# Thermodynamic Cycle Simulator - Carnot, Otto, Diesel, Brayton, Stirling

> Interactive PV and TS diagrams for Carnot, Otto, Diesel, Brayton, Stirling and custom cycles. Drag states, see Q, W, ΔU, ΔS per step, efficiency and COP.

- URL: https://qilingtools.com/en/math/thermo-cycle
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Pick a cycle and both the P–V and T–S diagrams are drawn side by side, each process coloured by type, with the enclosed area shaded as net work. Drag the state points to change the compression ratio, peak temperature or volumes, or use the sliders for exact values; the working gas is an ideal gas with monatomic, diatomic, polyatomic or custom γ. An animated piston shows the cylinder volume, gas temperature and whether heat is flowing in or out right now, while the table lists Q, W, ΔU and ΔS for every step and compares the computed efficiency with the textbook formula and the Carnot limit. Switch to refrigerator or heat-pump mode to run the cycle backwards and get the COP, or build your own cycle from isothermal, adiabatic, isobaric, isochoric and straight-line steps.

## How to use

1. Choose Carnot, Otto, Diesel, Brayton, Stirling or a custom cycle, and set γ and the amount of gas
2. Drag the state points on the P–V diagram or use the sliders for ratios, temperatures and volumes
3. Watch the dot travel round the cycle and the piston take in or reject heat
4. Read the per-step Q, W, ΔU, ΔS table and the efficiency or COP

## Features

- Five classic cycles plus a custom builder (isothermal, adiabatic, isobaric, isochoric, straight line; auto-closing)
- Linked P–V and T–S diagrams with shaded net work and optional log axes
- Drag state points to edit the cycle; axes stay put while you drag
- Per-step Q, W, ΔU, ΔS table with ΣΔU = 0 and ΣΔS = 0 checks
- Efficiency vs textbook formula and Carnot limit; Stirling with an optional ideal regenerator
- Refrigerator and heat-pump modes with the Carnot COP bound from the actual heat-exchange temperatures

## FAQ

### Why does the Otto efficiency depend only on the compression ratio?

The air-standard Otto cycle has two adiabats and two constant-volume steps. Heat in is nCv(T₃ − T₂), heat out is nCv(T₄ − T₁), and the adiabats give T₂/T₁ = T₃/T₄ = r^(γ−1). Put these into η = 1 − Q_out/Q_in and every temperature cancels, leaving η = 1 − r^(1−γ): about 56% for r = 8 and γ = 1.4. Real petrol engines reach 25–35% because of friction, heat loss and incomplete combustion, and knock limits the compression ratio.

### At the same compression ratio a Diesel cycle is less efficient — so why are diesel engines more economical?

The Diesel cycle adds heat at constant pressure instead of constant volume, and the larger the cut-off ratio rc, the larger the penalty factor (rc^γ − 1)/(γ(rc − 1)), so at equal r it loses to Otto. But a diesel engine compresses only air and cannot knock, so it runs at compression ratios of 15–22 instead of 8–12, which more than makes up the difference.

### What does the area on a T–S diagram mean?

For a reversible process dQ = T dS, so the area under a curve on the T–S diagram is the heat absorbed. Over a full cycle ΔU = 0, so the net heat equals the net work and the area enclosed on the T–S diagram equals the area enclosed on the P–V diagram. A Carnot cycle is a rectangle on a T–S diagram — horizontal isotherms and vertical adiabats — which shows at a glance why it is the most efficient cycle between two temperatures.

### How can a refrigerator's COP be greater than 1?

COP is heat moved divided by work used, not an energy conversion efficiency. A refrigerator spends work W to remove Q_C from the cold side and dumps Q_H = Q_C + W on the hot side, so COP_R = Q_C/W can be well above 1, and a heat pump's COP_HP = Q_H/W = COP_R + 1 is higher still. The Carnot limit T_C/(T_H − T_C) grows as the temperature difference shrinks, which is why air conditioners and heat pumps work best when indoor and outdoor temperatures are close.

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