# Water Pressure Calculation (Limited Information)

Is it possible to find the pressure of water that leaves a pipe of certain area/diameter given only the flow rate of the water itself? OR the requires piping diameter/area given the pressure that I desire.

I'm trying to design a system that requires relatively high pressure (for spraying) however, I do not want to invest money in pumps, piping, gear etc. I'd rather try find a way to figure out a combination of criteria before I do this.

For this, I'm assuming that the flow regime is laminar. I have found an equation that I think may give me what I need, however I am unsure how I can apply it in this situation.

Hagen-Poiseuille equation:

Flow rate = πr4(P−P0)/8ηl

where r is the radius of the pipe or tube, P0 is the fluid pressure at one end of the pipe, P is the fluid pressure at the other end of the pipe, η is the fluid's viscosity, and l is the length of the pipe or tube.

I am unsure what pressure I should set the input (pump end) of the equation, P0?

Any help would be appreciated, thanks.

## Answers and Replies

Last edited:
Chestermiller
Mentor
Is it possible to find the pressure of water that leaves a pipe of certain area/diameter given only the flow rate of the water itself? OR the requires piping diameter/area given the pressure that I desire.

I'm trying to design a system that requires relatively high pressure (for spraying) however, I do not want to invest money in pumps, piping, gear etc. I'd rather try find a way to figure out a combination of criteria before I do this.

For this, I'm assuming that the flow regime is laminar. I have found an equation that I think may give me what I need, however I am unsure how I can apply it in this situation.

Hagen-Poiseuille equation:

Flow rate = πr4(P−P0)/8ηl

where r is the radius of the pipe or tube, P0 is the fluid pressure at one end of the pipe, P is the fluid pressure at the other end of the pipe, η is the fluid's viscosity, and l is the length of the pipe or tube.

I am unsure what pressure I should set the input (pump end) of the equation, P0?

Any help would be appreciated, thanks.
In the Hagen-Poiseuille equation, P0 is the outlet pressure, not the inlet pressure.

For flow coming out of a pipe into the air, the pressure at the outlet of the pipe is atmospheric pressure, ~100 kPa.

For the relationship between pressure drop and flow rate in turbulent flow of water in a pipe, see Transport Phenomena by Bird, Stewart, and Lightfoot.

Chet

Hagen-Poiseuille equation is theory result for laminar flow with very fine tube surface.
Meassures on real old iron tubes shows that flow rate related to ##r^{2.63}\left(\frac{dp}{dx}\right)^{0.54}##.
I think you must take these results as limits between them is the real behaviour.
https://www.uio.no/studier/emner/ma...rvisningsmateriale/modul-5/Pipeflow_intro.pdf

The piping in which I'd be using ideally will be clear vinyl tubing, which I'd assume has a much lower surface roughness than something like iron tubing which may have a rougher surface due to machining processes. But I will take into account what you have stated in the future, thank you mate.

In the Hagen-Poiseuille equation, P0 is the outlet pressure, not the inlet pressure.

For flow coming out of a pipe into the air, the pressure at the outlet of the pipe is atmospheric pressure, ~100 kPa.

For the relationship between pressure drop and flow rate in turbulent flow of water in a pipe, see Transport Phenomena by Bird, Stewart, and Lightfoot.

Chet

Thanks for the book reference Chet, I'll be sure to go find it at my university. Though, if we are talking about not extremely high pressured water which passes through a very smooth surfaced pipe, would it be correct to classify it as laminar? my project is based upon the premise of a weed sprayer which I'd think would have laminar flow up until its ejection through the nozzle which then would of course turn the flow into laminar spray.

Chestermiller
Mentor
The piping in which I'd be using ideally will be clear vinyl tubing, which I'd assume has a much lower surface roughness than something like iron tubing which may have a rougher surface due to machining processes. But I will take into account what you have stated in the future, thank you mate.

Thanks for the book reference Chet, I'll be sure to go find it at my university. Though, if we are talking about not extremely high pressured water which passes through a very smooth surfaced pipe, would it be correct to classify it as laminar? my project is based upon the premise of a weed sprayer which I'd think would have laminar flow up until its ejection through the nozzle which then would of course turn the flow into laminar spray.
I doubt if it will be laminar, but you will be calculating the Reynolds number, which will tell you for sure.

CWatters
Science Advisor
Homework Helper
Gold Member
I do not want to invest money in pumps, piping, gear etc. I'd rather try find a way to figure out a combination of criteria before I do this.

I made my own small spraying system for weedkiller some years ago. As I recall the company that sold me the spray nozzles had charts for the flow rate at a given pressure. Just had to ensure the pump could manage the combined flow rate of several nozzles at the required pressure. The nozzles had colour coded replaceable inserts so it's quite easy and cheap to change the flow rate if I got it slightly wrong. The flow rate was modest so I could ignore any pressure losses in the hose pipes etc.

Reconsider your assumptions. This will not be laminar flow in the tubing. Use the following equation for flow out the nozzle:

Flow rate (Volume/unit time) = Cd*A*sqrt(2*P/density)

For Cd (nozzle coefficient, no units) use 0.89, an approximation for most nozzles.

A is area of nozzles in length squared.

P is pressure force/area, pressure in the tubing just before the nozzle.

Density is the liquid density

If proper units & conversions are selected, the sqrt items end up being length/unit time, so when multiplied by A & Cd, flow rate is Volume/ unit time.

My. Reference book doesn't give provide the name of this equation, but I have used it many times.

V,