Numerical Comparison

The solution of LBV using numerical models depends on the properties of reactants and products which can be determined using equilibrium codes or NASA polynomials. The author utilized CANTERA software [169] for that purpose because of its simplicity and integrity with MATLAB. Equilibrium codes require a chemical kinetic mechanism that contains the thermal and

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transport properties as well as chemical reactions and reaction rate coefficients. Methane kinetic mechanisms developed and validated over the years by several studies [176-178] because of its extensive use in the scientific community. The most known methane kinetic mechanisms are GRI-Mech 3.0 [34] and ARAMCO-GRI-Mech 1.3 [56]. Table 3.2 provides the number of species and reactions considered in GRI-Mech 3.0 and ARAMCO-Mech 1.3; also the validated pressure and temperature range are tabulated. Recently, the Combustion Chemistry Centre at NUI Galway introduced an updated version - ARAMCO-Mech 2.0 [179], however, the number of species and reactions increased dramatically which resulted in a significant increase in the computational time.

Because of that reason, the author preferred to use ARAMCO-Mech 1.3 in this work.

Table 3.2. Specification of methane chemical kinetic mechanisms used in the dissertation.

GRI 3.0 ARAMCO 1.3

Number of chemical reactions 325 1542

Ignition delay pressure validation range [atm] 0.013-10 1.2-260 Ignition delay temperature validation range [K] 1000-2500 1200-2500

All LBV results shown before the Numerical Comparison section was made with the GRI-Mech 3.0 to reduce the computational cost. Figure 3.9 compares the useful range of LBV obtained using the Multilayer model and two chemical kinetic mechanisms. It is evident that the LBV values determined by ARAMCO-Mech 1.3 are slightly lower than those found with GRI-Mech 3.0.

However, the difference is insignificant compared to the uncertainties of spherical combustion chamber experiments. In this case, if the value extrapolated linearly to the initial temperature, the LBV of ARAMCO-Mech 1.3 and Mech 3.0 mechanisms are 36.8 cm/s and 36.95 cm/s. GRI-Mech 3.0 overestimates the value by only 0.4%, an order of magnitude smaller than the

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experimental uncertainty. Throughout the dissertation several mechanisms are utilized, the used mechanism will be specified in each section to eliminate any confusion.

Figure 3.9. Laminar burning velocity comparison based on two chemical kinetic mechanisms for stoichiometric CH4/air mixture at an initial temperature 300 [K] and initial pressure of 1 [atm].

In the constant volume method, the LBV fluctuates at the beginning of each experiment until an evident pressure change is recorded (see stage 1 in Figure 3.8). The fluctuation disappears approximately after 5 ms. Hence, an extrapolation method is needed to estimate the LBV at the initial condition. The linear extrapolation was validated by using the premixed flame model from CHEMKIN PRO [33] using the ARAMCO-Mech 1.3 [56] mechanism. The initial conditions of each data point are calculated based on the isentropic compression process before solving the premixed flame code. Note that the Soret and multicomponent mass diffusions are considered in the code to provide accurate LBV values. Figure 3.10 illustrates the numerical prediction results versus the unburned gas temperature. It is apparent that the LBV increases linearly with the

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unburned gas temperature in the tested range.The least square fit matches the numerical values, and only residuals of 0.0024 was found which might be due to the convergence criterion in the premixed flame model. To conclude, it is valid to extrapolate the LBV trend linearly to the initial condition in order to estimate the LBV value.

Figure 3.10. Validation of Linear extrapolation by using ARAMCO-Mech 1.3 [56] for stoichiometric CH4/air mixture at an initial temperature 300 [K] and initial pressure of 1 [atm].

The CHEMKIN PRO [33] solution was extended to study flame properties along one isentrope because of their importance in flame instability studies. The range of data points increased from 1-10 atm to provide a general trend for an experiment that starts with atmospheric pressure and room temperature. Three types of flame instabilities arise in a confined spherically expanding flame, body forces, hydrodynamic, and diffusional-thermal instabilities [58, 173, 180, 181]. The buoyancy effect can be neglected for most hydrocarbon fuels because LBV is higher than 15 cm/s.

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However, hydrodynamic and diffusional-thermal instabilities distort flame surface area by developing cracks and cells.

Hydrodynamic instability occurs if the effect of density variation across the flame (density jump) is higher than the stabilization factor. The density jump (𝜎) is defined as the density ratio between burned to unburned gases as shown in equation (3.42). This ratio increases along one isentrope and coincides with the increase of density difference between burned and unburned gases which in turn promotes flame instability. Conversely, fast expanding flames prevents cells formation if the rate of expansion is higher than the rate of cells formation. Namely, the flame positive curvature stabilizes the expanding spherical flame and reduces cells development [180].

Since the flame curvature is directly related to the flame thickness, thicker flame increases flame stabilization. However, cell development is expected as combustion proceeds due to the reduction of flame thickness along one isentrope. temperature and the temperature of unburned mixture respectively. A numerical derivative was used for the flame temperature to provide the maximum value after solving the premixed flame code.

As the name implies, diffusional-thermal instability is caused by the difference between thermal and mass diffusivities. A rigorous discussion has been made in the literature [58, 138-140], they found that the flame will be unstable if the mass diffusion is faster than the thermal diffusion.

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Lewis number (Le), defined as the ratio of thermal to mass diffusivity [58, 138]. is usually used to examine the mixture tendency toward diffusional-thermal instability. Since several binary diffusional coefficients exist for a mixture of gases, Lewis number was estimated for a mixture using the thermal diffusivity of the mixture relative to the mass diffusivity of the deficient reactant toward the diluent gas [58]. The diffusional-thermal instability delivers cracks or cells from the initial part of combustion process. Therefore, if no instability behavior was noticed from the beginning (flame visualization), the diffusional-thermal instability can be neglected for the tested mixture.

The variation of flame properties (density jump, flame thickness, and LBV) are shown in Figure 3.11 for CH4/air mixture. The author decided to plot the flame properties with respect to the pressure along one isentrope to allow for direct comparison with the LBV trend obtained earlier as a function of unburned gas temperature. Keep in mind that the range of data points was increased dramatically from 1-2 atm in Figure 3.10 to 1-10 atm in Figure 3.11 because the aim was to evaluate variation of properties along the entire combustion event. One notices that all obtained parameters vary nonlinearly with pressure. Also, it is evident that the density jump that reflects the difference between reactant and product densities increases along one isentrope. The density jump increased from 0.136 to 0.23 by increasing the equilibrium condition from 1-10 atm. Similarly, the density difference increased from 0.97 kg/m³ to 4.6 kg/m³ at those conditions.

The flame thickness was calculated using equation (3.43) after solving the premixed flame model. The domain was kept constant as 0.6 cm with 300 grid points, and adaptive grid curvature and gradient of 0.5 was chosen. Unlike the density jump, it was found that the flame thickness decreases alone one isentrope. The flame thickness of a stoichiometric CH4/air mixture was found to be 0.375 mm at atmospheric pressure and 0.08 mm at 10 atm. As stated earlier, the stabilization

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effect decreases as the flame gets thinner in expanding spherical propagation flame which in turn leads to hydrodynamic flame instability. This is the most important factor for hydrodynamic instabilities in constant volume experiments. Therefore, a stable flame in a confined volume develops flame instabilities as combustion proceeds (an example is shown in Figure 3.2). Care must be taken to eliminate such data points from the LBV fit to ensure reliable values.

Figure 3.11. Properties of flame along one isentrope obtained using CHEMKIN PRO [33] with the ARAMCO-Mech 1.3 [56] mechanism.

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