Contribution: Bilger Mixture Fraction Function Object

@andy @Sergio

Hi, If you can encapsulate the code into a function object and attached a small tutorial case with for testing, we can have a look and introduce your contribution. We'd need how you'd like to be referenced for your contribution. .ie. name, institute, reference, etc. Thanks

Thanks for the instructions.

Attached you find in contribution_zirwes.zip

a git patch file for the contribution (contribution_zirwes/feature-BilgerMixtureFraction.patch)
a stand-alone version of the function object and a test case

The only difference between the code in the patch file and the standalone version is that the patch modifies the existing Make/files and options files, while the stand-alone version provides its own Make directory. Additionally, the patch also changes an existing tutorial case to showcase this function object.

Compile and Run the Code

To compile the code, go to standalone/BilgerMixtureFraction and compile with wmake.
To run the code, go to testcase/counterFlowFlame2D and run the case with blockMesh && reactingFoam

This testcase is an exact copy of $FOAM_TUTORIALS/combustion/reactingFoam/laminar/counterFlowFlame2D, the only difference being the functions entry in system/controlDict

There will be an additional field written called f_Bilger. The test case also contains a picture (results.png) which shows the timestep t=0.5s. On the right of the picture, the mixture fraction field is zero for the oxidizer (on the right, inlet for air) and 1 for the fuel (on the left, inlet for fuel). The stoichiometric mixture fraction for methane/air is at f=0.055. The white line shows the iso-line of f=0.055, which marks the position of the flame (same position as highest temperature on the left).

What the Code does

The function object computes the (Bilger) mixture fraction from:

f_{\mathrm{Bilger}} \equiv \frac{\beta-\beta_{\mathrm{ox}}} {\beta_{\mathrm{fuel}}-\beta_{\mathrm{ox}}}

with

\beta = 2\frac{Y_C}{W_C}+2\frac{Y_S}{W_S}+\frac{1}{2}\frac{Y_H}{W_H} -\frac{Y_O}{W_O}

where

Y_e

is the elemental mass fraction of element

e

and

W_e

its atomic weight. The subscribt

\mathrm{ox}

denotes the oxidizer composition,

\mathrm{fuel}

the fuel composition and the one without subscript is the value of

\beta

for the current mixture in the cell. The code considers carbon, hydrogen, sulfur and oxygen. A reference to the original literature by Bilger is included in the header file.

The elemental mass fraction is defined as

Y_e\equiv\sum_k \frac{a_{e,k}W_e}{W_k}Y_k

, where the subscript

k

denotes the chemical species and

a_{e,k}

is the number of atoms of element

e

in species

k

. Substituting this in the expression for

\beta

, it can be written as:

\beta = 2\sum_k\frac{a_{C,k}Y_k}{W_k} + 2\sum_k\frac{a_{S,k}Y_k}{W_k} + \frac{1}{2}\sum_k\frac{a_{H,k}Y_k}{W_k}-\sum_k\frac{a_{O,k}Y_k}{W_k}

which is what is implemented in the code. The user has to provide a composition for the fuel and oxidizer (which is usually given by the inlets for fuel and oxidizer, but there are some cases where they might be defined differently) in the following way:

mixtureFraction
{
    type BilgerMixtureFraction;

    fuel
    {
        composition moleFractions; // optional entry
        CH4 1;
    }

    oxidiser
    {
        composition massFractions; // optional entry
        O2 0.23;
        N2 0.77;
    }

    //result "my_f_Bilger"; // optional alternative name for the field
    //phase gas;            // optional entry to support multi-phase solvers
    //region fluid;         // optional entry to support multi-region solvers (as part of OpenFOAM's function object implementation, not my code)
}

The sub dicts fuel and oxidizer contain the composition of fuel and oxidizer. They can contain an arbitrary number of species. The optional entry composition determines if the specified values are interpreted as either mass or mole fractions. The input is automatically normalized to 1, so O2 2.3; N2 7.7; would be a valid entry too.

If you want to use this function object with a multi-phase solver, include the phase keyword. If you want to use it with a multi-region solver, include the region keyword. If you want to change the name of the field, include the result keyword. Of course, the information about the elemental composition for each species has to be provided in the case setup (for some tutorial cases, depending on the combustion model, this information might not be present but can easily be added by the user. No changes necessary for the counterflow tutorial case).

All details given here are also included in the description at the top of the header file.

