Diesel Cycle Simulator: Interactive Compression Ignition Engine Thermodynamics Calculator

✓ Verified Content: All equations, formulas, and reference data in this simulation have been verified by the Simulations4All engineering team against authoritative sources including MIT OpenCourseWare thermodynamics courses, NIST thermophysical properties, and Engineering Toolbox. See verification log

The diesel cycle simulator is your gateway to understanding compression ignition engines [1]. Unlike gasoline engines that rely on spark plugs, diesel engines achieve combustion through pure compression, squeezing air until it's hot enough to ignite fuel spontaneously. The second law tells us that higher compression means higher peak temperatures and better thermal efficiency.

Introduction

Here's the efficiency gap that explains why diesel dominates heavy transport: a typical diesel engine achieves 40-45% thermal efficiency compared to 25-30% for gasoline. That's 10-15 percentage points of fuel you're not wasting. No real Otto cycle achieves Carnot efficiency because knock limits compression ratio, but diesel engines have no knock limit. They can squeeze air to 20:1 or higher, reaching temperatures so extreme the fuel ignites on contact [2].

Energy in must equal energy out, plus whatever work you extract. In the diesel cycle, constant-pressure combustion means the piston moves down while fuel is still burning, extracting work during heat addition. Thermal engineers find this explains diesel's efficiency advantage: you add heat at higher average temperature than Otto's constant-volume explosion, even though the theoretical cutoff ratio penalty would suggest otherwise.

In practice, you lose energy to the exhaust stream, cooling system, friction, and the incomplete expansion that leaves hot gases trapped at exhaust valve opening. Experienced diesel engineers know that marine slow-speed engines achieve 50%+ efficiency by using extreme stroke/bore ratios and turbocompounding, recovering exhaust energy that smaller engines simply dump overboard.

Our interactive diesel cycle simulator lets you explore the fundamental thermodynamics of compression ignition engines. Adjust the compression ratio, cutoff ratio, and inlet conditions to see how they affect efficiency, work output, and the characteristic P-V and T-S diagrams that make diesel the energy accountant's favorite engine cycle.

How to Use This Simulation

Energy in must equal energy out. The diesel cycle converts fuel energy to work through compression ignition. The second law tells us that higher compression ratios enable higher efficiency, and diesel exploits this advantage with compression ratios Otto engines cannot match.

Main Controls

Input Parameters

Output Display

The results panel tracks complete cycle performance:

• Thermal Efficiency (%): The cutoff ratio penalty appears in the formula: η = 1 - (1/r^(γ-1)) × [(α^γ - 1)/(γ(α-1))]
• Net Work Output (kJ/kg): Work per unit mass of air. Diesel typically produces more work per cycle than Otto
• Heat Added Qin (kJ/kg): Energy during constant-pressure combustion (2→3)
• Heat Rejected Qout (kJ/kg): Exhaust losses (4→1). The second law requires Qout > 0
• Cutoff Ratio (α): Displayed for reference. Higher load = higher α = lower efficiency
• MEP (kPa): Mean Effective Pressure. Diesel MEP is higher than Otto, explaining torque advantage

State Point Table

Diesel vs Otto Comparison Panel

The simulation shows side-by-side efficiency comparison:

The Otto formula gives higher efficiency at same r, but Otto cannot reach diesel compression ratios. In practice, you lose energy to the cutoff ratio penalty, but you gain it back through higher r.

Tips for Exploration

1. Start with Standard Diesel preset (r = 15:1, α = 2.0): See efficiency around 55%. Compare to Otto at same r (impossible in reality due to knock)
2. Increase compression ratio from 15:1 to 20:1: Watch efficiency climb. This explains why truck diesels use extreme compression
3. Vary cutoff ratio from 1.5 to 3.0: Lower α gives higher efficiency but less power. High-load conditions require higher α
4. Try Marine Engine preset: r = 20:1 with optimized parameters. No real diesel cycle achieves Carnot efficiency, but marine slow-speed engines approach 55%
5. Check compression temperature T₂: At r = 18:1 and T₁ = 300 K, T₂ exceeds 900 K. This is well above diesel autoignition temperature (~480 K)

Why Diesel Beats Otto on Efficiency

Despite the cutoff ratio penalty, diesel wins in practice:

The second law tells us that higher compression temperature enables more work extraction. Diesel's ability to reach extreme compression ratios more than compensates for the constant-pressure heat addition penalty.

