Temperature ratio across normal shockwave

[roldy]

Homework Statement

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.

$$U_2=3000m/s$$
$$T_2=300K$$
$$\Theta_{V,N_2}=3390K$$

Homework Equations

Continuity equation
Momentum equation
Energy equation

$$\epsilon=\frac{\rho_1}{\rho_2}$$

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 $$\epsilon$$
2) Calculate $$U_1=\frac{U_2}{\epsilon}$$
3) Calculate $$h_1=h_2+\frac{U_2^2-U_1^2}{2}$$
For diatomic nitrogen,
$$h_2=7/2RT_2+\left[\frac{\Theta_V/T_2}{e^{\Theta_V/T_2}-1}\right]RT_2$$
4) Calculate $$T_1=\frac{h_1}{7/2R}$$

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/" , I can solve for h₁ 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.

$$\epsilon_{i+1}=\epsilon_i-\frac{\Delta h_i}{\frac{\Delta h_i-\Delta h_{i-1}}{\epsilon_i-\epsilon_{i-1}}}$$

$$\Delta h_i=|\bar{h}_i-h_i|$$
$$\Delta h_{i-1}=|\bar{h}_{i-1}-h_{i-1}|$$

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?

 

[LawrenceC]

You have a 3390K number in your post. What is it?

As I recall there are textbook arithmetic relations for normal shock computation. They provide pressure ratios, density ratios, and entropy changes across the shock as a function of the upstream mach number and specific heat ratio. The one for entropy change requires constant volume specific heat in addition to mach number and specific heat ratio. Aren't they applicable to this problem?

 

[roldy]

3390 K is the characteristic temperature for molecular vibration of N₂. I know what the equations you are talking about are. Those would work if I didn't have to deal with this molecular vibration. Because this shock wave is in thermodynamic equilibrium and I'm only given the velocity and temperature after the shock, I have to solve iteratively. Assume a value for the density ratio, solve for the velocity before the shock, calculate the temperature, etc. Unfortunately I can not figure out the next step.

 

[LawrenceC]

I'm guessing here but with molecular vibration of nitrogen is stagnation temperature constant across the shock as it is when vibration is not considered? Or is stagnation temperature plus the kinetic energy of the vibration a constant? There is no work being done.

Whenever involved with an iterative process, a wrong estimate produces a violation of some basic law so the estimated is adjusted until the law is no longer violated. I'm guessing again but would an energy balance of some sort be appropriate as a check?

 

[roldy]

The stagnation temperature of an ideal gas is constant across a normal shock wave. I have no way of calculating the energy from the molecular vibration since I do not know the frequency of the vibrations. I don't think it's as easy as adding the vibration and the stagnation temperature, I could be wrong though. Also, the gas constant on both sides of the shock wave are equal to each other.

On other homework assignments involving an iterative process, we would use the following procedure. Note, the following steps are for solving the state variable AFTER a normal shock wave with known values of P₁ and h₁ known.

1) start the iteration with ε₁=0.001 and ε₂=0.1.

2) Solve for P₂ using ε₂
$$P_2=P_1+\rho_1U_1^2(1-\epsilon_2)$$

3) Solve for h₂ using ε₂
$$h_2=h_1+1/2U_1^2(1-\epsilon_2^2)$$

4) Using the Eqair applet, find h₂' as a function of the pressure and density. Meaning, in the applet select pressure and density and type in the value for pressure found in step 2. ρ₂=ρ₁/ε₂.

5) With the value found in step 4, find the absolute value of the difference of the two. Δh=|h₂'-h₂|

6) Check to see if the difference found in step 5 is less than the tolerance of .0001. If it is, Then the calculated values of pressure, enthalpy, and density (as well as anything else that can be calculated from these) are correct. If however, the difference is greater than the tolerance. I go onto step 7

7) Use the secant method to find my next value of ε which is then used in 2, then repeat the procedure.

Sorry for a long post but I wanted you to see where I was coming from for my derivation of the iterative process. I rewrote this procedure in terms of solving for the state variables before the shock wave.

 

[roldy]

So apparently my assumption the problem was wrong. The problem is of a thermally perfect gas. I believe I figured it out. I will post my solution later if anyone cares to confirm it.