Water is flowing through a horizontal pipe of varying cross section.At any two places the diameter of the tube is 4cm and 2cm. If the pressure difference between these two places be equal to 4.5cm(water),then determine the Rate of Flow of Water in the Tube?
To determine the rate of flow of water in the tube, you can use Bernoulli's equation, which states that the total mechanical energy per unit mass of a fluid along a streamline is constant.
The equation can be expressed as:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Where:
- P₁ and P₂ are the pressures at two different points along the streamline
- ρ is the density of the fluid (water in this case)
- v₁ and v₂ are the velocities of the fluid at the two points
- g is the acceleration due to gravity
- h₁ and h₂ are the heights of the fluid at the two points
In this case, let's assume point 1 is where the diameter is 4 cm and point 2 is where the diameter is 2 cm.
Given:
- Diameter at point 1 (D₁) = 4 cm
- Diameter at point 2 (D₂) = 2 cm
- Pressure difference (P₁ - P₂) = 4.5 cm of water
To solve for the rate of flow of water (Q), you need to determine the velocities at each point (v₁ and v₂).
To find the velocities, you can use the equation for the flow rate (Q) of an incompressible fluid through a pipe:
Q = A₁v₁ = A₂v₂
Where:
- A₁ and A₂ are the cross-sectional areas of the pipe at points 1 and 2, respectively.
The cross-sectional area can be calculated using the formula:
A = πr²
Where:
- r is the radius of the pipe.
To find the radius of the pipe at points 1 and 2, you can use the formula:
r = D/2
Where:
- D is the diameter of the pipe.
Let's calculate the values step by step:
Step 1: Calculate the cross-sectional area at point 1 (A₁):
- D₁ = 4 cm
- r₁ = D₁/2 = 2 cm
- A₁ = πr₁²
Step 2: Calculate the cross-sectional area at point 2 (A₂):
- D₂ = 2 cm
- r₂ = D₂/2 = 1 cm
- A₂ = πr₂²
Step 3: Calculate the velocities at points 1 and 2 (v₁ and v₂) using the flow rate equation:
A₁v₁ = A₂v₂
Step 4: Calculate the pressure at point 1 (P₁) and point 2 (P₂) using Bernoulli's equation:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Given:
- Pressure difference (P₁ - P₂) = 4.5 cm of water
Step 5: Once you have the values for v₁ and v₂, you can calculate the rate of flow of water (Q):
Q = A₁v₁ = A₂v₂
By following these calculations, you should be able to determine the rate of flow of water in the tube.