MECHANICAL ENGINEERING • THERMODYNAMICS & POWER SYSTEMS

Rankine Cycle T-s Diagram: Complete Engineering Guide

By Anil Sharma • LiveSimulators Engineering

Understand the fundamental vapor power cycle driving modern electric generation. Track isobaric boiler heat addition, superheated steam expansion, isentropic turbine efficiency, liquid pump compression, and exhaust condenser dynamics on the temperature-entropy plane.

01. Definition

The Rankine cycle is the idealized thermodynamic closed-loop cycle used to convert thermal energy into mechanical shaft work in steam turbine power generation facilities. Governed by the First and Second Laws of Thermodynamics, the working fluid (water and steam) undergoes phase changes between liquid and vapor states under two isobaric pressure levels: a high boiler pressure and a deep vacuum condenser backpressure.

On a Temperature-Entropy (T-s) diagram, the cycle is characterized by four distinct physical processes: isentropic or non-isentropic liquid pumping (State 3 to 4), constant-pressure subcooled heating and vaporization in the boiler steam drum (State 4 to 1), adiabatic turbine expansion producing shaft work (State 1 to 2), and constant-pressure heat rejection within the cooling water condenser (State 2 to 3). Plotting these processes against the water-steam vapor dome illustrates net cycle power, Carnot efficiency limits, and moisture condensation risks.

02. Governing Equations

Steady-flow energy equations applied across the four major plant control volumes determine specific enthalpy transfers (kilojoules per kilogram):

1. First-Law Thermal Efficiency

η_{th} = \frac{w_{net}}{q_{in}} = \frac{w_{t} - w_{p}}{q_{in}} = \frac{(h_{1} - h_{2}) - (h_{4} - h_{3})}{h_{1} - h_{4}}

Net work output divided by total heat added in the boiler economizer, evaporator, and superheater tubes.

2. Turbine Work Output and Isentropic Efficiency

w_{t} = h_{1} - h_{2} = η_{t} (h_{1} - h_{2 s})

Actual enthalpy drop across turbine blading accounting for aerodynamic friction and blade leakage.

3. Boiler Feed Pump Compression Work

w_{p} = h_{4} - h_{3} \approx v_{3} (P_{high} - P_{low})

Incompressible liquid pump work where v3 is saturated liquid specific volume at condenser pressure.

4. Back Work Ratio and Steam Quality

BWR = \frac{w_{p}}{w_{t}}, x_{2} = \frac{s_{2} - s_{f}}{s_{f g}}

Steam quality x2 must strictly satisfy ASME PTC 6 criteria (x2 > 0.88) to protect low-pressure turbine blades.

03. Worked Numerical Example

Evaluate a subcritical utility steam power generation unit operating under the standard ASME PTC 6 acceptance trial conditions modeled in LiveSimulators:

Plant Operating Parameters:
Boiler Operating Pressure: P₁ = 80 bar = 8.0 MPa
Turbine Inlet Superheat Temp: T₁ = 480 °C
Condenser Vacuum Backpressure: P₂ = 0.08 bar = 8.0 kPa (T_sat ≈ 41.5 °C)
Turbine Isentropic Efficiency: η_t = 85% = 0.85

Step	Thermodynamic State	Steam Table Evaluation / Formulation	Computed Result
Step 1	State 1: Turbine Inlet Steam	Superheated steam at 80 bar, 480 °C (IAPWS-IF97):	h₁ = 3348.4 kJ/kg s₁ = 6.658 kJ/(kg·K)
Step 2	State 2s: Isentropic Expansion	s_2s = s₁ = 6.658 kJ/(kg·K) at 0.08 bar: s_f = 0.5926, s_fg = 7.6361 kJ/(kg·K) x_2s = (6.658 - 0.5926) / 7.6361 = 0.7943	h_2s = 2108.5 kJ/kg
Step 3	Actual Turbine Work Output	w_t = η_t × (h₁ - h_2s) = 0.85 × (3348.4 - 2108.5)	w_t = 1053.9 kJ/kg h₂ = 2294.5 kJ/kg
Step 4	Exhaust Vapor Quality x₂	x₂ = (h₂ - h_f) / h_fg = (2294.5 - 173.8) / 2403.1	x₂ = 0.882 (88.2%) (Meets ASME >88% limit)
Step 5	Feed Pump Compression Work	w_p = v₃ × (P₁ - P₂) = 0.001008 m³/kg × (80 - 0.08) × 100 kPa	w_p = 8.06 kJ/kg h₄ = h₃ + w_p = 181.86 kJ/kg
Step 6	Boiler Heat Input q_in	q_in = h₁ - h₄ = 3348.4 - 181.86	q_in = 3166.54 kJ/kg
Step 7	Cycle Thermal Efficiency η_th	w_net = w_t - w_p = 1053.9 - 8.06 = 1045.84 kJ/kg η_th = 1045.84 / 3166.54	η_th = 33.03% BWR = 0.76% (TODO: human verification for exact IAPWS-IF97 interpolation)

04. Common Engineering Mistakes

1. Forgetting Liquid Pump Work in Cycle Energy Balances Engineers sometimes assume pump work is negligible (w_p ≈ 0) because water is a liquid. In ultra-high pressure and supercritical utility boilers (P₁ > 150 bar), feed pump work exceeds 25 kJ/kg and consumes significant auxiliary station power. Neglecting w_p produces artificially inflated net efficiency reports.
2. Assuming Turbine Expansion Is 100% Isentropic Drawing a purely vertical downward line on the T-s diagram ignores fluid aerodynamic friction, boundary layer shear, and tip leakage across turbine stages. Real turbines operate with isentropic efficiencies between 80% and 92%, curving the expansion path to the right (entropy increases, s₂ > s₁) and dramatically altering exhaust enthalpy.
3. Allowing Turbine Exhaust Quality to Fall Below 88% (x₂ < 0.88) When moisture content at the low-pressure turbine exhaust exceeds 12% (steam quality drops below 0.88), high-velocity liquid water droplets strike spinning titanium and nickel blades at supersonic relative speeds. This causes severe blade leading-edge pitting, erosion, and catastrophic mechanical failure.

Interact with the Dynamic T-s Vapor Dome

Adjust boiler pressures from 20 to 160 bar, increase turbine superheat temperatures to 600 °C, and observe live updates to the saturated vapor dome, enthalpy drops, and cycle thermal efficiency.

Launch Interactive Rankine Cycle Simulator →

05. References & Standards

Moran, M. J., Shapiro, H. N., Boettner, D. D., & Bailey, M. B. (2018). Fundamentals of Engineering Thermodynamics (9th ed.). Wiley. Chapter 8: Vapor Power Systems.
Cengel, Y. A., & Boles, M. A. (2019). Thermodynamics: An Engineering Approach (9th ed.). McGraw-Hill Education. Chapter 10: Vapor and Combined Power Cycles.
ASME PTC 6-2004 (R2014): Steam Turbines Performance Test Codes. American Society of Mechanical Engineers.
IAPWS-IF97: Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam. International Association for the Properties of Water and Steam.