Derive equation 7.55 in Turns starting from the following energy equation: $$\frac{\partial \rho e}{\partial t} = -\nabla\cdot(\rho\vec{v} e)-\nabla\cdot\vec{q}-\nabla\cdot(\mathbf{\tau}\cdot\vec{v})-\nabla\cdot(P\vec{v})+\rho\vec{g}\cdot\vec{v}$$
The mixture fraction can be used to express a species composition like CO$_2$ in a steady laminar flame using the equation: $$\chi\frac{\partial^2y_i}{\partial\xi^2} = -\frac{\omega_iM_i}{\rho}.$$
This equation balances reaction and diffusion (diffusion in the mixture fraction coordinate, where $\chi$ acts like a diffusion coefficient.
Use the Cantera premixed flame code to compute the flame thickness for a stoichiometric propane-air flame at P=1, 10, and 100 atm. Take 300 K reactants. What are the implications for the design of explosion-proof housings for electrical devices?