OCR Text |
Show are varied. The results of a bifurcation analysis can give infonnation about the steady state behavior; whether it is a stable or unstable operating point, for a range of physical conditions. One such study of stability behavior is with continuously stirred tank reactors with recycles (CSTR). Bifurcation analysis on CSTR's have shown instability behavior where conventional stability and stability upset analysis did not. One type of behavior could be the influence of an unstable point ( saddle) which would cause the system to dynamically move away from the point, either on to an other stable point, or out to infinity. Another possibility, in a bifurcation analysis, is the determination of Hopf bifurcations, if any exist. These types of bifucations imply the existence of time-periodic solutions with detenninable frequencies, which is, as in our case, pulsating combustion. The set of equations presented in the Baker and Essenhigh paper, were incomplete, which resulted in incorrect time dependent solutions. An additional balance was needed in the development, which gives an extra variable, oxygen concentration. The equations that we developed paralleled the development mentioned in the earlier paper but, with the additional oxygen variable, the system was expanded from a two dimensional to a three dimensional system. The development included an energy, and two species balance. The equations were also non-dimesionalized using the same constants as the ones used by in the Baker and Essenhigh paper, so that a comparison of the results of a bifurcation analysis could be done. However, in the process, there were two additional parameters that were developed, that are not present in the Baker and Essenhigh paper. DEVELOPMENT OF EQUATIONS Discussion of Baker and Essenhighs equations The equations presented in the Baker and Essenhigh paper were limited to two variables; the fuel concentration and temperature only. The fuel concentration was nondimensionalized, as the fuel efficiency, defined as: llr = l-Cr ICt Their equations then reduced to a set of two first order time derivatives. Unfortunately there were problems with these equations, which were discovered when the time-dependent solutions were 3 |