Otto Cycle - Cycle Analysis

Cycle Analysis

Processes 1-2 and 3-4 do work on the system but no heat transfer occurs during adiabatic expansion and compression. Processes 2-3 and 4-1 are isochoric; therefore, heat transfer occurs but no work is done. No work is done during an isochoric (constant volume) because work requires movement; when the piston volume does not change no shaft work is produced by the system. Four different equations can be derived by neglecting kinetic and potential energy and considering the first law of thermodynamics (energy conservation). Assuming these conditions the first law is rewritten as:

Applying this to the Otto cycle the four process equations can be derived:

Since the first law is expressed as heat added to the system and work expelled from the system then and will always produce positive values. However, since work always involves movement, processes 2-3 and 4-1 will be omitted because they occur at a constant volume. The net work can be expressed as:

The net work can also be found by evaluating the heat added minus the heat leaving or expelled.

Thermal efficiency is the quotient of the net work to the heat addition into system. Upon rearrangement the thermal efficiency can be obtained (Net Work/Heat added):

Equation 1:

Alternatively, thermal efficiency can be derived by strictly heat added and heat rejected.

In the Otto cycle, there is no heat transfer during the process 1-2 and 3-4 as they are reversible adiabatic processes. Heat is supplied only during the constant volume processes 2-3 and heat is rejected only during the constant volume processes 4-1.

Equation 1 can now be related to the specific heat equation for constant volume. The specific heats are particularly useful for thermodynamic calculations involving the ideal gas model.

Rearranging yields:

Inserting the specific heat equation into the thermal efficiency equation (Equation 1) yields.

Upon rearrangement:

Next, noting from the diagrams, thus both of these can be omitted. The equation then reduces to:

Equation 2:

Since the Otto cycle is an isentropic process the isentropic equations of ideal gases and the constant pressure/volume relations can be used to yield Equations 3 & 4.

Equation 3:

Equation 4:

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The derivation of the previous equations are found by solving these four equations respectively (where is the gas constant):

Further simplifying Equation 4, where is the compression ratio :

Equation 5:

Also, note that

where is the specific heat ratio

From inverting Equation 4 and inserting it into Equation 2 the final thermal efficiency can be expressed as:

Equation 6:

From analyzing equation 6 it is evident that the Otto cycle efficiency depends directly upon the compression ratio . Since the for air is 1.4, an increase in will produce an increase in . However, the for combustion products of the fuel/air mixture is often taken at approximately 1.3. The foregoing discussion implies that it is more efficient to have a high compression ratio. The standard ratio is approximately 10:1 for typical automobiles. Usually this does not increase much because of the possibility of autoignition, or "knock", which places an upper limit on the compression ratio. During the compression process 1-2 the temperature rises, therefore an increase in the compression ratio causes an increase in temperature. Autoignition occurs when the temperature of the fuel/air mixture becomes too high before it is ignited by the flame front. The compression stroke is intended to compress the products before the flame ignites the mixture. If the compression ratio is increased, the mixture may auto-ignite before the compression stroke is complete, leading to "engine knocking". This can damage engine components and will decrease the brake horsepower of the engine.