Adjust compression ratio and heat input to draw the ideal air-standard Otto cycle on a P-V diagram using real isentropic and constant-volume relations, and compute thermal efficiency live.
This simulator draws the ideal air-standard Otto cycle — the theoretical basis for spark-ignition internal combustion engines — on a pressure-volume (P-V) diagram, using the real isentropic (Pv^γ = constant) and constant-volume ideal-gas relationships for each of the four processes. Adjusting compression ratio and heat input redraws the cycle and recomputes thermal efficiency instantly.
1→2, isentropic compression: the piston compresses the air-fuel mixture from bottom dead center (BDC) to top dead center (TDC) with no heat transfer, so T2 = T1·r^(γ−1) and P2 = P1·r^γ, where r = V1/V2 is the compression ratio. 2→3, constant-volume heat addition: combustion occurs so fast (idealized as instantaneous) that volume does not change; pressure and temperature both jump, with T3 = T2 + Qin/cv. 3→4, isentropic expansion: the power stroke, where high-pressure combustion gas pushes the piston back to BDC with no heat transfer, T4 = T3/r^(γ−1). 4→1, constant-volume heat rejection: the exhaust/intake stroke is idealized as the working fluid rejecting heat back to the surroundings at constant volume, closing the cycle.
An isentropic (reversible, adiabatic) process for an ideal gas follows Pv^γ = constant, a power-law relationship, not a straight line. Since γ (1.4 for air) exceeds 1, the isentropic curve is steeper than a comparable isothermal curve (Pv = constant) at the same point — this is why the compression and expansion legs of the P-V diagram bow outward. This simulator computes each curve point-by-point directly from the isentropic relation, rather than approximating it as a straight segment.
The ideal Otto cycle thermal efficiency depends only on the compression ratio: η = 1 − 1/r^(γ−1) — notably independent of how much heat is added. This is why real engine designers push compression ratios as high as knock (pre-ignition) and material limits allow. In practice, real engines fall well short of this ideal efficiency because of non-instantaneous combustion, heat losses through cylinder walls, friction, and the fact that intake/exhaust are not truly constant-volume — real indicator diagrams are rounded rather than sharp-cornered like the ideal cycle shown here.
The Otto cycle efficiency formula η = 1 − 1/r^(γ−1) falls directly out of the four process equations: the T3 and T4 terms in the heat-rejected/heat-added ratio Qout/Qin both scale by the same factor when you substitute the constant-volume relations, and that factor cancels algebraically, leaving efficiency dependent only on the compression ratio r and the gas property γ. Adding more heat (richer fuel mixture, higher temperature) increases net work output and peak pressure/temperature, but not the fraction of heat converted to work.
Compression ratio r = V1/V2 is the ratio of cylinder volume at bottom dead center to volume at top dead center. Real gasoline (spark-ignition) engines are typically limited to roughly 8:1 to 12:1 because higher compression raises the end-of-compression temperature enough to risk auto-ignition ("knock") of the fuel-air mixture before the spark fires, which can severely damage the engine. Higher-octane fuels resist knock better and allow somewhat higher compression ratios.
The Otto cycle adds heat at constant volume (an idealized instantaneous spark-ignited combustion), while the Diesel cycle adds heat at constant pressure (combustion occurs gradually as fuel is injected into already-compressed, hot air, so the piston moves outward during combustion). Diesel engines typically use much higher compression ratios (16:1 to 22:1) since they compress air alone (no knock risk from fuel), which is part of why diesel engines are generally more thermally efficient despite the Diesel cycle formula itself being slightly less efficient than Otto at the same compression ratio.
State 1 is bottom dead center at the end of the intake stroke (cylinder full of air-fuel mixture at ambient conditions). State 2 is top dead center after the compression stroke. State 3 is still at top dead center, immediately after the (idealized instantaneous) spark-ignited combustion — the pressure and temperature spike shown as the vertical 2→3 line. State 4 is bottom dead center at the end of the power stroke, just before the exhaust valve opens.
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