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Thermodynamics - Quasi 1-D Air Mixture

  1. Sep 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Constant area mixing tube with stations (1) and (2) (i.e., (1)-->-->--(2))
    At station (1):
    - Mixture of two air streams, defined as Primary (p) and Secondary (s)
    u_p = 300 m/s
    T_p = 900K
    u_s = 30 m/s
    T_s = 300K
    p_1 = 0.1 MPa
    (A_p)/(A_s) = 1/3

    1-D flow assumed

    Find: Pressure, Velocity, and Temperature at station (2)

    2. Relevant equations

    [itex]\dot{m}_{1}=\dot{m}_{2}[/itex] (Continuity)
    [itex]\dot{m}_{1}h_{01}=\dot{m}_{2}h_{02}[/itex] (1st Law)
    [itex]\sum \vec{F}=\frac{d}{dt}\int{\rho\vec{u}(\vec{u}\hat{n})dV}+\int{\rho\vec{u}(\vec{u}\hat{n})dA} [/itex](Conservation of Momentum)
    [itex]p=\rho R T[/itex] (perfect-gas law)

    3. The attempt at a solution
    From Continuity:
    [itex]\dot{m}_{p}+\dot{m}_{s}=\dot{m}_{2}[/itex]
    [itex]\rho_{p}u_{p}A_{p}+\rho_{s}u_{s}A_{s}=\rho_{2}u_{2}A_{2}[/itex]
    [itex]\frac{1}{4}\rho_{p}u_{p}+\frac{3}{4}\rho_{s}u_{s}=\rho_{2}u_{2}[/itex]

    From 1st law:
    [itex]\dot{m}_{p}(h_{p}+\frac{1}{2}u_{p}^{2})+\dot{m}_{s}(h_{s}+\frac{1}{2}u_{s}^{2})=\dot{m}_{2}(h_{2}+\frac{1}{2}u_{2}^{2})[/itex] (incidentally, anyone know why this LaTeX isn't working properly?)
    [itex]\frac{1}{4}\rho_{p}u_{p}(c_{p_{p}}T_{p}+\frac{1}{2}u_{p}^{2})+\frac{3}{4}\rho_{s}u_{s}(c_{p_{s}}T_{s}+\frac{1}{2}u_{s}^{2})=\rho_{2}u_{2}(c_{p_{2}}T_{2}+\frac{1}{2}u_{2}^{2})[/itex]

    From Conservation of Momentum:
    [itex]\int{\rho\vec{u}(\vec{u}\hat{n})dA}=p_{1}-p_{2}[/itex]
    [itex]\frac{1}{4}\rho_{p}u_{p}^{2}+\frac{3}{4}\rho_{s}u_{s}^{2}-\rho_{2}u_{2}=p_{1}-p_{2}[/itex]

    I can get [itex]\rho_{2}u_{2}[/itex] and the stagnation enthalpy at station 2, but beyond that I'm not sure how to proceed. I feel like there's an assumption that I'm missing which would allow me to calculate [itex]p_{2}[/itex] or some other condition at station 2, and then use the perfect-gas law to fill things out, but I'm not sure what that is. I freely admit that there's probably something about how the conditions necessary to apply these equations interact which I'm not understanding.
     
  2. jcsd
  3. Oct 3, 2012 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    Sorry you have no replies yet. I personally don't know what a mixing tube is. A diagram would help.
     
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