- #1
roldy
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Homework Statement
Calculate the percent difference of P2/P1, T2/T1, and ρ2/ρ1 between the CPG assumption and the thermal-chemical equilibrium assumption. Which percent difference is the lowest and what could be the possible reason?
U1=4000 m/s
Altitude=60 km
R=8314 N m/(kmol K)
P1=21.96 Pa
ρ1=.0003097 kg/m3
Homework Equations
CPG:
[tex]\frac{P_2}{P_1}=1+\frac{2\gamma}{\gamma+1}(M^2-1)[/tex]
[tex]\frac{T_2}{T_1}=1+\frac{2\gamma}{\gamma+1}(M^2-1)\frac{2+(\gamma-1)M^2}{(\gamma+1)M^2}[/tex]
[tex]\frac{\rho_2}{\rho_1}=\frac{\gamma+1}{2+(\gamma-1)M^2}[/tex]
TCE:
Used an iterative process describe as follows:
Pick an ε1=.001 and ε2=.1
1)[tex]U_{2_i}=\epsilon_iU_1[/tex]
2)[tex]\rho_{2_i}=\frac{\rho_1}{\epsilon_i}[/tex]
3)[tex]P_{2_i}=P_1+\rho_1U_1^2(1-\epsilon_i)[/tex]
4)[tex]T_{2_i}=\epsilon_i \left(T_1+\frac{U_1^2}{R}(1-\epsilon_i)\right)[/tex]
5)[tex]h_{2_i}=h_1+1/2U_1^2(1-\epsilon_i^2)[/tex]
6) The pressure and density found in steps 2 and 3 are used as inputs in the supplied function to calculate the enthalpy[tex]\bar{h}_{2_i}=f(P_{2_i},\rho_{2_i})[/tex]
7)[tex]\Delta h_i=|\bar{h}_{2_i}-h_{2_i}|[/tex]
8) If Δhi≤.0001, then end the iteration. If hi> .0001, find a new ε in step 9.
9) [tex]\epsilon_{i+1}=\epsilon_i-\frac{\Delta h_i}{\frac{\Delta h_i-\Delta h_{i-1}}{\epsilon_i-\epsilon_{i-1}}}[/tex]
10) Repeat steps 1-9 until convergence
The Attempt at a Solution
I programmed this in MATLAB and the results are as follows
Percent difference between CPG and TCE:
P2/P1=3100.2
T2/T1=64.452
ρ2/ρ1=259.61
Assuming these values are correct, the temperature ratio difference is the lowest out of the three. I cannot figure out the reason behind this. I've been trying to compare the dependencies of the CPG functions to the TCE ones. For CPG the pressure, temperature, and density ratios are a function of the Mach number. For TCE, the dependence is on U1 and T1. Could someone explain to me why the temperature ratio is the lowest? I can't find anything in my book or online. I know it's some basic concept I'm overlooking.