| OCR Text |
Show corresponding to stable (sink) solutions and a lower (saddle) unstable solutions, and the trivial stable solutions along the 4> axis (again not shown on the figure). As the parameter, :I, is decreased, the loop begins to shrink. That is to say, for the case of:l = 9.SeS, the limits of 4> (.77,8.3) for :I =S.Oe6, change to (0.99,4.2), decreasing the range; and the maximum attainable reaction efficiency occurs at (<!>max,11f,max) ~ (1.2,6.6). Whereas for :I = 9.Se6. the maximum attainable reaction efficiency occurs at (4)max,11t:rnax) ~ (0.6,.96). In summary, Figure 2 shows that as :I is decreasing, the ranges of the closed loop also decrease. In addition, the occurrence of the maximum attainable reaction efficiency decreases, while the equivalence ratio corresponding to that maximum is increasing, infact, for a 4> greater than one. Figure 3, shows the variation of the loop for different temperatures, which determined the value of t{, which has Cp as one of its factors, and thus is dependent on temperature. As temperature increases, Cp increases, and thus ~ decreases. However, the value of ~ is held constant for a given computer run. The figure shows that at low temperatures, the range of loop is very wide, and it decreases as the temperature increases. This was done for very low firing rates, or low:l. The figure also shows that the maximum reaction efficiency decreas~s, as the temperature increases. DISCUSSION AND CONCLUSIONS The steady state analysis of PSR's shows that the concentration of the fuel and of the oxygen are proportional to each other by the factor, 4>. However, when considering the unsteady state behavior, the rate of change of the oxygen concentration is not proportional to the rate of change of the fuel concentration. Thus, the oxygen's concentration rate must be taken into account, which results in a third equation, expanding the original equations to include a third variable, 11ox. This new expanded system, fixes the original problem of the time dependent solution of the fuel reaction efficiency going negative. The bifurcation analysis shows the variation of the steady state solutions, with varying parameter, 4>. This analysis resulted in a closed loop, for which as in other studies 9 |
