Integral methods general procedure

The concentration of each chemical species, as a function of time, during cure can be calculated numerically from Equations 3-6 using the Euler-Romberg Integration method if the initial concentrations of blocked isocyanate and hydroxyl functionality are known. It is a self-starting technique and is generally well behaved under a wide variety of conditions. Details of this numerical procedure are given by McCalla (12). 

A General Integral Method for the Analysis of Kinetic Data—Graphical Procedure. 

A more general method of using integrated forms involves using the standard procedure where one waits until the TSR experiment is completed and then proceeds to smooth the (X, T, t) surface as described in previous discussion. On the smoothed surface one identifies the operating lines for the system and uses them in an integral method of data interpretation. This removes the restriction employed in the hydrolysis study, that the system be at quasi-isothermal conditions, and makes the method more general. 

Although this equation is a solvable quadratic (owing to the second order reaction), we will solve it nonetheless by numerical means to demonstrate the general procedure of the implicit integration method. The iteration scheme for the Newton-Raphson procedure is (Appendix A)... 

As discussed previously, the relation between stress and strain for linear viscoelastic materials involves time and higher derivatives of both stress and strain. While the differential equation method can be quite general, a hereditary integral method has proved to be appealing in many situations. This hereditary integral equation approach is attributed to Boltzman and was only one of his many accomplishments. In the late nineteenth century, when the method was first introduced, considerable controversy arose over the procedure. Now, it is the method of choice for the mathematical expression of viscoelastic constitutive (stress-strain) equations. For an excellent discussion of these efforts of Boltzman, see Markovitz (1977). 

Work practice controls are administrative controls that define the methods used to perform a task in order to reduce potential hazards. They are integrated into job procedures and general safety work rules. A comprehensive JHA is used to develop work practices controls and the implementing of specific training. Work practices are used in conjunction with and not as a substitute for engineering controls. 

---