The Shell autoignition model: A new mathematical formulation

The equations of the Shell model are reexamined with a view to their more e ffective implementation into a computational fluid dynamics code. The simpl ification of the solution procedure without compromising accuracy is achiev ed by replacing time as an independent variable with the fuel depletion, wh ich is the difference between the initial fuel concentration and the curren t one. All the other variables used in this model, including temperature, c oncentration of oxygen, radicals, intermediate and branching agents are exp ressed as functions of fuel depletion. Equations for the temperature and co ncentration of the intermediate agent are of the first order and allow anal ytical solutions. The concentrations of oxygen and fuel are related via an algebraic equation which is solved in a straightforward way. In this case t he numerical solution of five coupled first-order ordinary differential equ ations is reduced to the solution of only two coupled first-order different ial equations for the concentration of radicals and branching agent. It is possible to rearrange these equations even further so that the equation for the concentration of the radicals is uncoupled from the equation for the b ranching agent. In this case the equation for the concentration of radicals becomes a second-order ordinary differential equation. This equation is so lved analytically in two limiting cases and numerically in the general case . The solution of the first-order ordinary differential equation for the co ncentration of the branching agent and the solution of the first-order diff erential equation for time are presented in the form of integrals containin g the concentration of the radicals obtained earlier. This approach allows the central processing unit (CPU) time to be more than halved and makes the calculation of the autoignition process using the Shell model considerably more effective. (C) 1999 by The Combustion Institute.