Freshwater (𝜌 = 1,020 kg/m3) is flowing in a horizontal pipe whose cross section widens from 9.0 cm2 to 13.0 cm2. The water speed at the wide section of the pipe is 1.25 m/s. What pressure difference (absolute value) is there between narrow and wide section?

Between 600 Pa and 1,200 Pa

Teri maa ki

125

To find the pressure difference between the narrow and wide sections of the pipe, you can use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid at two different points in a flowing system.

Bernoulli's equation states:

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

Where:
P₁ and P₂ are the pressures at two different points in the flow (narrow and wide sections),
ρ is the density of the fluid (1,020 kg/m³ for freshwater),
v₁ and v₂ are the velocities at the two points (unknown and 1.25 m/s, respectively),
g is the acceleration due to gravity (9.8 m/s²),
h₁ and h₂ are the heights at the two points (we can assume they are at the same level, so h₁ = h₂).

As the height is the same for both sections, the term ρgh cancels out from the equation. Rearranging the equation, we get:

P₁ + ½ρv₁² = P₂ + ½ρv₂²

Since we are interested in finding the pressure difference, we can subtract P₁ from P₂:

P₂ - P₁ = ½ρv₁² - ½ρv₂²

Now we can plug in the values given in the problem:

P₂ - P₁ = ½ × 1,020 kg/m³ × (v₁² - v₂²)

To find v₁, we can use the principle of continuity, which states that the mass flow rate is constant in an incompressible fluid flowing through a pipe. The mass flow rate is given by:

m₁ = m₂

Where m is mass flow rate and ρ is density. Rearranging the equation, we get:

A₁v₁ = A₂v₂

Where A is the cross-sectional area of the pipe.

We know the values for A₁ (9.0 cm² = 0.0009 m²), A₂ (13.0 cm² = 0.0013 m²), and v₂ (1.25 m/s). Substituting these values into the equation, we can solve for v₁:

0.0009 m² × v₁ = 0.0013 m² × 1.25 m/s

v₁ = (0.0013 m² × 1.25 m/s) / 0.0009 m² ≈ 1.8056 m/s

Now we can substitute this value back into the pressure difference equation:

P₂ - P₁ = ½ × 1,020 kg/m³ × (1.8056 m/s)² - ½ × 1,020 kg/m³ × (1.25 m/s)²

Calculating this expression gives us the pressure difference between the narrow and wide sections of the pipe.