5.2 Vapor Power Cycles (Rankine, Superheat, Reheat & Regeneration)
Key Takeaways
- The baseline Rankine cycle consists of four continuous processes: isentropic liquid pumping, constant-pressure boiler addition, isentropic turbine expansion, and constant-pressure condenser heat rejection.
- Turbine moisture erosion limits turbine exhaust quality to $x \ge 0.88 - 0.90$; superheating and reheating safely elevate exhaust quality while simultaneously increasing thermal efficiency.
- Reheat cycles increase overall thermal efficiency by 4–5% by raising the average temperature of heat addition without requiring metallurgical upgrades above the primary superheat temperature.
- Regenerative Feedwater Heating (using Open and Closed Feedwater Heaters) extracts turbine steam to preheat boiler feedwater, decreasing external heat addition and serving as a deaerator.
- Cogeneration (Combined Heat and Power) systems achieve utilization efficiencies of 75–85% by simultaneously producing electricity and industrial process steam from a single fuel input.
Vapor Power Cycles (Rankine, Superheat, Reheat & Regeneration)
The Rankine cycle is the fundamental thermodynamic operating model for vapor-driven power stations worldwide, including fossil fuel, biomass, nuclear, and concentrated solar thermal plants. Unlike the ideal Carnot vapor cycle—which suffers from severe mechanical limitations such as compressing two-phase liquid-vapor mixtures and handling severe turbine droplet impingement—the Rankine cycle operates entirely within practical engineering constraints by condensing steam to saturated liquid before pumping.
1. The Ideal Rankine Cycle: Architecture & Energy Balances
The standard ideal Rankine cycle consists of four steady-flow processes executed in series:
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| IDEAL RANKINE CYCLE FLOW DIAGRAM |
| |
| Boiler / Steam Generator |
| +------------------------+ |
| | q_in (Q_H) | |
| +------------------------+ |
| ^ | |
| State 2 | High-P Subcooled | State 3: High-P Superheated Steam |
| Liquid | v |
| +---------+ | +---------+ |
| w_p,in --> | PUMP |----+ | TURBINE | ----> w_t,out |
| +---------+ +---------+ |
| ^ | |
| State 1 | Saturated Liquid | State 4: Low-P Wet Mixture |
| Condensate v |
| +------------------------+ |
| | q_out (Q_L) | |
| +------------------------+ |
| Condenser |
+-----------------------------------------------------------------------------------------+
T-s Diagram: Ideal Rankine Cycle
T ^
| Boiler Heat Addition (2->3)
| 3 (Superheated Vapor)
| /|
| / |
| 2 / |
| /|__________/ | Turbine Expansion (3->4s, Isentropic)
| / | Liquid- |
| 1--+--Vapor Dome--+-- 4s (Low-P Mixture)
| | | |
| +--+--------------+--------> s
| Pump (1->2s) Condenser Heat Rejection (4->1)
Mathematical Formulation of Process States
| Process | Equipment | Governing Equation | Physical Description |
|---|---|---|---|
| 1 $\to$ 2 | Feed Pump | $w_{p,in} = h_2 - h_1 = v_1(P_2 - P_1)$ | Isentropic liquid compression ($s_2 = s_1$) |
| 2 $\to$ 3 | Boiler | $q_{in} = h_3 - h_2$ | Isobaric heat addition ($P_2 = P_3$) |
| 3 $\to$ 4 | Steam Turbine | $w_{t,out} = h_3 - h_4$ | Isentropic vapor expansion ($s_4 = s_3$) |
| 4 $\to$ 1 | Condenser | $q_{out} = h_4 - h_1$ | Isobaric heat rejection ($P_4 = P_1$) |
Net Work, Thermal Efficiency & Back Work Ratio
The net specific work produced by the cycle is:
The Thermal Efficiency ($\eta_{th}$) is defined as:
The Back Work Ratio ($BWR$) for a Rankine cycle is remarkably small compared to gas turbine cycles:
Because pumping an incompressible liquid requires minimal work compared to the work extracted by expanding vapor, the pump work is often negligible in preliminary estimates, though critical in exact power plant balances.
