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Discussion about thermal chemical-equilibrium gas and calorically perfect gas

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Calculate the percent difference of P2/P1, T2/T1, and ρ21 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

    2. Relevant 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

    3. 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
    ρ21=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.
     
  2. jcsd
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