Thermodynamic Cycles and Steam Turbines

Power cycles are a staple of the GATE ME and XE papers. The Rankine cycle dominates steam turbine questions, while the Brayton, Otto, and Diesel cycles appear in gas turbine and IC engine contexts. Expect 3–8 marks from this topic annually, with cycle analysis and efficiency calculations being the most frequent question types.

---

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Rankine Cycle — The Steam Power Cycle

Four processes: Pump → Boiler → Turbine → Condenser → Pump (back to boiler)

Component	Process	Energy Change
Pump	Isentropic compression	Work input (small)
Boiler	Constant pressure heat addition	$q_{in} = h_3 - h_1$
Turbine	Isentropic expansion	Work output $w_T = h_3 - h_4$
Condenser	Constant pressure heat rejection	$q_{out} = h_4 - h_1$

$$\eta_{Rankine} = \frac{w_{net}}{q_{in}} = \frac{(h_3 - h_4) - (h_2 - h_1)}{h_3 - h_2}$$

Key Modifications:

• Reheating: Increases average temp of heat addition → higher efficiency
• Regeneration: Feedwater heating → reduces fuel requirement
• Supercritical: Boiler pressure above critical point (22.06 MPa) → no boiling transition

Brayton Cycle (Gas Turbine) $$\eta = 1 - \frac{T_1}{T_2} = 1 - r_p^{(\gamma-1)/\gamma}$$ Where $r_p = P_2/P_1$ is pressure ratio.

Otto Cycle (SI Engines): $\eta = 1 - \frac{1}{r^{\gamma-1}}$ where $r = V_1/V_2$ (compression ratio)

Diesel Cycle: $\eta = 1 - \frac{1}{\gamma}\cdot\frac{r^\gamma - 1}{r - 1}\cdot\frac{1}{cutoff\text{-}ratio^{\gamma-1}}$

⚡ Exam Tip: Rankine cycle is almost always paired with steam tables in GATE — memorize how to read $h, s$ values.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Rankine Cycle — Detailed Analysis

The Rankine cycle is the ideal cycle for steam power plants. Unlike Carnot, it uses turbines that expand steam isentropically (rather than isothermally, which is impractical).

T-s Diagram Key Points

• Pump work is small relative to turbine work (visually, the pump leg is nearly vertical)
• Boiler heat addition appears as area under the curve between saturated liquid and superheated steam
• Turbine work is the drop across the turbine
• Condenser rejection is the horizontal line at low pressure

Rankine vs Carnot Efficiency

Rankine	Carnot
Turbine exhaust is wet steam	Requires isothermal expansion (impractical)
Pump work is small	Pump work would be large
Easier to construct	Ideal but not realizable

The Carnot efficiency $\eta = 1 - T_L/T_H$ is a theoretical maximum. Rankine approaches it but cannot equal it because the turbine exhaust contains moisture (wet steam) rather than saturated vapor.

Reheat Cycle

Purpose: Reduce turbine exit moisture and increase efficiency.

Two turbine stages with reheating between them:

• High-pressure turbine expands steam to an intermediate pressure
• Steam is reheated in the boiler (or reheater)
• Low-pressure turbine expands to condenser pressure

Efficiency improvement: Reheat increases average temperature of heat rejection → higher cycle efficiency, but with diminishing returns beyond 2–3 reheat stages.

Regenerative Cycle

Feedwater heating uses extracted steam to preheat the feedwater, reducing the amount of heat required in the boiler.

Open feedwater heater: Extracted steam mixes directly with feedwater (isenthalpic mixing).

Closed feedwater heater: Extracted steam condenses on tube exterior, transferring heat without mixing.

Efficiency impact: Regeneration increases cycle efficiency by reducing the $q_{in}$ requirement, but reduces net work output since some steam is bled before doing full work in the turbine.

Supercritical Rankine Cycle

Operating above the critical pressure (22.06 MPa, 374°C):

• No distinct phase transition (no boiling)
• Water directly goes from liquid to supercritical fluid
• Higher thermal efficiency (typically 40–45% for modern plants)
• Requires specialized materials due to high pressures and temperatures

Brayton Cycle — Gas Turbines

The Brayton cycle consists of:

1. Isentropic compression (compressor)
2. Constant pressure heat addition (combustion chamber)
3. Isentropic expansion (turbine)
4. Constant pressure heat rejection (exhaust)

$$\eta_{Brayton} = 1 - \left(\frac{P_2}{P_1}\right)^{(\gamma-1)/\gamma} = 1 - \frac{T_1}{T_2}$$

Key observations:

• Higher pressure ratio → higher efficiency, but also higher turbine inlet temperature
• $T_3$ (turbine inlet temp) is limited by material constraints (~1200–1500 K)
• Gas turbine requires regeneration for efficiency gains at low pressure ratios

Deviation from Ideal Brayton

Component efficiencies:

• Compressor: $\eta_c = (T_{2s} - T_1)/(T_2 - T_1)$
• Turbine: $\eta_t = (T_3 - T_4)/(T_3 - T_{4s})$

Where $s$ denotes isentropic conditions.

