Appendix B: Otto Cycle

Earlier, we discussed the cyclic process. But without a concrete example, the cyclic process can feel like a meaningless concept. So let’s give the Otto cycle a brief study to help you get a better grasp of a cyclic process.

Four-Stroke Engine

Perhaps you are aware that a gasoline engine is an internal combustion engine designed to run on petrol. Four-stroke gasoline engines power the vast majority of automobiles, small trucks, buses and medium-large motorbikes.

If a car has a 1600 cc engine, it means that the total volume of the cylinders containing the fuel-air mixture is 1600 cm3. It always amazes me that heavy vehicles and machinery are powered by such a small volume of gas.

These cylinders are then put through these four-strokes repetitively (at a few thousand RPM):

  1. compression stroke
  2. power stroke
  3. exhaust stroke
  4. intake stroke

Otto Cycle

The Otto Cycle is basically an idealized cyclic process of the 4-stroke gasoline engine. On the P-V diagram, it is represented by four thermodynamic processes:

Adiabatic compression (process AB), which is initiated by the compression stroke.

Isochoric heating (process BC), which occurs when the gasoline-air mixture is ignited.

Adiabatic expansion (process CD) during the power stroke, when the heated mixture pushes on the piston, performing work in the process.

Isochoric cooling (process DA), which occurs when the used mixture is dumped (during the exhaust stroke) and new fuel and cool air is drawn into the cylinder (during the intake stroke).

Note that the engine does positive work during the adiabatic expansion (power stroke) but negative work during the adiabatic compression (compression stroke). However, since the expansion occurred at higher pressure than the contraction, overall, it is the engine that is doing the net work. In fact, the area of the enclosed loop represents WBY, the net work done by the gasoline engine in one complete cycle.

From the first law

\displaystyle \displaystyle \begin{aligned}\overset{0}{\mathop{{\Delta U}}}\,&=\overset{{}}{\mathop{Q}}\,-{{\overset{{\text{+ve}}}{\mathop{W}}\,}_{{BY}}}\\Q&={{W}_{{BY}}}\end{aligned}

we can deduce that net Q is positive in one complete cycle. This makes sense since this is after all a heat engine, whose purpose is to convert heat into mechanical work. From energy consideration, the net work done by the engine must be equal to the net heat supply to the engine.

Efficiency of a Heat Engine

There is one more insight to the Otto cycle. Since \displaystyle Q=0  for adiabatic processes, heat transfer occurs only during the two isochoric processes of the Otto cycle: the heat supplied QH during the fuel combustion (process BC) and the heat lost QC to the surrounding when the used fuel is dumped (process DA). So

\displaystyle \displaystyle {{W}_{{BY}}}=\text{net }Q=\left| {{{Q}_{H}}} \right|-\left| {{{Q}_{c}}} \right|

Noting that useful work output of the engine is \displaystyle \displaystyle {{W}_{{BY}}}=\left| {{{Q}_{H}}} \right|-\left| {{{Q}_{c}}} \right| , while the energy input is \displaystyle \left| {{{Q}_{H}}} \right| , we can actually write the efficiency of a heat engine as

\eta =\frac{{\text{output}}}{{\text{input}}}=\frac{{\left| {{{Q}_{H}}} \right|-\left| {{{Q}_{C}}} \right|}}{{\left| {{{Q}_{H}}} \right|}}=1-\frac{{\left| {{{Q}_{C}}} \right|}}{{\left| {{{Q}_{H}}} \right|}}

Unfortunately QC is an unavoidable heat loss. While our heat engine took in QH amount of heat, it cannot produce this amount of work. A portion of it, QC, must always be lost as heat to the surrounding. It is a reminder of the unfortunate fact that while mechanical work can be completely converted into heat (e.g. brake a car), heat cannot be completely converted into mechanical work. 100% efficiency is impossible in principle.

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