Carnot vs. Real Cycles — Compare Carnot cycle to real cycles conceptually - Thermodynamics

SECTION 1

Historical Context & Motivation

The quest to understand the fundamental limits of heat engines began during the Industrial Revolution, when steam engines were transforming manufacturing, mining, and transportation across Europe. Engineers knew that some engines performed far better than others, yet no one had established a theoretical ceiling on how efficiently heat could be converted into work. The central question was deceptively simple: is there a maximum possible efficiency for any engine operating between two thermal reservoirs? Answering this question would reshape physics and engineering for centuries to come.

1824 • Carnot Publishes Réflexions

Sadi Carnot published his landmark treatise Réflexions sur la puissance motrice du feu, introducing the idealized cycle that bears his name. He argued that no engine operating between two temperatures can surpass the efficiency of a reversible engine, establishing the concept of an upper bound on thermal efficiency.

1850–1865 • Clausius and Kelvin Formalize the Second Law

Rudolf Clausius and Lord Kelvin independently formulated the Second Law of Thermodynamics, providing the rigorous mathematical foundation for Carnot's intuition. Clausius introduced the concept of entropy, which made irreversibility quantifiable.

1876 • Otto Cycle Developed

Nikolaus Otto built the first practical four-stroke internal combustion engine. The idealized Otto cycle—with constant-volume heat addition—became a standard benchmark for spark-ignition engines, revealing substantial departures from Carnot efficiency due to irreversibilities.

1893 • Diesel Cycle Patented

Rudolf Diesel patented a compression-ignition engine modeled on constant-pressure heat addition. Diesel engines approached higher efficiencies than Otto engines by operating at higher compression ratios, yet still remained well below the Carnot limit due to friction, finite-rate heat transfer, and combustion irreversibilities.

1930s–Present • Vapor-Compression & Gas Turbine Refinement

The Rankine cycle for steam power plants and the Brayton cycle for gas turbines became the workhorses of modern power generation. Advanced techniques—reheat, regeneration, intercooling—pushed real cycles closer to the Carnot ideal, but irreversibilities inherent in every physical process guarantee that the gap can never fully close.

Carnot's insight crystallized a profound truth: the efficiency of a heat engine depends only on the temperatures of its thermal reservoirs, not on the working fluid or the mechanical details. Every real cycle—Otto, Diesel, Rankine, Brayton—falls short of this ideal, and understanding why and by how much they fall short is the central objective of this lesson.

SECTION 2

Core Principles & Definitions

Before comparing the Carnot cycle to real cycles, we must establish several foundational ideas that govern the behavior of thermodynamic cycles. These principles explain why the Carnot cycle occupies its privileged theoretical position and why no physical engine can replicate it exactly.

Reversibility

A reversible process can be reversed without leaving any net change in the system or surroundings. The Carnot cycle consists exclusively of reversible processes—two isothermal and two adiabatic (isentropic) steps. Real processes always involve friction, unrestrained expansion, or finite temperature gradients, making true reversibility physically unattainable.

Thermal Reservoirs

The Carnot cycle exchanges heat with exactly two thermal reservoirs at fixed temperatures T_H (hot) and T_L (cold). Heat transfer across a finite temperature difference—unavoidable in real engines—generates entropy and degrades performance.

Entropy Generation

Entropy generation (S_gen ≥ 0) quantifies irreversibility. In the Carnot cycle, S_gen = 0 for every process. In every real cycle, S_gen > 0, which directly reduces the net work output and, consequently, the thermal efficiency.

Carnot Efficiency

The Carnot efficiency η_Carnot = 1 − T_L/T_H represents the absolute maximum efficiency for any heat engine operating between T_H and T_L. It is independent of the working fluid or engine design.

Sources of Irreversibility

Real cycles depart from Carnot behavior through irreversibilities including fluid friction, throttling, non-quasi-static compression/expansion, heat transfer across finite ΔT, mixing, and chemical non-equilibrium during combustion. Each source contributes positively to S_gen.

✦

KEY TAKEAWAY

Think of the Carnot cycle as a perfectly frictionless, infinitely slow race on a theoretically perfect track—it sets the lap record that no real car can ever beat. Real engines are like actual race cars navigating a real track with tire friction, aerodynamic drag, and imperfect cornering. Every physical imperfection (irreversibility) adds time to the lap (reduces efficiency). The Carnot efficiency tells you the best possible lap time so you can measure how much room your real engine has for improvement.