Otto and Diesel Cycles

Both are air-standard cycles for reciprocating engines.

Otto Cycle (Spark Ignition)

• Constant volume heat addition
• Compression ratio $r = V_1/V_2$ determines efficiency
• $\eta_{Otto} = 1 - r^{-(\gamma-1)}$
• Higher compression ratio → higher efficiency (but limited by knocking in SI engines)

Diesel Cycle (Compression Ignition)

• Constant pressure heat addition (idealized)
• Cut-off ratio $\rho = V_3/V_2$ affects efficiency
• $\eta_{Diesel} = 1 - \frac{1}{\gamma}\cdot\frac{r^\gamma - 1}{(\rho - 1)r^{\gamma-1}}$
• Diesel engines typically have higher thermal efficiency than Otto engines

⚡ Common Mistake: Students confuse compression ratio and cut-off ratio. In Diesel cycle, efficiency decreases with increasing cut-off ratio (more heat added at constant pressure).

Cycle	Heat Addition	Heat Rejection	Typical Use
Otto	Constant V	Constant V	SI engines
Diesel	Constant P	Constant V	CI engines
Dual	V + P	Constant V	Modern engines

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Combined Cycles and Advanced Concepts

Combined Gas-Vapor (Rankine + Brayton)

Combined cycle power plant uses Brayton cycle topping and Rankine cycle bottoming:

• Gas turbine exhaust heat generates steam in HRSG (Heat Recovery Steam Generator)
• Steam turbine produces additional power
• Combined efficiency: 50–60% (vs ~35% for standalone cycles)

This is the dominant technology in modern power generation.

Binary Vapor Cycles (Mercury-Water)

Historical concept using mercury (boiling at higher temp) as working fluid in a topping cycle, with steam as bottoming cycle. Largely superseded by combined gas-vapor cycles.

Rankine Cycle with Superheat, Reheat, and Regeneration

For a complete Rankine cycle with superheat, reheat, and feedwater heating:

Steam properties needed from steam tables:

• Saturated liquid line: $h_f$, $s_f$
• Saturated vapor line: $h_g$, $s_g$
• Superheated steam: interpolate in tables for given $P, T$

Isentropic expansion in turbine: $$s_3 = s_4 \implies \text{Find } h_4 \text{ using steam tables}$$

If $s_3 > s_g$ at turbine exit pressure → mixture (use $x_4$ quality) $$h_4 = h_f + x_4(h_g - h_f)$$

Brayton Cycle — Regeneration Effects

With regeneration (heat exchanger using turbine exhaust to preheat compressed air):

• Reduces fuel consumption
• Effective at low pressure ratios
• Maximum benefit when $T_4 \approx T_2$ (turbine exit temp equals compressor exit temp)

Optimum pressure ratio for maximum specific work (no regeneration): $$\left(\frac{P_2}{P_1}\right)_{opt} = \left(\frac{T_3}{T_1}\right)^{\gamma/[2(\gamma-1)]}$$

Example Problem

GATE 2021 (ME) Style: In a Rankine cycle, steam enters the turbine at 10 MPa, 500°C and condenses at 0.1 bar. Find the cycle efficiency and turbine exit quality.

Solution approach:

1. At turbine inlet (state 3): $h_3, s_3$ from superheated steam tables at 10 MPa, 500°C
2. Isentropic expansion: $s_3 = s_4$ at 0.1 bar → find $h_4$
   • If $s_4 < s_g$ at 0.1 bar → quality $x_4 = (s_4 - s_f)/s_{fg}$
   • $h_4 = h_f + x_4 \cdot h_{fg}$
3. Pump work: $h_2 - h_1 \approx v_f \Delta P$ (approximately, for incompressible liquid)
4. Efficiency: $\eta = [(h_3 - h_4) - (h_2 - h_1)]/(h_3 - h_2)$

⚡ GATE Tip: Always check if turbine exit quality is acceptable (should be > ~85-90% for most designs). Low quality causes cavitation in the condenser.

Summary Table — Cycle Efficiencies

Cycle	Maximum Efficiency	Limiting Factor
Carnot	$1 - T_L/T_H$	Requires isothermal expansion
Rankine	Lower than Carnot	Moisture in turbine exhaust
Brayton	$1 - T_1/T_2$	Material temp limit $T_3$
Otto	$1 - r^{-(\gamma-1)}$	Knocking, compression ratio
Diesel	Lower than Otto	Cut-off ratio

---

Previous Year GATE Pattern

Year	Topic Focus	Marks
2023	Rankine with reheat, efficiency	5
2022	Brayton cycle, pressure ratio	2
2021	Otto vs Diesel efficiency	3
2020	Combined cycle	2

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.