A 1 diameter pipe with volumetric flow rate of 1 m³/s has a constriction down to half the original diameter.what is the velocity in the constriction.
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To find the velocity in the constriction, we can apply the principle of continuity. According to this principle, the volumetric flow rate remains constant in an incompressible fluid flowing through a pipe.
Given:
- Initial pipe diameter (D1) = 1 m
- Volumetric flow rate (Q) = 1 m³/s
- Constriction diameter (D2) = 1/2 D1 (half the original diameter)
The formula for volumetric flow rate (Q) is defined as:
Q = A × V
Where:
- Q is the volumetric flow rate
- A is the cross-sectional area of the pipe
- V is the velocity of the fluid
Let's calculate the cross-sectional area of the initial pipe (A1):
A1 = π × (D1/2)²
A1 = π × (1/2)²
A1 = π × 1/4
A1 = π/4
Since the volumetric flow rate remains constant, the volumetric flow rate at the constriction (Q2) will also be 1 m³/s.
Let's calculate the cross-sectional area of the constriction (A2):
A2 = π × (D2/2)²
A2 = π × (1/4)²
A2 = π × 1/16
A2 = π/16
Now, we can calculate the velocity in the constriction (V2) using the formula:
Q2 = A2 × V2
Substituting the values:
1 = (π/16) × V2
To isolate V2, divide both sides of the equation by (π/16):
V2 = 1 / (π/16)
V2 = 16/π
So, the velocity in the constriction is approximately 5.09 m/s (rounded to two decimal places).
To find the velocity in the constriction, we can use the principle of conservation of mass. According to this principle, the mass flow rate must remain the same before and after the constriction.
The mass flow rate can be calculated using the formula:
Mass flow rate = density * volumetric flow rate
Given that the volumetric flow rate is 1 m³/s, we need to determine the density of the fluid flowing through the pipe. Let's assume the fluid is water.
The density of water at room temperature is approximately 1000 kg/m³. So, the mass flow rate before and after the constriction can be calculated as follows:
Mass flow rate before constriction = 1000 kg/m³ * 1 m³/s
Since the diameter is reduced to half the original diameter, the cross-sectional area of the pipe at the constriction will be one-fourth (πr²) of the original cross-sectional area (πr²).
Mass flow rate after constriction = Mass flow rate before constriction
To calculate the velocity in the constriction, we can use the formula:
Velocity = Flow rate / Area
The flow rate before constriction is 1 m³/s, and the cross-sectional area of the pipe at the constriction is one-fourth of the original area.
Velocity = 1 m³/s / (1/4) * π * (1/2)²
Simplifying the equation:
Velocity = 1 m³/s / (1/4) * π * (1/4)
Velocity = 1 m³/s / (1/16) * π
Therefore, the velocity in the constriction is 16π m/s or approximately 50.27 m/s.