# Pipe diameter, Flow Rate, Pressure & Velocity for water

I am hoping for some help with calculating the size of pipe for a given length. The system design is a 10000L supply fills a 5 tonne water tank that then feeds a variable speed pump. The pump sends the water around a 150M loop/ring of pipe then returns to the 5 tonne water tank. The return pipe will be approximately 600mm from the inside base of the tank, so it will be in the water creating a closed loop.

I need to move the water at a velocity of above 4.5 m/s to decrease biofilm on the pipe surface. along the 150M loop there will be 4 take off points, 1 of which requires 2000 l/h output, the other 3 require 500 l/h output. So if they were all open at the same time a total of 3500 l/h is required. However in the future the loop/ring will be increased to double its size to 300M with similar outputs, so a total of 7000 l/h.

How can I calculate the pipe size for: A - 3500 l/h @ 150M lenght (with bends) B - 7000 l/h @ 300M length (with bends)

Also once A is calculated, would increasing the pump speed and pressure increase the flow rate to achieve B?

I think the math will be a bit beyond me, so please can you go gentle with me.

This is a complicated question - it may be useful to add a diagram if you can, since each output and input to the pipe system will influence your question's answer, and by reading we may miss one.

The solution to problems like yours (where we know our requirements about the inputs and outputs of a pipe system) are often solved using the Hardy Cross method: https://en.wikipedia.org/wiki/Hardy_Cross_method.

We set each pipe input and output with their fixed conditions and solve for the flow inside the "network" of pipes using the Darcy-Weisbach equation and some initial guesses. Then we vary the non-fixed conditions until the solution converges.

You can use the Darcy-Weisbach equation for a simple one-input one-output pipe system but you must use something more like Hardy-Cross for a complex system as the one you've suggested. The Darcy-Weisbach equation essentially relates flow speed ($v$) to pressure loss ($\delta_{P}$) and characteristic length of the pipe ($L$, $D$):

$$\frac{\delta_{P}}{L} = F_d \rho \left(\frac{v^2}{2 D}\right)$$

where:

• $\delta_P$ - pressure loss [$ML^{-1}T^2$]
• $L$ - unit length of pipe [$L$]
• $F_d$ - friction factor [$1$]
• $\rho$ - density of fluid [$ML^{-3}$]
• $v$ - speed of fluid [$LT^2$]
• $D$ - diameter of pipe [$L$]

So you can see from this that since you know your required flow rates, you will perhaps need to vary your upstream pressure (by changing your pump) or your pipe sizes / friction factors to achieve the velocities you want. And you'll definitely need to know your upstream pressure at your pump or reservoir!

This means you will provide an initial guess of your pipe size and friction factors and vary them until you get a network that has continuity of flow (i.e. mass conservation). For complex pipe shapes (such as bends, junctions, etc) you may need to look into using the resistance coefficient method (also known as the $k$-value method) to find your pressure loss. There are very good sets of experimental fits for loss factors in Robert Blevin's "Applied Fluid Dynamics Handbook" which I highly suggest you pick up if you commonly encounter these kinds of problems.

Description of the $k$-value method can be found here: https://neutrium.net/fluid_flow/pressure-loss-from-fittings-excess-head-k-method/

Another good reference would be the Crane TP-410 publication on pipe flow. The math isn't super hard for pipe flow, but doing complicated pipe networks can be tricky.