water is flowing through a horizontal pipe of varying cross section at any two places the diameters of the tube are 4cm and 2cm if the pressure difference between these two places be equal to 4.5cm then determine the rate of flow of water in the tube
To determine the rate of flow of water in the tube, we can apply Bernoulli's equation which relates the pressure, velocity, and height of a fluid flowing through a pipe.
Bernoulli's equation: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Where:
P₁ and P₂ are the pressures at two different points in the pipe,
v₁ and v₂ are the velocities at those two points,
ρ is the density of the fluid (in this case, water),
g is acceleration due to gravity, and
h₁ and h₂ are the heights (relative to some reference point) at which the pressures are measured.
In this case, the tube is horizontal, so the height difference (h₁ - h₂) is zero. Thus, the equation simplifies to:
P₁ + ½ρv₁² = P₂ + ½ρv₂²
Since the pressure difference (P₁ - P₂) is given as 4.5 cm, we can rewrite the equation as:
(½ρv₁²) - (½ρv₂²) = 4.5 cm
Now, we'll use the fact that the cross-sectional area of the pipe is related to its diameter:
A₁ = π*(d₁/2)² = π*(2 cm)² = 4π cm²
A₂ = π*(d₂/2)² = π*(4 cm)² = 16π cm²
We can also relate the velocities to the cross-sectional areas:
v₁ = Q/A₁
v₂ = Q/A₂
Where Q is the rate of flow of water in the tube (what we want to find).
Substituting these values into the Bernoulli's equation, we get:
(½ρ*(Q/A₁)²) - (½ρ*(Q/A₂)²) = 4.5 cm
Simplifying further:
(Q²/A₁²) - (Q²/A₂²) = 9 cm²
Now, we substitute the known values:
A₁ = 4π cm²
A₂ = 16π cm²
(1/16π²)*(Q²) - (1/256π²)*(Q²) = 9 cm²
Combining terms, we have:
(256 - 16)*(π²)*(Q²) = 9*(16π²)
240*(π²)*(Q²) = 9*(16π²)
Q² = (9/240)*(16/π²)
Q² = 4/5
Q = √(4/5) cm³/s
Therefore, the rate of flow of water in the tube is approximately 0.894 cm³/s.