Answer:
P₂= 74 kPa under constant density and ρ₂ = 1.06 kg/m³ (-38.6% of error compared with incompresible assumption) . Thus Bernoulli’s equation should not be applied
Explanation:
Assuming ideal gas behaviour of air , then
P*V= n*R*T = m / M * R *T
since
ρ= m/V = P*M /( R *T)
where
n= moles , V= volume , m= mass
ρ= density
P= pressure = 120 kPa= 120000 Pa
M= molecular weight of air = 0.21*32+0.79*28= 28.24 gr/mol = 0.02824 kg/mol
T= absolute temperature = 10°C + 273 = 283 K
R= ideal gas constant = 8.314 J/mol K
solving for ρ
ρ= P*M /( R *T) = 120000 Pa*0.02824 kg/mol/(8.314 J/mol K*283 K) = 1.47 kg/m³
then from Bernoulli's equation
P₁ + ρ*v₁²/2 = P₂ + ρ*v₂²/2
where 1 denotes inlet and 2 denotes other point , p = pressure and v= velocity . Then solving for p₂
P₁ + ρ*v₁²/2 = P₂ +ρ*P₂²/2
P₂= P₁ +ρ*v₁²/2 - ρ*v₂²/2 = P₁ +ρ/2*(v₁² - v₂²)
replacing values
P₂= P₁ +ρ/2*(v₁² - v₂²) = 120000 Pa + 1.47 kg/m³/2*[(30 m/s)²-(250 m/s)²] = 74724 Pa = 74 kPa
P₂= 74 kPa
then if the temperature remains constant
ρ₁= P₁*M /( R *T) and ρ₂= P₂*M /( R *T)
dividing both equations
ρ₂/ρ₁ = P₂/ P₁
ρ₂ = (P₂/ P₁)*ρ₁
then from Bernoulli's equation
P₁ + ρ₁*v₁²/2 = P₂ + ρ₂*v₂²/2
P₂ = P₁ + ρ₁*v₁²/2 - ρ₂*v₂²/2
therefore
ρ₂ = (P₂/ P₁)*ρ₁ = (P₁ + ρ₁*v₁²/2 - ρ₂*v₂²/2 ) /P₁ *ρ₁
P₁ * ρ₂ = P₁ *ρ₁ + ρ₁²*v₁²/2 - ρ₂*ρ₁ * v₂²/2
P₁ * ρ₂ + ρ₂*ρ₁ * v₂²/2 = P₁ *ρ₁ + ρ₁²*v₁²/2
ρ₂* (P₁ + ρ₁ * v₂²/2) = P₁ *ρ₁ + ρ₁²*v₁²/2
ρ₂ = (P₁ *ρ₁ + ρ₁²*v₁²/2)/(P₁ + ρ₁ * v₂²/2) = (P₁ + ρ₁*v₁²/2)/(P₁/ρ₁ + v₂²/2)
replacing values
ρ₂ = ( 120000 Pa + 1.47 kg/m³/2*(30 m/s)²)/(120000 Pa/1.47 kg/m³+1/2*(250 m/s)²)
ρ₂ = 1.06 kg/m³
the error of assuming constant ρ would be
e = (ρ₂ - ρ)/ρ₂= 1- ρ/ρ₂= 1- 1.47 kg/m³/1.06 kg/m³ = -0.386 (-38.6%)
thus Bernoulli’s equation should not be applied
