Answer:
1.8 m/s
Step-by-step explanation:
Here is the complete question
The diameters of a water pipe gradually changes from 5 cm at point A to 15 cm at point B. Point A is 5 m lower than point B. The pressure is 700 kPa at point A and 664 kPa and point B. Friction between the water and the pipe walls is negligible.
What is the rate of discharge at point B?
Solution
Using Bernoulli's equation,
P₁ + ρgh₁ + 1/2ρv₁² = P₂ + ρgh₂ + 1/2ρv₂² where P₁ = pressure at point A = 700 kPa, h₁ = height at point A, v₁ = speed at point A and P₂ = pressure at point B = 664 kPa, h₂ = height at point B, v₂ = speed at point B, ρ = density of water = 1000 kg/m³
P₁ - P₂ + ρgh₁ - ρgh₂ = 1/2ρv₂² - 1/2ρv₁²
P₁ - P₂ - ρg(h₂ - h₁) = 1/2ρ(v₂² - v₁²)
(h₂ - h₁) = 5 m
Substituting the values of the variables, we have
700000 Pa - 664000 Pa - (1000 kg/m³ × 9.8 m/s² × 5 m) = 1/2 × 1000 kg/m³(v₂² - v₁²)
36000 Pa - 49000 Pa = 500 kg/m³(v₂² - v₁²)
- 13000 Pa = 500 kg/m³(v₂² - v₁²)
(v₂² - v₁²) = - 13000 Pa/500 kg/m³
(v₂² - v₁²) = -26 m²/s²
By mass flow rate, A₁v₁ = A₂v₂ where A₁ = cross-sectional area at point A and A₂ = cross-sectional area at point B
πd₁²v₁/4 = d₂²v₂/4 where d₁ = diameter at point A = 5 cm = 0.05 m and d₂ = diameter at point B = 15 cm = 0.15 m
d₁²v₁ = d₂²v₂
v₁ = v₂(d₂/d₁)²
= v₂(0.15/0.05)²
= v₂(3)²
= 9v₂
So, substituting v₁ = 9v₂ into the above equation, we have
(v₂² - v₁²) = -26 m²/s²
v₂² - 9v₂² = -26 m²/s²
- 8v₂² = -26 m²/s²
v₂² = -26 m²/s² ÷ (-8)
v₂² = 3.25 m²/s²
taking square root of both sides,
v₂ = √(3.25 m²/s²)
= 1.8 m/s
So, the rate of discharge at point B is 1.8 m/s