In a continuous pipe system, two points are being considered. Water is flowing at 2.5 m/s at point A where the cross-sectional area is 0.02 m. Point B has a cross-sectional of 0.01 m², and is 1.5 m higher than point A. Find the pressure difference between points A and B.
In this case, put complete solution with formula used.
The difference is 1.784, glad to help
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To find the pressure difference between points A and B in a continuous pipe system, we can use Bernoulli's equation, which states that the total pressure at a point in a fluid system is equal to the sum of the pressure due to height, the pressure due to velocity, and the pressure due to any other forces acting on the fluid.
Mathematically, Bernoulli's equation can be written as:
P1 + 1/2 * ρ * V1^2 + ρ * g * h1 = P2 + 1/2 * ρ * V2^2 + ρ * g * h2
where:
P1 and P2 are the pressures at points A and B respectively,
V1 and V2 are the velocities at points A and B respectively,
ρ is the density of the fluid, and
g is the acceleration due to gravity.
In this case, as point A is at a higher velocity, we can ignore the pressure due to height (as the elevation difference is small compared to the velocity difference). Therefore, the equation simplifies to:
P1 + 1/2 * ρ * V1^2 = P2 + 1/2 * ρ * V2^2
Given:
V1 = 2.5 m/s (velocity at point A),
A1 = 0.02 m² (cross-sectional area at point A),
A2 = 0.01 m² (cross-sectional area at point B),
h = 1.5 m (height difference between points A and B).
Now, let's calculate the pressure difference between points A and B step-by-step.
Step 1: Calculate the velocity at point B
Using the equation of continuity (A1 * V1 = A2 * V2), we can find the velocity at point B.
V2 = (A1 * V1) / A2
V2 = (0.02 m² * 2.5 m/s) / 0.01 m²
V2 = 5 m/s
Step 2: Calculate the pressure difference
Plugging the values into Bernoulli's equation:
P1 + 1/2 * ρ * V1^2 = P2 + 1/2 * ρ * V2^2
Since we are interested in the pressure difference (P2 - P1), we can rearrange the equation:
P2 - P1 = 1/2 * ρ * (V1^2 - V2^2)
Since the pipe is filled with water, we can assume ρ (density of water) to be 1000 kg/m³.
P2 - P1 = 1/2 * 1000 kg/m³ * (2.5^2 - 5^2)
P2 - P1 = 1/2 * 1000 kg/m³ * (-12.5)
Simplifying the equation further:
P2 - P1 = -6,250 Pa
or
P1 - P2 = 6,250 Pa
Therefore, the pressure difference between points A and B is 6,250 Pa.
To find the pressure difference between points A and B in the continuous pipe system, we can use Bernoulli's equation, which relates the pressure, velocity, and height at two different points in a fluid flow.
Bernoulli's equation states: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Where:
P₁ and P₂ are the pressures at points A and B respectively,
ρ is the density of the fluid (water in this case),
v₁ and v₂ are the velocities at points A and B respectively,
g is the acceleration due to gravity, and
h₁ and h₂ are the heights at points A and B respectively.
In this problem, we are given:
v₁ = 2.5 m/s (velocity at point A)
A₁ = 0.02 m² (cross-sectional area at point A)
A₂ = 0.01 m² (cross-sectional area at point B)
h₂ - h₁ = 1.5 m (height difference between points B and A)
Since we need to find the pressure difference (P₂ - P₁), we can rearrange Bernoulli's equation as follows:
P₂ - P₁ = ½ρ(v₁² - v₂²) + ρg(h₁ - h₂)
First, let's calculate the velocity at point B (v₂). We can use the continuity equation, which states that the product of the cross-sectional area and velocity at different points in a flow remains constant, assuming no leaks or changes in flow rate.
A₁v₁ = A₂v₂
Substituting the given values:
0.02 m² * 2.5 m/s = 0.01 m² * v₂
v₂ = (0.02 m² * 2.5 m/s) / 0.01 m²
v₂ = 5 m/s
Now we can substitute the known values into the pressure difference equation:
P₂ - P₁ = ½ρ(v₁² - v₂²) + ρg(h₁ - h₂)
Substituting the values:
P₂ - P₁ = 0.5 * ρ * (2.5 m/s)² - (5 m/s)² + ρ * 9.8 m/s² * (1.5 m)
Now we need to know the density of water (ρ). The density of water is approximately 1000 kg/m³.
P₂ - P₁ = 0.5 * 1000 kg/m³ * (2.5 m/s)² - (5 m/s)² + 1000 kg/m³ * 9.8 m/s² * 1.5 m
Now we can calculate the pressure difference:
P₂ - P₁ = 0.5 * 1000 kg/m³ * (2.5 m/s)² - (5 m/s)² + 1000 kg/m³ * 9.8 m/s² * 1.5 m
P₂ - P₁ = 6250 Pa - 25000 Pa - 14700 Pa
P₂ - P₁ = -33450 Pa
Therefore, the pressure difference between points A and B in the continuous pipe system is -33450 Pa.