Abstract: A method including converting a predetermined number of catalytic zones into a number of finite elements, where the number of finite elements include a number of collocation points represented by a number of mathematical roots in an algebraic system, modeling a catalyst volume including a length of a continuous reactor to have the predetermined number of catalytic zones, representing the first ordinary differential equation, second ordinary differential equation, and third ordinary differential equations in the algebraic system, and performing orthogonal collocation on the number of finite elements in the algebraic system while simultaneously varying at least one of the first number of polynomials and simultaneously varying a percentage of active catalyst and a length of each of the predetermined number of catalytic zones in the second number of polynomials to obtain the mass flow rate of the product for the given chemical reactions.