Bernouilli in vertical pipes

[adrich91]

Good morning,

I have a doubt about the calculation of the installation curve for dimensioning the pump necessary to pump the fluid flow I want.

The system is as shown below, where practically the source and end tank would be at the same height, but in between, apart from the elbows, valves, etc., there is a 20 meters pipe up and then down (it has to cross a street over it).



If I apply Bernouilli in points 1-2 it will give me an ‘erroneous’ or incorrect value for my installation, as it will not take into account the 20 m of ascent pipe. My question is:

When calculating the head loss, only the friction losses are taken into account, either in pipes and fittings, but in this type of case, it would obviously be necessary to add the manometric height that has to be saved by that vertical pipe, right?

And now... the height of 20 metres that I have to save, wouldn't the downward section give me "free energy" to me later? because there I would have that manometric height that makes me ‘gain’ pressure for the descent, wouldn't it?

In this particular case, it is difficult for me to interpret the system in order to get the curve of the installation.

Thank you very much in advance for your comments

Regards,

 

[LittleInch]

Is there a height difference between 1 and 2?

Dont use bernoulli for this.

Your issue is whether there is enough frictional losses to avoid getting a vacuum or low pressure at the top of the loop.

The devil is in the detail here.

There are a number of ways to address this, but having a back pressure on the end might be needed to avoid slack flow.

 

[pierreick]

a few questions:
Why don't you dig a trench?
how do you start your system, if you don't take into account the 20 m height? apply Bernoulli between the right points.
pieces of equipment are missing in this sketch, like change check valve, vacuum breaker.
Good luck anyway.

 

[adrich91]

Good evening

My question is not whether it goes through a trench or overhead, but how to approach the calculation of head loss when there is an important vertical section in the system but it does not end at the top, but goes down.

If I count the head losses in pipes (valves, elbows...etc) I should add the manometric height too, right? If so, shouldn't I subtract the manometric head because it is the static head that I use afterwards?
That's my question: Pressure loss = pipe friction + friction loss in fittings + maximum manometric head?

If I go up 20 metres, do I have to take into account this rise but not the descent?

I need to know how you resolve this type of examples. If you need numbers, just consider:

Height 1-2: 2 m
Pressure drop in all pipe: 5 bar (with elbows, tees, valves...etc)
Vertical pipe: 20 m

Thanks

 

[georgeverghese]

At point 2, is the exit in free air or is it always dipped in the water ?

 

[pierreick]

Your downhill pipe should be self-vented, check Froud number. Apply Bernoulli to the top of the system to ensure fluid will be transferred. Continuity applies from the top to the point 2, head loss should be compensated with the static head.
For your consideration.

Pipe Elevation Changes and effect on pressure loss

www.pipeflow.com

 

[Snickster]

I checked calculations for 4 cases attached. I assumed 4 inch ID pipe at 350 gpm just for setting up a reasonable example. Based on the total friction loss of 5 bar I calculated a total line length to produce 5 bar (167.3 feet) head loss. I then broke up line into segments based on the known lengths you provided and some assumptions.

Case 1 is for total length of piping calculated of 2420 feet with 2182 feet in horizontal past high point loop. With this length of piping downstream of loop the pressure in the loop is always above atmospheric due to the friction loss, so this would be a valid system curve with pressure at outlet of pump Hg = 161.56 feet head. So pump differential head required at flow would be this discharge pressure minus the suction pressure. .

Case 2 is shorter length in horizontal downstream 600 feet. In this case the lowest pressure in the loop at point Hd is -24.86 feet which is below atmospheric but still above the vapor pressure of water assumed at 0.5 psia. This would also be a valid system curve with pressure at the pump outlet Hg = 52.18 feet. However to get over the hump initially the pump must output 65.6 static head plus friction loss at flowrate until the line is packed.

Case 3 is for even shorter length downstream of 490 feet when pressure at Hd just reaches the vapor pressure of the water at 0.5 psia (-32.4 feet). This would also be a valid system curve with pressure at pump outlet Hg = 44.58 feet. So pump differential head at flow would be this discharge pressure minus the suction pressure. However to get over the hump initially the pump must output greater than 65.6 feet static height plus friction loss at a lower flowrate until the line is packed. Although this design may work it is a poor operating point and pressure at Hg should be kept significantly higher than vapor pressure. This would be a unstable point where the pressure at Hd may fluctuate between vacuum and atmospheric causing the pump flow to cycle.

Case 4 is for a length even shorter than when vapor pressure occurs at Hd, however since the lowest possible pressure at Hd is the vapor pressure of water 0.5 psia, then this will be the vapor pressure minimum. In this case with shorter piping downstream so less friction losses there will be a greater driving pressure at Hd than the pressure loss downstream and the velocity in the pipe will increase such that the flow free falls and the pipe flows partially full. If this occurs then air will enter the pipe and the pressure at Hd will be atmospheric. This is not a valid point to size the pump and develop a system curve. You could purposely size the downstream piping such that you have free flow partially filled piping at a given flowrate but then you would need to size your pump accordingly for the head loss to the elbow at Hd plus the static height to Hd plus the velocity head at the flowrate. Then once it makes it over the hump it will free flow partially full from there.

