Thermal Efficiency — Compute thermal efficiency of heat engines - Thermodynamics

SECTION 1

Historical Context & Motivation

The concept of thermal efficiency arose from a profoundly practical question: how much useful work can we extract from a given quantity of heat? During the Industrial Revolution, steam engines were transforming manufacturing, mining, and transportation, yet engineers lacked a rigorous framework for comparing one engine design against another. The fuel consumed by early Newcomen engines was staggering relative to the work they delivered, and incremental improvements proceeded largely by trial and error. It was not until the early nineteenth century that natural philosophers began formalizing the relationship between heat input and mechanical output, laying the groundwork for what we now call the second law of thermodynamics and the science of thermodynamics itself.

1712 • Newcomen Engine

Thomas Newcomen builds the first commercially successful atmospheric steam engine for pumping water from mines. Its thermal efficiency is roughly 0.5%, but it marks the beginning of the heat-engine era.

1769 • Watt's Separate Condenser

James Watt patents the separate condenser, dramatically reducing wasted heat and roughly tripling the efficiency of the Newcomen design. Watt's innovations motivate the need for a quantitative efficiency metric.

1824 • Carnot's Réflexions

Sadi Carnot publishes Réflexions sur la Puissance Motrice du Feu, introducing the concept of an ideal reversible cycle and establishing that no engine can surpass the efficiency of a Carnot engine operating between the same thermal reservoirs.

1850s • Clausius & Kelvin Formalize the Second Law

Rudolf Clausius and Lord Kelvin independently articulate the second law of thermodynamics, introducing entropy and providing a rigorous mathematical foundation for thermal efficiency and its upper bound.

1876–1897 • Otto & Diesel Cycles

Nikolaus Otto patents the four-stroke internal combustion engine (1876), and Rudolf Diesel demonstrates the compression-ignition engine (1897). Each cycle's thermal efficiency depends on compression ratio and specific-heat characteristics, making the efficiency formula a central design tool.

From Newcomen's smoky boilers to modern combined-cycle gas turbines operating above 60% efficiency, the central question has remained the same: what fraction of the heat we supply actually becomes work? Answering this question precisely requires the definition of thermal efficiency and an understanding of the thermodynamic limits that govern it—topics we develop in the sections that follow.

SECTION 2

Core Principles & Definitions

A heat engine is any device that operates in a thermodynamic cycle, receives heat from a high-temperature source, converts part of that heat into net work, and rejects the remaining energy to a low-temperature sink. The thermal efficiency (η_th) of such an engine quantifies the fraction of the heat input that is successfully transformed into work. Before computing η_th, it is essential to internalize several foundational ideas that underpin the definition and govern its limits.

Energy Conservation (First Law)

For a cyclic device, the net work output equals the difference between the heat absorbed from the hot reservoir (Q_H) and the heat rejected to the cold reservoir (Q_L): W_net = Q_H − Q_L.

Thermal Efficiency Definition

η_th = W_net / Q_H. It is a dimensionless ratio between 0 and 1 (or 0% and 100%). An efficiency of 1 would mean all heat is converted to work—forbidden by the second law.

Kelvin–Planck Statement

No heat engine operating in a cycle can convert 100% of the heat it receives into work; some heat must always be rejected. This is the second law in its heat-engine form, guaranteeing η_th < 1 for every real or ideal cycle.

Carnot Upper Bound

The maximum possible efficiency between reservoirs at temperatures T_H and T_L is η_Carnot = 1 − T_L / T_H, where temperatures are in absolute units (K or R).

Irreversibilities Reduce Efficiency

Friction, unrestrained expansion, heat transfer across finite temperature differences, and mixing are all irreversible processes. Every real engine contains such losses, causing its thermal efficiency to fall below the Carnot limit for the same reservoir temperatures.

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KEY TAKEAWAY

Think of a heat engine as a financial investor who receives income (Q_H) and must pay taxes (Q_L). Thermal efficiency is the investor's net return on income: the fraction of heat income that actually becomes useful work. The second law of thermodynamics is the 'tax code' that guarantees the tax bill is always positive—no engine keeps 100% of its heat income.

SECTION 3

Visual Explanation — The Heat-Engine Energy Flow

Energy-flow diagram of a generic heat engine. Heat Q_H enters from the hot reservoir at temperature T_H. The engine converts a portion into net work W_net (shown exiting to the right in green) and rejects the remainder Q_L to the cold reservoir at T_L. The first-law energy balance and thermal efficiency definition are summarized at the bottom.

