A horizontal pipe contains water at a pressure of 108 kPa flowing with a speed of 1.5 m/s. The pipe narrows to one-half its original diameter. What are the speed and the pressure of the water when the pipe narrows?
To find the speed and pressure of the water when the pipe narrows, we can use the principle of continuity and Bernoulli's equation.
1. Continuity equation:
The principle of continuity states that the mass flow rate of fluid is constant in a pipe. In other words, the product of the cross-sectional area of the pipe (A) and the fluid velocity (v) should remain constant.
The equation is given as:
A1 * v1 = A2 * v2
Where,
A1 = Cross-sectional area of the pipe before narrowing
v1 = Fluid velocity before narrowing
A2 = Cross-sectional area of the pipe after narrowing
v2 = Fluid velocity after narrowing
Since the pipe narrows to one-half its original diameter, the cross-sectional area is reduced to one-fourth (A2 = 1/4 * A1).
2. Bernoulli's equation:
Bernoulli's equation relates the pressure, velocity, and height (potential energy) of a fluid flowing through a pipe without any losses. It states:
P1 + 1/2 * ρ * v1^2 + ρ * g * h1 = P2 + 1/2 * ρ * v2^2 + ρ * g * h2
Where,
P1 = Pressure of the water before narrowing
ρ = Density of the water
v1 = Fluid velocity before narrowing
P2 = Pressure of the water after narrowing
v2 = Fluid velocity after narrowing
g = Acceleration due to gravity
h1 = Height of the water before narrowing
h2 = Height of the water after narrowing
Given:
P1 = 108 kPa
v1 = 1.5 m/s
Now we can solve for the unknowns, v2 and P2, using the equations above.
First, we need to find the relationship between A1 and A2:
A2 = 1/4 * A1
Next, we can use the continuity equation to find v2:
A1 * v1 = A2 * v2
v2 = A1 * v1 / A2
v2 = (A1 * v1) / (1/4 * A1)
v2 = 4 * v1
v2 = 4 * 1.5 m/s
v2 = 6 m/s
Finally, we can calculate P2 using Bernoulli's equation. Assuming the pipes are at the same height:
P1 + 1/2 * ρ * v1^2 = P2 + 1/2 * ρ * v2^2
P2 = P1 + 1/2 * ρ * v1^2 - 1/2 * ρ * v2^2
P2 = 108 kPa + 1/2 * ρ * (1.5 m/s)^2 - 1/2 * ρ * (6 m/s)^2
To calculate the pressure, we need the density of water (ρ). At room temperature, the density of water is approximately 1000 kg/m^3.
Plugging in the values:
P2 = 108 kPa + 1/2 * 1000 kg/m^3 * (1.5 m/s)^2 - 1/2 * 1000 kg/m^3 * (6 m/s)^2
Simplifying the equation will give us the pressure P2.
Therefore, the speed of the water when the pipe narrows is 6 m/s, and the pressure of the water is P2, which can be obtained by calculating the equation mentioned above.