Forward Finite Differences

To solve the problem a sequential quadratic programming code was used in the outer loop of calculations. Inner loops were used to evaluate the physical properties. Forward-finite differences with a step size of h = 10 7 were used as substitute for the derivatives. Equilibrium data were taken from Holland (1963). The results shown in Table E12.1B were essentially the same as those obtained by Sargent and Gaminibandara. 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

Working in the same way, with different backward-forward finite difference schemes for the second-order finite differences, the solution of Equation 11.6 is as follows ... 

In this calculation scheme, the first term of Eq. 10.72 is replaced by a forward finite difference while the second term is replaced by a backward finite difference. We obtain the equation... 

The backward-forward finite difference scheme is identical to the Craig model if we choose the time and space increments such that = H. The Craig model has been used by many authors, including Eble et ah [45], Czok and Guiochon [49,50], and El Fallah and Guiochon [55]. This model affords a good numerical solution of the gradient elution problem, which is very difficult to solve numerically with the forward-backward finite difference scheme [55,56]. 

With the backward-forward finite difference scheme, the space increment, h, and the time increment, t, are chosen such that (1 — a) = D<, /w or... 

With the backward-forward finite difference scheme ... 

Selcuk et al. [5,123] describe the simulation of a shelf-type dryer. Close [124] worked out the simulation of an air-type solar drying system equipped with gravel-bed heat storage using a 10-node discretized model for the heat storage and forward finite difference technique for solving the equation system. Imre et al. [25,125] presented a simnlation of solar dryer for alfalfa. 

Carter and Husain<"> Ext. fluid + Pore Diffusion Modified Langmuir Crank Nicolson + Forward finite difference 2 + carrier CO2-H2O on 4A sieve... 

In general, it is found that the forward finite difference approximation prodnces an unstable particle number. The backward approximation, on the other hand, eliminates the oscillation and predicts a more stable PSD. 

The development of forward finite differences follows a course parallel to that used in the development of backward differences. 

In similarity to the backward finite differences, the forward finite differences also have coefficients which corre.spond to those of the binomial expansion (a - b)". Therefore, the general formula of the nth-order forward finite difference can be expressed as... 

In MATLAB, the function diff y) returns forward finite differences of y. Values of nth-order forward finite difference may be obtained from diffiy, n). 

The complete set of relationships between central difference operators and differential operators is summarized in Table 3.3. These relationships will be used in Chap. 4 to develop a set of formulas expressing the derivatives in terms of central finite differences. These formulas will have higher accuracy than those developed using backward and forward finite differences. 

The Gregory-Newton forward interpolation formula can be derived using the forward finite difference relations derived in Secs. 3.2 and 3.4. Eq. (3.17), written for the function/... 

The relationships between forward difference operators and differential operators, which are summarized in Table 3.2, enable us to develop a variety of formulas expressing derivatives of functions in terms of forward finite differences and vice versa. As was demonstrated in Sec. 4.2, these formulas may have any degree of accuracy desired, provided that a sufficient number of terms are retained in the manipulation of these infinite series. A set of expressions, parallel to those of Sec. 4,2, will be derived using the forward finite differences. 

First-Order Derivative in Terms of Forward Finite Differences with Error of Order h... 

Eq. (4.33) enables us to evaluate the first-order derivative ofy at position i in terms of forward finite differences with error of order h. 

---