Revisica
Sign in
ENGR5003 / Week 2
L4 slide 18: Show the Brayton work ratioFrom lectureCurrent spec2:16 marks8 min

ENGR5003 Lecture 4, slide 18 (Imperfect Brayton cycle - task left for students)

The work ratio rwr_w is the net work output divided by the gross work output: rw=wout−winwout=1−winwoutr_w = \frac{w_{out} - w_{in}}{w_{out}} = 1 - \frac{w_{in}}{w_{out}} The lecturer leaves it as a task to show that, for the ideal Brayton cycle (air as an ideal gas with constant cpc_p and γ\gamma, pressure ratio rp=p2/p1=p3/p4r_p = p_2/p_1 = p_3/p_4), rw=1−T1T3 rp(γ−1)/γr_w = 1 - \frac{T_1}{T_3}\, r_p^{(\gamma-1)/\gamma}

Formulas you may need
  • SFEE for adiabatic compressor and turbine: win=cp(T2−T1)w_{in} = c_p(T_2 - T_1), wout=cp(T3−T4)w_{out} = c_p(T_3 - T_4) (on the formula sheet)
  • Isentropic relation: T2T1=(p2p1)(γ−1)/γ\dfrac{T_2}{T_1} = \left(\dfrac{p_2}{p_1}\right)^{(\gamma-1)/\gamma} (on the formula sheet)
  • Brayton work ratio: rw=1−T1T3rp(γ−1)/γr_w = 1 - \dfrac{T_1}{T_3}r_p^{(\gamma-1)/\gamma} (on the formula sheet; derivation left as a task in Lecture 4)
  1. (a)
    Show this result.
    [4]
  2. (b)
    Explain why the practical efficiency of a real gas turbine improves as rwr_w approaches 1, and hence why T3T_3 should be as high as possible.
    [1]
  3. (c)
    [Added check] Evaluate rwr_w for the Lecture 4 Exercise 1 data: T1=300T_1 = 300 K, T3=1200T_3 = 1200 K, rp=8r_p = 8, γ=1.4\gamma = 1.4.
    [1]