Water Falls

I spent 25 years working for a world leader in large-scale flood modelling.  In true Aussie fashion, jovial belittlement of each other’s passion drove part of the company’s continuous improvement.  The ongoing one-upmanship produced a stream (pun intended) of world-class products.  Being part of the mechanical engineering team, one of my favorite quips about flood modelling was, “Flood modelling?  Big whoop!  Water runs downhill.  How hard can it be?”  The comeback of course remains unprintable.

Confining things to the inside of a conduit doesn’t change the situation much; water still falls, flowing from regions of high pressure to regions of low pressure.  That said, it doesn’t like doing so, and it fights back the whole way.  How much of a fight it puts up depends on the conduit dimensions.

Longwall hydraulic systems often include kilometres of hose between the supply and the demand.  Contrary to first impressions, the long hoses often represent a relatively small part of the overall circuit resistance; the valves used in longwall machinery have far greater impact on system performance than hoses.  This post aims to provide some simple tools to help evaluate longwall hydraulic circuit flow restrictions.  For those with little interest in longwalls or hydraulics, the tools are transferable to other water reticulation systems.

Like aerodynamic forces, which vary with the square of the air speed, the water pressure lost along a hose varies with the square of the flow rate.  As such, the pressure drop along a hose will quadruple if the flow rate doubles.  This is expressible using a very simple equation, ΔP = kQ².  ΔP is the pressure loss, k is the conduit ‘resistance’, and Q is the fluid flow rate.  The trick is to estimate k using readily available information.

A simplification I like for longwalls is k ≅ 70 L/Dˆ5.  The units are a little odd, but convenient nonetheless; L is the hose length in metres, D is hose bore in millimetres, and the pressure drop will be in bar for flow rates in litres per minute (LPM).  That means k has units of bar/LPM², but a quick example should help distract you from the poor SI compliance.  For a DN50 hose 400m long, the pressure drop at 800 LPM would be 70 × 400 ÷ 50ˆ5 × 800² ≅ 57 bar.  It can’t get much simpler than that!  Note that k is proportional to the conduit length, but it’s much more sensitive to diameter – doubling the hose bore reduces the resistance by a factor of thirty-two!  When it comes to hose diameter, bigger is better.  Much better.

Many longwall monorail systems include several parallel hose runs.  With the same pressure differential across all hoses, the flow in each hose is the same.  Two hoses carry twice the flow of a single hose, and three hoses 3 times the flow etc.  As such, the effective combined path resistance (k) reduces with the square of the number of identical paths.

For example, 2 × DN50 hoses 400m long would have k = 70 × 400 ÷ 50ˆ5 ÷ 2² ≅ 2.2E-5 bar/LPM², and the pressure drop at 1,600 LPM would be 2.2E-5 × 1,600² ≅ 57 bar.  Guess what the pressure drop along 3 × DN50 hoses 400m long would be at 2,400 LPM?  Yep!  About 57 bar.

For your next trick, you’ll need to remember the number “five hundred”.  Many longwall system valves have flow resistance equivalent to something like 500 ‘diameters’ of the same bore hose.  For a DN20 valve, the equivalent hose length would be 500 × 20mm = 10,000mm, so L = 10m.  For a DN10 valve, the equivalent hose length would be 500 × 10mm = 5000mm, or L = 5m.

You’re now able to estimate k for many valves.  For example, a single DN20 valve would have k = 70 × 10 ÷ 20ˆ5 ≅ 2.2E-4 bar/LPM².  Did you notice this resistance is 10 times that of the 2 × 400m × DN50 hoses above?

I’ll leave it to you as an exercise, but a single 20mm valve creates the same pressure drop as nearly 4km of dual DN50 monorail, or nearer to 9km of triple DN50 monorail passing fluid at the same rate.  I expect you might now see how valves can be bigger drivers of longwall performance than long hose runs.

Without undermining the usefulness of simple relationships like those above, it’s often difficult to accurately calculate real system resistances.  If at all possible, physical measurement on the real installation is prudent.  Of course, during the design phase, physical testing is only possible if nominally identical installations already exist and are accessible.  More robust calculation practices might be the only option for some new designs.

That covers ‘waterfalls’.  I’ll be sticking with an ‘outdoor’ theme; stay tuned for an upcoming post about spring water.