The diagram above captures the essential energy accounting for any heat engine cycle. Notice that all three energy quantities are taken as positive magnitudes; the direction of energy transfer is indicated by the arrows. The hot-reservoir arrow points downward into the engine, indicating Q_H is received by the working fluid. The cold-reservoir arrow points downward out of the engine, indicating Q_L is rejected. The work arrow exits to the right, representing the net mechanical output delivered to the surroundings. This sign convention—where Q_H, Q_L, and W_net are all positive—is the standard convention in engineering thermodynamics and simplifies the efficiency formula to a straightforward ratio.

SECTION 4

Mathematical Framework

We now formalize the relationships introduced qualitatively in the preceding sections. The derivation proceeds from the first law applied to a cyclic process and culminates in the Carnot efficiency, which sets the absolute upper bound on thermal efficiency for any engine operating between two thermal reservoirs.

FIRST LAW FOR A CYCLE

W_net = Q_H − Q_L

For a system executing a complete thermodynamic cycle, the change in internal energy is zero (ΔU = 0). Therefore, the net work equals the net heat transfer: W_net = Q_H − Q_L, where Q_H is the heat absorbed from the high-temperature reservoir and Q_L is the heat rejected to the low-temperature reservoir (both positive magnitudes).

THERMAL EFFICIENCY DEFINITION

η_th = W_net / Q_H = (Q_H − Q_L) / Q_H = 1 − Q_L / Q_H

η_th is a dimensionless ratio, 0 < η_th < 1. The form 1 − Q_L/Q_H is particularly useful because it shows that efficiency improves when the ratio of rejected heat to input heat decreases.

CARNOT EFFICIENCY (UPPER BOUND)

η_Carnot = 1 − T_L / T_H

T_H and T_L are the absolute temperatures (in kelvin) of the hot and cold reservoirs, respectively. This expression follows from the Carnot cycle, where Q_L/Q_H = T_L/T_H for reversible isothermal-adiabatic processes. All real engines satisfy η_th < η_Carnot.

EFFICIENCY IN TERMS OF POWER

η_th = Ẇ_net / Q̇_H

When analyzing steady-state power plants, it is convenient to express efficiency in terms of rates: Ẇ_net is the net power output (kW or MW) and Q̇_H is the rate of heat input. The numerical value of η_th is identical whether computed from total energies per cycle or from power rates.

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Important Note on Units

When using the Carnot formula η_Carnot = 1 − T_L/T_H, temperatures must be in absolute units—kelvin (K) or Rankine (°R). Using Celsius or Fahrenheit will produce incorrect results because ratios of those scales are not physically meaningful.

SECTION 5

Efficiency of Common Power Cycles

The general thermal-efficiency formula applies to every heat-engine cycle, but each ideal cycle yields a specific expression that depends on its characteristic process parameters. Understanding these expressions allows engineers to identify the primary design levers—compression ratio, pressure ratio, peak temperature—that govern efficiency for a particular engine type. The diagram below compares the pressure-volume (P-v) behavior and thermal efficiency expressions for four foundational power cycles.

Comparison of the Carnot and Otto cycles on P-v diagrams, with thermal efficiency expressions for the Carnot, Otto, and Diesel cycles. The compression ratio r is the dominant parameter controlling efficiency in internal-combustion cycles.

Cycle	Ideal Efficiency Expression	Primary Lever	Typical Real η_th
Carnot	1 − T_L/T_H	Temperature ratio T_L/T_H	Theoretical upper bound only
Otto	1 − 1/r^γ−1	Compression ratio r	25–35%
Diesel	1 − (r_c^γ − 1) / [γ(r_c − 1) × r^γ−1]	Compression ratio r and cutoff ratio r_c	35–45%
Rankine	(h₁ − h₂ − w_pump) / q_in	Boiler pressure & superheat temperature	33–45% (steam)
Brayton	1 − 1/r_p^(γ−1)/γ	Pressure ratio r_p	30–40% (simple); >60% (combined)

Thermal efficiency expressions and typical real-world efficiencies for common power cycles.

SECTION 6

Worked Example — Steam Power Plant Efficiency

Consider a coal-fired steam power plant operating on a simple ideal Rankine-like cycle. The boiler supplies heat at a rate of Q̇_H = 500 MW, and the condenser rejects heat to a cooling river at a rate of Q̇_L = 320 MW. The hot-source temperature is T_H = 550 °C and the condenser temperature is T_L = 30 °C. Determine (a) the net power output, (b) the thermal efficiency, and (c) the maximum possible (Carnot) efficiency between the given reservoir temperatures.

Steam Power Plant — Thermal Efficiency Calculation

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Step 1 — Identify Given Values

Q̇_H = 500 MW (rate of heat input from the boiler). Q̇_L = 320 MW (rate of heat rejection in the condenser). T_H = 550 °C = 550 + 273.15 = 823.15 K. T_L = 30 °C = 30 + 273.15 = 303.15 K.

