Phenomenological analysis of a deflagration wave

The burning velocity Vq can be related to this wave thickness as follows. The mass of combustible material per unit area per second flowing into the wave is PqVq, where po is the density of the initial combustible gas mixture. The deflagration wave consumes these reactants at a rate wd (mass per unit area per second). Hence mass conservation implies that PqVq = wd, which, in conjunction with equation (1), yields 

Equation (2) shows that the burning velocity is proportional to the square root of a reaction rate and to the square root of the ratio of the thermal conductivity to the specific heat. The factor on the right-hand side of equation (2) with the strongest temperature dependence is usually w, which varies roughly as where is an activation energy, and 

A and Cp are independent of the pressure py Po P according to the ideal gas law [equation (1-9)], and w p , where n is the order of the reaction [see equation (1-8)]. Hence the pressure and temperature dependences of Vq are given approximately by 

In principle, equation (2) should provide a numerical estimate for Vq. However, in practice, values of w are so uncertain (see Appendix B) that equation (2) is more useful for estimating w from experimental values of Vq. A useful way to deduce an overall order and an overall activation energy for a reaction is to measure the T and p dependences of Vq and to correlate these empirically with equation (3) by adjusting n and E.  

Empirical laminar burning velocities lie between 1 and 1000 cm/s. Since these velocities are small compared with the speed of sound, equation (1-25) is valid for laminar flames. Thus laminar flames are nearly isobaric, we were justified in not attaching a subscript to p in equation (3), and the quantity Cp appearing above is the specific heat at constant pressure. 

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