Isocline method

Teorell studied the system (6.1-1)—(6.1.5) graphically, by the isocline method, and also numerically. He recovered most of the features observed experimentally. This study was further elaborated by several investigators. Thus, Kobatake and Fujita [5], [6] criticized the original model for invoking the ad hoc equation (6.1.5). These authors assumed instead instantaneous relaxation of the resistance to its stationary value while preserving the overall order of the relevant ordinary differential equation (ODE) system by including consideration of the mechanical inertia of the liquid column in the manometer tube. 

The above remark forms the basis for an approximate method of constructing the phase portraits for equation (A35), called the isocline method. At the points (x, y) where the slope of the trajectory is a, the equality tan(a) = f(x, y) is fulfilled, since /[x, y(x)] = y (x). All such points lie on a line called the isocline, defined by the formula tan (a) = /(x,y) isoclines connect the points at which the sections tangent to the trajectory are identically inclined with respect to the x-axis. 

By a brilliant physical intuition B. van der Pol succeeded finally (1920) in establishing his equation (which is given in Section 6.11) but, not having any mathematical theory at his disposal, he determined the nature of Ike solution by the graphical method of isoclines. It became obvious that the problem, which was a real stumbling block for many years, had been finally solved, at least in principle. 

Using the method isoclinics, Checker and coworkers [116] studied the flow of polyethylene and polypropylene melts through two dimensional contractions and used birefringence measurements to determine the relaxation behavior of these materials subject to complex flows. 

Bilous and Amundson [1] were the first to describe the phenomenon of parametric sensitivity in cooled tubular reactors. This parametric sensitivity was used by Barkelew [2] to develop design criteria for cooled tubular reactors in which first order, second order and product- inhibited reactions take place. He presented diagrams from which for a certain tube diameter dt the required combination of CAO and Tc can be derived to avoid runaway or vice versa. Later van Welsenaere and Froment [3] did the same for first order reactions, but they also used the inflexion points in the reactor temperature T versus relative conversion XA trajectories, which describe the course of the reaction in the tubular reactor. With these trajectories they derived a less conservative criterion. Morbidelli and Varma [4] recently devised a method for single order reactions based on the isoclines in a temperature conversion plot as proposed by Oroskar and Stern [5]. 

The runaway limits determined by Morbidelli and Varma [1982] are based on the occurrence of an inflection point in the temperature profile before the hot spot. They used the method of isoclines, which requires the numerical integration of a differential equation. The method is also applicable to reaction orders different from 1, as shown in Fig. 11.5.3-1. The runaway region becomes more important as the order decreases. Tjahjadi et al. [1987] developed a new approach, applicable to more complex reactions, for example, the radical polymerization of ethylene in a tubular reactor. Hosten and Froment [1986]... 

Rogers utilized the equations developed by Harvey and Foust in an effort to predict the behavior of liquid hydrogen flowing in relatively well-insulated pipelines his results have not been experimentally verified, A difficulty with the equations of Harvey and Foust is the large amount of computation required to obtain numerical results by the method of isoclines. In his work, Rogers used a high speed computer. One of the reasons for this complexity is that Harvey and Foust substituted the complicated results of Lockhart and Martinelli into their differential equations. 

The birefringence and isoclinic angles may be determined using the above methods. However, the rheologist usually desires the stresses. To do this, the stress-optic relation must be applied, as discussed in Section 20.2. The above discussion deals with steady flow situations. Transient flows (except those with very slow changes) need other methods and recently a number of new ideas have emerged. 

---