SECTION 3

Visual Explanation — T-s and P-v Diagrams

The most instructive way to compare the Carnot cycle with real cycles is on the temperature–entropy (T-s) diagram. On this diagram, the Carnot cycle appears as a perfect rectangle bounded by two horizontal isotherms and two vertical isentropes. The enclosed area equals the net work produced per unit mass. Real cycles, by contrast, have rounded corners, sloped heat-addition paths, and bulging profiles that always enclose less area relative to the bounding Carnot rectangle, visually revealing their lower efficiency.

The cyan rectangle represents the ideal Carnot cycle with vertices at states 1–4: two horizontal isotherms at T_H and T_L, and two vertical isentropes. The pink dashed curve shows a typical Rankine (real) cycle. Notice how the real cycle's area—representing net work—is smaller than the Carnot rectangle, reflecting lower thermal efficiency.

Several features of the T-s diagram deserve close attention. First, the Carnot cycle's horizontal top and bottom edges indicate that all heat addition occurs at a single temperature T_H and all heat rejection occurs at T_L. In a real Rankine cycle, heat addition occurs over a range of temperatures as the working fluid is preheated, boiled, and superheated, which means the effective average temperature of heat addition is lower than T_H. This reduction in the mean heat-addition temperature is one of the primary reasons real cycles have lower efficiency than the Carnot cycle.

SECTION 4

Mathematical Framework

The mathematical comparison between Carnot and real cycles rests on the interplay between heat addition, heat rejection, entropy generation, and the resulting thermal efficiency. We begin with the Carnot efficiency and then introduce the tools for quantifying the departure of real cycles from this ideal.

CARNOT EFFICIENCY

η_Carnot = 1 − T_L / T_H

Where T_H = absolute temperature of the hot reservoir (K), T_L = absolute temperature of the cold reservoir (K). Both temperatures must be in Kelvin. This expression follows directly from the Second Law and is valid for any reversible heat engine operating between these two reservoirs.

GENERAL THERMAL EFFICIENCY

η_th = W_net / Q_H = 1 − Q_L / Q_H

W_net = net work output, Q_H = heat input from the hot source, Q_L = heat rejected to the cold sink. For real cycles, Q_L/Q_H > T_L/T_H due to irreversibilities, so η_th < η_Carnot.

CLAUSIUS INEQUALITY

∮ (δQ / T) ≤ 0

The cyclic integral of δQ/T is zero for a reversible (Carnot) cycle and strictly negative for any irreversible (real) cycle. This inequality is the mathematical embodiment of the Second Law applied to cycles.

ISENTROPIC EFFICIENCY (TURBINE)

η_s,turbine = (h₁ − h₂ₐ) / (h₁ − h₂s)

Where h₁ is the inlet enthalpy, h_2a is the actual exit enthalpy, and h_2s is the isentropic exit enthalpy. This ratio measures how closely a real turbine approximates the ideal (isentropic) expansion. Typical values for large steam turbines range from 0.80 to 0.92.

The isentropic efficiency concept extends to pumps and compressors as well. For a pump, the isentropic efficiency is defined as η_s,pump = (h_2s − h₁)/(h_2a − h₁), the reciprocal arrangement, because a real pump requires more work than the ideal pump. These component-level efficiencies are how engineers bridge the gap between the idealized Carnot/cycle analysis and the performance of actual hardware. When every component has η_s < 1, entropy generation accumulates throughout the cycle, and the overall thermal efficiency drops further below the Carnot value.

SECTION 5

Detailed Comparison of Real Cycles to the Carnot Ideal

Each major power and refrigeration cycle departs from the Carnot ideal in characteristic ways dictated by its heat-addition and rejection processes. The following diagram places four common ideal cycles alongside the Carnot cycle on a single T-s diagram to highlight how the shape of each cycle's profile translates into a different efficiency gap relative to the Carnot rectangle.

Comparison of four ideal cycles overlaid with the bounding Carnot rectangle (dashed cyan). The Otto cycle has two isochoric (constant-volume) heat processes; the Diesel cycle has constant-pressure heat addition; the Rankine cycle involves phase change, which lowers the mean temperature of heat addition; and the Brayton cycle operates with constant-pressure processes. None fills the full Carnot rectangle.

