Laminar flames equations

Using this graph and the relationship it contains, one can now address the question of whether and under what conditions a laminar flame can exist in a turbulent flow. As before, if allowance is made for flame front curvature effects, a laminar flame can be considered stable to a disturbance of sufficiently short wavelength however, intense shear can lead to extinction. From solutions of the laminar flame equations in an imposed shear flow, Klimov [50] and Williams [51] showed that a conventional propagating flame may exist... 

Si is the laminar flame velocity, the function Z(co) is the heat response function Equation 5.1.16, whose real part is plotted in Figure 5.1.10. The function f(r, giJ is a dimensionless acoustic structure factor that depends only on the resonant frequency, a , the relative position, r, of the flame, and the density ratio Pb/Po-... 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

Then it became apparent that certain physical principles could be used to simplify the complete equations so they could be solved relatively easily. Such a simplification was first carried out by von Karman and Penner [9], Their approach was considered one of the more significant advances in laminar flame propagation, but it could not have been developed and verified if it were not for the extensive work of Hirschfelder and his collaborators. The major simplification that von Karman and Penner introduced is the fact that the eigenvalue solution of the equations is the same for all ignition temperatures, whether it be near T or not. More recently, asymptotic analyses have been developed that provide formulas with greater accuracy and further clarification of the wave structure. These developments are described in detail in three books [10-12],... 

As implied in the previous section, the Russian investigators Zeldovich, Frank-Kamenetskii, and Semenov derived an expression for the laminar flame speed by an important extension of the very simplified Mallard-Le Chatelier approach. Their basic equation included diffusion of species as well as heat. Since their initial insight was that flame propagation was fundamentally a thermal mechanism, they were not concerned with the diffusion of radicals and its effect on the reaction rate. They were concerned with the energy transported by the diffusion of species. 

To determine the laminar flame speed and flame structure, it is now possible to solve by computational techniques the steady-state comprehensive mass, species, and energy conservation equations with a complete reaction mechanism for the fuel-oxidizer system which specifies the heat release. The numerical... 

The Mallard-Le Chatelier development for the laminar flame speed permits one to determine the general trends with pressure and temperature. When an overall rate expression is used to approximate real hydrocarbon oxidation kinetics experimental results, the activation energy of the overall process is found to be quite high—of the order of 160kJ/mol. Thus, the exponential in the flame speed equation is quite sensitive to variations in the flame temperature. This sensitivity is the dominant temperature effect on flame speed. There is also, of course, an effect of temperature on the diffusivity generally, the dif-fusivity is considered to vary with the temperature to the 1.75 power. 

FlameMaster v3.3 A C+ + Computer Program for OD Combustion and ID Laminar Flame Calculations. FlameMaster was developed by H. Pitsch. The code includes homogeneous reactor or plug flow reactors, steady counter-flow diffusion flames with potential flow or plug flow boundary conditions, freely propagating premixed flames, and the steady and unsteady flamelet equations. More information can be obtained from http //www.stanford.edu/group/pitsch/Downloads.htm. 

This section concentrates on laminar premixed flames, which serve to illustrate many attributes of steady-state one-dimensional reacting systems. The governing equations themselves can be written directly from the more general systems derived in Chapter 3. Referring to the cylindrical-coordinate summary in Section 3.12.2, and retaining only the axial components, the one-dimensional flame equations reduce immediately to... 

The general equations are simplified for a one-dimensional laminar flame in the steady state (Hi). 

The difference between RANS and LES is depicted in Figure 20.1, which shows the temperature fields of a pool fire flame. While the RANS result shows smooth variations and looks like a laminar flame, the LES result clearly illustrates the large-scale eddies. Both results are the correct solutions of the corresponding equations. However, the time accuracy of LES is also essential for the quantitative accuracy of the buoyancy-driven flows. As Rehm and Baum have shown [10], the dynamic motions or eddies are responsible for most of the air entrainment into the fire plumes. Because these motions cannot be captured by RANS, LES is usually better suited for fire-driven flow. LES typically requires a finer spatial resolution than RANS. Examples of RANS-based fire CFD models are JASMINE, KAMELEON [11], SMARTFIRE [12], SOFIE [13], ISIS [14], and ISIS-3D [15]. Examples of LES models are the FDS [4,5] and SMAFS [16], developed at Lund University. Fire simulations using LES have also been performed by Cheung et al. [17] and Gao et al. [18],... 

Validation of the Global Rates Expressions. In order to validate the global rate expressions employed in the model, temperature and concentration profiles determined by probing the flames on a flat flame burner were studied. Attention was concentrated on Flames B and C. The experimental profiles were smoothed, and the stable species net reaction rates were determined using the laminar flat-flame equation described in detail by Fristrom and Westenberg (3) and summarized in Reference (8). 

Empirical laminar burning velocities lie between 1 and 1000 cm/s. Since these velocities are small compared with the speed of sound, equation (1-25) is valid for laminar flames. Thus laminar flames are nearly isobaric, we were justified in not attaching a subscript to p in equation (3), and the quantity Cp appearing above is the specific heat at constant pressure. 

A consequence of the equivalence of the two problems is that any of the procedures discussed in Section 5.3 for determining the laminar flame-speed eigenvalue A of equation (5-45) can be applied without modification for finding A as a function of t., a, and P [defined in equation (5-44)] in the present problem. In using these procedures, one should recognize that co(t) is given by equation (5-43) only for t > ct = 0 for t < t-. Results... 

Unlike diffusion flames, premixed laminar flames are appropriately characterized in terms of a laminar flame speed Vq [for example, equation... 

V / oX / )]" a relationship that enables lines of constant values of Ri to be plotted, as shown. Turbulence Reynolds numbers quoted in the literature are often based on the Taylor scale, equation (31), instead of the integral scale these are directly related to and are denoted by in Figure 10.5. In addition to the ratio IJd) of the smallest turbulence scale to the laminar-flame thickness, the ratio of the largest scale (the integral scale) to the flame thickness, //<5, is a relevant parameter. Lines of constant values of //(5, generated from equation (30), also are shown in Figure 10.5. 

Although strain and curvature effects can be combined as in equation (55), it cannot be concluded that they are of equal importance for wrinkled laminar flames in turbulent flows. If it is assumed that the flame shape is affected mainly by the large eddies, then in terms of the flame thickness 3 and the integral scale /, the nondimensional curvature is of order 3/i This may be compared with the corresponding relevant nondimensional strain... 

This result can conveniently be expressed in terms of the laminar flame speed Vq by employing equation (5-2) the result is... 

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