Understanding the Diesel Cycle

What Makes Diesel Different?

The key difference between diesel and Otto cycles lies in a single word: isobaric. In the Otto cycle (gasoline engines), heat is added at constant volume: boom, instant combustion. In the diesel cycle, fuel is injected gradually, and combustion occurs at essentially constant pressure as the piston moves down [1, 4].

The Four Processes

Process 1→2: Isentropic Compression

Air is compressed from state 1 (intake) to state 2 (TDC). This is where diesel engines shine. Compression ratios of 15:1 to 25:1 are typical, compared to 8:1-12:1 for gasoline engines. The temperature rises dramatically:

$T_2 = T_1 \cdot r^{\gamma-1}$

For a compression ratio of 18:1 and γ = 1.4, inlet air at 300 K reaches about 950 K (677°C)—well above diesel fuel's auto-ignition temperature of ~210°C.

Process 2→3: Constant Pressure Heat Addition

Here's the magic. Fuel is injected directly into the hot compressed air and ignites immediately. The piston moves down while combustion continues, maintaining approximately constant pressure. The cutoff ratio α defines when fuel injection stops:

$\alpha = \frac{V_3}{V_2} = \frac{T_3}{T_2}$

Typical cutoff ratios range from 1.5 to 4.0, depending on engine load.

Process 3→4: Isentropic Expansion

The high-pressure, high-temperature gases expand, pushing the piston down and producing work. This is the power stroke:

$T_4 = T_3 \cdot \left(\frac{V_3}{V_4}\right)^{\gamma-1} = T_3 \cdot \left(\frac{\alpha}{r}\right)^{\gamma-1}$

Process 4→1: Constant Volume Heat Rejection

The exhaust valve opens, and pressure drops rapidly as hot gases exit. In the idealized cycle, this is modeled as constant-volume heat rejection.

Key Parameters

Key Equations and Formulas

Thermal Efficiency

Formula: $\eta_{\text{Diesel}} = 1 - \frac{1}{r^{\gamma-1}} \cdot \frac{\alpha^\gamma - 1}{\gamma(\alpha - 1)}$

Where:

• η = thermal efficiency (dimensionless)
• r = compression ratio (V₁/V₂)
• α = cutoff ratio (V₃/V₂)
• γ = specific heat ratio (typically 1.4 for air)

Key insight: The bracketed term [α^γ - 1]/[γ(α-1)] is always greater than 1, which is why diesel efficiency is lower than Otto efficiency at the same compression ratio. But here's the catch: diesel engines can use much higher compression ratios without knock, so they end up more efficient overall [4].

Heat Addition (Constant Pressure)

Formula: $Q_{\text{in}} = c_p(T_3 - T_2) = \gamma c_v(T_3 - T_2)$

Where:

• Q_in = heat added per unit mass (kJ/kg)
• c_p = specific heat at constant pressure (≈1.005 kJ/kg·K for air)
• T₂, T₃ = temperatures before and after heat addition (K)

Heat Rejection (Constant Volume)

Formula: $Q_{\text{out}} = c_v(T_4 - T_1)$

Where:

• Q_out = heat rejected per unit mass (kJ/kg)
• c_v = specific heat at constant volume (≈0.718 kJ/kg·K for air)

Mean Effective Pressure (MEP)

Formula: $\text{MEP} = \frac{W_{\text{net}}}{V_1 - V_2}$

Where:

• MEP = mean effective pressure (kPa)
• W_net = net work output per cycle (kJ/kg)
• V₁ - V₂ = displacement volume (m³/kg)

MEP is the constant pressure that would produce the same net work over the displacement volume. It's a key metric for comparing engine performance [5].