Some Remarks about the Code

I put the function object in $FOAM_SOURCE/thermophysicalModels/reactionThermo/functionObjects, as I feel this makes the most sense.
I added the function object to the counterflow tutorial case of reactingFoam to showcase its functionality, because I think this is the most likely place users would search for this function object ($FOAM_TUTORIALS/combustion/reactingFoam/counterFlowFlame2D).
Following the naming scheme in OpenFOAM (e.g. ft for the turbulent mixture fraction) and to avoid clashes with other types of existing mixture fractions (singleMixtureFraction, SootMixtureFraction), I named the function object BilgerMixtureFraction and the resulting field f_Bilger.
The getElementInformation member function is quite verbose, but I think there might be no other way to support all possible types of reacting mixtures.
Both "oxidizer" and "oxidiser" spellings are supported to avoid confusion.
I tried to include as many "sanity checks" as possible to catch invalid entries.

Attribution

Regarding the reference of the contribution: I looked at the code to see what other people did. As written in the patch file in contribution_zirwes/feature-BilgerMixtureFraction.patch, I added my name to

the headers of the two source files of the function object
the CONTRIBUTORS.md file in the main directory

My full reference is:

Thorsten Zirwes, Karlsruhe Institute of Technology (Steinbuch Centre for Computing & Engler-Bunte-Institute)

I used the code in the following publication:

T. Zirwes, F. Zhang, P. Habisreuther, M. Hansinger, H. Bockhorn, M. Pfitzner, and D. Trimis, “Quasi-DNS dataset of a piloted flame with inhomogeneous inlet conditions,” Flow, Turbulence and Combustion, vol. 104, pp. 997–1027, 2020 (DOI:10.1007/s10494-019-00081-5)

Let me know if you want me to do any changes to the code or if I should provide more information.

Kind regards,

Thorsten

contribution_zirwes.zip

Hi Thorsten, Thanks for the very detailed examples and documentation. It is a very nice contribution. @kuti Please proceed with this contribution and let me know if you have any questions

assigned to @kuti

changed milestone to %v2012

added contribution label

Dear @g3,

Will be doing few style and consistency changes, if you don't mind. You can follow the branch (but just started to checks - and Friday night :) ), and please feel free to suggest anything you want along the way: feature-Bilger-mixture-fraction-fo

Few questions (definitely more will come), if possible:

Is there any minimal verification example, preferably a comparison to an analytical solution, a high-fidelity simulation or an experiment? (e.g. f=0.05 plot is an analytical solution?)
Does \beta have a descriptive name? its units?

Many thanks.

Great to see the quick progress :)

Sure, feel free to modify the code.

I think
\beta
is usually called "coupling function". I double-checked it in the textbook by Peters (ISBN-10 : 0521660823). Since I cannot link the contents of the book here, you can see this lecture (slide 67)
the unit of
\beta
is kmol / kg

There are plenty analytical solutions:

Example 1: stoichiometric mixture fraction of methane/air

For methane, we have

\mathrm{CH}_4 + 2\mathrm{O}_2 \rightarrow \mathrm{CO}_2 + 2\mathrm{H}_2\mathrm{O}

. Or in other words: the molar ratio of methane to oxygen is 1:2 for stoichiometric combustion. Let's assume the air composition in terms of mass fractions is

Y_{\mathrm{O}_2}=0.23,\ Y_{\mathrm{N}_2}=0.77

. We can convert the mass fractions

Y

to mole fractions

X

with

X_k=Y_k\frac{W}{W_k},\ W=\left(\sum_k \frac{Y_k}{W_k}\right)^{-1}

where

W

is the mean molecular weight of the mixture. So we can get the following equations:

X_{\mathrm{O}_2}=2X_{\mathrm{CH}_4} \rightarrow Y_{\mathrm{O}_2}\frac{W}{W_{\mathrm{O}_2}}=2Y_{\mathrm{CH}_4}\frac{W}{W_{\mathrm{CH}_4}}\\ \frac{Y_{\mathrm{O}_2}}{Y_{\mathrm{N}_2}} = \frac{0.23}{0.77}\\ Y_{\mathrm{O}_2}+Y_{\mathrm{CH}_4}+Y_{\mathrm{N}_2}=1

The first equation is the 2:1 ratio of methane and air in a stoichiometric mixture, the second equation is the ratio of oxygen and N2 in the air, and the third equation is that the sum of all mass fractions have to be unity. Solving the equations yields:

Y_{\mathrm{CH}_4} = \left(1+2\frac{W_{\mathrm{O}_2}}{W_{\mathrm{CH}_4}}\left(1+\frac{0.77}{0.23}\right)\right)^{-1}

Using the atomic masses as defined here, the result is:

Y_{\mathrm{CH}_4} = 0.0545137147773594210...

