Bernoulli's Equation, Find gauge pressure at 2nd point

[Ishraq Begum]

Homework Statement

At a certain point in a pipeline, the velocity is 1 m/s and the gauge pressure is 3 x 10⁵ N/m². Find the gauge pressure at a second point in the line 20 m lower than the first if the cross-section at the second point is one half that at the first. The liquid in the pipe is water.

Homework Equations

Bernoulli's equation and Equation of continuity.

The Attempt at a Solution

I don't really know where to start

 

[stockzahn]

Homework Statement

At a certain point in a pipeline, the velocity is 1 m/s and the gauge pressure is 3 x 10⁵ N/m². Find the gauge pressure at a second point in the line 20 m lower than the first if the cross-section at the second point is one half that at the first. The liquid in the pipe is water.

Homework Equations

Bernoulli's equation and Equation of continuity.

The Attempt at a Solution

I don't really know where to start

Try to start writing down the equations with symbols. What will be the velocity of the water at the second point?

 

[Ishraq Begum]

Try to start writing down the equations with symbols. What will be the velocity of the water at the second point?

The velocity of the water at the second point will be double of that at the first point, i.e., 2 m/s
How do you find the gauge pressure??

 

[stockzahn]

The velocity of the water at the second point will be double of that at the first point, i.e., 2 m/s
How do you find the gauge pressure??

What does Bernoulli's law state (in case of a water stream without frictional pressure losses)?

 

[Ishraq Begum]

What does Bernoulli's law state (in case of a water stream without frictional pressure losses)?

It states that total energy per unit volume remains constant throughout the flow.
Mathematically,
P + ρgh + 1/2 ρv² = constant
P = pressure, ρ = density of fluid, h = height of fluid, v = velocity with fluid is flowing

 

[stockzahn]

It states that total energy per unit volume remains constant throughout the flow.
Mathematically,
P + ρgh + 1/2 ρv² = constant
P = pressure, ρ = density of fluid, h = height of fluid, v = velocity with fluid is flowing

Well, that means, that the sum of static, dynamic and hydrostatic pressure at both points are equal. How could you describe this mathematically (e.g. by using subscripts $_1$ and $_2$ for the two different points)?

 

[Ishraq Begum]

Well, that means, that the sum of static, dynamic and hydrostatic pressure at both points are equal. How could you describe this mathematically (e.g. by using subscripts $_1$ and $_2$ for the two different points)?

or P₁ + ρgh₁ + 1/2 ρv₁² = P₂ + ρgh₂ + 1/2 ρv₂²

Here, P₁ = 3 x 10⁵
ρgh₁ = 9.8 x 10³ x h
1/2 ρv₁² = 1/2 x 1³ x 1 = 500
P₂ = ? (to be found)
ρgh₂ = 9.8 x 10³ x (h - 20)
1/2 ρv₂² = 2 x 10²

⇒ (3x10⁵) + 9.8 x 10³ (h - h +20) + (500 - 2 x 10³) = P₂
⇒ (3 x 10⁵) + (1.96 x 10⁵) + (-1500) = P₂
Therefore, P₂ = 4.95 x 10⁵ N/m²

That's the answer! Thanks a lot!