Numerical Solution of the Two-Dimensional Premixed Laminar Flame Equations

Abstract

The present paper presents a numerical solution method for the two-dimensional elliptic partial differential equations modeling a premixed laminar flame freely propagating at constant pressure between parallel plates having a constant, cool wall temperature. The technique developed is an efficient, semidirect (semi-iterative) method termed "multistep damped Picard iteration." Picard linearization of the nonlinear terms together with variable damping as the solution progressed and a sparse matrix elliptic solver allowed rapid solution to be obtained as a limit of a convergent sequence of solutions. The flame-quenching limit was determined by iterating the solution with respect to the distance separating the plates and approaching the limit state to within a specific criterion. Global kinetics and average transport properties were used for a simulated propane-air mixture; radiation loss was neglected throughout the analysis. The method was used to study the effect of the "ignition temperature" on the flame parameters as well as to reveal the two-dimensional structure of the flame at the quenching limit. Nomenclature= cross-sectional area, Sec. IV B, cm 2 = specific heat at constant pressure of the mixture, cal/g K = diffusion coefficient for the fuel species cm 2 /s = hydraulic (equivalent) diameter of the parallel plates, cm = activation energy, cal/mole = mass flow rate (mass burning rate), g/cm 2 s; eigenvalue of the flame equations = heat of combustion of the fuel, -cal/g = frequency factor (reaction rate constant), units depend on reaction order = distance separating the parallel plates, cm = reaction order = perimeter, Sec. IV B, cm = pressure, atm = quenching Peclet number = quenching distance, cm = universal gas constant, 1.986 cal/mole K = surface of the upstream boundary, cm 2 , Sec. Ill B = surface of the downstream boundary cm 2 , Sec. Ill B = temperature, K = ignition (cutoff) temperature, K = ambient temperature, K = x component of the mixture velocity, cm/s = volume in Sec. Ill B cm 3 = flame speed in Eq. (22), cm/s = velocity of the mixture at the downstream boundary (_L toS out ),cm/s = rate of consumption of fuel per unit volume and time, g/cm 3 s = mass of reactant (fuel) per unit mass of mixture -mass of reactant (fuel) in unburned mixture per unit mass of that mixture A = thermal conductivity of the mixture cal/cm s K p = local mixture density, g/cm P 0 = density of unburned mixture, g/cm 3 Pom = density of the mixture at the downstream boundary, Sec. Ill B Subscript 0 = upstream boundary (reactants or cold)

Laminar Flame Structure and Heat Release Study of Diesel Fuel

“…Upon passing inside the flame region the and chemical reaction takes place between the fuel and mixture. The following global kinetic rate equation is the oxidation of diesel fuel in air [ Neglecting the y component of the mixture velocity [9] the continuity equation yields the following relation p u = G = constant (2) Considering the usual approximation made in deflagration waves (laminar flame wave), namely 1:i constant, the system under study would be governed by the following energy and fuel mass species conservation equations…”

“…The solution method is based on linearizing the non-linear terms in the energy equation using Picard method [9]. The second and first derivatives are then discretized using central and upwind differencing, respectively.…”

Analysis of Flame Propagation in Small Adiabatic Tubes Characterized by Different Degrees of the End Opening

“…The flame propagation in small channels has also been analysed numerically, for example, in the papers [3,4]. The flame propagation mechanism in a tube with adiabatic and isothermal (cold) wall can be found in the paper [5], and for flames propagating between parallel plane walls in the work [6].…”

Theoretical and experimental studies on flame propagation and quenching of powdered fuels

“…These limitations were eliminated in the method of Aly et al [1979], and Aly and Hermance [1981]. Unfortunately, their assumption of a plane flame propagating freely between parallel plates at a speed equal to the burning velocity is unrealistic.…”