Theory — Basic Rankine/Brayton Cycle Simulator
1) What this tool computes
This simulator compares two classic ideal power cycles:
Rankine (steam power plant) and Brayton (gas turbine).
For each cycle it computes:
- Stage state estimates (1→2→3→4),
- Heat transfers \(q_{in}\), \(q_{out}\),
- Work terms \(w\) (pump/compressor and turbine),
- Thermal efficiency \( \eta = \dfrac{w_{net}}{q_{in}} \).
The diagram shown is a T–s diagram (Temperature–Entropy) with stage labels.
In a reversible process, entropy generation is zero; real devices have irreversibilities so real efficiencies are lower.
2) Rankine cycle (steam power plant)
The ideal Rankine cycle has four main components:
pump (1→2), boiler (2→3),
turbine (3→4), and condenser (4→1).
States (basic interpretation)
- 1: saturated liquid leaving the condenser at \(T_{cond}\) (low pressure).
- 2: compressed liquid after the pump at boiler pressure.
- 3: superheated steam at turbine inlet (boiler outlet), often specified by \(T_3\) and \(P_{boiler}\).
- 4: turbine exhaust at condenser pressure (often a wet mixture).
Energy relations (per unit mass)
In steady-flow devices, neglecting KE/PE changes, the first law gives the familiar enthalpy work/heat relations:
- Pump work (approx. incompressible): \( w_{p,ideal} \approx v_f (P_{boiler}-P_{cond}) \).
- Boiler heat input: \( q_{in} = h_3 - h_2 \).
- Turbine work: \( w_t = h_3 - h_4 \).
- Condenser heat rejection: \( q_{out} = h_4 - h_1 \).
Net work and efficiency:
\[
w_{net} = w_t - w_p,\qquad
\eta = \frac{w_{net}}{q_{in}}.
\]
Isentropic efficiencies (optional)
- Pump: \( w_p = \dfrac{w_{p,ideal}}{\eta_p} \).
- Turbine: \( h_4 = h_3 - \eta_t\,(h_3 - h_{4s}) \), where \(4s\) is the isentropic exhaust state.
Steam properties: A full Rankine analysis needs steam tables (or IF97). This “basic” simulator uses a small built-in saturation table near typical condenser temperatures and an approximate superheat model (or manual \(h_3,s_3\)).
3) Brayton cycle (gas turbine)
The ideal Brayton cycle has:
compressor (1→2), combustor/heater (2→3),
turbine (3→4), and cooler/exhaust (4→1).
Ideal-gas, constant-property model
With constant \(c_p\) and \(\gamma\), and pressure ratio \(r_p = P_2/P_1\):
Isentropic compression:
\[
T_{2s} = T_1\, r_p^{(\gamma-1)/\gamma},\qquad
T_2 = T_1 + \frac{T_{2s}-T_1}{\eta_c}.
\]
Isentropic expansion:
\[
T_{4s} = \frac{T_3}{r_p^{(\gamma-1)/\gamma}},\qquad
T_4 = T_3 - \eta_t\,(T_3 - T_{4s}).
\]
Work and heat (per unit mass)
- Compressor work: \( w_c = c_p (T_2 - T_1) \).
- Turbine work: \( w_t = c_p (T_3 - T_4) \).
- Heat input (constant pressure): \( q_{in} = c_p (T_3 - T_2) \).
- Net work and efficiency: \( w_{net}=w_t-w_c \), \( \eta=\dfrac{w_{net}}{q_{in}} \).
Real Brayton cycles have pressure losses, variable properties, and finite combustion efficiency, so real performance is lower than the ideal model.
4) Why the T–s diagram is useful
The T–s diagram helps you visualize where heat is added and rejected:
- Area under a reversible heat-transfer curve relates to heat: \( q_{rev}=\int T\,ds \).
- Vertical lines are isentropic (ideal) processes (constant \(s\)).
- For Rankine, the “dome” region indicates two-phase mixture (wet steam).
5) Practical note on “real engine” efficiency
This tool reports air-standard/idealized efficiencies. Real plants are lower due to:
irreversibilities (friction, finite \(\Delta T\) heat transfer), pressure drops, non-ideal component efficiencies,
and auxiliary loads.
Typical real-world ranges (very rough):
steam Rankine plants ~30–45%, simple gas turbines ~25–40%, combined cycles higher.