Cycle	Heat Addition	Heat Rejection	Key Departure from Carnot	Typical η_th Range
Carnot	Isothermal at $T_H$	Isothermal at $T_L$	None — ideal reference	Depends on $T_H$ , $T_L$
Otto	Constant volume	Constant volume	Non-isothermal heat addition/rejection; T varies during both processes	25–35% (gasoline)
Diesel	Constant pressure	Constant volume	Higher compression ratio helps, but extended heat addition at varying T still lowers mean T of heat input	30–45% (diesel)
Rankine	Constant pressure (liquid → vapor)	Constant pressure (condenser)	Large portion of heat added during phase change at constant T, but subcooled preheating lowers average $T_H$ ,mean	33–42% (steam)
Brayton	Constant pressure	Constant pressure	Both heat exchange processes span wide T ranges; mean temperatures far from $T_H$ and $T_L$	30–40% (gas turbine)

Comparison of ideal power cycles to the Carnot benchmark

A unifying observation emerges: any cycle whose heat-addition temperature varies inevitably has a lower mean temperature of heat addition than the peak temperature T_H, and therefore a lower efficiency than a Carnot cycle operating between T_H and T_L. Techniques such as superheating, reheating, and regeneration aim to raise this mean temperature, nudging the cycle profile closer to the Carnot rectangle.

SECTION 6

Worked Example — Comparing Carnot and Real Rankine Efficiencies

Consider a steam power plant operating on a simple ideal Rankine cycle. The boiler supplies steam at 10 MPa and 500 °C (T_H,peak ≈ 773 K), and the condenser operates at 10 kPa (T_sat ≈ 318.8 K = T_L). We wish to compute the Carnot efficiency between these extreme temperatures and compare it to the Rankine cycle efficiency using steam-table values.

Carnot vs. Ideal Rankine Efficiency

1
Step 1 — Compute Carnot Efficiency

Using T_H = 773 K (the maximum temperature in the cycle) and T_L = 318.8 K (condenser saturation temperature):

η_Carnot = 1 − 318.8/773 = 1 − 0.4125 = 0.5875 or 58.75%
2
Step 2 — Identify Rankine Cycle State Points

From steam tables: State 1 (turbine inlet, 10 MPa, 500 °C): h₁ = 3373.7 kJ/kg, s₁ = 6.5966 kJ/(kg·K). State 3 (condenser exit, saturated liquid at 10 kPa): h₃ = h_f = 191.83 kJ/kg, v₃ = 0.00101 m³/kg. State 2 (turbine exit, isentropic to 10 kPa): s₂ = s₁ = 6.5966. Use quality x₂ = (s₂ − s_f)/(s_fg) = (6.5966 − 0.6493)/7.5009 = 0.7930. h₂ = h_f + x₂ × h_fg = 191.83 + 0.7930 × 2392.8 = 2089.3 kJ/kg.

h₁ = 3373.7, h₂ = 2089.3, h₃ = 191.83 kJ/kg
3
Step 3 — Pump Work

The ideal pump work per unit mass is w_pump = v₃ × (P₄ − P₃) = 0.00101 × (10000 − 10) = 10.09 kJ/kg. Therefore h₄ = h₃ + w_pump = 191.83 + 10.09 = 201.92 kJ/kg.

w_pump = 10.09 kJ/kg
4
Step 4 — Compute Rankine Efficiency

q_in = h₁ − h₄ = 3373.7 − 201.92 = 3171.8 kJ/kg. w_turbine = h₁ − h₂ = 3373.7 − 2089.3 = 1284.4 kJ/kg. w_net = w_turbine − w_pump = 1284.4 − 10.09 = 1274.3 kJ/kg. η_Rankine = w_net/q_in = 1274.3/3171.8 = 0.4018.

η_Rankine = 40.18%
5
Step 5 — Compare and Interpret

The Carnot efficiency between the same extreme temperatures is 58.75%, while the ideal Rankine efficiency is only 40.18%. The ratio η_Rankine/η_Carnot = 0.4018/0.5875 ≈ 0.684, meaning the ideal Rankine cycle captures only about 68.4% of the Carnot potential. The gap arises primarily because the working fluid absorbs heat over a range of temperatures (from subcooled liquid at ~319 K up to superheated steam at 773 K), reducing the mean temperature of heat addition well below T_H. In a real plant with component irreversibilities (η_s,turbine ≈ 0.85, η_s,pump ≈ 0.80, pressure drops, heat losses), the actual efficiency would fall even further, to perhaps 33–36%.