2. Real Cycle Deviations & Component Irreversibilities
Actual vapor power cycles deviate from the idealized model due to internal irreversibilities:
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| ACTUAL CYCLE LOSS MECHANISMS |
| |
| 1. TURBINE IRREVERSIBILITIES: Fluid friction and blade turbulence cause an entropy |
| increase (s_4a > s_3), reducing actual work output: |
| eta_t = (h_3 - h_4a) / (h_3 - h_4s) ==> h_4a = h_3 - eta_t * (h_3 - h_4s) |
| |
| 2. PUMP IRREVERSIBILITIES: Mechanical and fluid shear losses increase pump work: |
| eta_p = w_p,s / w_p,a = v_1*(P_2 - P_1) / (h_2a - h_1) |
| ==> h_2a = h_1 + [v_1*(P_2 - P_1) / eta_p] |
| |
| 3. CONDENSER SUBCOOLING: Cooling condensate below T_sat(P_cond) requires extra boiler |
| heat to reach saturation, degrading cycle thermal efficiency. |
| |
| 4. PRESSURE DROPS: Viscous friction in steam lines and boiler tubes drops P_inlet. |
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3. Superheating & The Reheat Rankine Cycle
To increase cycle efficiency and protect turbine blades from moisture erosion, modern utility boilers superheat steam well above the saturation temperature. However, if boiler pressure is increased to improve efficiency, the expansion path shifts leftward, resulting in dangerously wet steam at the turbine exhaust.
[!IMPORTANT] The 10% Moisture Rule: Water droplets moving at high velocities erode turbine blades and degrade aerodynamic stage efficiency. Power plant turbines require exhaust quality $x \ge 0.88 \text{ to } 0.90$ (moisture content $\le 10-12%$).
T-s Diagram: Reheat Rankine Cycle
T ^
| Primary Boiler Addition
| 3 (HP Inlet)
| /| 5 (LP Inlet, Reheated)
| / | /|
| 2 / | 4 / |
| /|__________/ |/____/ | LP Expansion (5->6)
| / | Liquid- | Reheat|
| 1--+--Vapor Dome--+--------+-- 6 (Safe Quality x > 0.90)
| +--+--------------+--------+--------> s
| HP Expansion (3->4) Condenser (6->1)
The Reheat Thermodynamic Configuration
In a reheat cycle, steam expands through a High-Pressure (HP) Turbine ($3 \to 4$), exhausts back to the boiler's reheater tubes where it is reheated at constant pressure to approximately the original inlet temperature ($T_5 \approx T_3$), and then expands through a Low-Pressure (LP) Turbine ($5 \to 6$).
[!TIP] Optimum Reheat Pressure Rule: Peak cycle efficiency occurs when the reheat pressure is set to approximately $20% \text{ to } 25%$ of the maximum boiler pressure ($P_{reheat} \approx 0.20 - 0.25 \times P_{boiler}$). Reheating increases overall thermal efficiency by $4% \text{ to } 5%$.
4. Regenerative Feedwater Heating (FWH)
In standard cycles, cold condensate from the condenser enters the boiler, lowering the average temperature of heat addition ($T_{mean,in}$). Regeneration bleeds intermediate-pressure steam from turbine stages to preheat the feedwater before it enters the boiler.
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| OPEN VS CLOSED FEEDWATER HEATERS (FWH) |
| |
| OPEN FEEDWATER HEATER (OFWH / Direct Contact): |
| - Bleed steam mixes directly with feedwater condensate. |
| - Saturated liquid exits at heater pressure: h_out = h_f(P_heater). |
| - Serves as a DEAERATOR to strip dissolved O2 and CO2, preventing boiler tube pitting.|
| - Requires a separate pump after each heater. |
| |
| CLOSED FEEDWATER HEATER (CFWH / Shell-and-Tube Indirect Contact): |
| - Feedwater flows through tubes; extraction steam condenses on shell exterior. |
| - Streams do not mix; can operate at different pressures. |
| - Condensate drains are either pumped forward or throttled (trapped) backward. |
| - Terminal Temperature Difference: TTD = T_sat(P_bleed) - T_fw,out (typically 2-5 °F).|
+-----------------------------------------------------------------------------------------+
Open Feedwater Heater Mass & Energy Balance
Extraction Steam: y [kg], h_extract, P_FWH
|
v
Feedwater from LP Pump: +----------+
(1 - y) [kg], h_in, P_FWH | OFWH | ---> Saturated Liquid Out to HP Pump:
------------------------> | (Direct) | 1.0 [kg], h_out = h_f(P_FWH)
+----------+
Extraction Fraction ($y$) Calculation for an OFWH
Setting a control volume around the Open Feedwater Heater with 1 kg total flow exiting:
Total turbine work per unit mass entering the HP stage:
5. Cogeneration & Combined Heat and Power (CHP)
Industrial plants require vast quantities of both electrical power and process steam (for chemical refining, paper manufacturing, district heating). Standalone electricity generation rejects $55-65%$ of fuel energy to cooling towers. Cogeneration (CHP) captures this thermal energy for process heating.