All temperatures converted to kelvin for the Carnot calculation.
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Step 2 — Compute Net Power Output

Apply the first law for a cyclic steady-state device: Ẇ_net = Q̇_H − Q̇_L = 500 MW − 320 MW = 180 MW.

Ẇ_net = 180 MW
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Step 3 — Compute Thermal Efficiency

η_th = Ẇ_net / Q̇_H = 180 MW / 500 MW = 0.36. Converting to a percentage: η_th = 36%.

η_th = 36.0%
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Step 4 — Compute Carnot (Maximum) Efficiency

η_Carnot = 1 − T_L / T_H = 1 − 303.15 / 823.15 = 1 − 0.3682 = 0.6318.

η_Carnot = 63.2%
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Step 5 — Interpret Results

The actual thermal efficiency (36%) is well below the Carnot limit (63.2%), which is expected for a real power plant. The ratio η_th / η_Carnot = 0.36 / 0.632 ≈ 0.57, meaning this plant achieves about 57% of the theoretically maximum efficiency. Irreversibilities in the boiler, turbine, pump, and condenser account for the gap.

The plant operates at roughly 57% of the Carnot limit.

SECTION 7

Strengths, Limitations & Practical Considerations

The thermal efficiency metric is one of the most widely used performance indicators in energy engineering, but like any single metric it has both strengths and limitations that must be understood in context. The table below summarizes the key advantages of using η_th alongside practical caveats that arise in real-world applications.

Strengths	Limitations
Universal applicability: the definition W_net/Q_H applies to every heat engine regardless of working fluid or cycle type.	Does not capture cost of fuel, capital, or environmental impact. Two plants at the same η_th may differ enormously in economic viability.
Provides a clear theoretical ceiling via the Carnot efficiency, guiding R&D priorities toward reducing irreversibilities.	The Carnot bound uses average reservoir temperatures; real cycles exchange heat over temperature ranges, making the 'effective' Carnot limit somewhat ambiguous.
Dimensionless ratio allows easy comparison across scales—from a 1 kW Stirling engine to a 1 GW nuclear plant.	Ignores exergetic (second-law) considerations. A cycle may have reasonable η_th but destroy large amounts of available work (exergy) internally.
Directly tied to fuel consumption: η_th ↑ ⇒ fuel per kWh ↓, making it a practical operational benchmark.	Part-load efficiency can differ significantly from design-point efficiency, so a single η_th value may be misleading for variable-demand applications.

Strengths and limitations of thermal efficiency as a performance metric.

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KEY TAKEAWAY

Thermal efficiency is analogous to a fuel-economy rating for a car: it tells you how many kilometers you get per liter of fuel but says nothing about purchase price, insurance, maintenance costs, or emissions. In engineering practice, thermal efficiency is a necessary but not sufficient metric—designers also evaluate second-law (exergetic) efficiency, economic dispatch, lifecycle emissions, and reliability before selecting a power-cycle configuration.

SECTION 8

Connection to Second-Law (Exergetic) Efficiency

While thermal efficiency measures the fraction of heat converted to work, it does not reveal how wisely the available energy (exergy) is utilized. Two engines may have identical thermal efficiencies yet vastly different levels of internal irreversibility. The second-law efficiency (also called exergetic efficiency) compares an engine's actual performance not to Q_H but to the maximum work theoretically obtainable from that Q_H given the reservoir temperatures.

Attribute	Thermal (First-Law) Efficiency	Exergetic (Second-Law) Efficiency
Definition	η_th = W_net / Q_H	η_II = W_net / W_rev,max = η_th / η_Carnot
Reference benchmark	Total heat input (Q_H)	Maximum reversible work (Carnot work)
Range	0 < η_th < 1	0 < η_II ≤ 1 (equals 1 only for a fully reversible cycle)
Physical insight	How much heat becomes work	How close the engine operates to the thermodynamic ideal; identifies where exergy is destroyed
Typical use	Quick performance rating; fuel-economy comparisons	Advanced cycle optimization; identifying the most lossy components

First-law vs. second-law efficiency: a conceptual comparison.

In subsequent courses on advanced thermodynamics or power-plant engineering, you will learn to apply exergy analysis to individual components—boilers, turbines, compressors, heat exchangers—pinpointing exactly where the greatest irreversibilities occur. The thermal efficiency you have learned here serves as the essential first step; exergetic efficiency builds directly upon it by normalizing W_net to the Carnot work rather than to Q_H. Mastering η_th therefore equips you with the foundational ratio from which all second-law analyses are derived.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A classmate claims that a heat engine receiving 400 kJ of heat and producing 400 kJ of work is possible because energy is conserved. Explain, citing a specific law of thermodynamics, why this engine cannot exist.

