Answer:
The pressure P1 at the entrance end of the pipe = 444,001.4 Pa = 4.382 atm
Explanation:
The Bernoulli's principle is basically a law of conservation of energy, it theorizes that the sum of the energy due to pressure, kinetic energy and potential energy of a fluid is constant all through the points of the fluid motion.
Or that the losses due to pressure effects, kinetic effects and potential effects is equal to 0 all through the fluids motion.
It is given mathematically as
[(P₂ - P₁)/ρg] + [(v₂² - v₁²)/2g] + [z₂ - z₁] = 0
P₂ = Pressure at the exit = 1.3 atm = 1.3 × 101325 = 131,722.5 Pa
P₁ = Pressure at the entrance end = ?
ρ = density of the fluid = 1237 kg/m³
g = acceleration due to gravity = 9.8 m/s²
v₁ = velocity of the fluid at the entrance end = 1.84 m/s
v₂ = velocity of the fluid at the exit = ?
But we can obtain this using the continuity equation, the continuity equation stated mathematically is
ρ₁ A₁ v₁ = ρ₂A₂v₂
Since the density of the fluid is constant all through, the continuity equation becomes
A₁ v₁ = A₂ v₂
A₁ = cross sectional Area At the entrance end = (πd₁²/4)
A₂ = cross sectional Area At the exit = (πd₂²/4)
The continuity equation then further becomes
d₁² v₁ = d₂² v₂
0.18² × 1.84 = 0.05² × v₂
v₂ = 23.8464 m/s
z₂ = elevation of the exit end
z₁ = elevation of the entrance end
z₂ - z₁ = -3.08 m (minus sign because the exit end is lower than the entrance end)
[(P₂ - P₁)/ρg] + [(v₂² - v₁²)/2g] + [z₂ - z₁] = 0
Multiplying through by ρg, the Bernoulli equation becomes
[(P₂ - P₁)] + [ρ(v₂² - v₁²)/2] + ρg[z₂ - z₁] = 0
ΔP + [1237 (23.8464² - 1.84²)/2] + (1237×9.8×-3.08) = 0
ΔP + 349,616.522 - 37,337.608 = 0
ΔP + 312,278.914 = 0
ΔP = - 312,278.914 Pa
ΔP = (P₂ - P₁) = -312,278.914
P₂ = 131,722.5 Pa
P₁ = ?
131,722.5 - P₁ = -312,278.914
P₁ = 131,722.5 + 312,278.914 = 444,001.414 Pa
P₁ = (444,001.414) ÷ (101325) = 4.382 atm
Hope this Helps!!!
