Ordinary differential equation integrating factor

To analyze this phenomenon further, 2D numerical simulations of (49) and (50) were performed using a central finite difference approximation of the spatial derivatives and a fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. Details of the simulation technique can be found in [114, 119]. The material parameters of the polymer blend PDMS/PEMS were used and the spatial scale = (K/ b )ll2 and time scale r = 2/D were established from the experimental measurements of the structure factor evolution under a homogeneous temperature quench. 

As a consequence of the oscillatory phase factor in (7.7), contributions to the integral over t stem mainly from the region around t t so that an t ) on the right-hand side may be approximated by its value at time t. With this substitution Equation (7.7) becomes an ordinary differential equation,t... 

Obviously, numerical methods are required to calculate the effectiveness factor, so the problem is reformulated in terms of three coupled first-order ordinary differential equations. This is required for numerical integration. 

For this we first need to establish the correspondence between the Y- and IF-coordinates in order to calculate the factor exp (— kF ) for each F value. This can be done by integration of the following ordinary differential equation ... 

The final two-dimensional mathematical model thus consists of one partial parabolic differential mass balance equation (3.12) with boundary and initial conditions in (3.14) for each of the j reactions and one partial parabolic differential heat transfer equation (3.15) with boundary conditions in (3.17), (3.18) and initial conditions in (3.20). Simultaneously the pressure drop ordinary differential equation (3.7) and the differential equations for the temperature and pressure in each of the surrounding channels in (3.22) must be integrated. Catalyst effectiveness factors in the catalyst bed must be available in all axial and radial integration points using the methods in Section 3.4. 

Equation (7-9) is a linear, first-order, ordinary differential equation that can be solved by the integrating factor approach. The solution is... 

The final model hence includes one ordinary second-order differential equation (3.32) with an integral boundary condition at the surface (3.41) for each of the reactions. The numerical solution of the catalyst effectiveness factor can be carried out using the orthogonal collocation method by Villadsen and Michelsen [512]. The steam reforming reaction takes place mainly in the outer shell of the catalyst particle, since large particles are used to limit pressure drop. In this case it is advantageous to divide the catalyst pellet into two sections, an inner section and an outer section, divided by a spline point and with an appropriate coupling between the two sections. The spline collocation method has been used by [525] and [181]. A description of the method, spline collocation, can be found in [512]. 

---