 

[Snickster]

I should have mentioned that yes the height does cancel. In my calculations I found Hd by taking pressure Hc and subtracting the friction from c to d and the static height, however the same static height is added back from e to f. The important thing is to trace the pressure in the system and to keep it significantly above the vapor pressure of the fluid. As Little Inch indicated if there is not enough back pressure to keep above vapor pressure then you could install a back pressure regulator at the end of the line or a manual throttle valve.

 

[LittleInch]

I haven't looked through snicksters work, but normally you need to plot horizontal length versus metres head for your flow rate as though the up and over wasn't there.

Then you superimpose the profile on the same graph. If the profile of the pipe is below the head loss line then you can ignore the height issue other than add on 40m of pipe for your frictional losses.

If the height of the pipe is higher than the the frictional loss line then you're in trouble and it all gets a bit more complex.

The exact length, location of this high point, pipe size, flow rate all make a difference as to whether this is a simple head loss calculation or something more complex.

 

[katmar]

In order to be able to offset the pressure loss in the up-leg against the pressure recovery in the down-leg you have to ensure that the pressure at the top of the down-leg remains above the vapor pressure of the liquid. There are basically 2 ways of doing this. You either install a back pressure valve near the discharge (as per LittleInch) to maintain the pressure at the top of the down-leg at all times or you rely on the friction loss after the down-leg (as per Snickster). The disadvantage of relying on the friction loss is that it is dependent on the flow rate and the installation will not work when the flow rate is low.

If the pressure at the top of the down-leg does go below the fluid's vapor pressure you will not get the anticipated pressure recovery but also you will get boiling and potentially damaging vibration in the pipe.

The safe, old-fashioned way to do it was to install a vent or vacuum release at the top of the down-leg. This maintains the pressure at the top of the down-leg at atmospheric. This method is simple, reliable and works for a wide range of flow rates. But it does have two disadvantages. Firstly you will never get any pressure recovery in the down-leg and secondly you would have to make the down-leg a larger diameter to ensure self-venting flow or else air will be entrained.

 

[LittleInch]

The real bottom line here is that your pump is going to have to be capable of at least 2 barg (20m head) to work. If the required frictional losses mean that the pressure at the top of the downward riser is below atmospheric pressure then at least at start up you will need to have a column of liquid able to rise to the top of the high point before falling like a waterfall down the other side. This is termed slack flow and has a number of issues associated with it as noted above. If you don't close the outlet valve before stopping pumping, then that downwards vertical leg will pull a vacuum, start boiling the water or liquid and create a large void in the pipe. On restart this void can collapse suddenly and create very large pressure spikes which can easily break your pipe over time.

If the fluid frictional losses are such that your pressure at the top of the downward riser is at least atmospheric pressure or more then you will need at least 2 bar discharge pressure to achieve your flow rate.

 

[adrich91]

Good,

I thank you all (especially Snickster) for the great class on fluid mechanics.

I have always applied bernouilli for this kind of pumping problems (without an equal loop up and down) of pumping at X meters height and so on... but of course, now I had the doubt of being at the same level but with a pronounced loop...

In short: This is an example where bernoilli cannot be applied between the initial and final points as easy as that, right?

Add: What would be the equation of the system curve for this drawing (for case 1 for example)? I want to know how to get that curve to plot vs pump curve:






Thanks again

 

[LittleInch]

In short, it's not easy to use bernoulli in such a system.

Equation for the system curve is a simple frictional loss curve which is promotional to velocity ^2 plus a fixed head or pressure difference between inlet and exit of the pipe.

Pump curve is a performence curve, not a formula.

 

[Snickster]

I am not sure why you say you can't apply Bernoulli. All fluids is an application of the balance of energy terms which are found in the Bernoulli equation:

P1/gamma + V1^2/2g +H1 = P2/gamma + V2^2/2g +H2 + H friction

The only issue I see is that once you get to where below vapor pressure is predicted in the piping then the conditions above predicted by the energy equation is no longer valid. This is because when you get below vapor pressure of the liquid, the equation above will keep on predicting lower and lower negative pressures if you vary say flowrate or friction loss or static head available, etc.. However this is erroneous since the lowest pressure possible is the vapor pressure of the liquid, since if you had a perfect vacuum with a fluid in it equilibrium will always exists when the fluid evaporates so that the vapor space is filled with the liquid at its vapor pressure. So if you try to produce an absolute vacuum in the pipe you cannot since the fluid will vaporize at pressure corresponding to its vapor pressure at the actual temperature.