η_Rankine/η_Carnot ≈ 68.4% of theoretical maximum

SECTION 7

Sources of Irreversibility — Why Real Cycles Fall Short

The gap between the Carnot ideal and a real operating cycle is not a single deficiency but the cumulative result of multiple irreversibilities occurring simultaneously throughout the system. Understanding each source individually is essential for engineers seeking to prioritize design improvements.

Source of Irreversibility	Effect on Cycle	Mitigation Strategy
Fluid Friction	Pressure drops in pipes, valves, heat exchangers require additional pump/compressor work and reduce available pressure for expansion.	Optimize pipe diameters, reduce fittings, use streamlined flow paths.
Heat Transfer Across Finite ΔT	Requires the working fluid temperature to be lower than $T_H$ during heat addition and higher than $T_L$ during rejection, reducing effective cycle temperatures.	Increase heat exchanger area, use counterflow arrangements, employ phase-change processes.
Non-Isentropic Compression/Expansion	Real turbines and compressors have isentropic efficiencies < 1, generating entropy and wasting potential work.	Improve blade aerodynamics, reduce tip clearances, use multistage expansion with reheating.
Combustion Irreversibility	Chemical reactions in IC engines and gas turbines are highly irreversible, producing entropy as fuel energy is converted to thermal energy at finite rates.	Lean premixed combustion, higher combustion temperatures, exhaust gas recirculation.
Mixing & Throttling	Mixing of streams at different temperatures and throttling through valves (especially in refrigeration) destroy exergy without producing work.	Replace throttling valves with expansion turbines; minimize mixing of dissimilar streams.
Heat Loss to Surroundings	Unintended heat loss from hot components reduces q_in that actually contributes to work, effectively lowering η_th.	Insulate high-temperature components; recover waste heat via regeneration or cogeneration.

Major sources of irreversibility in real power and refrigeration cycles

✦

KEY TAKEAWAY

Imagine trying to carry water from a well to your house using a series of leaky buckets. The Carnot efficiency tells you how much water you'd deliver with a perfectly sealed bucket (an upper bound). Each source of irreversibility—friction, heat leaks, imperfect machinery—is like a different hole in the bucket. Patching one hole helps, but you'll never eliminate all of them. In engineering practice, identifying which 'hole' loses the most water (performing an exergy destruction analysis) guides where to invest design effort for the greatest efficiency gain.

SECTION 8

Connection to Exergy Analysis and Advanced Cycles

The Carnot-versus-real comparison naturally leads to exergy analysis (also called availability analysis), which extends the Carnot framework by quantifying the maximum useful work extractable from a system interacting with a specified environment at T₀. While the Carnot efficiency provides a single-number ceiling for the overall cycle, exergy analysis pinpoints exactly where, within the cycle, the greatest thermodynamic losses occur. This component-level insight is invaluable for optimizing combined-cycle plants, cogeneration systems, and advanced configurations.

Concept	Carnot / Second-Law Framework	Exergy / Advanced Framework
Efficiency Metric	η_Carnot = 1 − $T_L$ / $T_H$ gives the overall upper bound	Second-law (exergetic) efficiency η_II = W_actual/W_reversible for each component
Loss Quantification	Total entropy generation S_gen for the cycle	Exergy destruction rate Ė_D = T₀ × Ṡ_gen for each component individually
Design Guidance	Indicates how far the cycle is from ideal; does not localize losses	Ranks components by exergy destruction, guiding targeted improvements
Cycle Modifications	Motivates raising $T_H$ or lowering $T_L$	Motivates reheat, regeneration, intercooling, combined cycles to reduce exergy destruction in the largest-loss components

From Carnot analysis to exergy-based optimization

Modern combined-cycle gas turbine (CCGT) plants, which pair a Brayton topping cycle with a Rankine bottoming cycle, achieve thermal efficiencies exceeding 60%—remarkably close to the Carnot limit for their operating temperatures. This demonstrates that while no single cycle can match the Carnot ideal, creative engineering that reduces the largest sources of exergy destruction can dramatically narrow the gap. Advanced topics in this direction include supercritical CO₂ cycles, organic Rankine cycles (ORC) for low-temperature waste heat, and thermoelectric generators that bypass mechanical components entirely.

