Oil (sp. gr.= 0.8) flows smoothly through the circular reducing section shown at 3 ft^3/s. If the entering and leaving velocity profiles are uniform, estimate the force which must be applied to the reducer to hold it in place.
When Fluid is entering the pipe: P= 50 psig
Diameter of the pipe= 12 in.
Fluid leaving the pipe: P= 5 psig
Diameter of the pipe= 2.5 in.
To estimate the force required to hold the reducer in place, we can use the principle of conservation of momentum. The force required will be equal to the change in momentum of the fluid as it passes through the reducing section.
Step 1: Calculate the mass flow rate:
Mass flow rate (m_dot) = density (ρ) × volumetric flow rate (Q)
The volumetric flow rate is given as 3 ft^3/s.
To convert it to ft^3/min, multiply by 60.
Volumetric flow rate (Q) = 3 ft^3/s × 60 s/min = 180 ft^3/min
The density is given as specific gravity (Sp. gr.) × density of water.
Since the specific gravity is given as 0.8, and the density of water is 62.4 lb/ft^3, we can calculate the density (ρ):
Density (ρ) = 0.8 × 62.4 lb/ft^3 = 49.92 lb/ft^3
Now we can calculate the mass flow rate:
Mass flow rate (m_dot) = 49.92 lb/ft^3 × 180 ft^3/min = 8,985.6 lb/min
Step 2: Calculate the change in velocity:
The change in velocity can be calculated using the equation of continuity, which states that the mass flow rate is constant.
m_dot = ρ × A × V, where A is the cross-sectional area and V is the velocity of the fluid.
For the entering section:
A1 = π × (d1/2)^2, where d1 is the diameter of the entering pipe.
d1 = 12 in
A1 = π × (12 in/2)^2 = 113.097 in^2
For the leaving section:
A2 = π × (d2/2)^2, where d2 is the diameter of the leaving pipe.
d2 = 2.5 in
A2 = π × (2.5 in/2)^2 = 4.9087 in^2
Using the equation of continuity, we can calculate the velocity of the entering section (V1):
m_dot = ρ × A1 × V1
V1 = m_dot / (ρ × A1) = 8,985.6 lb/min / (49.92 lb/ft^3 × 113.097 in^2) ≈ 12.68 ft/min
Using the equation of continuity, we can calculate the velocity of the leaving section (V2):
m_dot = ρ × A2 × V2
V2 = m_dot / (ρ × A2) = 8,985.6 lb/min / (49.92 lb/ft^3 × 4.9087 in^2) ≈ 366.93 ft/min
The change in velocity (ΔV) = V1 - V2 = 12.68 ft/min - 366.93 ft/min ≈ -354.25 ft/min (negative sign indicates direction)
Step 3: Calculate the force required:
The force required to hold the reducer in place will be equal to the change in momentum of the fluid.
Force = mass flow rate × change in velocity = m_dot × ΔV
Force = 8,985.6 lb/min × -354.25 ft/min = -3,183,311.2 lb·ft/min ≈ -3,183,311.2 lb·min^2/s^2
So, the estimated force required to hold the reducer in place is approximately -3,183,311.2 lb·min^2/s^2. Note that the negative sign indicates that the force is in the opposite direction of fluid flow.
To estimate the force required to hold the reducer in place, we need to consider the change in momentum of the oil as it flows through the reducing section. We can use Bernoulli's equation and the principle of conservation of mass to calculate the force.
Step 1: Calculate the mass flow rate of the oil:
The mass flow rate (ṁ) can be calculated using the formula:
ṁ = ρ * A * V
where ρ is the density of the oil, A is the cross-sectional area of the pipe, and V is the velocity of the oil.
Given:
Density of oil (ρ) = specific gravity * density of water = 0.8 * 62.43 lb/ft³
Flow rate (Q) = 3 ft³/s
Diameter of the pipe (D1) = 12 in = 1 ft (convert inches to feet)
Area of the pipe (A1) = (π * D1²) / 4
Using these values, we can calculate the mass flow rate:
ṁ = (0.8 * 62.43 lb/ft³) * (π * (1 ft)²) * (3 ft³/s)
Step 2: Calculate the change in momentum:
The change in momentum (∆P) can be calculated using the formula:
∆P = ṁ * (V2 - V1)
where V1 and V2 are the velocities at the inlet and outlet of the reducer, respectively.
Given:
Velocity at the inlet (V1) = Q / A1
Velocity at the outlet (V2) = Q / A2
Diameter of the pipe at the outlet (D2) = 2.5 in = 0.2083 ft (convert inches to feet)
Area of the pipe at the outlet (A2) = (π * D2²) / 4
Using these values, we can calculate the change in momentum:
∆P = (ṁ * (Q / A1)) * ((Q / A2) - (Q / A1))
Step 3: Calculate the force required:
The force required (F) can be calculated using the formula:
F = ∆P * t
where t is the time interval for which the force is being applied.
Given:
Time interval (t) = 1 second
Using these values, we can calculate the force required:
F = ∆P * t
By substituting the values calculated in the previous steps, we can find the force required to hold the reducer in place.