Respuesta :
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;
[tex]\frac{T_{2}}{T_{1}}=\left (\frac{v_1}{v_2} \right )^{\gamma -1}[/tex]
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;
[tex]\frac{900}{300}=\left (\frac{v_1}{v_2} \right )^{1.4 -1} \Rightarrow 3 = \left (\frac{v_1}{v_2} \right )^{0.4}[/tex]
log(3) ÷ 0.4 = log(v₁/v₂)
∴ The compression ratio is given as follows;
[tex]\left (\frac{v_1}{v_2} \right ) = 10^{\frac{log(3)}{0.4} }= 15.59[/tex]
2) The final pressure is found as follows;
[tex]\frac{P_{2}}{P_{1}}=\left (\frac{v_1}{v_2} \right )^{\gamma}[/tex]
Where:
P₁ = Initial pressure = 1 bar
P₂ = Final pressure = Required
[tex]\left (\frac{v_1}{v_2} \right ) = Compression \ ratio= 15.59[/tex]
γ = 1.4
Plugging in the values, we have;
[tex]\frac{P_{2}}{1}=15.59^{1.4} \Rightarrow P_{2} = 46.765 \ bar[/tex]
Therefore the final pressure = 46.765 bar.
