Problem 1
Read the notes on the derivation of the transport equations. Write a summary of key points from this, such as steps in using the RTT, assumptions made, or observation, tricks, or key points.
Problem 2
List five assumptions used in deriving the Shvab-Zeldovich form of the energy equation.
Problem 3
The mixture fraction can be used to express a species composition, like CO$_2$, in a steady laminar flame using the flamelet equation
$$\chi\frac{\partial^2y_i}{\partial\xi^2} = -\frac{\dot{m}^{\prime\prime\prime}}{\rho}.$$
This equation balances reaction and diffusion (diffusion in the mixture fraction coordinate, where $\chi$ acts like a diffusivity, or a mixing rate).
Part A
Sketch the profile of a species like CO$_2$ versus:
- $\xi$ for a typical $\xi_{st}$, and
- for $\xi_{st}=0.5$. You have already created profiles like this before in your homework and exam.
Part B
Based on the shape of these profiles and the flamelet equation, which of the two cases would you expect to have a higher $\chi$ when blowout occurs? Turbulence intensity is directly related to $\chi$ so the case with the higher $\chi$ at blowout will be more resistent to blowout and can handle higher mixing rates (actually, $\chi$ is a mixing rate, similar to $1/\tau$ in a PSR). Higher mixing rates give higher combustion rates give smaller required combustion volumes.
Part C
Based on the above equation, if the right-hand side is fixed, then if $\chi$ increases, what happens to the shape of the term it multiplies? Sketch the $y_i$ profile qualitatively versus $\xi$ as $\chi$ changes.
Problem 4
Use the Cantera premixed flame code to compute the flame thickness for a stoichiometric methane-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?
Problem 5
Turns 8.6
Problem 6
Turns 8.9
Problem 7
Turns 8.15