The Shell autoignition model: a new mathematical formulation

The equations of the Shell model are reexamined with a view to their more effective implementation into a computational fluid dynamics code. The simplification of the solution procedure without compromising accuracy is achieved by replacing time as an independent variable with the fuel depletion, which is the difference between the initial fuel concentration and the current one. All the other variables used in this model, including temperature, concentration of oxygen, radicals, intermediate and branching agents are expressed as functions of fuel depletion. Equations for the temperature and concentration of the intermediate agent are of the first order and allow analytical solutions. The concentrations of oxygen and fuel are related via an algebraic equation which is solved in a straightforward way. In this case the numerical solution of five coupled first-order ordinary differential equations is reduced to the solution of only two coupled first- order differential 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 branching agent. In this case the equation for the concentration of radicals becomes a second-order ordinary differential equation. This equation is solved analytically in two limiting cases and numerically in the general case. The solution of the first-order ordinary differential equation for the concentration of the branching agent and the solution of the first-order differential equation for time are presented in the form of integrals containing 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.