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Although such an engine would satisfy the first law (energy is conserved if Q_L = 0), it violates the Kelvin–Planck statement of the second law, which states that no heat engine operating in a cycle can convert all of the heat it receives from a single reservoir into work without rejecting some heat to a lower-temperature reservoir. If Q_L = 0, the thermal efficiency would be 100%, which the second law prohibits. Thus, the proposed engine is a perpetual-motion machine of the second kind and is physically impossible.

Question 2: BASIC CALCULATION

A heat engine absorbs 800 kJ of heat per cycle from a furnace and rejects 520 kJ to the ambient air. Calculate (a) the net work per cycle and (b) the thermal efficiency.

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(a) W_net = Q_H − Q_L = 800 − 520 = 280 kJ. (b) η_th = W_net / Q_H = 280 / 800 = 0.35 or 35%. Alternatively, η_th = 1 − Q_L/Q_H = 1 − 520/800 = 0.35, confirming the result.

Question 3: INTERMEDIATE

A Carnot engine operates between a high-temperature reservoir at 727 °C and a low-temperature reservoir at 27 °C. (a) Determine the Carnot efficiency. (b) If the engine produces 150 kW of power, calculate the rate of heat input and the rate of heat rejection.

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(a) Convert to kelvin: T_H = 727 + 273 = 1000 K; T_L = 27 + 273 = 300 K. η_Carnot = 1 − 300/1000 = 0.70 or 70%. (b) η_th = Ẇ_net / Q̇_H ⇒ Q̇_H = 150 / 0.70 = 214.3 kW. Then Q̇_L = Q̇_H − Ẇ_net = 214.3 − 150 = 64.3 kW. The engine rejects about 64.3 kW to the cold reservoir.

Question 4: APPLIED

A gasoline engine operates on an air-standard Otto cycle with a compression ratio r = 9 and uses air with γ = 1.4. (a) Compute the ideal Otto-cycle thermal efficiency. (b) If friction and other irreversibilities reduce the actual efficiency to 60% of the ideal value, and the engine consumes fuel releasing 45 MJ per kg at a rate of 0.008 kg/s, determine the actual power output.

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(a) η_Otto = 1 − 1/r^γ−1 = 1 − 1/9^0.4. We compute 9^0.4 = e^{0.4 × ln 9} = e^{0.4 × 2.197} = e^0.879 ≈ 2.408. So η_Otto = 1 − 1/2.408 = 1 − 0.415 = 0.585 or 58.5%. (b) Actual η = 0.60 × 0.585 = 0.351. The heat input rate is Q̇_H = 0.008 kg/s × 45 × 10⁶ J/kg = 360 kW. Actual power = 0.351 × 360 = 126.4 kW. The engine delivers approximately 126 kW of shaft power.

Question 5: CRITICAL THINKING

An inventor proposes an engine that operates between reservoirs at T_H = 600 K and T_L = 300 K, absorbing Q_H = 1000 kJ per cycle and producing W_net = 600 kJ. Is this engine thermodynamically possible? If not, identify which law is violated, and determine the maximum work output that a reversible engine could produce under the same conditions.

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First, check the claimed efficiency: η_claim = 600/1000 = 0.60 or 60%. Next, compute the Carnot limit: η_Carnot = 1 − 300/600 = 0.50 or 50%. Since 60% > 50%, the inventor's claim violates the second law of thermodynamics: no engine operating between these reservoirs can exceed 50% efficiency. The first law is not violated (Q_L = 1000 − 600 = 400 kJ, so energy is conserved), but the second law imposes a stricter constraint. The maximum work from a reversible engine would be W_rev = η_Carnot × Q_H = 0.50 × 1000 = 500 kJ, with Q_L = 500 kJ rejected. This problem illustrates the essential role of the Carnot bound as a feasibility check for any claimed heat-engine performance.

SUMMARY

Lesson Summary

Summary

A heat engine absorbs heat Q_H from a hot reservoir, converts part of it into net work W_net = Q_H − Q_L, and rejects the remainder Q_L to a cold reservoir. The thermal efficiency η_th = W_net / Q_H = 1 − Q_L / Q_H measures the fraction of heat successfully converted to work. By the Kelvin–Planck statement of the second law, η_th is always less than 1 for any real or ideal engine.

The Carnot efficiency η_Carnot = 1 − T_L / T_H (absolute temperatures) establishes the upper bound on thermal efficiency for any engine operating between two given reservoir temperatures. Specific cycle expressions—such as η_Otto = 1 − 1/r^γ−1 for the Otto cycle—reveal which design parameters (compression ratio, pressure ratio, peak temperature) engineers can manipulate to approach the Carnot limit. For a deeper assessment of internal losses, the second-law (exergetic) efficiency η_II = η_th / η_Carnot normalizes performance to the reversible ideal, guiding targeted improvements in real power-plant components.