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 consider the change in momentum of the fluid as it flows through the reducer.
The force required to hold the reducer in place can be determined using Newton's second law of motion, which states that the force acting on an object is equal to the rate of change of momentum of the object.
1. Calculate the mass flow rate:
The mass flow rate (m_dot) can be calculated using the formula:
m_dot = ρ * Q
Where:
- m_dot is the mass flow rate
- ρ is the density of the fluid
- Q is the volumetric flow rate
Given the density (ρ) of the oil as 0.8 (sp. gr.), and the volumetric flow rate (Q) as 3 ft^3/s, we can compute the mass flow rate.
2. Calculate the change in momentum:
The change in momentum (∆P) of the fluid as it flows through the reducer can be calculated using the formula:
∆P = m_dot * ∆V
Where:
- ∆P is the change in momentum
- m_dot is the mass flow rate
- ∆V is the change in velocity
Since the velocity profiles are assumed to be uniform, the change in velocity (∆V) can be calculated by subtracting the exit velocity (V_exit) from the entrance velocity (V_entrance).
3. Calculate the force:
The force required to hold the reducer in place is equal to the rate of change of momentum, which can be calculated as:
Force = ∆P / ∆t
Where:
- Force is the force required to hold the reducer in place
- ∆P is the change in momentum
- ∆t is the time interval over which the change in momentum occurs
Since the problem does not provide the time interval, we cannot calculate the exact force required to hold the reducer in place without this information.