Since methane is the only fuel, the mass fraction of methane in the mixture is also the mass fraction of the fuel in the mixture, and therefore

Y_{\mathrm{CH}_4}=f_{\mathrm{Bilger}}

.

Example 2: Specifying a mixture fraction

Let's say we choose an arbitrary composition for the fuel and mixture in mass fractions (e.g. for a case where some exhaust gas is directed back into an engine):

Y_{k,\mathrm{fuel}} = \begin{cases} \mathrm{CH}_4=0.6\\ \mathrm{O}_2=0.02\\ \mathrm{N}_2=0.2\\ \mathrm{CO}=0.15\\ \mathrm{CO}_2=0.03\\ \end{cases} ,\ Y_{k,\mathrm{oxidiser}} = \begin{cases} \mathrm{CH}_4=0.01\\ \mathrm{O}_2=0.2\\ \mathrm{N}_2=0.7\\ \mathrm{CO}=0.05\\ \mathrm{CO}_2=0.04\\ \end{cases}

Let's say we want to create a mixture with mixture fraction

f=0.1

. This means, we have to mix 1 kg of the fuel with 9 kg of the oxidizer. So the resulting mixture has the composition:

Y_{k,\mathrm{mixture}} = \frac{1}{10}\begin{cases} \mathrm{CH}_4=1\times 0.6+9\times 0.01 = 0.69\\ \mathrm{O}_2=1\times0.02 +9\times 0.2 = 1.82\\ \mathrm{N}_2=1\times 0.2+9\times 0.7 = 6.5\\ \mathrm{CO}=1\times 0.15 + 9\times 0.05 = 0.6\\ \mathrm{CO}_2=1\times 0.03+9\times 0.04 = 0.39\\ \end{cases}

The

\frac{1}{10}

in front is to normalize all mass fractions to add up to unity, because

0.69+1.82+6.5+0.6+0.39=10

. If the function object is used for the mixture above, it should figure out that the mixture was created from 1 part fuel and 9 parts oxidizer and output

f_{\mathrm{Bilger}}=0.1

Example 3: pure air or pure fuel

The trivial analytic solutions are if the mixture composition is equal to either the fuel or oxidizer composition. In this case,

f_{\mathrm{Bilger}}=1

for pure fuel and

f_{\mathrm{Bilger}}=0

for pure oxidizer.

Test cases

I think the quickest way to validate the results is to use chemFoam and the tutorial case from $FOAM_TUTORIALS/combustion/chemFoam/gri.

Example 1:

A testcase for example 1 is here: example1.tar.gz

Just run the case with chemFoam. The relevant entries are in system/controlDict

    mixtureFraction
    {
        type BilgerMixtureFraction;

        fuel
        {
            CH4 1; // fuel is pure methane
        }

        oxidiser // as specified in example 1 above
        {
            O2 0.23;
            N2 0.77;
        }
    }

and in constant/initialProperties:

fractionBasis   mole;

fractions
{

    CH4             1;
    O2              2; // stoichiometric mixture; CH4:O2 = 1:2
    N2              #eval "2*(0.77/0.23)*(15.99940/14.00670)";
}

The relevant output is in postProcessing/fieldMinMax1/0/fieldMinMax.dat. For example:

# Time    CO2         O2         f_Bilger         CH4
1e-07     1.517e-18   2.175e-01  5.451e-2         5.451e-2
...
0.01      8.268e-2    2.293e-2   5.451e-2         1.091e-2

OpenFOAM result vs. analytical solution for f_Bilger:

0.05451371477735941 (OpenFOAM)
0.054513714777359421 (analytical)

Also, the value of f_Bilger is the same for each time step, even though the mixture burns (CO2 goes up, O2 and CH4 go down), because the elemental composition of the mixture stays constant (as it should).

