Shock heating versus channel heating for NO_x formation

It has been suggested that NO_x formation occurs in the shock front that is generated by the rapid release of energy that occurs when an electrical discharge is fired [Tuck, 1976; Chameides et al., 1977; Chameides, 1979; Levine et al., 1979]. The amount of NO_x formed was calculated by assuming that the bulk of the gas entrained by the shock front was compressed into a region just behind the shock front, so the temperature and rate of cooling of the bulk of the gas could be approximated by the temperature behaviour at the shock front [Tuck, 1976; Chameides et al. 1977; Chameides, 1979]. The behaviour of strong shock waves has been calculated by Lin [1954] for an idealised, instantaneous release of energy along a line source and this allowed Chameides [1979] to find the rate of cooling of the gas at the shock front (tau_cooling). Thermochemical and kinetic calculations of [NO]_equil. and tau_equil. allowed the freeze out temperature (T_{freeze out}) and NO mixing ratio, and hence the amount of NO_x formed by the discharge (P), to be calculated via equations 1-2.

However, Hill et al. [1979, 1980] have, on the basis of numerical simulation of lightning, suggested that due to the relatively long duration of the energy release (10 - 100 µs), the resulting shock front is not well represented by the analysis of Lin, which assumes an instantaneous release of energy, and is not energetic enough to reach the temperatures required to fix nitrogen. It was instead suggested that NO_x was formed predominantly in the hot channel region of high temperature and low density well behind the shock front.

The question of whether the resulting shock fronts do raise the air temperature sufficiently to fix nitrogen can be addressed by monitoring the velocity of the front as it propagates away from the discharge. The close relationship that exists between the shock velocity and the temperature jump across the shock, given by the Rankine-Hugoniot equations [eg. Chorlton, 1967], allows the shock front temperature used in Chameides analysis to be determined experimentally:

    Ts            [2gamma M2 - (gamma - 1)][2 + (gamma - 1)M2]

    --    =       ------------------------------------------               (3)

    To                           [(gamma + 1)2M2]

where T_o is the ambient temperature and T_s is the temperature just behind the shock front, gamma is the ratio of specific heats, and M = U/c, where U is the velocity of shock front and c is the velocity of sound in unperturbed gas.

The maximum shock front velocity from figure 2 is 2.2×10³ m s^-1 and given that gamma = 1.335 at ca. 2000 K and a = 330 m s^-1 at 300 K [Zel'dovich and Raizer, 1966], the maximum temperature immediately behind the shock front would be 2480 K. This temperature would be sufficient to produce NO_x if it was maintained for a sufficiently long time. However, from figure 6, tau_equil.(2480 K) 6×10^-2 s whereas tau_cooling < 10^-5 s for the shock front (from the decrease in the front velocity with time from figure 2), the air entrained by the shock front is not at 2480 K for long enough for significant NO_x production to occur. A temperature of greater than 4500 K is required to give tau_equil. < 10^-5 s, corresponding to a shock velocity of greater than 3600 m s^-1. The measured velocities are well below this value, so for these experiments, NO_x formation is unlikely to be by shock heating. Nevertheless, NO_x is still produced in significant quantities by these discharges, suggesting that formation is occurring in the hot channel region as suggested by Hill et al.

Also shown for comparison on figure 2 is the radial propagation of the shock front as calculated by Hill [1971] using a channel heating model, and by Lin [1954], which assumes an instantaneous linear release of energy, and on which Chameides theory of NO_x formation by lighting shock fronts is based. Lin's theory greatly overestimates the radial velocity of the front, while the calculations of Hill give velocities closer to the experimental values. The comparison is reasonable considering that both the energy per unit length and ambient pressure used in Hill's calculation are twice as high as used in the experiments. However, because no information has been obtained in this work on the energy or density distribution behind the shock front, it is not possible to determine from these experiments the fraction of the total discharge energy that is released as kinetic energy of the shock front. Nevertheless, it is not necessary to have a detailed description of the discharge development to refute the suggestion that NO_x is formed by the shock front in these experiments, as the temperature immediately behind the shock front can be determined solely from the Rankine-Hugoniot relationship which depends only on gamma and the front velocity.

Some further evidence for this model was presented by Picone et al. [1981] who used Schlieren photography to examine the time development of the hot channel of a 15 cm, 600 J m^-1 discharge. As the shock front was recorded in the first few photographs, the maximum observed front velocity can be determined as approximately 500 m s^-1 (ca. 30µs after start of discharge), again, too slow for NO_x formation in the front.