Learning Objectives

After completing this simulation, you will be able to:

1. Calculate diesel cycle thermal efficiency using the compression ratio and cutoff ratio formula
2. Explain why diesel engines can achieve higher compression ratios than gasoline engines without knock
3. Analyze how the cutoff ratio affects both efficiency and power output
4. Compare diesel and Otto cycle efficiencies under various conditions
5. Interpret P-V and T-S diagrams for compression ignition cycles
6. Predict the effect of turbocharging on diesel engine performance

Exploration Activities

Activity 1: Compression Ratio Investigation

Objective: Discover why diesel engines use high compression ratios

Setup:

• Set cutoff ratio α = 2.0
• Set γ = 1.40 (air-standard)
• Start with compression ratio r = 12:1

Steps:

1. Note the thermal efficiency at r = 12:1
2. Gradually increase r to 15:1, 18:1, 21:1, and 24:1
3. Observe the efficiency increase at each step
4. Check the compression temperature T₂ at each setting

Observe: How does efficiency change? What happens to T₂?

Expected Result: Efficiency increases from about 48% to 60% as r goes from 12 to 24. Compression temperature T₂ exceeds 800 K at high compression ratios, well above diesel's auto-ignition temperature (~480 K or 210°C).

---

Activity 2: The Cutoff Ratio Trade-off

Objective: Understand the efficiency-power trade-off of cutoff ratio

Setup:

• Set compression ratio r = 18:1
• Set inlet temperature T₁ = 300 K
• Set inlet pressure P₁ = 100 kPa

Steps:

1. Set cutoff ratio α = 1.5 and record efficiency and heat input
2. Increase α to 2.0, 2.5, 3.0, and 3.5
3. Note how efficiency decreases while heat input (power) increases

Observe: The trade-off between efficiency and power output

Expected Result: At α = 1.5, efficiency is highest (~56%) but Q_in is lowest. At α = 3.5, efficiency drops to ~48% but Q_in nearly doubles. This is why heavy-duty diesels at full load operate at higher cutoff ratios.

---

Activity 3: Diesel vs. Otto Comparison

Objective: Compare diesel and Otto cycles at the same compression ratio

Setup:

• Note the Otto efficiency displayed in the comparison panel
• Set r = 15:1 for direct comparison

Steps:

1. At r = 15:1 and α = 2.0, compare diesel efficiency with Otto efficiency shown
2. Gradually reduce α toward 1.0 and observe what happens

Observe: The diesel efficiency approaches Otto efficiency as α → 1

Expected Result: Otto efficiency at r = 15 is about 63%. Diesel efficiency is lower (51-55%) due to the cutoff ratio factor. As α approaches 1.0, the diesel cycle mathematically approaches the Otto cycle (constant pressure becomes constant volume).

---

Activity 4: Turbocharging Effects

Objective: Explore how turbocharging affects diesel performance

Setup:

• Use the "Turbodiesel" preset
• Compare with "Standard Diesel" preset

Steps:

1. Load "Standard Diesel" preset and note all parameters
2. Switch to "Turbodiesel" and observe changes
3. Pay attention to inlet pressure P₁ and power output

Observe: Inlet pressure and mass flow increase, while efficiency may slightly decrease

Expected Result: The turbo preset has P₁ = 180 kPa vs. 100 kPa naturally aspirated. This roughly doubles the air mass (and fuel) per cycle, increasing power output proportionally. The slightly higher inlet temperature (320 K) reduces efficiency marginally.

Real-World Applications

Understanding the diesel cycle is essential in many fields:

1. Marine Engineering: The world's largest diesel engines power container ships. The Wärtsilä RT-flex96C produces 80,080 kW at 102 RPM with thermal efficiency exceeding 50%, making it the most efficient heat engine ever built [6]. These two-stroke monsters have compression ratios around 18:1 and stroke lengths measured in meters.

