**PROBLEM DESCRIPTION AND RESOLUTION**

A methane (CH4) / air flame in a closed vessel is investigated. The studie is based on FLUENT 8^{th}Tutorial (FLUENT convection model).The generalized finite-rate chemistry model is used to analyze the combustion.

The combustion is modeled using a global one-step reaction mechanism, assuming complete conversion of the fuel to CO2 and H2O. The reaction equation is :CH4 + 2O2 ---> CO2 +2H2O

This reaction is defined in terms of stoichiometric coefficients, formation enthalpies, and parameters that control the reaction rate. The reaction rate will be determined assuming that turbulent mixing is the rate-limiting process, with the turbulence-chemistry interaction modeled using the eddy-dissipation model.

**I - FLUENT CONVECTION MODEL
(STEADY)**

A cylindrical combustor burning methane (CH4) in air is studied using the finite-rate chemistry model in FLUENT.

The flame considered is a turbulent diffusion flame. A small nozzle in the center of the combustor introduces methane at 80 m/s. Ambient air enters the combustor coaxially at 0.5 m/s. The overall equivalence ratio is approximately 0.76 (about 28% excess air). The high-speed methane jet initially expands with little interface from the outer wall, and entrains and mixes with the low-speed air. The Reynolds number based on the jet diameter is approximately 28000.

Grid used by FLUENT to study air/methane combustion

This model was found to be helpful to read about chemical reaction modeling in the user guide. It was used firsteval to get experience whith chemical reaction and combustion modeling. And in the second hande to test influence of diffrent parameters used in this kind of modelisation (initial values, under-relaxation factors ...etc).The initial calculation was performed assuming that all properties except density are constant. Using constant transport properties (viscosity, thermal conductivity, and mass diffusion coefficients) is acceptebal here because the flow is fully turbulent. The molecular transport properties will play a minor role compared to turbulent transport.

Temperature values : Constant C

_{p}

The assumption of constante specific heat , in contrast, has a strong effect on the combustion solution, and was changed in the second calculation to a non-constant value. In fact the strong temperature and composition dependence of the specific heat have a significant impact on the predicted flame temperature. So the solution is recomputed using the temperature-varying property information in the FLUENT database.

Temperature values : Variable C

_{p}

Remarques

- The flame temperatures computed by this model are found to be the same of those calculated theoretically in the BEI.

- Changing the initial conditions ( temperature, massic fraction of methane and oxygene ) seems to have not affect on the solution results and iteration number needed to reache the convergence. Whereas under-relaxation factors increase considerably iterations number and stabilize numerical scheme.

Initial temperature (K) Convergence Nbr Iterations Under-relaxation factors

2000 yes 300 0.9

300 yes 300 0.9

200 yes 300 0.9

20 yes 300 0.9

0 yes 300 0.9

300 yes 300 1

300 yes 650 0.5

300 yes 3700 0.1

In order to compute flame methane/air combustion temperature, a two dimension closed vessel domaine is meshed. A cartesien structured PREBFC mesh was created.Because of idependance between domaine dimenstions and flame caracteristics, only a 7 by 1 millimetre region was meshed (70x10). Four adiabatic walls was defined as limit conditions of the domain containing premixed gase to be burn.

Structured mesh for modelig 2D transie combustion

As was been done in FLUENT convection model, constant and variable specific heat were used to compute flame temperature in the closed vessel model. In this part unsteady solver was selected to invetigate transie combustion regime. This choice have the advantage to increase combustion control. Nevertheless, it has complicated the the study and more time was been spend to compute teh solution.Time step size was fixed to 0.0001 second and iterations was done untel convergence before going on the new time step. But unfortunately, computing was for a long time divergent. In fact, the temperature increased very rapidely and no convergence was been obtaned.

Different solutions was been tested. The first ones dealed whith initial conditions (temperature, initial mass fraction of CH

_{4 }and O_{2}) but no significant results. Many other testes was been done to reache converge by changing the solver, changing limit conditions and so on, but no solution was found.Changing under-relaxation factor (URF) on FLUENT convection model gaves the an idea on the value of this factor capable performing scheme convergence.

So with 600 K initial temperature and a 0.4 under-relaxation factor, the first step time converged in 70 iterations. Whereas the second step time step diverged rapidely. Thas whay, this factor was put to 0.1 in the second step time. When divergence appeares, the relaxation factor was put to 0.04 but it was very late and divergence appears again.

Coputing divergence : Temperature reaches code limites (5000K)

Under-relaxation factor : 0.4 then 0.1 then 0.04 (divergence)

With 2000 K initial temperature (accelerate convergence) and a 0.4 URF, the first step time converged in 70 iterations. In the second step time URF was changed to 0.01. Convergence is reached after 475 iterations.

Convergence Temperature with varible C

_{p}

Convergence number of Iterations with varible C

_{p}

Convergence velocity with varible C

_{p}The Figure of convergence velocity shows onde propagation after combustion inside the closed vessel. In fact in time=2.0000e-4 seconds gaseous are moving from the center to the vessel sides and from the sides to its center due to combustion.

Convergence Pressur with varible C

_{p}

The same URF (0.4) and initialtemperature (2000K) were applied to constant specific heat model. The tempeature diverged after a small number of iterations, as in first cases studied. URF have to be decreased in order to more stabilize the numerical sheme in this circumstance.

Divergence Temperature with constant C

_{p}

Divergence number of Iterations with constant C

_{p}

Different Objectives was reached in this study :

- Familiarization with industrial code (FLUENT)

- Familiarization with combustion and transie numerical studies

- Particpating in the BEI engine design