The 4 cases I looked at varied the outlet pipe length so that the pressure at Hd varied to Case 1 above atmosphere, Case 2 to vacuum but above vapor pressure, Case 3 to exactly when vapor pressure at Hd is reached, to Case 4 to when below vapor pressure exists per the above equation – however since vapor pressure is the actual minimum possible pressure in the pipe, the pressure predicted by the equation at Hd (in Case 4 ) is erroneous when it predicts pressures below vapor pressure of the liquid.

That being the case you can plot the system curve by sticking to plotting only parts of the curve that no points in the piping is below the vapor pressure of the liquid as predicted by the above equation (Bernoulli).

Looking at Case 1 – where sufficient piping exists to keep pressure in elevated loop above vapor pressure and also above atmospheric pressure at a rated flow of 350 gpm. To plot the system curve you would plug in lower flowrates and plot the corresponding system Head required to get that flow. Using my spreadsheet I see that when I plug in a flow of 5750 Barrels of Water Per Day (BWPD), 167.7 GPM see attached Case 5, this just reaches vapor pressure of water (assume 0.5 psia = -32.5 feet) at Hd. If I plug into my spreadsheet 3000 BWPD (87.5 GPM) I get a pressure at Hd of -59.69 feet see Case 6 attached. This is not possible as this would be below vapor pressure of liquid and also way below absolute vacuum. Therefore the equation is not valid when it predicts below vapor pressure in the system and especially when it predicts below absolute vacuum. In this case you would just plot the system curve for 167.7 GPM and above where pressure is just at and above the vapor pressure of the liquid. So the system curve would be the pressure at the pump discharge Hg minus the suction pressure at the given flowrate.

So what happens when the equation predicts vapor pressure below actual vapor pressure of the liquid? Imagine that you have a constant flow source such as a PD pump at a given RPM and you keep on reducing the RPM and flow to when you just get vapor pressure developed at Hd (167.7 GPM in my Case 5 example). Then suddenly you reduce the speed and flow of the pump to 87.5 GPM. Although the equation predicts a pressure of -59.69 feet per the attached Case 6, this is not possible as the minimum possible pressure is vapor pressure of 0.5 psia = -32.5 feet so the actual pressure will be -32.5 feet in the pipe. Actually since I included the velocity head in the total head then when pressure reaches -32.5 feet the total head on spreadsheet at Hd would be -33.74 feet since 1.24 feet is velocity head – see Ha.

The equilibrium pressure of -32.5 feet or 0.5 psia at point Hd was the equilibrium pressure that balanced with the friction losses and static head gains, etc. of the Bernoulli equation in the downstream piping at the flow of 167.7 GPM. Now when you decreased the flow instantaneously the pressure remains the same at Hd at vapor pressure since it cannot go any lower, but since the flow decreased the losses downstream decreased from what they were when the flow was at 167.7 GPM. There is now an imbalance of forces and energy between point Hd and end of pipe with higher energy/force available at Hd than the losses in the downstream piping, so the velocity (not flow since the assumption is constant flow at given RPM) must increase until the forces/energy losses are balanced again. If the velocity increases it can reach a point where the pipe does not flow full anymore like in a sewer pipe. It may be that the flow will free fall in the vertical down until it reaches a lower elevation that the static height of the fluid is equalized with the pressure drop in the piping downstream to the end. In this case there will be vapor space in the vertical run at vapor pressure of liquid and there be a resulting lower static liquid height in the down leg above Hc determined by when the forces/energy balance is re-established.

At this point it is difficult to predict exactly what will happen and not really of concern as far as your actual design. So if you just stick to plotting the system curve for flows above point where vapor pressure of liquid occurs you should have a valid system curve.

 

[georgeverghese]

If the exit is in free air, then the downcomer will be in 2 phase air / water flow

 

[LittleInch]

The reason I've never used it in anger is because for a flat pipe all the elements cancel each other out apart from friction. So just work friction out and add or subtract the head difference.

I don't have any issue in your explanation, but I would just do it differently.

The practical issue becomes how do you start and stop this system. To get any flow starting with an empty pipe, which could easily happen if the pipe slopes away from the high point and the end becomes exposed to atmosphere, you will need to generate at least 20m head with your desired flow. Then you can operate in slack flow mode, but the flow will be unstable going into the other tank.

When you stop flow, at whatever flowrate you're pumping at, even more than 167 gpm, the down leg will pull a vacuum down to liquid vapour pressure and leave a void of at least 10m drop and maybe more. When your pipe starts again at more than 170 gpm, this void collapses violently and you get significant pressure pulses which eventually break your pipe or make it jump a lot.

 

[pierreick]

LI ,
"The practical issue becomes how do you start and stop this system", that's the main point not addressed in the replies above.
Not considered by the OP.

Pierre

 

[LittleInch]

Maybe not, but there is no point in ignoring it during the design period.

Snickster has done an amazing job looking at the options and issues for a steady state solution, but you also need to look at how something starts and stops.