2. Heavy Transport: Long-haul trucks use turbodiesel engines with compression ratios of 15:1-17:1. The higher efficiency translates directly to fuel savings. A 10% efficiency improvement can save $10,000+ annually in fuel costs for a fleet truck driving 100,000 miles/year.

3. Power Generation: Diesel generators power hospitals, data centers, and remote locations. Their rapid start capability (full power in seconds) makes them essential for emergency backup power. Efficiency here means both lower operating costs and smaller fuel storage requirements.

4. Railway Locomotives: Diesel-electric locomotives dominate freight rail outside electrified corridors. A modern EMD SD70ACe produces 4,500 HP and can pull trains over a mile long across mountain grades. The diesel-electric hybrid approach optimizes the engine for constant-speed, peak-efficiency operation.

5. Agricultural Equipment: Large tractors and combines use diesels for their high torque at low speeds, which is essential for pulling plows or operating harvesting equipment. Compression ratios of 17:1-19:1 are common, with turbocharging standard on larger equipment.

Reference Data

Standard Air Properties

Diesel Fuel Properties

Typical Engine Parameters by Application

Challenge Questions

Level 1: Basic Understanding

1. Why can diesel engines use higher compression ratios than gasoline engines without experiencing knock?

2. In the P-V diagram, why is the heat addition process (2→3) drawn as a horizontal line at the top, while in the Otto cycle it's vertical?

Level 2: Intermediate Analysis

1. Calculate the thermal efficiency of an ideal diesel cycle with r = 20:1, α = 2.5, and γ = 1.35. How does this compare to an Otto cycle with the same compression ratio?

2. A diesel engine has a compression ratio of 18:1. At what compression ratio would a gasoline (Otto cycle) engine need to operate to match the diesel's thermal efficiency (assuming α = 2.0 and γ = 1.4)?

3. Explain why diesel efficiency decreases as the cutoff ratio increases, even though more fuel is being burned and more power is produced.

Level 3: Advanced Applications

1. A marine diesel engine operates at r = 20:1 with inlet conditions of 320 K and 200 kPa (turbocharged). The peak cycle temperature is 2200 K. Calculate: (a) the cutoff ratio, (b) thermal efficiency, and (c) net work output per kg of air.

2. For the dual cycle (a realistic model combining features of both Otto and Diesel), derive an expression for thermal efficiency in terms of compression ratio r, pressure ratio rp (constant volume heat addition), and cutoff ratio α.

3. A truck engine manufacturer wants to improve fuel economy by 5% without changing displacement. Analyze whether this is more effectively achieved by increasing compression ratio (limited by structural strength to r = 19:1) or reducing cutoff ratio (limited by power requirements to α ≥ 1.8).

Common Misconceptions

Misconception 1: "Diesel engines are less efficient because the efficiency formula has an extra term"

Reality: At the same compression ratio, yes—the Otto cycle formula gives higher efficiency. But diesel engines operate at much higher compression ratios (18:1 vs 10:1 typical), which more than compensates for the cutoff ratio penalty. Real diesel engines achieve 35-45% efficiency vs. 25-35% for gasoline engines [4].

Misconception 2: "Higher cutoff ratio always means more power"

Reality: Higher cutoff ratio does increase heat input and power per cycle, but it also reduces efficiency. The sweet spot depends on the application—light-load cruising uses low α for economy, while full-throttle acceleration uses high α for power. Modern diesels vary effective cutoff ratio through injection timing and duration.

Misconception 3: "Diesel engines don't have knock problems"

Reality: Diesel engines don't experience spark knock (pre-ignition before the spark), but they can experience "diesel knock"—a different phenomenon caused by rapid pressure rise during the ignition delay period. This is why cetane number (ignition quality) matters for diesel fuel, just as octane number matters for gasoline.

