A large tank is initially filled with water 10.4 m above the center of sharp-edged 9.5 cm diameter orifice.The tank water surface is open to the atmosphere, and the orifice drains to the atmosphere.Assume the kinetic energy correction factor at the orifice to be 1.15.If the total irreversible head loss in the system is 0.82 m, determine the initial discharge velocity of water in m/s from the tank.
Question
A large tank is initially filled with water 10.4 m above the center of sharp-edged 9.5 cm diameter orifice.The tank water surface is open to the atmosphere, and the orifice drains to the atmosphere.Assume the kinetic energy correction factor at the orifice to be 1.15.If the total irreversible head loss in the system is 0.82 m, determine the initial discharge velocity of water in m/s from the tank.
Solution
To solve this problem, we can use the Bernoulli's equation which is a statement of the conservation of energy principle for flowing fluids. It accounts for gravitational potential energy, kinetic energy, and fluid pressure.
The Bernoulli's equation is given by:
P₁/ρg + V₁²/2g + z₁ = P₂/ρg + V₂²/2g + z₂ + hL
where: P₁ and P₂ are the pressures at the two points, ρ is the fluid density, g is the acceleration due to gravity, V₁ and V₂ are the velocities at the two points, z₁ and z₂ are the heights at the two points, hL is the head loss.
In this case, the tank water surface is open to the atmosphere, so P₁ = P₂ = atmospheric pressure. The orifice drains to the atmosphere, so z₂ = 0. The velocity at the water surface is negligible compared to the velocity at the orifice, so V₁ = 0.
Substituting these values into the Bernoulli's equation gives:
z₁ = V₂²/2g + hL
We can solve this equation for V₂, the initial discharge velocity:
V₂ = sqrt[2g(z₁ - hL)]
Substituting the given values:
V₂ = sqrt[29.81(10.4 - 0.82)] = sqrt[29.819.58] = 13.8 m/s
However, we are given a kinetic energy correction factor of 1.15. This means that the actual velocity is 1.15 times the ideal velocity calculated using Bernoulli's equation. So, we multiply the calculated velocity by the square root of the kinetic energy correction factor to get the actual velocity:
V₂ = 13.8 * sqrt[1.15] = 15.0 m/s
So, the initial discharge velocity of water from the tank is 15.0 m/s.
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