Derivative Approach - Pyrolytic Dehydrogenation of Benzene

In this section we shall only present the derivative approach for the solution of the pyrolytic dehydrogenation of benzene to diphenyl and triphenyl regression problem. This problem, which was already presented in Chapter 6, is also used here to illustrate the use of shortcut methods. As discussed earlier, both state variables are measured and the two unknown parameters appear linearly in the governing ODEs which are also given below for ease of the reader. 

The above equation yields to the following integral equation for kd, 

Here X denotes lb-moles of benzene per lb-mole of pure benzene feed and x, denotes lb-moles of diphenyl per lb-mole of pure benzene feed. The parameters k, and k2 are unknown reaction rate constants whereas K, and K2 are known equilibrium constants. The data consist of measurements of Xi and x2 in a flow reactor at eight values of the reciprocal space velocity t. The feed to the reactor was pure benzene. The experimental data are given in Table 6.2 (in Chapter 6). The governing ODEs can also be written as  

As we mentioned, the first and probably most crucial step is the computation of the time derivatives of the state variables from smoothed data. The best and easiest way to smooth the data is using smooth cubic splines using the IMSL routines CSSMH, CSVAL CSDER. The latter two are used once the cubic splines coefficients and break points have been computed by CSSMH to generate the values of the smoothed measurements and their derivatives (rj, and t] )- 

Estimates based on Gauss-Newton method for ODEs 354.61 400.23 

---