3.The diameter of a pipe changes gradually from 150mm at point A, 6m above datum, to 75 mm at B, 3 m above the datum.
The pressure at point A is 103KN/m^2 and the velocity of flow is 3.6m/s .
Neglecting loses, determine the pressure at point B.
To determine the pressure at point B, we can use the principle of conservation of energy for fluid flow. This principle states that the total energy in a fluid flow system remains constant between any two points.
In this case, we can use the Bernoulli's equation to analyze the change in pressure with respect to the change in diameter and elevation.
Bernoulli’s equation is given as:
P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂
Where:
P₁ and P₂ are the pressures at points A and B respectively,
ρ is the density of the fluid (which we assume as constant),
V₁ and V₂ are the velocities at points A and B respectively,
g is the acceleration due to gravity,
h₁ and h₂ are the elevation of points A and B respectively.
Given:
P₁ = 103 KN/m²
V₁ = 3.6 m/s
d₁ = 150 mm = 0.15 m
d₂ = 75 mm = 0.075 m
h₁ = 6 m
h₂ = 3 m
Let's calculate the pressure at point B using Bernoulli’s equation:
P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂
The first term is the pressure at point A, which is already given as 103 KN/m².
Next, we need to calculate the velocity V₂ at point B. Since the pipe is gradually changing diameter, the velocity will change accordingly. We can use the equation of continuity to find V₂.
Continuity equation states:
A₁V₁ = A₂V₂
Where:
A₁ and A₂ are the cross-sectional areas of the pipe at points A and B respectively.
The cross-sectional area of a pipe can be calculated using the formula:
A = πr²
Let's calculate the cross-sectional area A₁ at point A:
A₁ = π(0.15/2)²
Now, let's calculate the cross-sectional area A₂ at point B:
A₂ = π(0.075/2)²
Using the continuity equation, we can find V₂:
V₂ = (A₁V₁) / A₂
Now, substitute the known values into the equation and calculate V₂.
Finally, substitute all the known values into Bernoulli's equation and solve for P₂, which is the pressure at point B.
Note: Neglecting losses implies we are assuming the system is ideal without any energy losses due to friction, turbulence, or other factors.
I hope this helps you understand how to determine the pressure at point B!