# Solving Continuity Equation in Water Flow

• R Power
In summary, the conversation discusses the relationship between velocity and pressure in a vertical pipe with water flowing from top to bottom. The continuity equation states that the velocity at the top and bottom should be equal, but the group discusses how gravity and pressure can affect this relationship. The concept of hydraulic cross-section, which is the cross-section of the fluid rather than the pipe, is also brought up. Ultimately, the conversation concludes that the velocity and pressure are not necessarily equal at the top and bottom, and that the velocity can increase at the bottom due to acceleration from gravity.

#### R Power

Hi
Friends, consider a pipe of uniform cross section held vertical and water flows through it top to bottom. Now if we apply equation of continouty here we find that velocity at enter or at top must be equal to velocity at exit (or bottom) because of constant cross section. O.K?
But due to gravitational acceleration velocity at bottom (exit) should be greater than that of top by an addition of (2gh)^2 (if h is differnce between enter and exit).
i.e V(bottom) = V(top) + (2gh)^2

so is continouty equation invalid here? or if I am wrong somewhere please correct me!

Hi R Power!

No, because of the work done by the water against the pressure …

think how exhausted a little drop of water must get fighting its way through to the bottom, with all that high pressure in front of it, and only low pressure behind to help it!

tiny-tim said:
No, because of the work done by the water against the pressure …

think how exhausted a little drop of water must get fighting its way through to the bottom, with all that high pressure in front of it, and only low pressure behind to help it!
How is work done against pressure? When did I say that pressure at exit is greater?

reconsider, pressure lower at bottom (exit) and water flows top to bottom.

R Power said:
How is work done against pressure? When did I say that pressure at exit is greater?

reconsider, pressure lower at bottom (exit) and water flows top to bottom.
The pressure at any point in water column is ρgh, with h as the height of the column above
gravity is nullified by drag force, it is called terminal velocity

Yes you are right!
Just appy bernouli's equation and we get all potential energy gets converted iinto pressure energy. It was so simple though!
Though! initially when I put water in the pipe first time , then velocity at bottom will be maximum but once pipe is completely filled and flow has started pressue at bottom will be more which will compemsate gravity.

R Power said:
Hi
Friends, consider a pipe of uniform cross section held vertical and water flows through it top to bottom. Now if we apply equation of continouty here we find that velocity at enter or at top must be equal to velocity at exit (or bottom) because of constant cross section. O.K?

Not OK. The velocities are not equal at the top and bottom. The hydraulic cross-sectional area isn't constant. It becomes smaller at the bottom due to the increased velocity even though the pipe cross-sectional area is constant.

CS

stewartcs said:
Not OK. The velocities are not equal at the top and bottom. The hydraulic cross-sectional area isn't constant. It becomes smaller at the bottom due to the increased velocity even though the pipe cross-sectional area is constant.

CS

What's hydraulic cross-section?

tiny-tim said:
What's hydraulic cross-section?

The cross-section of the fluid (as opposed to the pipe).

CS

Not OK. The velocities are not equal at the top and bottom. The hydraulic cross-sectional area isn't constant. It becomes smaller at the bottom due to the increased velocity even though the pipe cross-sectional area is constant.

But according to continouty equation velocity should be same and that was only my doubt that velocity should increase due to gravity but i got answer as it doesn't change due to greater pressure at bottom which compensates for gravity.
Now you say velocity will increase at bottom.Why? And tiny tim asked a good question.

R Power said:
But according to continouty equation velocity should be same and that was only my doubt that velocity should increase due to gravity but i got answer as it doesn't change due to greater pressure at bottom which compensates for gravity.
Now you say velocity will increase at bottom.Why? And tiny tim asked a good question.

The velocity increases since gravity is accelerating it (think a free-falling body - it falls faster as time goes on).

The continuity equation is just a relation between the two quantities (area and velocity). Since you know that the velocity must increase, the continuity equation dictates that the area must decrease. This is precisely why water falling from a faucet will "neck down". The diameter of the fluid stream is reduced since the velocity is increased.

CS

stewartcs said:
Not OK. The velocities are not equal at the top and bottom. The hydraulic cross-sectional area isn't constant. It becomes smaller at the bottom due to the increased velocity even though the pipe cross-sectional area is constant.

Really? The kilograms per second has to be equal at the top and bottom. The mean velocity has to be equal at the top and bottom. If the viscosity changes, the velocity profile could change. Read the following from Wikipedia:
For shapes such as squares, rectangular or annular ducts (where the height and width are comparable) the characteristic dimension for internal flow situations is taken to be the hydraulic diameter, DH, defined as 4 times the cross-sectional area, divided by the wetted perimeter. The wetted perimeter for a channel is the total perimeter of all channel walls that are in contact with the flow.

For a circular pipe, the hydraulic diameter is exactly equal to the inside pipe diameter, as can be shown mathematically.

See

http://en.wikipedia.org/wiki/Reynolds_number

http://en.wikipedia.org/wiki/Hagen-Poiseuille_equation

The velocity profile in round pipes is primarily determined by the stagnant layer at the pipe wall, and the radial velocity profile due to fluid viscosity.

Bob S

Bob S said:
Really? The kilograms per second has to be equal at the top and bottom. The mean velocity has to be equal at the top and bottom. If the viscosity changes, the velocity profile could change. Read the following from Wikipedia:

See

http://en.wikipedia.org/wiki/Reynolds_number

http://en.wikipedia.org/wiki/Hagen-Poiseuille_equation

The velocity profile in round pipes is primarily determined by the stagnant layer at the pipe wall, and the radial velocity profile due to fluid viscosity.

Bob S

No. The hydraulic areas in a vertical section of pipe are not the same. Presume for a moment that we have an ideal fluid and there are no viscous effects and thus no pressure losses (i.e. P1 = P2).

