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ENGR5003 / Week 1
Old spec L7 worked example - isochoric / isentropic / isothermal cycleTutorialOld spec ENGR2172:115 min

ENGR217 Lecture 7 worked example (slides 17-23)

Air (an ideal gas with constant cvc_v) is heated in a closed volume to increase its temperature from T1T_1 to T2T_2. The air then expands reversibly and adiabatically (i.e. isentropically) until its temperature returns to T1T_1. It finally returns to its initial state (pressure and volume) by an isothermal compression.

Formulas you may need
  • First law: ΔU=Q−W\Delta U = Q - W; ideal gas dU=mcv dTdU = mc_v\,dT (on the formula sheet)
  • Adiabatic process: TVγ−1=constTV^{\gamma-1} = \text{const}; adiabatic work W=−ΔUW = -\Delta U (on the formula sheet)
  • Isothermal ideal gas: W=mRgTln⁡(V2/V1)W = mR_gT\ln(V_2/V_1), ΔU=0\Delta U = 0 so Q=WQ = W (on the formula sheet)
  • Rg=cv(γ−1)R_g = c_v(\gamma - 1) (on the formula sheet)
  • Reversible heat at constant TT: ΔS=Q/T\Delta S = Q/T (on the formula sheet)
  • Carnot efficiency: η=1−TC/TH\eta = 1 - T_C/T_H (learn this)
  1. (a)
    Draw the p-V diagram for this sequence of events.
  2. (b)
    Show that the thermal efficiency of this cycle is ηth=net workheat supplied=1−T1T2−T1ln⁡T2T1\eta_{th} = \dfrac{\text{net work}}{\text{heat supplied}} = 1 - \dfrac{T_1}{T_2 - T_1}\ln\dfrac{T_2}{T_1}.
  3. (c)
    Find the change in entropy (per unit mass) during the isothermal compression in terms of cvc_v, T1T_1 and T2T_2.
  4. (d)
    Evaluate the efficiency and the entropy change in (c) for T1=300 KT_1 = 300\ \mathrm{K}, T2=600 KT_2 = 600\ \mathrm{K}, cv=0.718 kJ/(kg K)c_v = 0.718\ \mathrm{kJ/(kg\,K)}, and compare with a Carnot engine between the same temperatures.