Parametric instabilities

Stability plots for two methane-air flames using the analytical solution of Bychkov [17] with parameters for equivalence ratios of 0.8 and 1.3 are shown in Fig. 7.4. The y-axis corresponds to the acoustic velocity normalized by the laminar-flame speed, and the x-axis is the wave number scaled by the flame thickness. However, within this model the Markstein parameter (Eq. (8) in [17]) was modified to incorporate an effective Lewis number to allow use of the model for mixtures at and near stoichiometric composition [11]. A frequency of 920 radians per second is selected, which corresponds to the quarter-wavelength frequency of the TC burner during the parametric instability, and the activation... 

The model predicts that methane-air flames with an equivalence ratio of 0.8 propagating downward are always unstable as a result of either the Darrieus-Landau instability or the secondary pyroacoustic instability, also known as the parametric instability, yet flames with an equivalence ratio of 1.3 will be stabilized by a small band of normalized acoustic-velocity amplitudes between 3.7 and 4. Although it was found experimentally that a methane-air flame with an equivalence ratio of 0.8 can be stabilized, the results of the model agree qualitatively with the experimental findings, specifically that a methane-air flame with an equivalence ratio of 1.3 is stable over a larger range of acoustic amplitudes than one with an equivalence ratio of 0.8. 

For flames that exhibit the parametric instability, the velocity at which the exponential growth of velocity fluctuations started for each experiment was noted. These critical velocities are shown in Fig. 7.5, normalized by the laminar-flame speeds reported in [13]. All points shown on this plot represent the ensemble average of measurements from five experiments, and the error bars indicate the standard deviation about the mean value. The other curve on this plot was calculated using the analytical model of a premixed flame under the influence of an oscillating gravitational field by Bychkov [17], ris described above. Each point represents the smallest normalized acoustic velocity at the most unstable reduced wave number that resulted in the parametric instability. The experimental results show the same trend as the theoretical model mixtures with an equivalence ratio of 0.9, which require the smallest normalized acoustic velocity to trigger the parametric instability while flames on either side require larger values. 

Figure 7.5 Normalized critical axial acoustic velocity for onset of the parametric instability plotted vs. equivalence ratio 1 — experiment and 2 — analytical [17].

Thornton, B. H. and Bogy, D. B., "A Parametric Study of Head-Disk Interface Instability Due to Intermolecular Forces, IEEE Trans. Magn., Vol. 40, No. (1), 2004, pp. 337-343. 

G. Searby and D. Rochwerger. A parametric acoustic instability in premixed flames. Journal of Fluid Mechanics, 231 529-543,1991. 

The following parametric effects on density wave instability are summarized, as these effects have been often observed in the most common type of two-phase flow instability (Boure et al., 1973) ... 

Recently, Razumovskid441 studied the shape of drops, and satellite droplets formed by forced capillary breakup of a liquid jet. On the basis of an instability analysis, Teng et al.[442] derived a simple equation for the prediction of droplet size from the breakup of cylindrical liquid jets at low-velocities. The equation correlates droplet size to a modified Ohnesorge number, and is applicable to both liquid-in-liquid, and liquid-in-gas jets of Newtonian or non-Newtonian fluids. Yamane et al.[439] measured Sauter mean diameter, and air-entrainment characteristics of non-evaporating unsteady dense sprays by means of an image analysis technique which uses an instantaneous shadow picture of the spray and amount of injected fuel. Influences of injection pressure and ambient gas density on the Sauter mean diameter and air entrainment were investigated parametrically. An empirical equation for the Sauter mean diameter was proposed based on a dimensionless analysis of the experimental results. It was indicated that the Sauter mean diameter decreases with an increase in injection pressure and a decrease in ambient gas density. It was also shown that the air-entrainment characteristics can be predicted from the quasi-steady jet theory. 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

The use of feedback-control techniques to modulate combustion processes in propulsion systems has recently received extensive attention [1-3]. Most of the previous studies involved direct implementation of existing control methods designed for mechanical devices, with very limited effort devoted to the treatment of model and parametric uncertainties commonly associated with practical combustion problems. It is well established that the intrinsic coupling between flow oscillations and transient combustion responses prohibits detailed and precise modeling of the various phenomena in a combustion chamber, and, as such, the model may not accommodate all the essential processes involved due to the physical assumptions and mathematical approximations employed. The present effort attempts to develop a robust feedback controller for suppressing combustion instabilities in propulsion systems. Special attention is given to the treatment of model uncertainties. Various issues related to plant... 

The Villermaux criterion and the Da/Si criterion are dynamic stability criteria, meaning that with a cooling medium temperature above the limit level, 20 resp. 30 °C, the reactor will be operated in the instable region and present the phenomenon of parametric sensitivity. If instead of B12, B is used, both criteria lead to the same result. This should not be surprising since they derive from the same heat balance considerations, that is, the heat release rate of the reaction increases faster with temperature than the heat removal does. 

Chemburkar et al. [3]. An extensive discussion of the parametric sensitivity of the CSTR is presented by Varma [4]. In the example in Figure 8.4, A is a working point on the cold branch, B is an instable point, and C is a working point on the hot branch. The consequences of this multiplicity are explained in more detail in Section 8.2.6.1. 

Parametric Sensitivity. One last feature of packed-bed reactors that is perhaps worth mentioning is the so-called "parametric sensitivity" problem. For exothermic gas-solid reactions occurring in non-adiabatic packed-bed reactors, the temperature profile in some cases exhibits extreme sensitivity to the operational conditions. For example, a relatively small increase in the feed temperature, reactant concentration in the feed, or the coolant temperature can cause the hot-spot temperature to increase enormously (cf. 54). This sensitivity is a type of instability, which is important to understand for reactor design and operation. The problem was first studied by Bilous and Amundson (55). Various authors (cf. 57) have attempted to provide estimates of the heat of reaction and heat transfer parameters defining the parametrically sensitive region for the plug-flow pseudohomogeneous model, critical values of these parameters can now be obtained for any reaction order rather easily (58). 

INFLUENCE OF MARKSTEIN NUMBER ON THE PARAMETRIC ACOUSTIC INSTABILITY... 

Zhang, C., W. Zhao, T. Ye, S. H. Frankel, and J. P. Gore. 2002. Parametric effects on combustion instability in a lean premixed dump combustor. AIAA Paper No. 02-4014. 

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