Homework 4

Problem 1

Calculate the flame length for a propane-air and hydrogen-air jet flame. Assume ambient and fuel temperatures of 300 K, and a jet exit diameter of 5 mm. Assume a momentum dominated jet.

Problem 2

For a momentum dominated jet, if you double the exit diameter, what happens to the flame length? What happens if you double the stoichiometric mixture fraction? Do these results make sense?

Problem 3

In the notes, we gave the equation $\eta = (\nu^3/\epsilon)^{1/4}$. Derive this expression. Also, what is the relation for $u_\eta$?

Problem 4

Estimate the Kolmogorov length scale and Re during combustion in an autombile engine. (Yep, that’s it.)

Now, if that feels like you don't have enough information, here are a few hints. Take the integral scale as the space between the top of the cylinder head and the top of the cylinder at "top dead center" (TDC). Take the integral velocity as the piston velocity for some appropriate engine RPM and some appropriate piston stroke length. Use some appropriate kinematic viscosity (like the average of the value before and after burning. I made this up. You might think or or find better assumptions.

Problem 5

Compute the flame speed and flame thickness for a premixed propane-air flame at an equivalence ratio of 0.8. For the flame speed, use the correlation in the notes and the online tool. For the flame thickness, just use the online tool. Take reactants at 298 K and 1 atm.

Problem 6

For the conditions of Problem 5, compute the turbulent flame speed using the Damkohler, Klimov, and Clavin/Williams expressions. Use $u^\prime_{rms}=500$ cm/s and $u^\prime_{rms}$ of half the velocity you used in Problem 4 (where we are assuming the engine is operating in the wrinkled flame regime, which isn’t a great assumption). Take $S_L$ from the correlation.

Problem 7

The notes indicated typical soot volume fractions are 1 ppmv and have a particle diameter of approximately 50 nm. What is the number density of soot (number of particles per cm$^3$)?