Question

Diesel engines do not use spark plugs to ignite the fuel-air mixture. Instead, they rely on the compression stroke to raise the temperature of the air in the cylinders before injecting the fuel.

Calculate the compression ratio (final/initial volume) required to raise the temperature of air, initially at 1 bar and 300 K, to 900 K. What is the final pressure?
Air can be considered an ideal gas with Cv = 5R/2 at all conditions of interest.

Answer 1

1) The compression ratio required to raise the the temperature of air initially at 1 bar and 300 K to 900 K is 15.59

2) The final pressure is 46.765 bar

Step-by-step explanation:

1) Here we have the relationship between the compression ratio and temperature given as follows;

`(T_(2))/(T_(1))=\left ((v_1)/(v_2) \right )^(\gamma -1)`

Where:

T₁ = Initial temperature = 300 K

T₂ = Final temperature = 900 K

v₁ = Initial volume

v₂ = Final volume

γ = Ratio of specific heat capacities Cp/Cv

Cp - Cv = R

∴ Cp = R + Cv = R + 5·R/2 = 7·R/2

∴ γ = Cp/Cv = 7·R/2 ÷ 5·R/2 = 7/5 = 1.4

Plugging in the values, we have;

`(900)/(300)=\left ((v_1)/(v_2) \right )^(1.4 -1) \Rightarrow 3 = \left ((v_1)/(v_2) \right )^(0.4)`

log(3) ÷ 0.4 = log(v₁/v₂)

∴ The compression ratio is given as follows;

`\left ((v_1)/(v_2) \right ) = 10^{(log(3))/(0.4) }= 15.59`

2) The final pressure is found as follows;

`(P_(2))/(P_(1))=\left ((v_1)/(v_2) \right )^(\gamma)`

Where:

P₁ = Initial pressure = 1 bar

P₂ = Final pressure = Required

`\left ((v_1)/(v_2) \right ) = Compression \ ratio= 15.59`

γ = 1.4

Plugging in the values, we have;

`(P_(2))/(1)=15.59^(1.4) \Rightarrow P_(2) = 46.765 \ bar`

Therefore the final pressure = 46.765 bar.