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

Answer:

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

Explanation:

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

$\frac{T_{2}}{T_{1}}=\left (\frac{v_1}{v_2} \right )^{\gamma -1}$

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;

$\frac{900}{300}=\left (\frac{v_1}{v_2} \right )^{1.4 -1} \Rightarrow 3 = \left (\frac{v_1}{v_2} \right )^{0.4}$

log(3) ÷ 0.4 = log(v₁/v₂)

∴ The compression ratio is given as follows;

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

2) The final pressure is found as follows;

$\frac{P_{2}}{P_{1}}=\left (\frac{v_1}{v_2} \right )^{\gamma}$

Where:

P₁  = Initial pressure = 1 bar

P₂  = Final pressure = Required

$\left (\frac{v_1}{v_2} \right ) = Compression \ ratio= 15.59$

γ = 1.4

Plugging in the values, we have;

$\frac{P_{2}}{1}=15.59^{1.4} \Rightarrow P_{2} = 46.765 \ bar$

Therefore the final pressure = 46.765 bar.