A diesel engine does not use spark plugs to ignite the fuel and air in the cylinders. Instead, the temperature required to ignite the fuel occurs because the pistons compress the air in the cylinders. Suppose that air at an initial temperature of 18° C is compressed adiabatically to a temperature of 733° C. Assume the air to be an ideal gas for which

γ = 7/5

Find the compression ratio, which is the ratio of the initial volume to the final volume.

Compression ratio = 24.42

Explanation:

From Thermodynamic,

Adiabatic Equation ⇒ TV^{γ-1} = Constant.

⇒T₁V₁^{γ-1} = T₂V₂^{γ-1} ... (1)

Where T₁ = initial Temperature, V₁ = Initial Volume, T₂ = final Temperature, V₂ = Final Volume.

Given: T₁ = 18°C we convert to Kelvin (K) by adding 273.

∴ T₁ = 18°C + 273 = 291K

T₂ = 733° C also, we convert to Kelvin (K) by adding 273

∴ T₂ = 733° C + 273 = 1046K

γ = 7/5

∴ Rearranging equation (1), we have

(T₁/T₂) = (V₂/V₁) ^{γ-1} ... (2)

also rearranging equation (2) we have

(V₁/V₂) ^{γ-1} = (T₂/T₁).

Where (V₁/V₂) = Compression ratio.

∴ (V₁/V₂) ^{ (7/5) - 1} = (1046/291)

simplifying the index in the equation

I. e (7/5) - 1 = (7-5) / 5 = 2/5.

(V₁/V₂) ^2/5 = (1046/291) ^2/5

Multiplying the power on both side of the equation by 5/2.

∴ (V₁/V₂) ^ (2/5) * (5/2) = (1046/291) ^ (5/2)

⇒ V₁/V₂ = (1046/291) ^2.5 = (3.59) ^2.5

V₁/V₂ = 24.42.

∴ Compression ratio = 24.42