💡

Looking Ahead

In your advanced thermodynamics or energy systems courses, you will learn to construct Grassmann (exergy flow) diagrams that visually map the flow and destruction of exergy throughout a complete power plant. These diagrams are the natural evolution of the Carnot comparison: instead of asking 'how far am I from ideal?', they answer 'where exactly am I losing the most work potential, and what can I do about it?'

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why the Carnot cycle, despite being the most efficient possible cycle, is not used as the basis for real power plant design. Identify at least three practical reasons.

Show answer

The Carnot cycle is impractical for at least three reasons. First, isothermal heat addition and rejection require infinitely slow processes (quasi-static heat transfer with an infinitesimal temperature difference), which would produce zero power output despite high efficiency. Second, the isentropic expansion of a two-phase mixture in a turbine is problematic because liquid droplets cause blade erosion—real Rankine cycles superheat the steam to avoid this. Third, isentropic compression of a two-phase mixture is mechanically difficult to achieve; pumps work best with incompressible liquids, so the Rankine cycle condenses the fluid completely before compression. These practical constraints force engineers to adopt modified cycles that sacrifice some theoretical efficiency for feasibility and acceptable power density.

Question 2: BASIC CALCULATION

A heat engine operates between a hot reservoir at 800 K and a cold reservoir at 300 K. (a) Calculate the Carnot efficiency. (b) If the engine actually produces 350 kJ of work per 1000 kJ of heat input, find its actual thermal efficiency and the ratio η_actual/η_Carnot.

Show answer

(a) η_Carnot = 1 −

T_L

T_H

= 1 − 300/800 = 0.625 or 62.5%. (b) η_actual = W_net/

Q_H

= 350/1000 = 0.35 or 35%. The ratio η_actual/η_Carnot = 0.35/0.625 = 0.56, meaning the real engine captures only 56% of the thermodynamically available work. The remaining 44% is lost to irreversibilities such as friction, non-isentropic compression and expansion, and finite-rate heat transfer.

Question 3: INTERMEDIATE

An ideal Rankine cycle operates with a boiler pressure of 6 MPa, a turbine inlet temperature of 450 °C, and a condenser pressure of 20 kPa. The Carnot efficiency between T_H,peak = 723 K and T_L = 333.2 K is 53.9%. If the ideal Rankine cycle efficiency is calculated to be 37.5%, determine the percentage of the Carnot potential captured. Then, if the turbine isentropic efficiency drops to 0.82 and the pump isentropic efficiency is 0.78, qualitatively explain how the actual cycle efficiency will change.

Show answer

The ideal Rankine cycle captures 37.5%/53.9% = 69.6% of the Carnot potential. When component irreversibilities are introduced (η_s,turbine = 0.82, η_s,pump = 0.78), the actual turbine work decreases because h₂a > h₂s (the actual exit enthalpy is higher due to entropy generation), and the actual pump work increases because the pump must do more work to achieve the same pressure rise. Both effects reduce w_net while q_in changes only slightly (since the pump work correction is small relative to boiler heat input). The combined effect typically reduces η_th by 5–8 percentage points, bringing it to roughly 30–33%, which captures only about 56–61% of the Carnot potential. Each percentage point of isentropic efficiency lost in the turbine has a proportionally larger impact than the same loss in the pump, because the turbine handles a much larger enthalpy change.

Question 4: APPLIED

A combined-cycle power plant uses a Brayton topping cycle with a turbine inlet temperature of 1500 K and a Rankine bottoming cycle rejecting heat at 310 K. The measured overall thermal efficiency is 58%. (a) What is the Carnot efficiency between these extreme temperatures? (b) What fraction of the Carnot potential does the plant achieve? (c) An engineer proposes adding a regenerator to the Brayton cycle to preheat combustion air. Explain, using the concept of mean temperature of heat addition, why this modification could increase the combined cycle efficiency.

