Differentiation Formulas

Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 18.5) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points ... 

Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful. 

In eq 4 the rate is the time derivative of the conversion curve, which can be constructed from the observed conversion-time behavior by mathematical treatment, such as differentiation formulae or polynomial or spline interpolation, provided the product analysis is fast enough to follow the reaction. The same approach can be followed in principle for the plug flow reactor if data is collected at various space-time values. 

Application of the Newton-Cotes differential formula for = 2 to the linear Hamiltonian system (1) gives... 

In addition to the graphical technique used to differentiate the data, two other methods are commonly used differentiation formulas and polynomial fitting. 

Numerical Method. Numerical differentiation formulas can be used when the data points in the independent variable are equally spaced, such as — — tj At. 

Finding a and k. Now, using either the graphical method, differentiation formulas or the polynomial derivative, the following table can be set up ... 

The flow velocity within the porous panicle is estimated by the continuity equation assuming constant outflow of the produced gases from the particle. The one dimensional conservation equations are discretized in space by a finite volume approach and a backward differentiation formula is applied to time integration. 

Thus, we plot a dP/dt) versus hi[3 —Pit)/P Qi)] to determine the value of a. We calculate the derivatives numerically, using second-order forward, center, and backward numerical differentiation formulas, and obtain the following ... 

The NONLIN module is responsible for intializing the concentration vector, C(t), for l i NRCT. Here NRCT is the number of reactants. If there are no equilibrium reactions, then C i) is set to IC i), the initial concentration vector, for 1 < f < NRCT. If equilibrium reactions do exist, then the type (2) equations (with derivatives set to zero) and the Type (1) and Type (3) equations are all solved simultaneously for the equilibrium concentrations of all reactants. Because the equilibrium equations are generally nonlinear, the Newton-Raphson iteration method is used to solve these equations. Also, since there is no symbol manipulation capability in the current version of CRAMS, numerical differentiation is used to calculate the required partial derivatives. That is, the rate expressions cannot at this time be automatically differentiated by analytical methods. A three point differentiation formula is used 27) ... 

The derivatives and their associated maximum tolerances for the known reactants are calculated by using numerical differentiation formulas. 

In the final step DHPCG calculates the NSIM derivatives for the reactants that are being simulated. Since the derivative forms of T37pe (1), (2) and (3) equations are all linear with respect to the CP (i) s, they are solved simultaneously for the CP(iys by the Gaussian elimination method. The partial derivatives used in the evaluation of the derivative form of the Type (1) equation are calculated with the same numerical differentiation formula that is used in the NONLIN module. 

By differentiating formula (2) with respect top, the concentration (c) and, therefore, the value of irw being maintained constant, we get—... 

To find the rale law parameters and K,. we first apply the differential formulas in Chapter 5 to columns 1 and 2 of Table E7-5.1 to find r,. Because C, K, initially, it is best to regress the data using the Hanes-Woolf form of the Monod equation... 

This equation is analogous to Eq. (11.21). It will be our working equation for the propagation of errors through formulas. Since it is based on a differential formula, it becomes more nearly exact if the errors are small. 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

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