Misconception 4: "Turbocharging increases diesel cycle efficiency"

Reality: Turbocharging increases power density but typically reduces thermal efficiency slightly. The higher inlet temperature from compression heating increases T₁, which reduces the temperature ratio and thus efficiency. The real benefit is more power from a smaller, lighter engine—the efficiency per unit power is similar.

Misconception 5: "The diesel cycle accurately models real diesel engines"

Reality: The ideal diesel cycle is a simplification. Real diesel engines exhibit:

• Finite combustion time (not instantaneous)
• Heat transfer losses to cylinder walls
• Pumping losses during intake/exhaust
• Variable specific heats with temperature
• Incomplete combustion

The dual cycle (combined constant-volume and constant-pressure heat addition) better models real diesel combustion [7].

Misconception 6: "Diesel engines are always more efficient than gasoline"

Reality: For equivalent power output and typical driving conditions, yes. But for some specialized applications (high-RPM sports cars, small displacement engines), modern gasoline engines with variable valve timing and direct injection can match or exceed diesel efficiency while offering other advantages (lower emissions equipment, lighter weight).

Frequently Asked Questions

What is the thermal efficiency of a diesel cycle?

The thermal efficiency of an ideal diesel cycle is given by η = 1 - (1/r^(γ-1)) × [(α^γ - 1)/(γ(α-1))], where r is the compression ratio (typically 14-22 for diesels) and α is the cutoff ratio (1.5-4.0) [1, 4]. For typical values of r = 18 and α = 2.0 with γ = 1.4, the ideal efficiency is about 52%. Real diesel engines achieve 35-45% due to irreversibilities, heat transfer losses, and incomplete combustion [3].

Why do diesel engines have higher compression ratios than gasoline engines?

Diesel engines can use compression ratios of 14:1 to 25:1 because they compress pure air, not an air-fuel mixture [2]. Gasoline engines are limited to about 12:1 because higher compression would cause the pre-mixed fuel-air charge to auto-ignite prematurely (knock). Since diesel fuel is injected after compression, there's nothing to knock [4].

What is the cutoff ratio and how does it affect efficiency?

The cutoff ratio α = V₃/V₂ is the ratio of cylinder volume after heat addition to before heat addition [1]. It determines how long fuel injection continues. Higher cutoff ratios mean more fuel burned and more power, but lower efficiency—the efficiency factor [(α^γ - 1)/(γ(α-1))] increases as α increases, reducing overall efficiency [4].

How does the diesel cycle compare to the Otto cycle?

At the same compression ratio, the Otto cycle is more efficient because it adds heat at constant volume (more "explosive" combustion). However, diesel engines can use compression ratios of 18:1+ vs. 10:1 for gasoline, and the efficiency gain from higher r more than compensates [2, 3]. Additionally, diesels have no throttle losses at part load.

What is the significance of Mean Effective Pressure (MEP)?

MEP is the constant pressure that, acting on the piston over the displacement volume, would produce the same net work as the actual cycle [5]. It's a normalized measure of engine performance, independent of displacement. Higher MEP means more power from a given engine size. Typical diesel MEP values range from 800-2500 kPa.

Why are marine diesel engines so efficient?

Large marine diesels achieve 50%+ thermal efficiency through: (1) slow-speed operation allowing complete combustion, (2) very high compression ratios (18:1-22:1), (3) two-stroke operation eliminating pumping losses, (4) turbocompounding and waste heat recovery, and (5) optimized constant-speed operation [6]. The Wärtsilä RT-flex96C holds the record at 51.7% thermal efficiency.

