ChEn 433 Thermodynamic Cycles

David Lignell

Class 6

Outline

  • Cycles
    • Carnot
    • Otto
    • Diesel
    • Brayton
  • Conversion efficiencies

Carnot Cycle

  1. Isothermal heat transfer: hot reservoir to working fluid at \(T_h\).
  2. Adiabatic expansion of working fluid to \(T_c\).
  3. Isothermal heat transfer: working fluid to cold reservoir at \(T_c\).
  4. Adiabatic compression of working fluid to \(T_h\).

Carnot efficiency \(\eta_c = 1-\frac{T_c}{T_h}\)

Otto Cycle

  • Gas power cycle (uses a gas as the working fluid)
  • Spark ignition engines (gasoline)
  • The working fluid is not fixed (intake, exhaust)
  • Four strokes:
    1. Intake
    2. Compression
    3. Expansion
    4. Exhaust

Otto Cycle: steps

  • Steps
    • 0-1: Intake, const P
    • 1-2: \(W_{in}\): Adiabatic compression
    • 2-3: \(Q_{in}\): Heat addition, combustion
      • fast: constant V
    • 3-4: \(W_{out}\): Adiabatic expansion
    • 4-1: \(Q_{out}\): Heat rejection, valve open
      • fast: constant V
    • 1-0: Exhaust
  • How do these correspond to the 4 strokes?
  • Note, the area under the PV diagram is the work; the area under the TS diagram is the heat.

Otto cycle: net work

Net work is area under curve 3-4 minus area under curve 1-2

Question: how to best compute the net work?

Write the net work in terms of the internal energies at points 1, 2, 3, 4.

  • The compression and expansion steps are adiabatic.
    • \(W_{in} = u_2-u_1\)
    • \(W_{out} = u_3-u_4\)
    • \(W_{net} = W_{out}-W_{in} = (u_3-u_4) - (u_2-u_1)\)
    • there is a simplification here for combustion…

Adiabatic compression/expansion

  • Ideal gas compression and expansion relations
  • Use \(γ = c_p/c_v\)
    • See this link for a list of values.
    • \(γ = 1.4\) for air \[PV^γ = \mbox{const.}\] \[TV^{γ-1} = \mbox{const.}\] \[TP^{(1-γ)/γ} = \mbox{const.}\]

Otto cycle: efficiency

  • Assume air as a working fluid
  • Assume constant heat capacity

\[η = \frac{W_{net}}{Q_{in}}\] \[W_{net} = (u_3-u_4) - (u_2-u_1)\] \[W_{net} = c_v(T_3-T_4) - c_v(T_2-T_1)\] \[Q_{in} = u_3 - u_2 = c_v(T_3-T_2)\] \[η = \frac{T_3 - T_4 - T_2 + T_1}{T_3-T_2} = 1 - \frac{T_4 - T_1}{T_3 - T_2}\] Factor out \(T_1\) from the numerator and \(T_2\) from the denominator: \[η = 1 - \left(\frac{T_1}{T_2}\right)\left(\frac{T_4/T_1 - 1}{T_3/T_2 - 1}\right)\]

Otto cycle: efficiency

\[η = 1 - \left(\frac{T_1}{T_2}\right)\left(\frac{T_4/T_1 - 1}{T_3/T_2 - 1}\right)\] Now, the compression and expansion steps are adiabatic, so \(TV^{γ-1}=\mbox{const}\). Hence,

\[\frac{T_4}{T_3} = \left(\frac{V_3}{V_4}\right)^{γ-1}\] But \(V_4 = V_1\) and \(V_3 = V_2\), so \[\frac{T_4}{T_3} = \left(\frac{V_3}{V_4}\right)^{γ-1} = \left(\frac{V_2}{V_1}\right)^{γ-1} = \frac{T_1}{T_2}\] Hence, \(T_4/T_1 = T_3/T_2\), giving \[η = 1 - \frac{T_1}{T_2} = 1 - \left(\frac{V_2}{V_1}\right)^{γ - 1}\]

\[η = 1 - \frac{1}{r^{γ-1}}\]

How does this efficiency compare to Carnot?

Diesel cycle

Diesel cycle: thermodynamics

Key difference with the Otto cycle is the constant pressure heat addition rather than constant volume.

Diesel cycle: efficiency

Question: do you expect the efficiency of the Diesel cycle to be higher or lower than for the Otto cycle?

\[η = 1 - \frac{1}{r^{γ-1}}\left(\frac{α^γ-1}{γ(α-1)}\right),\] where \(r=V_1/V_2\) is the compression ratio and \(α=V_3/V_2=T_3/T_2\) is the cut-off ratio. \(\alpha\) can be written in terms of the high and low temperatures and the compression ratio: \(\alpha=T_3/(T_1r^{\gamma-1})\).

Brayton cycle: gas turbines

Brayton cycle: thermodynamics

  • Gas turbine combustors
  • Like the Otto cycle, but heat transfer steps are constant \(P\) instead of constant \(V.\)
  • Same efficiency as the Otto cycle, but often written in terms of the pressure ratio.

Conversion efficiencies