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?
To find the pressure difference between the narrow and wide sections of the pipe, we can use Bernoulli's equation, which states that the total pressure at any point in a fluid flow is the sum of the static pressure and the dynamic pressure.
The equation can be written as:
P1 + 1/2 * 𝜌 * v1^2 = P2 + 1/2 * 𝜌 * v2^2
Where:
P1 = pressure at the narrow section (unknown)
P2 = pressure at the wide section (unknown)
𝜌 = density of water (1,020 kg/m^3)
v1 = velocity at the narrow section (unknown)
v2 = velocity at the wide section (1.25 m/s)
We also know the cross-sectional areas:
A1 = 9.0 cm^2 = 9.0 * 10^-4 m^2
A2 = 13.0 cm^2 = 13.0 * 10^-4 m^2
By the principle of conservation of mass, the water flow rate (Q) in the pipe remains constant. Therefore, we can write:
A1 * v1 = A2 * v2
Substituting the values, we have:
9.0 * 10^-4 * v1 = 13.0 * 10^-4 * 1.25
v1 = (13.0 * 10^-4 * 1.25) / (9.0 * 10^-4)
Simplifying, we find:
v1 = 1.8056 m/s
Substituting all the values into Bernoulli's equation, we have:
P1 + 1/2 * 𝜌 * v1^2 = P2 + 1/2 * 𝜌 * v2^2
P1 + 1/2 * 1,020 * (1.8056)^2 = P2 + 1/2 * 1,020 * (1.25)^2
P1 + 1/2 * 1,020 * 3.2592 = P2 + 1/2 * 1,020 * 1.5625
P1 + 1,667.9616 = P2 + 802.5
P1 - P2 = 802.5 - 1,667.9616
P1 - P2 = -865.4616
Taking the absolute value:
|P1 - P2| = 865.4616
Therefore, the pressure difference between the narrow and wide sections of the pipe is approximately 865.46 Pa.
To find the pressure difference between the narrow and wide sections of the pipe, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a flowing fluid.
The equation can be written as: P₁ + (1/2)ρv₁² + ρgh₁ = P₂ + (1/2)ρv₂² + ρgh₂
In this equation:
P₁ and P₂ are the pressures at the narrow and wide sections of the pipe, respectively.
ρ is the density of the fluid.
v₁ and v₂ are the velocities at the narrow and wide sections of the pipe, respectively.
g is the acceleration due to gravity.
h₁ and h₂ are the heights of the fluid at the narrow and wide sections, respectively. (Since the pipe is horizontal, the height difference will be zero.)
Now let's plug in the given values and solve the equation step by step:
P₁ + (1/2)ρv₁² = P₂ + (1/2)ρv₂²
Since the height difference (Δh) is zero, ρgh₁ = ρgh₂ = 0, we can eliminate the terms involving height.
P₁ + (1/2)ρv₁² = P₂ + (1/2)ρv₂²
We are interested in the pressure difference, so we can rearrange the equation:
P₂ - P₁ = (1/2)ρv₁² - (1/2)ρv₂²
Now let's substitute the given values:
P₂ - P₁ = (1/2)(1,020 kg/m³)(1.25 m/s)² - (1/2)(1,020 kg/m³)(0 m/s)²
P₂ - P₁ = (1/2)(1,020 kg/m³)(1.5625 m²/s²) - 0
P₂ - P₁ = (1/2)(1,020 kg/m³)(1.5625 m²/s²)
P₂ - P₁ = 802.5 N/m²
Therefore, the pressure difference (absolute value) between the narrow and wide sections of the pipe is 802.5 N/m².