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 fluid momentum equation. The force can be calculated by considering the change in momentum of the fluid as it passes through the reducer.
Step 1: Calculate the mass flow rate (ṁ) of the oil:
ṁ = ρ * Q
Where:
ρ = fluid density
Q = volumetric flow rate
Given:
Specific gravity (sp. gr.) = 0.8
Volumetric flow rate (Q) = 3 ft^3/s
To calculate the mass flow rate, we need to determine the fluid density. Since specific gravity is the ratio of the density of the fluid to the density of water, we can use the following relation:
ρ = (Specific gravity) * (Density of water)
The density of water is approximately 62.4 lb/ft^3.
ρ = (0.8) * (62.4 lb/ft^3)
Step 2: Convert the diameter of the pipe to feet:
Given:
Diameter of the entering pipe = 12 in.
Diameter of the leaving pipe = 2.5 in.
Converting both diameters from inches to feet:
Diameter of the entering pipe = 12 in. * (1 ft/12 in.) = 1 ft
Diameter of the leaving pipe = 2.5 in. * (1 ft/12 in.) = 0.2083 ft
Step 3: Calculate the velocity (v) of the fluid entering and leaving the pipe:
Given:
Volumetric flow rate (Q) = 3 ft^3/s
Diameter of the entering pipe = 1 ft
Diameter of the leaving pipe = 0.2083 ft
The formula to calculate the velocity of the fluid in a pipe is:
Q = A * v
Where:
A = Cross-sectional area of the pipe
v = Velocity of the fluid
For the entering pipe:
A = π * (diameter/2)^2 = π * (1/2)^2
v = Q/A = 3 ft^3/s / [π * (1/2)^2]
For the leaving pipe:
A = π * (diameter/2)^2 = π * (0.2083/2)^2
v = Q/A = 3 ft^3/s / [π * (0.2083/2)^2]
Step 4: Calculate the change in velocity (Δv):
Δv = v_entering - v_leaving
Step 5: Calculate the change in momentum (Δp):
Using the law of conservation of momentum, Δp can be calculated as:
Δp = ṁ * Δv
Step 6: Calculate the force (F):
The force required to hold the reducer in place is equal to the change in momentum. Mathematically:
F = Δp
Now, you can substitute the calculated values into the equations and solve for the force required to hold the reducer in place.