoam is a model based on Lewis numberand the mass diffusion flux is defined as:

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~ja = −ρDam∇ya , (2)

where

ρDam = a

Lea. (3)

The table 1 shows the possibilities of modelling the diffusion-flux andthe Diffusion coefficient.

Model type Diffusive-flux (~ja) Diffusion Coefficient RemarkFick’s model −ρDam∇ya Dam = (1−xa)∑

β=α

(xβDαβ

)Fick’s mo-del(verydiluted mix-ture)

−ρDam∇ya Dam = Dan Binary diffusioncoefficient of spe-cies α and carrierspecies n

Schmidtnumber

−ρDam∇ya ρDam = µSca

Sca is the Schmidtnumber

Lewis num-ber

−ρDam∇ya ρDam = αLea

Lea is the Lewisnumber

Maxwell-Stefan

∇xa =∑nα=1

xαxβDαβ

(~jβρ −

~jαρ

),

~ja = −∑n−1β=1 ρDαβ∇yβ

Dαβ = [D] = [A]−1[B] Modelling of [A]and [B] can be re-ferred in appen-ded document

Bosanquet’smodel

−ρDαeff∇ya 1

Daeff= 1−γxa

Dameff+ 1Dka

eff Dam is same asFick’s model, γ isa constant and ge-nerally zero

Tabelle 1: Methods of modelling the diffusion-flux

A library compiled by Valerio Novaresio facilitates the selection ofall the models mentioned in table 1 with the modified solver based on re-actingFoam is used to compare the mixture approach and the Eulerianapproach.

2 Initial comparisonThe solver reactingFoam is based on the Lewis number for modellingdiffusive-flux as mentioned earlier. The solver modifiedReactingFoam isintegrated with the diffusion library and various models can be selected ac-cording to the requirement. An initial comparison is made by simulating theloschmidt tube using reactingFoam and modifiedReactingFoam for a

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Wall

Left tube Right tube

50 x 10-6 50 x 10-6

Abbildung 1: Simulation model of loschmidt tube

ternary diffusion case and the variation of volume fractions on the left partof the diffusion tube is observed. The case setup is shown in figure 1. Thesame comparison criteria is used in our whole discussion.

The inital compositions of the three gases Methane(CH4), Argon(Ar)and Hydrogen(H2) are listed in table 2.

Component Left tube Right tubeCH4 0.295 0.405H2 0.4 0.3Ar 0.305 0.295

Tabelle 2: Initial composition

The case was simulated using both the solvers. The variation of ArgonMethane and Hydrogen can be seen in the figure 2.

It can be seen that, there is a slight deviation in the calculated resultsof both the solvers. This is attributed to the solution procedure in mo-difiedReactingFoam and other parameters that may affect the solutionunknown at this stage. A constant diffusion coefficient was used in themodi-fiedReactingFoam case whereas in the reactingFoam case, thermophy-sical properties are used to model the diffusion coefficiecnt. But the patternof variation of phase fractions is similar and quite accurate.

We can thus conclude that, the solvers modifiedReactingFoam andreactingFoam are similar. Now, we proceed further to the simulation ofthe ternary mixture using Maxwell-Stefan model for diffusive-flux.

3 Maxwell-Stefan modelThe solvermodifiedReactingFoam is used for simulating ternary diffusionbased on the mixture approach. The Maxwell-Stefan terms are modelled in

3

0.298

0.3

0.302

0.304

0.306

0.308

0.31

0 2e-07 4e-07 6e-07 8e-07 1e-06 1.2e-06 1.4e-06 1.6e-06

"Ar(modifiedReactingFoam).dat" using 1:3"Ar(reactingFoam).dat" using 1:3

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0 2e-07 4e-07 6e-07 8e-07 1e-06 1.2e-06 1.4e-06 1.6e-06

"CH4(modifiedReactingFoam).dat" using 1:3"CH4(reactingFoam).dat" using 1:3

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0 2e-07 4e-07 6e-07 8e-07 1e-06 1.2e-06 1.4e-06 1.6e-06

"H2(modifiedReactingFoam).dat" using 1:3"H2(reactingFoam).dat" using 1:3

Abbildung 2: Comparison of reactingFoam andmodifiedReactingFoam(Model based on Lewis number used for diffusive-flux)

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the species continuity equation which is run time slectable by the compiledlibrary mentioned earlier. The comparison is then drawn with the analyticalresults and the results obtained for the loschmidt tube case using multi-componentEulerFoam . The variations of the three gases with time is asshown in the following figure 3.

0.3

0.302

0.304

0.306

0.308

0.31

0.312

0.314

0.316

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 0.00016

"calculatedAr.dat" using 1:3"ArReacting.dat" using 1:3"analytical.dat" using 1:3

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 0.00016

"calculatedCH4.dat" using 1:3"CH4Reacting.dat" using 1:3

"analytical.dat" using 1:2

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 0.00016

"calculatedH2.dat" using 1:3"H2Reacting.dat" using 1:3

"analytical.dat" using 1:4

Abbildung 3: Comparison of mixture approach (modifiedReactingFoam)and Eulerian approach (multicomponentEulerFoam)

It is observed that, modifiedReactingFoam captures the behaviourof reverse diffusion, Diffusion barrier and normal diffusion in case of Argonwhich signifies the modelling of Maxwell-Stefan relations in the solver. Theslight deviation in results are still unknown at this stage and may be for thereasons mentioned previously. It is important to note that modifiedReac-tingFoam with its run time selectable library for diffusive-flux modellingcan be used and predicts the variation of fluxes quite closely as seen in 3.

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4 ConclusionAn intial comparison of reactingFoam and modifiedReactingFoam wasdone to check the similarity of the two solvers. The behaviour is accuratelycaptured in modifiedReactingFoam with a slight deviation. We may thusinfer that, by and large, both the solvers are quite similar. Then the valida-tion of Maxwell-Stefan relations implemented by the solver is done by usingpreviously simulated results using multicomponentEulerFoam and withanalytical solutions. There is a slight deviation in the results calculated butall the processes occuring in a ternary diffusion are accurately captured.

A more exhaustive study, ideally a line by line analysis of the code inmodifiedReactingFoam must be done and the effect of influencing para-meters must be analysed. This may result in a more accurate simulation ofthe process. By and large, the initial solution is quite close to the actualvalues, and gives a right direction to the simulation process.

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