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| COGENERATION EFFICIENCY METRIC |
| |
| Utilization Factor (Energy Utilization Efficiency): |
| |
| W_dot_net + Q_dot_process W_dot_net + m_dot_p * (h_supply - h_ret)|
| epsilon_u = --------------------------- = ------------------------------------------- |
| Q_dot_in Q_dot_in |
| |
| * Standalone Power Plants: eta_th = 35% - 42% |
| * Modern Cogeneration CHP: epsilon_u = 75% - 85%+ |
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Cogeneration Turbine Architectures
- Back-Pressure (Non-Condensing) Turbine: Steam expands through the turbine and exhausts directly at the required process pressure (e.g., $0.5\text{ MPa}$). All exhaust steam is routed to the process header. Electric power output is rigidly coupled to thermal process demand.
- Extraction-Condensing Turbine: High-pressure steam expands to an intermediate pressure where a variable fraction is extracted for process needs. The remainder continues expanding to the condenser, allowing decoupled, flexible electrical generation during process demand swings.
6. Step-by-Step Worked Problem: Reheat Rankine Cycle Analysis
Problem: A power plant operates on a reheat Rankine cycle. High-pressure steam enters the HP turbine at $P_3 = 10.0\text{ MPa}$ ($1450\text{ psia}$) and $T_3 = 550^\circ\text{C}$ ($1022^\circ\text{F}$). Steam expands to $P_4 = 2.0\text{ MPa}$ ($290\text{ psia}$) where it is reheated to $T_5 = 550^\circ\text{C}$. Steam then expands through the LP turbine to a condenser pressure of $P_6 = 10\text{ kPa}$ ($1.45\text{ psia}$). The HP and LP turbines each have an isentropic efficiency of $\eta_t = 85%$. The feed pump isentropic efficiency is $\eta_p = 80%$. Calculate:
- The actual total turbine work output ($w_{t,actual}$ in $\text{kJ/kg}$)
- The actual feed pump work required ($w_{p,actual}$ in $\text{kJ/kg}$)
- The total heat added in the boiler and reheater ($q_{in,total}$ in $\text{kJ/kg}$)
- The overall thermal efficiency ($\eta_{th}$)
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| REHEAT CYCLE CALCULATION STEPS |
| |
| STEP 1: Condensate & Actual Pump Work (P_1 = 10 kPa -> P_2 = 10 MPa) |
| h_1 = h_f@10kPa = 191.81 kJ/kg, v_1 = 0.001010 m^3/kg |
| w_p,s = v_1 * (P_2 - P_1) = 0.001010 * (10000 - 10) = 10.09 kJ/kg |
| w_p,a = w_p,s / eta_p = 10.09 / 0.80 = 12.61 kJ/kg |
| h_2a = h_1 + w_p,a = 191.81 + 12.61 = 204.42 kJ/kg |
| |
| STEP 2: HP Turbine Expansion (P_3 = 10 MPa, T_3 = 550 °C -> P_4 = 2.0 MPa) |
| From Superheated Tables: h_3 = 3502.0 kJ/kg, s_3 = 6.7585 kJ/(kg*K) |
| At P_4 = 2.0 MPa, isentropic state (s_4s = 6.7585): |
| Interpolating at 2.0 MPa gives h_4s = 2900.5 kJ/kg |
| w_t,HP,s = h_3 - h_4s = 3502.0 - 2900.5 = 601.5 kJ/kg |
| w_t,HP,a = eta_t * w_t,HP,s = 0.85 * 601.5 = 511.28 kJ/kg |
| h_4a = h_3 - w_t,HP,a = 3502.0 - 511.28 = 2990.72 kJ/kg |
| |
| STEP 3: Reheater Addition (P = 2.0 MPa, Reheated to T_5 = 550 °C) |
| From Superheated Tables at 2.0 MPa, 550 °C: |
| h_5 = 3579.0 kJ/kg, s_5 = 7.5725 kJ/(kg*K) |
