Exiting Pressure of Vertical Gravity Flow Pipe 1

If you have no nozzle, then the outlet area of the pipe = initial flow area of the "outside" of the released water jet.
Therefore, there will be no instantaneous change in area of the water flow, so, no - there is no change in velocity of water.
Now, as soon as the water hits the air, the water will slow down, spread out into average droplets, and the average water velocity will begin decreasing.

This is a problem that has many process engineers confounded (me included) at some time or other. There are 2 variables here that dont fit common perception:
a) P1 is not atmospheric. If you have a controlled "vent" at the top, air gets sucked in.
b) These lines dont run liquid full, much of the pipe is gas filled, so average rho is much less.
The air that gets sucked in makes up / balances for gas entrainment / carryunder as the firewater is released at point 2.
If you read up on gravity flow, general guidelines suggest keeping the Froude number (N_Fr)at less than 0.3 for water in gravity flow to minimise gas carryunder / reduce piping vibration.

As anyone familiar with common kitchen bottles and glassware: One cannot simply "turn the bottle over" and get predictable flow out of the spout.

There are 2 flow scenarios:

The first flow scenario would be if the overflow is flowing with enough capacity to fill the vertical pipe. This is probably an unlikely scenario, but your suggested theory is correct for this scenario. The pressure at the base will basically be the static head of the column and the static head will be converted into velocity energy.

The second flow scenario would be if there is not enough flow to fill the vertical pipe. This scenario can't be modeled, but has been determined from empirical studies of vertical wastewater stacks in buildings.

Water flowing down vertical pipes creates pneumatic disturbances of various degrees depending on the installation and rate of flow. These pneumatic disturbances are important since pressure differentials may be created.

Depending on the rate of flow out of the drain into the vertical pipe, the diameter of the vertical pipe, the type of pipe fitting, the discharge from the fixture drain may or may not fill the cross-section of the vertical pipe at the level of entry. In any event, as soon as the water enters the vertical pipe, the force of gravity rapidly accelerates it downward, and as it falls, the water assumes the form of a sheet around the wall of the vertical pipe, leaving the center of the pipe open for the flow of air. This sheet of water continues to accelerate until the frictional force exerted by the wall of the vertical pipe on the falling sheet of water equals the force of gravity. From that point on, if the distance the water falls is sufficient enough, the sheet remains unchanged in thickness and velocity until it reaches the bottom of the stack. The ultimate vertical velocity the sheet attains is called the terminal velocity. The distance the sheet must fall to attain this terminal velocity is called the terminal length.

Terminal velocity is approximately 10 to 15 fps (3.05 to 4.57 m/s), and this velocity is attained within 10 to 15 feet (3.05 to 4.57 m) of fall from the point of entry.

At the center of the vertical pipe is a core of air that is dragged along with the water by friction. A supply source of air must be provided to avoid excessive pressures in the stack. The usual means of supplying this air are through the vertical pipe vent. The entrained air in the vertical pipe causes a pressure reduction inside the stack, which is caused by the frictional
effect of the falling sheet of water dragging the core of air with it.

From empirical studies, the velocity at the base of a 100-story vertical pipe (not flowing full) is only slightly and insignificantly greater than the velocity at the base of a three-story vertical pipe.

Vertical overflow pipes typically end 12-Inches above grade onto a concrete splash pad which then drains off to a short large gravel runoff area to absorb the energy.

The attached plumbing article describes gravity flow from vertical pipes.

Further to bimr excellent post, see this one which has many links and discusses vertical flow.
An overflow will behave much more like a drainage pipe than a pipe full of liquid.

Remember - More details = better answers
Also: If you get a response it's polite to respond to it.

Hi bimr,
In my current situation, we have an existing firewater overboard dump line which has flowrate of 4300 m3/h and the Froude number is 4, so if I assume that flooding would occur and therefore line would be liquid filled, am I correct to say that then the pressure at P2 (i.e. at exit) would be more than atmospheric due to static? I understand that when it is flooding it will not be 100% liquid and the static density would be less, but for this discussion I somehow simplify the situation. What I want to understand is that if the pressure at P2 can in any case be larger than atmospheric and let the pressure at P1 be remained at atmospheric (assuming I have a adequately sized vent line).

