(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a stationary normal shock wave in pure diatomic nitrogen. The velocity and temperature upstream are known. Calculate the temperature ratio across the shock, assuming local thermodynamic equilibrium. Neglect any chemical reactions and electronic energy.

[tex]U_2=3000m/s[/tex]

[tex]T_2=300K[/tex]

[tex]\Theta_{V,N_2}=3390K[/tex]

2. Relevant equations

Continuity equation

Momentum equation

Energy equation

[tex]\epsilon=\frac{\rho_1}{\rho_2}[/tex]

3. The attempt at a solution

So this will be an iterative process. I am programming the solution in MATLAB but ran into a problem figuring out the last of the iterative solution steps. The steps I have found so far are as follows.

1) Assume a value for [tex]\epsilon[/tex]

2) Calculate [tex]U_1=\frac{U_2}{\epsilon}[/tex]

3) Calculate [tex]h_1=h_2+\frac{U_2^2-U_1^2}{2}[/tex]

For diatomic nitrogen,

[tex]h_2=7/2RT_2+\left[\frac{\Theta_V/T_2}{e^{\Theta_V/T_2}-1}\right]RT_2[/tex]

4) Calculate [tex]T_1=\frac{h_1}{7/2R}[/tex]

Now I need to figure out the check for the iterative process. Here is my idea. Using the Eqair applet here http://www.dept.aoe.vt.edu/~devenpor/tgas/" [Broken], I can solve for h_{1}if I new what the pressure and the density was. My professor posted MATLAB version of the Eqair applet. The inputs are pressure and density and the outputs are temperature and enthalpy. So the big question is how do I find what the pressure and density are before the shock. If only I could figure out what the density after the shock wave was I can then use the continuity equation to solve for the density before the shock wave and also the pressure.

I need to know what the enthalpy is from Eqair applet so I can use the sectant method to find my next ε value.

[tex]\epsilon_{i+1}=\epsilon_i-\frac{\Delta h_i}{\frac{\Delta h_i-\Delta h_{i-1}}{\epsilon_i-\epsilon_{i-1}}}[/tex]

[tex]\Delta h_i=|\bar{h}_i-h_i|[/tex]

[tex]\Delta h_{i-1}=|\bar{h}_{i-1}-h_{i-1}|[/tex]

This is the check that I will be implementing based off of some tolerance value.

if Δh_{i}≤.0001, then end the loop and calculate T2/T1.

I've been trying for the past few days now to figure out if I missed anything or if there is another way to go about the problem. Any suggestions or anything I messed up?

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# Temperature ratio across normal shockwave

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