Visualize Carnot, Otto, Diesel, and Brayton cycles on P-V and T-s diagrams in real time. Instantly compute thermal efficiency, net work, and heat exchange quantities.
Parameters
Cycle type
Heat capacity ratio γ
Air: 1.40 / monatomic ideal gas: 1.67
Cold reservoir T_L / inlet temperature T₁ [K]
K
Hot reservoir T_H / peak temperature T₃ [K]
K
Compression ratio r_c
Otto: 8–12 / Diesel: 14–22
Cutoff ratio r_CO
Volume ratio at the end of combustion, V₃/V₂
Pressure ratio r_p
Typical gas turbine: 10-40
Initial pressure p₁ [kPa]
kPa
Otto cycle
An ideal gasoline-engine cycle with constant-volume combustion. Theoretical efficiency is higher than the 30-35% typically seen in real engines.
Results
Thermal efficiency η
—
%
Net work W_net
—
kJ/kg
Heat input Q_in
—
kJ/kg
Heat rejection Q_out
—
kJ/kg
State point 1
—
State point 2
—
State point 3
—
State point 4
—
P-V diagram (equation of state)
T-s diagram (entropy)
What is a Thermodynamic Cycle?
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What exactly is a thermodynamic cycle, and why do we need to visualize it on different diagrams?
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Basically, it's a series of processes where a working fluid—like air or steam—returns to its initial state. We visualize it on Pressure-Volume (P-V) and Temperature-Entropy (T-s) diagrams to understand the work done and heat exchanged. In practice, the P-V diagram shows the net work output (the area inside the loop), while the T-s diagram clearly shows the heat added and rejected. Try switching between the Carnot and Otto cycles in the simulator to see how their shapes differ.
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Wait, really? So the area inside the loop is the work? That makes sense. But what's the difference between something like the Carnot cycle and the Otto cycle I see in the dropdown?
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Great question! The Carnot cycle is an ideal, theoretical limit with two isothermal and two adiabatic processes—it has the maximum possible efficiency. The Otto cycle, which models gasoline engines, replaces the isotherms with constant-volume heat addition and rejection. For instance, in a car engine, the spark plug ignites the fuel at nearly constant volume. You can see this difference clearly: adjust the "Cycle Type" to Otto and watch the top of the loop become a vertical line on the P-V diagram, representing that instant combustion.
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Okay, I see the shape change. The efficiency number updates too. So if I slide the "Compression ratio" control for the Otto cycle, why does the efficiency go up?
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Exactly! That's a key engineering trade-off. The compression ratio ($r_c$) is the ratio of maximum to minimum volume. A higher $r_c$ means the air-fuel mixture is squeezed more before ignition, leading to a higher peak temperature and pressure. This makes the cycle more efficient because you extract more work from the same amount of heat. In practice, there's a limit—too high, and the fuel pre-ignites, causing "knock." Move that slider up and watch the P-V loop get taller and skinnier, and see the efficiency formula update in real time.
Physical Model & Key Equations
The universal measure of performance for any heat engine cycle is its thermal efficiency, defined as the net work output divided by the total heat input.
$$\eta = 1 - \frac{Q_L}{Q_H}$$
Where $\eta$ is thermal efficiency, $Q_H$ is heat added (from the hot reservoir/fuel combustion), and $Q_L$ is heat rejected (to the cold reservoir/atmosphere). For the ideal Carnot cycle, this simplifies to a function of absolute temperatures only.
For practical cycles like Otto and Diesel, efficiency depends on geometric ratios and material properties. The Otto cycle efficiency is derived from the compression ratio and the heat capacity ratio of the working fluid.
Here, $r_c = V_1/V_2$ is the compression ratio (try the slider!), and $\gamma = c_p/c_v$ is the heat capacity ratio (around 1.4 for air). This shows why increasing $r_c$ improves efficiency—it raises the term $r_c^{\gamma-1}$.
Real-World Applications
Automotive Engines (Otto Cycle): This is the fundamental model for gasoline/petrol engines. Every time you adjust the "Compression Ratio" in the simulator, you're exploring the core design parameter that engineers balance against fuel octane rating to prevent engine knock while maximizing mileage.
Heavy-Duty & Marine Engines (Diesel Cycle): Diesel engines use a constant-pressure heat addition process, modeled by the "Cutoff Ratio" ($r_{CO}$) in the simulator. This allows for higher compression ratios and efficiency, which is why they are used in trucks and ships, but often at the cost of higher emissions.
Jet Engines & Power Plants (Brayton Cycle): This cycle models gas turbines. The key parameter here is the "Pressure Ratio" ($r_p$). By increasing this ratio in the simulator, you directly increase the efficiency of jet engines for aircraft and the gas turbines used in modern electricity power plants.
Ideal Benchmarking (Carnot Cycle): No real engine achieves Carnot efficiency, but it provides the absolute theoretical limit. Engineers use it as a gold standard to gauge how close their real-world designs (like the ones you can select) come to the ideal, highlighting where losses occur.
Common Misconceptions and Points to Note
There are a few key points you should be especially mindful of when starting to use this tool. First, understand that the calculation results are not the actual performance of a real engine. This is a "theoretical model" based on significant assumptions of the "air-standard cycle" (constant specific heats, ideal gas, combustion replaced by heat addition, no friction or heat losses). For example, even if the simulator shows an Otto cycle efficiency of 60%, the best actual thermal efficiency for a gasoline engine is around 40% at most. This gap illustrates the magnitude of real-world losses (wall heat losses, pumping losses, incomplete combustion, etc.).
Next, it's important to understand the practical reasons why parameters cannot be increased without limit. While increasing the compression ratio improves efficiency, gasoline engines face a knocking limit (e.g., 10-12:1), and diesel engines face limits of mechanical strength and maximum combustion pressure (e.g., 18-22:1). Raising the inlet temperature T₁ in a Brayton cycle also hits a wall—the limit of what turbine blade materials can withstand (even in advanced aircraft engines, this is around 1700°C). Don't just chase ideals with the tool; get into the habit of thinking, "Why can't we push this further?"
Finally, note that the "Carnot cycle is not always the best". While it indeed offers the highest efficiency for given temperature limits, realistically, its work output (power) is extremely small. Recall that the area inside the P-V diagram represents work output. In design, the perspective of trade-offs—"how to extract large amounts of work at high efficiency with a practical structure"—becomes critically important, not just pursuing efficiency alone.