Example 2:

Again, using the chemFoam case and setting fuel, oxidizer and mixture according to example 2: example2.tar.gz

    mixtureFraction
    {
        type BilgerMixtureFraction;
        fuel
        {
            CH4 0.6;
            O2 0.02;
            N2 0.2;
            CO 0.15;
            CO2 0.03;
        }
        oxidiser
        {
            CH4 0.01;
            O2 0.2;
            N2 0.7;
            CO 0.05;
            CO2 0.04;
        }
    }

and for the initial conditions:

fractionBasis   mass;

fractions
{
    CH4 0.69;
    O2  1.82;
    N2  6.5;
    CO  0.6;
    CO2 0.39;
}

The value of f_Bilger is:

0.1 (analytical)
0.09999999999999996 (OpenFOAM)

Example 3:

The trivial example cases are here: example_pure_air.tar.gz example_pure_fuel.tar.gz

In these cases, the initial conditions are either pure methane or pure air. Accordingly, the mixture fraction is 1 or 0, respectively.

Example 4: More complex case

If you want a more complex example, the counterFlowFlame2D case from the original contribution can be used. It has the special property that carbon only appears in the fuel (in form of CH4) but not in the oxidizer. Additionally, carbon only exists in CH4 and CO2. Therefore, the fuel can be detected by tracing the carbon atom by looking at CH4 and CO2 only (assuming constant mean molecular weight, which is not exactly true in this case. So technically it is not the analytical solution, but it is close enough):

f_{\mathrm{Bilger}}=f_{\mathrm{fuel}}\approx Y_{\mathrm{CH}_4} + \frac{W_{\mathrm{CH}_4}}{W_{\mathrm{CO}_2}}Y_{\mathrm{CO}_2}

In the counterFlowFlame2D case from my last post, add in system/controlDict

readFields
{
    type            readFields;
    libs            (fieldFunctionObjects);
    fields          (CO2 CH4);
}

Create a new dummy file:

cp 0.5/O2 0.5/fValidation

and set the alternative definition of the mixture fraction via setExprFields -withFunctionObjects -latestTime

expressions
(
    fValidation
    {
        field       fValidation;
        dimensions  [0 0 0 0 0 0 0];

        expression
        #
            // be careful: boundary conditions are not changed by this
            CH4 + (12.01115+4*1.00797)/(12.01115+2*15.99940)*CO2
        #};
    }
);

The full case is here: counterFlowFlame2D_validation.tar.gz

The result is shown below:

Closing remarks:

Now that I looked at the chemFoam cases, maybe it makes sense to streamline the "composition" option. This means, replacing the "composition" option name with the name "fractionBasis" and the names of "moleFractions" and "massFractions" with "mass" and "mole". Then it is consistent with chemFoam.

example1.tar.gz

example2.tar.gz

example_pure_air.tar.gz

example_pure_fuel.tar.gz

counterFlowFlame2D_validation.tar.gz

Dear @g3,

Many thanks for all these high-quality details.

I have changed composition entry to fractionBasis, moleFractions to mole and massFractions to mass.

Remarks:

If you don't mind, I will remove oxidiZer support (It was definitely good idea, but need to chop it for the uniformity with the rest of the code and simplify things).
I have moved some of the constructor-level functions into the read function to make the objects runtime modifiable.

yeah, sounds good to me

Dear @g3,

Would you also consider to submit a short communication/technical letter to https://journal.openfoam.com/index.php/ofj ? just a suggestion.

Sure, good idea. I will prepare a report about the implementation, validation and some examples from my own work

mentioned in merge request !393 (merged)

Dear @g3 , your contribution has been merged.

If possible, please let us know if anything is problematic, and reopen the ticket.

Thank you very much for your contribution!

closed

mentioned in issue #1977 (closed)

Contribution: Bilger Mixture Fraction Function Object

Child items

Activity

Compile and Run the Code

What the Code does

Some Remarks about the Code

Attribution

Example 1: stoichiometric mixture fraction of methane/air

Example 2: Specifying a mixture fraction

Example 3: pure air or pure fuel

Test cases

Example 1:

Example 2:

Example 3:

Example 4: More complex case

Closing remarks:

Admin message

Contribution: Bilger Mixture Fraction Function Object

Child items

Linked items

Related merge requests

Activity

Compile and Run the Code

What the Code does

Some Remarks about the Code

Attribution

Example 1: stoichiometric mixture fraction of methane/air

Example 2: Specifying a mixture fraction

Example 3: pure air or pure fuel

Test cases

Example 1:

Example 2:

Example 3:

Example 4: More complex case

Closing remarks: