Otto Cycle

The cycle of a highly idealized gasoline engine can be approximated by the Otto cycle (see Figure 1.1).1→2 and 3→4 are adiabatic compression and expansion, respectively; 2→3 and 4→1 are constant-volume processes. Treat the working medium as an ideal gas with constant γ = c_p/c_v.

Figure 1.1

a) Compute the efficiency of this cycle for γ = 1.4 and compression ratio r = V_i/V_f = 10.

b) Calculate the work done on the gas in the compression process 1→2, assuming initial volume V_i = 2L and P_i = 1 atm.

SOLUTION

a) The efficiency of the cycle is η = W/Q₁, where W is the work done by the cycle Q₁ and is the amount of heat absorbed by the gas. Because the working medium returns to its initial state W = Q₁ – Q₂, where Q₂ is the amount of heat transferred from the gas, therefore

(1)

Figure 1.2

Let us calculate Q₁ and Q₂. Since both processes are at constant volume (see Figure 1.2), we may write

(2)

and

(3)

We know that for an adiabatic process

So

(4)

Using

(5)

we find

(6)

and therefore the efficiency is

(6)

For γ = 1.4 and r = 10 the efficiency is

(7)

b) The work done on the gas in the compression process is

(8)

For V_i = 2 L and P_i = 1 atm,