Carnot Engine and Efficiency

Thermodynamics ka ek fundamental concept hai Carnot Engine, jo ideal heat engine ka model hai aur maximum possible efficiency ko define karta hai. Ye concept heat transfer, work, and energy conversion ko samajhne ke liye critical hai, aur modern engines, refrigerators, aur power plants ke liye base provide karta hai.

1. Introduction

Heat engines wo devices hain jo heat energy ko work me convert karte hain.

• Real engines: Steam engines, internal combustion engines, gas turbines
• Ideal engine: Carnot engine, proposed by Sadi Carnot (1824)
• Carnot engine maximum efficiency achieve karta hai under ideal conditions

Importance:

• Establishes thermodynamic limits of efficiency
• Basis of Second Law of Thermodynamics

---

2. Heat Engine: Basic Concepts

2.1 Definition

A heat engine is a device which absorbs heat from a high-temperature source, performs work, and rejects remaining heat to a low-temperature sink.

Components:

1. High-temperature source (T₁) → provides heat Q₁
2. Working substance → usually gas in cylinder or steam
3. Low-temperature sink (T₂) → absorbs waste heat Q₂
4. Engine mechanism → converts heat difference to work W

Energy balance: Q1=W+Q2Q_1 = W + Q_2Q1​=W+Q2​

Where:

• Q1Q_1Q1​ = heat absorbed from source
• Q2Q_2Q2​ = heat rejected to sink
• WWW = work done by engine

---

2.2 Work and Efficiency

• Work done by engine:

W=Q1−Q2W = Q_1 – Q_2W=Q1​−Q2​

• Efficiency (η\etaη) of engine:

η=WQ1=Q1−Q2Q1=1−Q2Q1\eta = \frac{W}{Q_1} = \frac{Q_1 – Q_2}{Q_1} = 1 – \frac{Q_2}{Q_1}η=Q1​W​=Q1​Q1​−Q2​​=1−Q1​Q2​​

Observation: Efficiency depends on heat rejection; lower Q₂ → higher efficiency.

---

3. Carnot Engine: Definition

Carnot Engine:

An ideal reversible heat engine operating between two heat reservoirs, which achieves the maximum possible efficiency allowed by thermodynamics.

Key features:

1. Reversible → no friction or dissipation
2. Operates between two constant temperature reservoirs
3. Maximum theoretical efficiency → basis for real engines

---

4. Carnot Cycle

Carnot engine works on Carnot cycle, consisting of 4 reversible processes:

1. Isothermal Expansion at T₁
   • Gas absorbs heat Q₁ from high-temperature reservoir
   • Temperature constant → ΔU = 0 → W = Q₁
2. Adiabatic Expansion
   • Gas expands without heat exchange (Q = 0)
   • Temperature falls from T₁ → T₂
   • Work done comes from internal energy decrease
3. Isothermal Compression at T₂
   • Gas releases heat Q₂ to low-temperature reservoir
   • Temperature constant → ΔU = 0 → W = -Q₂
4. Adiabatic Compression
   • Gas compressed without heat exchange
   • Temperature rises from T₂ → T₁
   • Work done on gas increases internal energy

Graphical Representation:

• P-V diagram → rectangular/elliptical path showing expansion and compression
• T-S diagram → area under curve represents heat transfer

---

5. Mathematical Analysis

5.1 Isothermal Processes

• For ideal gas during isothermal expansion:

W12=∫PdV=nRT1ln⁡V2V1W_{12} = \int P dV = nRT_1 \ln \frac{V_2}{V_1}W12​=∫PdV=nRT1​lnV1​V2​​

• Heat absorbed Q₁ = W₁₂ (ΔU = 0)
• During isothermal compression:

Q2=nRT2ln⁡V4V3=W34Q_2 = nRT_2 \ln \frac{V_4}{V_3} = W_{34}Q2​=nRT2​lnV3​V4​​=W34​

5.2 Adiabatic Processes

• Adiabatic expansion:

PVγ=constantP V^\gamma = \text{constant} PVγ=constant

• Work done:

Wad=P2V2−P1V1γ−1W_{ad} = \frac{P_2 V_2 – P_1 V_1}{\gamma – 1}Wad​=γ−1P2​V2​−P1​V1​​

• Temperature relation:

T1V2γ−1=T2V3γ−1T_1 V_2^{\gamma -1} = T_2 V_3^{\gamma -1}T1​V2γ−1​=T2​V3γ−1​

---

5.3 Efficiency of Carnot Engine

• Using Q₁ and Q₂:

