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.
Well, it seems like this reducer has some serious issues going on. It's like the fluid is going from a comfortable, spacious living room to a tiny bathroom. Talk about downsizing!
Anyway, let's get to the point. To estimate the force needed to hold this reducer in place, we can use the equation:
Force = (Pressure difference) * (Area)
First, we need to find the pressure difference. The pressure when the fluid enters the pipe is 50 psig, and when it leaves, it's 5 psig. So, the pressure difference is 45 psig.
Now, let's calculate the areas of the two sections. The diameter of the entering pipe is 12 inches, so the radius is 6 inches. The area is then π * (6 inches)^2 = 36π square inches.
Similarly, the diameter of the leaving pipe is 2.5 inches, so the radius is 1.25 inches. The area is π * (1.25 inches)^2 = 4.90625π square inches.
Now, let's plug these values into the equation:
Force = (45 psig) * (36π square inches - 4.90625π square inches)
Simplifying, we get:
Force ≈ (45 psig) * (31.09375π square inches)
I hope you have a calculator handy because this is going to be a big number. But hey, it's all about the force, right?
To estimate the force required to hold the reducer in place, we can use the principle of conservation of momentum, which states that the change in momentum of a fluid is equal to the force exerted on it.
Step 1: Find the mass flow rate of the fluid:
We know that the volume flow rate is given as 3 ft^3/s.
To find the mass flow rate, we need to convert the volume flow rate to gallons per minute (gpm) since the specific gravity is given.
1 cubic foot = 7.48052 gallons (approximately)
3 ft^3/s = 3 * 7.48052 gpm = 22.44156 gpm
Step 2: Calculate the velocities at the inlet and outlet:
Given that the fluid velocity profile is uniform, the velocities at the inlet and outlet can be calculated using the equation:
Q = A * V
where Q is the flow rate, A is the cross-sectional area, and V is the velocity.
For the inlet:
Diameter of the pipe = 12 inches
Radius (r) = diameter/2 = 12/2 = 6 inches = 0.5 feet
Area (A) = π * r^2 = 3.14159 * 0.5^2 = 3.14159 * 0.25 = 0.7854 ft^2
Velocity (V_in) = Q/A = 22.44156 gpm / 0.7854 ft^2 = 28.61 ft/s
For the outlet:
Diameter of the pipe = 2.5 inches
Radius (r) = diameter/2 = 2.5/2 = 1.25 inches = 0.1042 feet
Area (A) = π * r^2 = 3.14159 * 0.1042^2 = 3.14159 * 0.01084864 = 0.034027441 ft^2
Velocity (V_out) = Q/A = 22.44156 gpm / 0.034027441 ft^2 = 657.7645 ft/s
Step 3: Calculate the change in momentum:
The change in momentum is given by the equation:
Δp = m * (V_out - V_in)
where Δp is the change in momentum, m is the mass flow rate, and V_in and V_out are the velocities at the inlet and outlet.
Δp = 22.44156 gpm * (657.7645 ft/s - 28.61 ft/s)
Step 4: Convert the change in momentum to force:
To convert the change in momentum to force, we need to use the formula:
Force (F) = Δp / Δt
where Δp is the change in momentum and Δt is the time taken for the change in momentum.
Assuming the time taken is 1 second, we can calculate the force:
F = Δp / Δt
F = (22.44156) * (657.7645 - 28.61)
F = 14,825.77429 lb-ft/s
Therefore, the force required to hold the reducer in place is approximately 14,826 lb-ft/s.
To estimate the force required to hold the reducer in place, we can use the principle of conservation of momentum. According to this principle, the force required is equal to the change in momentum of the fluid as it passes through the reducer.
First, let's calculate the mass flow rate of the fluid using the given flow rate and the specific gravity of the oil:
Mass flow rate (ṁ) = Volumetric flow rate (Q) * Specific gravity (SG)
Given: Volumetric flow rate (Q) = 3 ft^3/s, Specific gravity (SG) = 0.8
ṁ = 3 ft^3/s * 0.8 = 2.4 ft^3/s
Since force is the product of mass flow rate and the change in velocity, we need to calculate the change in velocity.
The velocity of the fluid can be calculated using the volumetric flow rate and the cross-sectional area of the pipe:
Velocity (V) = Volumetric flow rate (Q) / Cross-sectional area (A)
Given: Diameter of the pipe (d1) = 12 in., Diameter of the pipe (d2) = 2.5 in.
The cross-sectional area (A) can be calculated using the formula for the area of a circle:
A = π * (d/2)^2
Where π = 3.14 and d is the diameter of the pipe.
Entering velocity:
A1 = π * (12/2)^2 = 3.14 * (6/12)^2 = 3.14 * 0.25^2 = 0.196 ft^2
V1 = 3 ft^3/s / 0.196 ft^2 = 15.31 ft/s
Leaving velocity:
A2 = π * (2.5/2)^2 = 3.14 * (2.5/4)^2 = 3.14 * 0.625^2 = 1.227 ft^2
V2 = 3 ft^3/s / 1.227 ft^2 = 2.45 ft/s
Now we can calculate the change in velocity:
ΔV = V2 - V1 = 2.45 ft/s - 15.31 ft/s = -12.86 ft/s
Note: The negative sign indicates a decrease in velocity.
Finally, we can calculate the force required:
Force = Mass flow rate (ṁ) * ΔV
Force = 2.4 ft^3/s * -12.86 ft/s = -30.864 ft^3 * ft/s
Since force is a vector quantity, the negative sign indicates that the force is acting in the opposite direction of the fluid flow.
Therefore, the estimated force required to hold the reducer in place is approximately -30.864 ft^3 * ft/s.