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 need to calculate the change in momentum of the fluid as it passes through the reducer. This change in momentum will be equal to the force exerted on the reducer.
First, let's calculate the mass flow rate (ṁ) of the fluid using the given flow rate (Q) and specific gravity (sp. gr.) of the oil:
ṁ = Q / sp. gr.
Given: Q = 3 ft^3/s, sp. gr. = 0.8
Convert the flow rate to gallons per minute (GPM) for convenience:
Q_GPM = Q * 448.832
Now we can calculate the mass flow rate:
ṁ = Q_GPM / sp. gr.
Next, let's calculate the velocity of the fluid at the inlet (V1) and outlet (V2) using the given diameter of the pipe (D1 and D2, respectively) and the mass flow rate (ṁ):
A1 = π * (D1/2)^2
A2 = π * (D2/2)^2
V1 = ṁ / A1
V2 = ṁ / A2
Now we can calculate the momentum change of the fluid:
Δp = ṁ * (V2 - V1)
Finally, the force required to hold the reducer in place is given by Newton's second law of motion:
F = Δp / Δt
Here, Δt represents the time it takes for the fluid to pass through the reducer, which is not given in the problem. Therefore, we cannot calculate the exact force without this information. However, we can calculate the force per unit time (Force/time) or the force per unit mass (Force/mass).
If you have information about the time or mass of the fluid passing through the reducer, you can plug in the values to calculate the force required.