The dynamics of compressor stall has been re-parameterized in a form that would facilitate the construction of a non-linear control law for the active non-linear control of compressor stall. The regions of stable performance in parameter space
Fig. 5.11 Root locus plot illustrating the effect of the negative feedback of D/c
(y = Jn, H > 0, J = Js = 0), unstable performance (y = yn or H < 0, J = 0) were identified. This has led to the belief that a control law that maintains both y = yn, H > 0 and J = Js = 0 would actively stabilize the compressor. One observes that by merely setting the throttle at its optimum equilibrium position does not maintain, J = Js = 0. An additional control input must aim to manipulate the transient and control pressure dynamics defined by Eq. 5.3.88 which would involve control inputs to the compressors inlet guide vanes or some other means of feedback control. That in turn points to a need for a better compressor pressure rise model incorporating the control input dynamics. Yet the relatively simple and systematic approach adopted in this section clearly highlights the main features on the controller that is capable of inhibiting compressor surge and rotating stall. Moreover, the method can be adopted for any axial compressor provided its steady-state compressor, and throttle maps are known. Furthermore, the linear perturbation controller synthesized in the previous section could be substituted by a non-linear controller synthesized by applying the backstepping approach as demonstrated by Krstic et al. (1995). Preliminary implementations of such a controller have supported the view that there is a need for an improved, matching, non-linear compressor pressure rise model including disturbance and uncertainty effects, and the results of this latter study involving a robust complimentary nonlinear H? optimal control law will be reported elsewhere. Coupled with the views
expressed by Greitzer (2009), the active structural control of surge and rotating stall could be effectively achieved by realistic low-order modeling of the compressor dynamics.
Fuel is introduced into the combustion chamber by fuel injectors in the form of a spray of fuel droplets atomized by a nozzle with a large pressure drop. The atomized liquid droplets injected in the combustion chamber may undergo one or more of the processes (following atomization), viz. collision, break-up, or evaporation. The evaporated fuel mixes into the air and finally combustion takes place. Combustion processes can be subdivided based on mixing as premixed, nonpremixed, and partially premixed. Combustion in homogeneous-charge spark – ignition engines and lean-burn gas turbines is under premixed conditions. From chemical thermodynamics, one can ascertain the end states following a thermochemical reaction. It is important to be able to assess how fast or slow the reactions will be completed which would depend on both the inherent rates of reaction and the rates of heat and mass transport to the reaction zones characterized by a flame. Combustion is the combination of chemical reactions with convective and diffusive transport of thermal energy and chemical species. Thus, combustion is a process involving both thermochemical reactions and the associated heat and mass transport. A flame is the visible, gaseous part of the region of combustion which is characterized by highly exothermic reactions taking place within a smaller zone. So the key parameters are the rates of flame zone propagation and heat generation. Broadly, flames can be classified into two groups: premixed—when reactants are intimately mixed on the molecular scale before combustion is initiated; and non – premixed—when reactants mix only at the time of combustion and so they have to mix first then burn. Premixed flames can be of three types:
1. Deflagration, in which case combustion wave or flame zone propagates at subsonic speeds. Deflagration involves a propagating subsonic front sustained by conduction of heat from the hot (burned) gases to the cold (unburned) gases which raises the temperature enough that chemical reaction can occur. However, since the chemical reaction rates are very sensitive to temperature, most of the reaction is concentrated in a thin zone near the high-temperature side. The flow within the zone may be laminar or turbulent. Almost all flames in practical combustion devices are turbulent because turbulent mixing increases burning rates, allowing more power per unit volume.
2. Detonation, when the combustion wave propagates at supersonic speeds. In this case, detonation is a shock wave sustained by energy released by combustion. The combustion process, in turn, is initiated by shock wave compression and results in high temperatures. Detonations involve interaction between fluid mechanic processes (shock waves) and thermochemical processes (combustion).
In this case, there are qualitative differences between upstream and downstream properties across the detonation wave which are similar to property differences across a normal shock although there also some significant differences between a normal shock wave and detonation wave. Detonation and deflagration velocities may be found by an analysis similar to normal shock analyses in Chap. 1.
3. Homogeneous flame, when one has a fixed mass (control mass) with uniform spatial distributions of temperature and pressure as well as a uniform composition. Furthermore, there is no “propagation” in space but is dynamic in time.
Chemical reactions relevant to combustion are generally very complicated and are approximated by a one-step overall reaction. To obtain quantitative model of the reaction rate, one can apply the law of mass action which states that the rate of reaction is proportional to the number of collisions between the reactant molecules, which in turn is proportional to the concentration of each reactant. Consider a reaction of the form,
nAA + nBB! nCC + nDD. (5.4.1)
The law of mass action states that the rate of reaction,
df Щ= d( Щ = _d( Щ = _d( [D]
dt nA dt nB dt nC dt nD
where [S] is the concentration of the species S in moles per unit volume. The proportionality constant kf is known as the forward reaction rate constant. The total number of moles of gas per unit volume for all reactants is calculated from the universal gas law, p = npRuT, where Ru is the universal gas constant. The concentration of a particular species is then found by multiplying the total concentration by the ratio of the number moles of the species in the reaction nS to the total number of moles n. The reaction rate constant kf is usually of the Arrhenius form
kf = kf 0Ta exp(-E/RuT) (5.4.3)
where E is the activation energy in calories/mole, and kf0 and a are constants which are determined experimentally. Boltzman showed in the 1800s that the fraction of molecules in a gas with translational kinetic energy greater than some value E is proportional to exp(—E/RuT). Thus, E represents the ‘‘energy barrier’’ that must be overcome for reaction to occur.