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Homework check (too easy) with on/off design analysis for variable area turbojet

  1. I'm studying on variable area turbojet problems in the book and I can't seem to get past this problem.

    Consider the performance of an ideal non-afterburning turbojet with flow at station 4(turbine entrance) and station 8 (nozzle throat) choked. A4 is fixed and A8 is varied in order to maintain constant compressor total pressure ratio ([tex]\pi_c[/tex]).

    On-design conditions are as follows:
    [tex]
    \pi_{cR}=15, M_{oR}=2.0, \tau_{\lambda{R}}=7.0
    [/tex]

    Find the required ratio of nozzle throat area (off-design) to nozzle throat area (on-design) for the engine operating at the same flight Mach number (2.0) but at the off-design condition such that [tex]\tau_\lambda=6.0[/tex].

    My work:

    Using Mach number 2

    On-design

    [tex]
    \tau_{rR}=0.8, \pi_{rR}=.458, \tau_{cR}=2.168, \tau_{tR}=.867, \pi_{tR}=.607
    [/tex]

    [tex]
    \tau_{tR}=1-\frac{\tau_{rR}}{\tau_{\lambda{R}}}(\tau_{cR}-1)=1-\frac{.8}{7}(2.168-1)=.867
    [/tex]


    Off-design

    Same values as on-design because Mach number does not change.
    [tex]
    \tau_{r}=0.8, \pi_{r}=.458, \tau_{c}=2.168, \tau_{t}=.867, \pi_{t}=.607
    [/tex]

    Using given [tex]\lambda=6[/tex] for off-design:

    [tex]
    \tau_t=1-\frac{\tau_r}{\tau_\lambda}(\tau_c-1)=1-\frac{.8}{6}(2.168-1)=.844
    [/tex]

    Hence, the area ratio:
    [tex]
    \frac{A_8}{A_{8R}}=\frac{\tau_t^\frac{1}{2}}{\pi_T}\frac{\pi_{tR}}{\tau_{tR}^\frac{1}{2}}=.9866.
    [/tex]

    Is this correct? This seems a little too easy.
     
    Last edited: Feb 12, 2011
  2. jcsd
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