Now apply Bernoulli's Equation and calculate the velocity at the bottom of the vertical pipe with YOUR assumption that the areas are equal. Let me know what you get.

CS

stewartcs said:
No. The hydraulic areas in a vertical section of pipe are not the same. Presume for a moment that we have an ideal fluid and there are no viscous effects and thus no pressure losses (i.e. P1 = P2).

Now apply Bernoulli's Equation and calculate the velocity at the bottom of the vertical pipe with YOUR assumption that the areas are equal. Let me know what you get.
So we have the same mass flow rate (kilograms per second), top and bottom, and the same pipe diameter, top and bottom, and the same fluid viscosity, top and bottom. The mean fluid velocity (meters per second) and fluid density (water) top and bottom has to be the same. So you are saying that the radial velocity profile is different, top and bottom?

Bob S

Bob S said:
So we have the same mass flow rate (kilograms per second), top and bottom, and the same pipe diameter, top and bottom, and the same fluid viscosity, top and bottom. The mean fluid velocity (meters per second) and fluid density (water) top and bottom has to be the same. So you are saying that the radial velocity profile is different, top and bottom?

Bob S

The pipe diameter in a vertical pipe section is irrelevant. The diameter of the fluid stream will reduce due to the increased velocity.

The hydraulic radius that you referenced previously is only for non-circular conduits. It makes the assumption that fluid completely fills the conduit. You can't apply it to your argument since it specifically assumes the conduit is full of fluid (which isn't true in a vertical pipe segment).

The mass flow rate is indeed the same at the top and bottom of the pipe, which is precisely why the velocity is increased at the bottom (again this due to the area decreasing as observed in practice). The mass flow rate for a fluid is simply pVA. Since the density (p) is constant for most liquids (nearly incompressible so it's an ok assumption) this reduces to VA. So the continuity equation is just a statement of the conservation of mass law, that is to say, V1A1 = C or V1A1 = V2A2.

The point you are missing is that the area of the fluid stream is not the same as the area of the pipe. It must be reduced if the velocity increases per the conservation of mass. Since it is extremely well known that objects (including fluid particles) have increased velocity as they fall due to gravitational acceleration, the area of the stream must decrease or it will violate the conservation of mass.

EDIT: If you want empirical proof go turn on your water faucet and observe that the fluid stream "necks down".

CS

CS
I think you are correct!
Moreover pressure at bottom is larger in case of stationary fluid only (unless moving fluid has perfect horizontal streamlines). When fluid is moving vertically i don't think pressure will increase at bottom and so it will not counter gravity.
So, definitely gravity will have it's effect on fluid velocity and thereby decrease the hydraullic cross section. Am I following you correct?
But why then all books write the other way. They all take same velocity at bottom which made me confused.

stewartcs said:
EDIT: If you want empirical proof go turn on your water faucet and observe that the fluid stream "necks down".

CS

I tried that …

it clearly "necks down" below the pipe (ie, outside it) …

but I couldn't see it doing so in the pipe.

R Power said:
CS
I think you are correct!
Moreover pressure at bottom is larger in case of stationary fluid only (unless moving fluid has perfect horizontal streamlines). When fluid is moving vertically i don't think pressure will increase at bottom and so it will not counter gravity.
So, definitely gravity will have it's effect on fluid velocity and thereby decrease the hydraullic cross section. Am I following you correct?
But why then all books write the other way. They all take same velocity at bottom which made me confused.

Yes you are following me correctly.

Most Fluid Mechanics books make the assumption that the pipe is completely full of fluid. This makes solving the problem easier since you can eliminate (in some cases) the velocity at the two points when using the general energy equation.

Since they assume the fluid's area is the same as the pipe, they thereby assume that the velocity is also the same (which is reasonable for most practical fluid flow problems).

In real life the physics of the vertical section dictate that since the velocity increases due to gravity accelerating it, the area must decrease in order to conserve mass. It would make the solution more difficult (using the techniques taught in most Fluid Mechanics courses) if they did not make that assumption.

CS

tiny-tim said:
I tried that …

it clearly "necks down" below the pipe (ie, outside it) …

but I couldn't see it doing so in the pipe.

Use your ex-ray glasses next time!...:rofl:

CS

stewartcs said:
Use your ex-ray glasses next time!...:rofl:

CS

But they don't see through metal!

## 1. What is the continuity equation in water flow?

The continuity equation in water flow is a fundamental principle in fluid mechanics that states that the mass of fluid entering a system must equal the mass of fluid leaving the system. This equation is based on the principle of conservation of mass.

## 2. How is the continuity equation used to solve problems in water flow?

The continuity equation is used to solve problems in water flow by providing a mathematical relationship between the flow rate and the cross-sectional area of a pipe or channel. This equation helps to determine the velocity of the fluid at different points in the system and can also be used to predict flow rates and pressure changes.

## 3. What are the assumptions made when using the continuity equation in water flow?

The continuity equation assumes that the fluid is incompressible, meaning that its density remains constant. It also assumes that the flow is steady and the flow is laminar, meaning that there is no turbulence. Additionally, the equation assumes that there are no external forces acting on the fluid, such as gravity or friction.

## 4. Can the continuity equation be applied to all types of fluid flows?

Yes, the continuity equation can be applied to all types of fluid flows, including both liquids and gases. However, it is most commonly used for incompressible fluids, as compressible fluids may have varying densities that can complicate the calculations.

## 5. How can the continuity equation be verified experimentally?

The continuity equation can be verified experimentally by using flow rate meters and measuring the velocity and cross-sectional area of the fluid at different points in the system. These measurements can then be compared to the values calculated using the continuity equation, and any discrepancies can be investigated to improve the accuracy of the equation.