OCR Text |
Show projection of the closed loop onto the reaction efficiency, llr, and equivalence ratio, cP (llr - ct» plane. Figure 1 shows the stability of the steady state solution, with the solid line representing the stable solutions, and the dotted line representing the unstable solutions. The dominant characteristic of this curve is a closed loop with a top side region of stable (sink) behavior [between (cp, llr,) of: (0.52,0.77) and (8 .3,0.1)], and a bottom side of unsteady-state (saddle) behavior. There is an additional line (not shown) of stable (spiral) solutions along the line llr = 0, which correspond to the near zero solution, or trivial solution, representing the uningnited fuel-air mixture. The physical significance of the closed loop and of the trivial solution, is that for any given value cp between 0.77 and 8.3, there are three possible solutions, which corresponds to the three steady-state solutions. These limits of cp correspond to the ignition and extinction limits, which can be calculated from the points of tangency of the heat loss line with the heat generated sigmoid curve. Within the loop, there are three steady-state solutions, which correspond to the intersection points of the heat loss line with the heat generated sigmoid curve. Outside these limits, there is only one intersection point. The curve also indicates that outside the loop, spontaneous ignition to a steadystate operation value is not possible. However, for conditions that are within the loop, ignition can take place and the system will move to a steady-state solution, a point on the top curve. The lower portion of the curve, are saddle points, and the system will move away from this point to a stable point. These points are often referred to as meta-stable conditions since, even though operation at that point, is stable, any small displacement away from this point will result in your system from moving to either the upper or lower stable points. This physically corresponds to either a combustor operation or to a flame extinction. Further noted from Figure 1, that the top side of the loop represents the stable reaction efficiency for a given cp, which is always less than unity. The maximum reaction efficiency (cpmax, llr.max) occurs at fuel-lean conditions, or for low cp. Figure 2 shows the same (CP,llr) loops for different values of the dimensionaless parameter, 3. Each curve is a closed curve on the (9, llr, llox;CP) space, and each have the same characteristics: a top curve 8 |