Show answer

(a) η_Carnot = 1 − 310/1500 = 1 − 0.2067 = 79.3%. (b) η_actual/η_Carnot = 58/79.3 = 73.1% of the Carnot potential. (c) A regenerator recovers exhaust heat from the gas turbine to preheat combustion air, which means less fuel energy is needed to bring the air to combustion temperature. From a T-s perspective, the regenerator shifts the heat-addition process so that external heat input begins at a higher temperature, raising the mean temperature of heat addition (

T_H

,mean). Since η is bounded by 1 −

T_L

T_H

,mean for an equivalent Carnot cycle, increasing

T_H

,mean brings the effective Carnot ceiling higher and, by restructuring where heat enters the cycle, allows the combined cycle to capture a greater fraction of the available work. The bottoming Rankine cycle also benefits because less exhaust heat is wasted, and the temperature match between the gas turbine exhaust and the steam cycle improves.

Question 5: CRITICAL THINKING

Consider two power cycles operating between the same reservoir temperatures (

T_H

= 1000 K,

T_L

= 300 K). Cycle A is an internally reversible cycle with a non-isothermal heat-addition process that raises the working fluid from 400 K to 1000 K at constant pressure. Cycle B is the Carnot cycle operating between the same reservoirs. (a) Without calculating exact efficiencies, argue using the concept of mean temperature of heat addition which cycle is more efficient. (b) Derive a general expression for the mean temperature of heat addition

T_H

,mean for a constant-pressure process in terms of outlet and inlet enthalpies and entropies, and explain why this quantity is always less than

T_H

for a non-isothermal process. (c) Discuss whether it is theoretically possible to design a non-Carnot cycle whose efficiency equals the Carnot efficiency between the same

T_H

and

T_L

Show answer

(a) Cycle B (Carnot) is more efficient. In Cycle A, heat is added over the range 400 K to 1000 K; the mean temperature of heat addition is the logarithmic mean or enthalpy-weighted average, which is strictly less than 1000 K. For the Carnot cycle, all heat addition occurs at exactly

T_H

= 1000 K, so

T_H

,mean =

T_H

. Since η ≤ 1 −

T_L

T_H

,mean and

T_H

,mean(A) <

T_H

,mean(B) =

T_H

, Cycle A has a lower efficiency ceiling. (b) The mean temperature of heat addition for a constant-pressure process is

T_H

,mean = (h_out − h_in)/(s_out − s_in) = q_in/Δs. This is the temperature that, if held constant during an isothermal process absorbing the same heat q_in, would produce the same entropy change Δs. For a non-isothermal process where T increases monotonically from T_in to T_out, the mean is always less than T_out because the lower-temperature portions of the process contribute disproportionately to entropy generation relative to the heat absorbed. Mathematically,

T_H

,mean = q_in/(∫dq/T) < T_max because 1/T is a convex function and Jensen's inequality applies. (c) A non-Carnot cycle can equal Carnot efficiency only if all heat addition occurs at

T_H

and all heat rejection occurs at

T_L

—meaning it would need isothermal heat exchange at both ends. Any departure from isothermal heat exchange reduces

T_H

,mean or raises

T_L

,mean, lowering efficiency. Therefore, a cycle with non-isothermal heat addition/rejection that still equals Carnot efficiency between the same extreme temperatures is thermodynamically impossible.

SUMMARY

Lesson Summary

Summary

The Carnot cycle establishes the absolute upper bound on thermal efficiency for any heat engine operating between two fixed-temperature reservoirs: η_Carnot = 1 − T_L/T_H. It achieves this maximum by consisting entirely of reversible processes—two isothermal and two isentropic steps—which produce zero entropy generation. Real cycles such as the Otto, Diesel, Rankine, and Brayton cycles depart from this ideal because they involve non-isothermal heat exchange, lowering the mean temperature of heat addition below T_H.

Beyond cycle-level differences, real hardware introduces irreversibilities including fluid friction, non-isentropic compression and expansion (captured by isentropic component efficiencies), heat loss, throttling, and combustion irreversibility, each generating entropy and further eroding efficiency. The ratio η_actual/η_Carnot quantifies how close a real system comes to the theoretical limit, and exergy analysis extends this comparison to the component level, guiding engineers toward the highest-impact improvements. Mastering the Carnot benchmark is essential for evaluating, designing, and optimizing every power and refrigeration cycle encountered in practice.