How to calculate the height of water column gushing that would be gushing out from a 100 mm diameter pipe opening if the velocity of the water flowing through the pipe is 1.6 meters / second.
To calculate the height of the water column gushing out from a pipe opening, you can use the principles of fluid dynamics and Bernoulli's equation.
First, we need to determine the cross-sectional area of the pipe opening. The formula to calculate the area of a circle is A = πr², where A is the area and r is the radius. In this case, the diameter of the pipe is given as 100 mm, so the radius would be half of that, which is 50 mm or 0.05 meters. Plugging this value into the formula, we get:
A = π(0.05)² = 0.00785 square meters
Next, we need to calculate the velocity of the water at the pipe opening. The given velocity is 1.6 meters/second.
To find the height of the water column, we can apply Bernoulli's equation, which states that the pressure at any point in a fluid is equal to the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume at another point. In this case, we can assume that the pipe opening is at the same height as the point where the water would be flowing out.
Bernoulli's equation can be expressed as:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Where:
P₁ and P₂ are the pressures at points 1 and 2,
ρ is the density of water,
v₁ and v₂ are the velocities at points 1 and 2,
g is the acceleration due to gravity,
h₁ is the height at point 1, and
h₂ is the height at point 2.
At point 1 (inside the pipe), the water is stationary, so the velocity (v₁) is equal to 0. Also, the height (h₁) can be considered as 0 since it is the reference point. Therefore, the equation simplifies to:
P₁ = P₂ + ½ρv₂² + ρgh₂
Now, we assume that the water is flowing freely out of the pipe without any external pressure acting on it. Therefore, we can say that P₂ is equal to atmospheric pressure, which is approximately 101,325 Pascals.
Given that the density of water (ρ) is approximately 1000 kg/m³ and the velocity (v₂) is 1.6 m/s, we can calculate the height (h₂) by rearranging the equation:
h₂ = (P₁ - P₂) / (ρg) + (v₂²/2g)
Substituting the values, we get:
h₂ = ((0 - 101325) / (1000 * 9.8)) + ((1.6²) / (2 * 9.8))
Simplifying further, we can calculate:
h₂ ≈ -10.38 meters
Since the height cannot be negative in this context, we take the absolute value:
h₂ ≈ 10.38 meters
Therefore, the water column gushing out from the 100 mm diameter pipe opening would have a height of approximately 10.38 meters.