For the second flow scenario, if I assume that my situation (i.e. Froude = 4) would not have liquid filled pipe, then am I correct to say that the pressure at P2 AND P1 are both at atmospheric and velocity at exit will be terminal velocity? (assume vertical pipe is longer than terminal length)

Hi georgeverghese,
Could you explain why P1 is not atmospheric?


Thank you all for your kind knowledge sharing!

P1 cannot be atmospheric because of the gas carryunder in the exiting firewater stream. So, by material balance, it has to sub atmospheric to allow for air to enter the dump line as makeup. This is also because water has a low vapor pressure at the operating pressure, and therefore the water cannot flash enough water vapor to make up for the loss in gas carryunder. By the way, just because NFr = 4 doesnt mean that the dump line is liquid flooded. It means that you've got excessive gas carryunder in the exiting water stream (due to high firewater velocity at point 2).

Hi georgeverghese,

Thank you for your response.
Can I not consider atmospheric pressure at point 1 if I have a properly sized vent line on top of it? I thought this is the purpose of the vent line to avoid vacuum / sub-atmospheric?

DoraemonS (Petroleum) said:

In my current situation, we have an existing firewater overboard dump line which has flowrate of 4300 m3/h and the Froude number is 4, so if I assume that flooding would occur and therefore line would be liquid filled, am I correct to say that then the pressure at P2 (i.e. at exit) would be more than atmospheric due to static? I understand that when it is flooding it will not be 100% liquid and the static density would be less, but for this discussion I somehow simplify the situation. What I want to understand is that if the pressure at P2 can in any case be larger than atmospheric and let the pressure at P1 be remained at atmospheric (assuming I have a adequately sized vent line).

Click to expand...

If you simplify it to that ideal scenario, the maximum pressure is the static pressure. The pressure acting in water at X feet of vertical water depth can be calculated as:

p = ρ g h
= (1.940 slugs/ft3) (32.17405 ft/s2) (__ ft)
= _____ lbf/ft2 (psf)

Having said that, it is probably not likely to occur because of the transition from an empty pipe to full pipe may take some time.

DoraemonS (Petroleum) said:

For the second flow scenario, if I assume that my situation (i.e. Froude = 4) would not have liquid filled pipe, then am I correct to say that the pressure at P2 AND P1 are both at atmospheric and velocity at exit will be terminal velocity? (assume vertical pipe is longer than terminal length)

Click to expand...

That is basically correct for an ideal scenario although in practice you may be some very slight pressure variations occurring (inches of water pressure) due to the entrance, exit, and air venting conditions with gurgling and splashing water.

The pressure at point 1 may approach atmospheric if you have a good size "vent" there. The actual pressure at point 1 will depend on the amount of gas carryunder at point 2 and the Cv of the vent at point 1.

In my previous response, am assuming that even at NFr = 4, the downpipe is still not liquid full due the large vertical height available - you havent said much about the configuration / dimensions of this downpipe to calculate if this will be liquid full.

From your stated values of 4300 m3/h and Fr=4 it seems that you have a pipe ID of about 18" (450 mm). Under these conditions the vertical pipe will not run full (unless it is throttled at the outlet) and Bernoulli does not apply. The question is therefore not relevant to the situation.

Katmar Software - AioFlo Pipe Hydraulics

"An undefined problem has an infinite number of solutions"

Hi Katmar,
Could you explain on why if the line is not liquid full, then Bernoulli equation does not apply?
Also, what would be the theory to be used for this situation in order to understand it?
Thank you.

As far I know, there are no quantitative expressions for the gas carryunder in to the exit liquid stream. One can however estimate at what flow (in this downpipe) the hydraulics will switch from single phase liquid to 2phase air/liquid. Obviously, Bernoulli only applies to single phase flow.