η=1−Q2Q1\eta = 1 – \frac{Q_2}{Q_1}η=1−Q1​Q2​​

• For reversible engine,

Q1T1=Q2T2  ⟹  η=1−T2T1\frac{Q_1}{T_1} = \frac{Q_2}{T_2} \implies \eta = 1 – \frac{T_2}{T_1}T1​Q1​​=T2​Q2​​⟹η=1−T1​T2​​

Key Points:

• Efficiency depends only on reservoir temperatures
• Maximum efficiency when T₂ → 0 K

Example:

• T₁ = 500 K, T₂ = 300 K

η=1−300500=0.4=40%\eta = 1 – \frac{300}{500} = 0.4 = 40\%η=1−500300​=0.4=40%

---

6. Reversibility and Irreversibility

• Reversible processes: Ideal → no entropy generation, maximum efficiency
• Irreversible processes: Real engines → friction, heat loss → lower efficiency
• Carnot cycle assumes ideal reversible processes → sets upper limit for efficiency

---

7. Carnot Theorem

Carnot Theorem:

1. No engine can be more efficient than a Carnot engine operating between same two temperatures.
2. All reversible engines operating between same temperatures have same efficiency.

Implication:

• Sets theoretical limit for real engine efficiency
• Real engines (steam, IC, gas turbines) always have η < η_Carnot

---

8. Real Engines vs Carnot Engine

Feature	Carnot Engine	Real Engine
Reversibility	Yes	No
Efficiency	Maximum (η_Carnot)	Lower than Carnot
Friction & losses	None	Present
Practicality	Ideal model	Realistic design
Heat transfer	Infinitesimal rate	Finite rate → ΔT required

---

9. Applications of Carnot Engine Concept

1. Heat Engines:
   • Steam turbines, IC engines → design efficiency benchmark
2. Refrigerators and Heat Pumps:
   • Reverse Carnot cycle → determine coefficient of performance (COP)
3. Thermodynamic Limits:
   • Maximum theoretical efficiency → limits real-world energy systems
4. Environmental Engineering:
   • Optimize energy use, reduce fuel consumption

---

10. Examples and Calculations

Example 1: Carnot Engine Efficiency

• Engine operating between T₁ = 600 K and T₂ = 300 K

η=1−T2T1=1−300600=0.5=50%\eta = 1 – \frac{T_2}{T_1} = 1 – \frac{300}{600} = 0.5 = 50\%η=1−T1​T2​​=1−600300​=0.5=50%

• If engine absorbs Q₁ = 2000 J from source, work done:

W=Q1η=2000×0.5=1000JW = Q_1 \eta = 2000 \times 0.5 = 1000 JW=Q1​η=2000×0.5=1000J

• Heat rejected:

Q2=Q1−W=2000−1000=1000JQ_2 = Q_1 – W = 2000 – 1000 = 1000 JQ2​=Q1​−W=2000−1000=1000J

---

Example 2: Reverse Carnot Cycle (Refrigerator)

• Coefficient of performance:

COP=Q2W=T2T1−T2COP = \frac{Q_2}{W} = \frac{T_2}{T_1 – T_2}COP=WQ2​​=T1​−T2​T2​​

• T₁ = 300 K, T₂ = 270 K

COP=27030=9COP = \frac{270}{30} = 9COP=30270​=9

• Implies 1 J work moves 9 J heat from cold reservoir

---

11. Graphical Representation

• P-V Diagram: Shows isothermal and adiabatic processes as closed loop
• T-S Diagram: Rectangle/loop → area = heat/work

T
|
|      -------------
|     |           |
|     |           |
|      -------------
|_________________ S

---

12. Limitations and Assumptions

1. Idealized process → frictionless, no heat loss
2. Infinite slow processes → reversible
3. Real engines have irreversibilities → lower efficiency
4. T₂ cannot reach absolute zero → efficiency < 100%

---

13. Summary Table

Concept	Formula/Relation	Notes
Efficiency (general)	η = W/Q₁ = 1 – Q₂/Q₁	Depends on heat transfer
Carnot efficiency	η = 1 – T₂/T₁	Maximum possible efficiency
Work done	W = Q₁ – Q₂	Heat converted to work
Heat transfer ratio	Q₁/T₁ = Q₂/T₂	Reversible process
Coefficient of performance	COP = Q₂/W	Refrigerators/heat pumps