| q_reheat = h_5 - h_4a = 3579.0 - 2990.72 = 588.28 kJ/kg |
| |
| STEP 4: LP Turbine Expansion (P_5 = 2.0 MPa -> P_6 = 10 kPa) |
| At P_6 = 10 kPa: s_f = 0.6492 kJ/(kg*K), s_fg = 7.4996 kJ/(kg*K) |
| h_f = 191.81 kJ/kg, h_fg = 2392.1 kJ/kg |
| x_6s = (s_5 - s_f) / s_fg = (7.5725 - 0.6492) / 7.4996 = 0.92315 |
| h_6s = 191.81 + 0.92315 * (2392.1) = 2400.07 kJ/kg |
| w_t,LP,s = h_5 - h_6s = 3579.0 - 2400.07 = 1178.93 kJ/kg |
| w_t,LP,a = eta_t * w_t,LP,s = 0.85 * 1178.93 = 1002.09 kJ/kg |
| h_6a = h_5 - w_t,LP,a = 3579.0 - 1002.09 = 2576.91 kJ/kg |
| |
| STEP 5: Net Performance Totals |
| Total Actual Turbine Work: |
| w_t,actual = w_t,HP,a + w_t,LP,a = 511.28 + 1002.09 = 1513.37 kJ/kg |
| |
| Net Specific Work: |
| w_net = w_t,actual - w_p,a = 1513.37 - 12.61 = 1500.76 kJ/kg |
| |
| Total Heat Added: |
| q_in = (h_3 - h_2a) + q_reheat = (3502.0 - 204.42) + 588.28 = 3885.86 kJ/kg |
| |
| Thermal Efficiency: |
| eta_th = w_net / q_in = 1500.76 / 3885.86 = 0.3862 = 38.62% |
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7. Common Exam Traps & PE Pro-Tips
- Trap 1 — Reheat Heat Addition Balance: When computing total cycle heat addition ($q_{in}$), never forget to add the reheater heat input ($q_{reheat} = h_5 - h_4$) to the primary boiler heat input ($q_{boiler} = h_3 - h_2$). Dividing net work by only primary boiler heat yields an erroneously high thermal efficiency.
- Trap 2 — OFWH Extraction Steam Balance: In regenerative cycles with an Open Feedwater Heater, remember that the low-pressure turbine produces work only on the unextracted steam fraction $(1 - y)$. The total turbine work must be weighted: $w_t = (h_{in,HP} - h_{extract}) + (1 - y)(h_{extract} - h_{exit,LP})$.
- Trap 3 — Condenser Subcooling Penalty: If the problem states that condensate exits the condenser subcooled by $\Delta T_{sub}$ below saturation, $h_1$ is evaluated at $T = T_{sat} - \Delta T_{sub}$. This requires extra pump and boiler heat, lowering cycle thermal efficiency.
What is the primary thermodynamic rationale for implementing a reheat cycle in modern high-pressure utility steam power plants?
In a regenerative Rankine cycle with a single Open Feedwater Heater (OFWH), high-pressure steam leaves the boiler at h3 = 3400 kJ/kg. Bleed steam is extracted at 0.8 MPa with h_extract = 2800 kJ/kg. The feedwater entering from the low-pressure condensate pump has h_in = 200 kJ/kg. If the liquid exiting the OFWH is saturated liquid at 0.8 MPa (h_f = 721 kJ/kg), what fraction y of the steam entering the turbine must be extracted?
An industrial cogeneration plant supplies 15 MW of electric power and delivers 30 MW of process steam heat while consuming fuel that provides a total thermal heat input of 60 MW in the boiler. What is the Energy Utilization Factor (utilization efficiency) of this cogeneration facility?
Which of the following describes the key operational and mechanical difference between an Open Feedwater Heater (OFWH) and a Closed Feedwater Heater (CFWH)?