water is flowing at 3 m/s in a horizontal pipe under a pressure of 200 kpa. the pipe, which narrows to half its original diameter, is laid out on a level ground? a) what is the speed of flow in the narrow section? b) what is the pressure in the narrow section? c)how do the volume flow rates in the two sections?
To solve this problem, we can use the principle of continuity and Bernoulli's equation.
a) To determine the speed of flow in the narrow section, we can apply the principle of continuity, which states that the volume flow rate (Q) is constant along a pipe.
The equation for the principle of continuity is:
A₁v₁ = A₂v₂
Where A₁ and A₂ are the cross-sectional areas and v₁ and v₂ are the velocities in the initial and narrow sections, respectively.
Since the pipe narrows to half its diameter, the cross-sectional area will be reduced to 1/4 of the original area (A₁). Therefore, A₂ = 1/4 * A₁.
We know that v₁ = 3 m/s, so we can substitute the given values into the continuity equation:
A₁ * v₁ = A₂ * v₂
A₁ * 3 = (1/4 * A₁) * v₂
3 = (1/4) * v₂
v₂ = 12 m/s
Therefore, the speed of flow in the narrow section is 12 m/s.
b) To calculate the pressure in the narrow section, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid.
The equation for Bernoulli's equation is:
P₁ + (1/2) * ρ * v₁² + ρ * g * h₁ = P₂ + (1/2) * ρ * v₂² + ρ * g * h₂
Where P₁ and P₂ are the pressures, v₁ and v₂ are the velocities, ρ is the density of the fluid, g is the acceleration due to gravity, and h₁ and h₂ are the heights in the initial and narrow sections, respectively.
Since the pipe is laid out on level ground, the height does not change, so h₁ = h₂.
Considering that the pressure in the initial section is given as 200 kPa, we can rewrite the Bernoulli's equation as:
200 kPa + (1/2) * ρ * v₁² = P₂ + (1/2) * ρ * v₂²
To find the pressure in the narrow section (P₂), we need to know the density (ρ) of water, which is approximately 1000 kg/m³. Also, we can square the velocities:
200 kPa + (1/2) * 1000 kg/m³ * (3 m/s)² = P₂ + (1/2) * 1000 kg/m³ * (12 m/s)²
Solving this equation will determine the pressure in the narrow section (P₂).
c) The principle of continuity states that the volume flow rate (Q) is constant along a pipe. Therefore, the volume flow rates in the two sections will be the same.
The equation for the volume flow rate is:
Q = A * v
Where Q is the volume flow rate, A is the cross-sectional area, and v is the velocity.
Since the initial section and narrow section have different cross-sectional areas and velocities, we can calculate their respective volume flow rates (Q₁ and Q₂) using the equations above.
However, since the cross-sectional area reduces to 1/4 of the original size in the narrow section and the velocity increases to 4 times the original speed, the volume flow rate remains the same:
Q₁ = Q₂