---

References and Further Reading

Historical Sources (Public Domain)

1. Diesel, R. (1897) "The Diesel Oil Engine and Its Industrial Importance Particularly for Great Britain." Institution of Mechanical Engineers Proceedings. — Historical primary source on diesel engine principles

Open Educational Resources (Free Access)

1. MIT OpenCourseWare — 2.43 Advanced Thermodynamics. Available at: ocw.mit.edu — Creative Commons, free thermodynamics course

2. MIT OpenCourseWare — 2.60J Fundamentals of Advanced Energy Conversion. Available at: ocw.mit.edu — IC engine thermodynamics

3. OpenStax — University Physics Volume 2, Chapter 4: The Second Law of Thermodynamics. Available at: openstax.org — Free textbook

4. HyperPhysics — Diesel Engine Thermodynamics. Georgia State University. Available at: hyperphysics.gsu.edu — Free educational resource

5. HyperPhysics — Diesel Engine. Available at: hyperphysics.gsu.edu — Free educational resource

Industry Sources (Free Access)

1. Wärtsilä Corporation — Marine Diesel Engine Technical Documentation. Available at: wartsila.com — Manufacturer specifications

Property Data Sources (Free Access)

1. NIST Chemistry WebBook — Thermophysical Properties of Fluids. National Institute of Standards and Technology. Available at: webbook.nist.gov — Public domain

---

About the Data

Property Data Sources

The thermodynamic property data used in this simulation comes from:

• Air properties (R, γ, cv, cp): NIST Chemistry WebBook [8], standard air-standard analysis assumptions
• Diesel fuel properties: Engineering Toolbox and NIST data
• Engine specifications: Manufacturer published data (Wärtsilä, Cummins public documentation)

Critical Values

The compression and cutoff ratio ranges in this simulation are based on:

• Compression ratios (12:1-24:1): Survey of production diesel engines from automotive to marine applications [2, 3]
• Cutoff ratios (1.5-4.0): Derived from typical combustion duration and engine load conditions

Accuracy Statement

This simulation is designed for educational purposes. The calculations use standard air-standard analysis assumptions accurate to within 5-10% for ideal cycle predictions. For critical engineering applications, always verify calculations with primary sources and consider:

• Temperature-dependent specific heats (actual γ varies from 1.4 at 300 K to 1.3 at 1500 K)
• Combustion kinetics and injection timing effects
• Heat transfer to cylinder walls
• Real gas behavior at high pressures

---

How to Cite This Simulation

Citation Guide

If you use this simulation in educational materials or research, please cite as:

APA Format:

Simulations4All. (2025). Diesel cycle simulator: Interactive compression ignition engine thermodynamics calculator. Retrieved from https://simulations4all.com/simulations/diesel-cycle-simulator

MLA Format:

"Diesel Cycle Simulator: Interactive Compression Ignition Engine Thermodynamics Calculator." Simulations4All, 2025, simulations4all.com/simulations/diesel-cycle-simulator.

IEEE Format:

Simulations4All, "Diesel Cycle Simulator," 2025. [Online]. Available: https://simulations4all.com/simulations/diesel-cycle-simulator

BibTeX:

@online{simulations4all_diesel,
  author = {{Simulations4All}},
  title = {Diesel Cycle Simulator: Interactive Compression Ignition Engine Thermodynamics Calculator},
  year = {2025},
  url = {https://simulations4all.com/simulations/diesel-cycle-simulator},
  urldate = {2025-12-25}
}

---

Verification Log

All scientific claims, formulas, and data have been verified against authoritative sources.

---

Summary

The diesel cycle represents one of thermodynamics' most practical achievements, an engine cycle that turns Rudolf Diesel's 1890s vision into today's most efficient heat engines. By understanding the interplay between compression ratio, cutoff ratio, and the fundamental difference of constant-pressure heat addition, you gain insight into why diesels dominate applications from container ships to long-haul trucks.

Key takeaways:

• Diesel efficiency depends on both compression ratio and cutoff ratio
• Higher compression ratios are possible because fuel is injected after compression
• The cutoff ratio represents a power-efficiency trade-off
• Real diesel engines achieve 35-45% thermal efficiency, with large marine diesels exceeding 50%