# Bernouli's equation to calculate flow velocites and pressure

• yugeci
In summary: If the pipe ends at the bottom of the reservoir, the pressure at the bottom of the reservoir would be atmospheric, not the pressure of 100 m of water.In summary, the problem involves using the incompressible version of Bernoulli's equation to estimate the flow-velocity of water from the top of a reservoir to the river via a pipe. The velocity at the bottom of the pipe can be found by assuming the velocity at the top is zero and using the equation of continuity. The pressure rise expected when the flow is suddenly stopped can be calculated by assuming the pressure at the bottom of the reservoir is atmospheric and using the equation for the pressure at a depth of 100 m of water. The answer to this question may be dependent
yugeci

## Homework Statement

Use the incompressible version of Bernoulli’s equation to estimate the flow-velocity when
water from the top of a reservoir, 100 m above the river, reaches the river via a pipe. (neglect
the atmospheric pressure change, and assume the water is flowing into the river at atmospheric
pressure). If this flow was inside a pipe, calculate the pressure rise expected, based on
Bernoulli’s equation, when the flow is stopped suddenly.

## Homework Equations

[/B]
Having a problem with this question assigned by my teacher. Using Bernouli's equation I simplified my expression from
to

V1^2 = 981 + V2^2 (neglecting the small chance in absolute pressure, and h at the bottom is zero).

## The Attempt at a Solution

[/B]
In the equation I wrote above V1 is the velocity at the bottom, and V2 at the top. Now my question is, how do we figure out the velocity at the top? Do we assume it to be zero? Or is there some way we can calculate it? 31.32m/s is my answer assuming it to be zero... but that's an assumption I can't be sure of.

How exactly do we solve the 2nd part of this quesstion as well? I guess P1 is our atmospheric pressure, P2 is what we need to find.. and V2 would be zero, but what about the initial velocity and height? Where do we get those from as well?

Help would be appreciated.

yugeci said:
V1^2 = 981 + V2^2
Where did you lose the factor ½ in the Bernoulli equation?

In (b) I think the scenario is that water is flowing freely through the pipe (with v1 = v2) when suddenly a valve closes at the exit, dropping v1 to zero.

Does the textbook provide the answer?

yugeci said:

## Homework Statement

Use the incompressible version of Bernoulli’s equation to estimate the flow-velocity when
water from the top of a reservoir, 100 m above the river, reaches the river via a pipe. (neglect
the atmospheric pressure change, and assume the water is flowing into the river at atmospheric
pressure). If this flow was inside a pipe, calculate the pressure rise expected, based on
Bernoulli’s equation, when the flow is stopped suddenly.

## Homework Equations

[/B]
Having a problem with this question assigned by my teacher. Using Bernouli's equation I simplified my expression from
to

V1^2 = 981 + V2^2 (neglecting the small chance in absolute pressure, and h at the bottom is zero).

## The Attempt at a Solution

[/B]
In the equation I wrote above V1 is the velocity at the bottom, and V2 at the top. Now my question is, how do we figure out the velocity at the top? Do we assume it to be zero? Or is there some way we can calculate it? 31.32m/s is my answer assuming it to be zero... but that's an assumption I can't be sure of.

How exactly do we solve the 2nd part of this quesstion as well? I guess P1 is our atmospheric pressure, P2 is what we need to find.. and V2 would be zero, but what about the initial velocity and height? Where do we get those from as well?

Help would be appreciated.

Reservoirs, by nature, tend to be quiescent bodies of water. Unless the water is being discharged over a spillway, it's safe to assume that the initial velocity is zero at the top of the pipe. If it isn't, then Bernoulli's equation, by itself, is insufficient to determine the velocity of the water at the top and the bottom of the pipe simultaneously. The equation of continuity of flow often is used in conjunction with the Bernoulli equation to provide the missing information.

The answer to that question wasn't provided unfortunately. But it does make sense to assume the velocity to be zero at the top otherwise I agree more information needs to be given.

I am having some trouble understanding the second part (b), if the final velocity is zero, we still need the height difference to calculate the change in pressure. And if the initial velocity at the top is zero, then both velocity terms cancel out (but I guess this is supposed to happen?)

My bad about leaving the 1/2 terms in the velocity. I canceled them without thinking about the PE term.

Help please? Just with the 2nd part (b).

If it is flowing through a pipe of uniform bore, the water velocity at top and bottom can only differ if the pipe is not filled to its full width at the bottom.
We are told to treat the pressure at the bottom as atmospheric. What does that give for the pressure 10m above the bottom?

Part b makes no sense to me. If you have an incompressible moving mass, it takes a certain impulse to arrest its momentum. If this is done 'suddenly', there is no upper limit to the force required. In the real world, you are only saved by a combination of a slight compressibility, some elasticity in the pipe, and limits on just how suddenly you can stop the flow.
If we remove the suddenness, then the answer is simply the pressure at the bottom of 100m of water.

Perhaps in (b) they mean "... when the pipe ends suddenly", meaning it discharges freely into the air?

Was the question translated from another language?

Yugeci,

## 1. What is Bernouli's equation?

Bernouli's equation is a fundamental equation in fluid dynamics that relates the flow velocities and pressures of a fluid in a streamline. It was developed by Swiss mathematician Daniel Bernouli in the 18th century.

## 2. How is Bernouli's equation used to calculate flow velocities and pressures?

Bernouli's equation states that the sum of the kinetic energy, potential energy, and pressure energy of a fluid remains constant throughout a streamline. This means that as the velocity of the fluid increases, the pressure decreases and vice versa. By solving for the unknown variables in the equation, we can calculate the flow velocities and pressures at different points in the streamline.

## 3. What are the applications of Bernouli's equation in real life?

Bernouli's equation is used in various engineering and scientific fields to analyze fluid flow. It is commonly used in the design of airplanes, pumps, turbines, and other systems that involve the movement of fluids. It is also used in weather forecasting, aerodynamics, and hydraulics.

## 4. What are the assumptions made in Bernouli's equation?

Bernouli's equation assumes that the fluid is incompressible, inviscid, and irrotational. This means that the fluid has a constant density, no internal friction, and no rotational motion. It also assumes that the flow is steady and along a streamline, and that there are no external forces acting on the fluid.

## 5. Are there any limitations to using Bernouli's equation?

Bernouli's equation is only applicable to idealized situations and cannot be used for all types of fluid flow. It does not take into account factors such as turbulence, viscosity, and compressibility, which can affect the accuracy of the calculations. Additionally, it is only valid for steady flow along a streamline and may not accurately predict the behavior of fluids in more complex systems.

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