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 mass and Bernoulli's equation.
1. Calculate the mass flow rate:
Mass flow rate (m_dot) = Density (ρ) * Volume flow rate (Q)
To find the mass flow rate, we need to know the density of the oil. Unfortunately, the density is not provided in the information you provided. Let's assume a typical density for oil, which is approximately 850 kg/m^3. However, keep in mind that this value may vary depending on the type and temperature of the oil.
2. Calculate the velocity at each section:
Velocity (V) = Volume flow rate (Q) / Cross-sectional area (A)
For the entering section, the diameter of the pipe is given as 12 inches. Convert it to feet (1 foot = 12 inches):
Diameter (D_1) = 12 inches = 1 foot
Area (A_1) = (π/4) * (D_1)^2
For the leaving section, the diameter of the pipe is given as 2.5 inches. Convert it to feet:
Diameter (D_2) = 2.5 inches = 2.5/12 feet
Area (A_2) = (π/4) * (D_2)^2
3. Apply the Bernoulli's equation:
Bernoulli's equation states that the total energy of a fluid flowing through a pipe is conserved. It can be expressed as:
P + 1/2 * ρ * V^2 + ρ * g * h = constant
where P is the pressure, ρ is the density, V is the velocity, g is the acceleration due to gravity, and h is the height (which we can ignore for this problem since the reducer is horizontal).
Let's apply this equation between the entering and leaving sections of the reducer:
P_1 + 1/2 * ρ * V_1^2 = P_2 + 1/2 * ρ * V_2^2
4. Find the pressure difference:
We are given the pressures at each section:
P_1 = 50 psig
P_2 = 5 psig
5. Calculate the force:
Force (F) = Pressure difference (ΔP) * Area (A_2)
ΔP = P_1 - P_2
Now, with the values you provided, we can calculate the force required to hold the reducer in place. However, please note that we need the density of the oil to obtain the precise answer.