Combustion waves, steady-state

The combustion wave of a premixed gas propagates with a certain velocity into the unburned region (with flow speed = 0). The velocity is sustained by virtue of thermodynamic and thermochemical characteristics of the premixed gas. Figure 3.1 illustrates a combustion wave that propagates into the unburned gas at velocity Mj, one-dimensionally under steady-state conditions. If one assumes that the observer of the combustion wave is moving at the same speed, Wj, then the combustion wave appears to be stationary and the unburned gas flows into the combustion wave at the velocity -Wj. The burned gas is expelled downstream at a velocity of-M2 with respect to the combustion wave. The thermodynamic characteristics of the combustion wave are described by the velocity (u), pressure (p), density (p), and temperature (T) of the unburned gas (denoted by the subscript 1) and of the burned gas (denoted by the subscript 2), as illustrated in Fig. 3.1. 

A deflagration wave formed by a reactive gas under one-dimensional steady-state flow conditions is illustrated in Fig. 3.7. In the combustion wave, the temperature increases from the initial temperature of the unburned gas to the ignition temperature and then reaches the flame temperature. The heat generated in the reaction zone is transferred back to the unbumed gas zone. 

In order to understand the fundamental concept of the cause of temperature sensitivity, in the analysis described in this section it is assumed that the combustion wave is homogeneous and that it consists of steady-state, one-dimensionally successive reaction zones. The gas-phase reaction occurs with a one-step temperature rise from the burning surface temperature to the maximum flame temperature. 

Fig. B-1 presents a steady-state flow in a combustion wave, showing mass, momentum, and energy transfers, including chemical species, in the one-dimensional space of Ax between Xj and %2- The viscous forces and kinetic energy of the flow are assumed to be neglected in the combustion wave. The rate of heat production in the space is represented by coQ, where ai is the reaction rate and Qis the heat release by chemical reaction per unit mass.

The velocity of advance of the front is super sonic in a detonation and subsonic in a deflagration. In view of the importance of a shock process in initiating detonation, it has seemed difficult to explain how the transition to it could occur from the smooth combustion wave in laminar burning. Actually the one-dimensional steady-state combustion or deflagration wave, while convenient for discussion, is not easily achieved in practice. The familiar model in which the flame-front advances at uniform subsonic velocity (v) into the unburnt mixture, has Po> Po> an[Pg.249]

The high flame front velocities prior to attainment of the steady state probably result from the transient conditions between the combustion front and shock front. Sufficient data were lacking to show whether the shock-heated gas ignited spontaneously, immediately behind the shock front, or whether the flame front overtook the shock front. In any event, the combustion wave finally moves along with the shock wave, thus forming a detonation wave... 

The instant at which combination of the combustion and shock waves begins is not necessarily the point at which the steady-state detonation velocity has been reached. Because of the turbulence generated by the flame, the velocity of the products relative to the initially forming detonation front may be subsonic. This state cannot last very long because rarefaction waves thru the burnt gas can move right up to the detonation wave and weaken it until its speed drops to unit Mach number relative to the products... 

No theoretical criterion for flammability limits is obtained from the steady-state equation of the combustion wave. On the basis of a model of the thermally propagating combustion wave it is shown that the limit is due to instability of the wave toward perturbation of the temperature profile. Such perturbation causes a transient increase of the volume of the medium reacting per unit wave area and decrease of the temperature levels throughout the wave. If the gain in over-all reaction rate due to this increase in volume exceeds the decrease in over-all reaction rate due to temperature decrease, the wave is stable otherwise, it degenerates to a temperature wave. Above some critical dilution of the mixture, the latter condition is always fulfilled. It is concluded that the existence of excess enthalpy in the wave is a prerequisite of all aspects of combustion wave propagation. 

Frequently in theoretical work on the subject, whether dealing with the steady or nonsteady state, the mathematical development of an adopted model is followed by a descriptive summary of the results which is rarely traced back clearly to the assumptions inherent in the model. This has often resulted in extravagant identification of the model with the actual phenomenon and has been an obstacle in the task of reconciling conflicting views. There is need, therefore, of a purely descriptive exposition, stripped as completely as possible of mathematical language, to clarify the physical concepts of the combustion wave phenomenon. 

It is found experimentally that limit mixtures, incapable of supporting combustion waves, nevertheless have theoretical thermodynamic flame temperatures of the order of 1000° C. or more. It is, therefore, not immediately clear why combustion waves, albeit slowly propagating, should not develop in mixtures possessing such substantial chemical enthalpy. The question arises whether the observed limits of flammability are true limits or whether such mixtures are actually capable of supporting combustion waves but are prevented from doing so by experimental limitations. Experimentalists believe that the limits are true. On the other hand, no theoretical criterion for the limit is obtained from the steady-state equations of the combustion wave. That is, the equations describe combustion waves without differentiating between mixtures that are known to be flammable and mixtures that are known to be nonflammable. Therefore, for nonflammable mixtures the combustion wave becomes unstable to perturbations and thus disappears (7). Conversely, for flammable mixtures the combustion wave can overcome perturbations—i.e., it returns to the steady state after being perturbed. 

The quantity, h, in Equation 5 is not likely to be greatly different from its value in a plane adiabatic combustion wave. Taking x as the coordinate normal to such wave, h becomes the integral of the excess enthalpy per unit volume along the x-axis, so that the differential quotient, dh/dx, represents the excess enthalpy per unit volume in any layer, dx. Assuming the layer to be fixed with respect to a reference point on the x-axis, the mass flow passes through the layer in the direction from the unbumed, w, to the burned, 6, side at a velocity, S, transporting enthalpy at the rate Sdh/dx. Because the wave is in the steady state, heat flows by conduction at the same rate in the opposite direction, so that... 

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