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I am considering the flow field around a hypersonic glide vehicle at $M=7$. As I understand it, air is in chemical equilibrium if the fluid time scale $t_f \ll t_c$, where $t_c$ is the chemical time scale. I can estimate $t_f$ very easily, but I have no clue how to estimate $t_c$.
Could someone tell me how to estimate $t_c$ for my problem ?
The chemical kinetics of air depend on both how fast you are flying and your altitude. Fortunately, NASA has studied these issues. The figures below are from NASA Report NACA-TN-4359. The predominant chemistry in the stagnation region of an airfoil as a function of flight speed and altitude are shown below:
You say $M=7$. If your vehicle is near sea level, then that is around 6,500 feet per second which, according to the above plot, would mean that vibrational kinetics of the molecules are important but that chemical dissociation reactions are not yet important.
You ask how to estimate chemical reaction time, $t_c$. The same concept can be characterized by chemical reaction length: $d_c = u \times t_c$ where $d_c$ is the characteristic length for the reactions and $u$ is the flow speed. This is handy because it is easy to compare with the dimensions of your vehicle. NASA calculates the characteristic lengths for vibration excitation as follows:
As you can see, for a given flow speed, the characteristic length varies strongly with altitude.
The air is in vibrational equilibrium when the the characteristic length from the plot above is much smaller than the characteristic lengths of your glide vehicle and it's airfoils.
At higher altitudes or higher speeds, molecular dissociation becomes important (see Fig 1 above). NASA calculates the characteristic lengths for oxygen dissociation as follows:
Again, the characteristic length depends strongly on both speed and altitude.
The air will be in (or near) equilibrium for oxygen dissociation when the characteristic length from the plot above is much smaller than the characteristic lengths of your vehicle.
