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ENGR5003 / Week 4
L10 slide 18: Show the dual-cycle efficiencyFrom lectureCurrent specFirst12 marks18 min

ENGR5003 Lecture 10, slide 18 (Dual cycle - exercise left for students)

The ideal dual cycle models a real diesel engine: 1-2 isentropic compression (compression ratio r=V1/V2r = V_1/V_2), 2-3 constant-volume heat addition, 3-4 constant-pressure heat addition, 4-5 isentropic expansion to V5=V1V_5 = V_1, 5-1 constant-volume heat rejection. Air is an ideal gas with constant cvc_v, cpc_p. The slide gives ηth=1−cv(T5−T1)cv(T3−T2)+cp(T4−T3)\eta_{th} = 1 - \frac{c_v(T_5 - T_1)}{c_v(T_3 - T_2) + c_p(T_4 - T_3)} and leaves it as an exercise to show that this can be written as ηth=1−1rγ−1[rprcγ−1(rp−1)+γrp(rc−1)],rp=p3p2, rc=V4V3\eta_{th} = 1 - \frac{1}{r^{\gamma-1}}\left[\frac{r_p r_c^{\gamma} - 1}{(r_p - 1) + \gamma r_p (r_c - 1)}\right], \qquad r_p = \frac{p_3}{p_2},\ r_c = \frac{V_4}{V_3}

Formulas you may need
  • Isentropic ideal gas: TVγ−1=constTV^{\gamma-1} = \text{const} (on the formula sheet)
  • Ideal gas: pV=mRgTpV = mR_gT; γ=cp/cv\gamma = c_p/c_v (on the formula sheet)
  • Otto: η=1−1rγ−1\eta = 1 - \dfrac{1}{r^{\gamma-1}}; Diesel: η=1−1rγ−1rcγ−1γ(rc−1)\eta = 1 - \dfrac{1}{r^{\gamma-1}}\dfrac{r_c^\gamma - 1}{\gamma(r_c - 1)} (on the formula sheet)
  • Dual cycle efficiency in terms of rr, rpr_p, rcr_c (NOT given: left as an exercise on the slide)
  1. (a)
    Show this result.
    [7]
  2. (b)
    Show that the dual-cycle result reduces to the Otto efficiency when rc=1r_c = 1 and to the Diesel efficiency when rp=1r_p = 1.
    [2]
  3. (c)
    [Added check] For r=16r = 16, rp=1.5r_p = 1.5, rc=1.6r_c = 1.6, γ=1.4\gamma = 1.4 and T1=300T_1 = 300 K, evaluate ηth\eta_{th} from the formula, and confirm it by finding T2T_2 to T5T_5 and using the heat-based expression.
    [3]