domestic water-supply system - theory - john hear field

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Water Flowing in Pipes - engineering and physics for plumbers (1)

Why do plumbers use so much half-inch copper pipe? This article explains why water pipes in houses are the

sizes they are. It shows how to choose the correct size of pipe, why it matters, and whether or not a house

water-supply system will work properly once it's installed.

Plumbing books say what to do, but not why. Building services design books offer design rules, but not where

they came from nor why they matter, and fluid mechanics textbooks are full of complicated theory that doesn't

seem relevant to real problems - Why does this pipe make a noise? - Can I add another radiator?

So I started from first principles, asking basic questions and following up the answers until I could see what

was going on. It turned out to be rather more complicated that I thought. Calculations for a real house are here

in Part 2.

Why does water flow?

This question seems almost too stupid to be worth asking, yet it leads to a way of heating the hot-water cylinder

without needing a pump.

A pint of water weighs a pound and a quarter…

When you turn on the tap, you expect water to flow out of the tank and down the pipe. Why does it do that?

Things don’t just start moving by themselves. There must be a force acting on the water in the pipe for it to

move, and the obvious one is its own weight. Water is quite heavy stuff – a litre of it weighs a kilogram. The

water in the tank pushes down on the water in the pipe.

So does the size of the tank matter? Common sense suggests that a big tank that holds more water must apply

more force to the water in the pipe than a little one would. But common sense is wrong. The force pushing the

water down the pipe has nothing to do with the volume of the tank, nor its surface area.

The force making the water flow down the pipe

must exist at the entrance to the pipe. How big is

this force? What controls it? Suppose the water is

stationary - any taps connected to the pipe are

turned off. Now think about the column of water

directly above the pipe entrance (shown as adotted line). It has weight, and its weight is a

force acting downwards. So,

Down-force on the water in the pipe

= Weight of water in the column above the pipe

But surely the weight of the water just outside the column affects the down-force too? No, it doesn’t. How can

it? Weight acts downwards, not sideways. The force due to the weight of the water outside the column acts

downwards too - but outside the column. So the down-force in the column stays the same no matter how much

water there is around it.

What about the down-force at some point further down inside the pipe? Since the down-force at any point is

due solely to the weight of water above it, this force must be bigger at a point further below the water surface.

There's a bigger - and heavier - column of water above it.

In the sketch above, both tanks hold the same amount of water, but one is twice as tall as the other. The water

surface in the taller tank is twice as far away from the pipe, so there is twice as much force pushing water out.

If you punctured both tanks near the bottom, water would squirt out much faster from the tall one.

Suppose the pipe runs vertically, then horizontally under a floor, then down vertically again. The water in the

horizontal section has weight too - won’t this add to the down-force? Well, no, it won’t, for the same reason as

before – weight acts downwards on that section of pipe, not sideways. So it turns out that the force that makes

the water flow has nothing to do with the size of the tank, nor with the length or shape of the pipe run. It’s

purely to do with the vertical distance between the water surface in the header tank and the point where the

water leaves the pipe at the tap. This distance is known as the head of water for the system.

Water squirts out of a hole

The force acts on the water in the pipe, and the pipe diameter is known, so it’s often convenient to think of the

Whatmakes water flow?

How fast will itflow?

Real pipesin real houses

How much pressure is needed?

Whatsize pipe do you need?

How fast will the water flow?





The force acts on the water in the pipe, and the pipe diameter is known, so it’s often convenient to think of the

force as a pressure. Pressure is simply force divided by area – in this case, the cross-sectional area of the pipe.

Think of a particle of water somewhere in the header tank. With no water moving, the particle is stationary.

There is no net force acting on it. If there were, it would move. But there is a force applied to it – the weight of

the water above it. And if the particle happened to be near the wall of the tank, and you poked a hole in the

tank wall, the particle would escape through it. So there must also be a sideways force from all the particles

next to it, or it wouldn't move sideways out of the hole. Yet if it's not moving, all these forces must be in

balance. In fact, in a stationary liquid, the pressure at a point is a force that acts in all directions at once: up,

down and sideways. The particle is ready to move in any direction at a moment’s notice, like the SAS. It’s

possible, and normal, to talk about the pressure at a point in a liquid.

The pressure at the bottom of a vertical pipe is ...

The weight depends on the volume of water (in m3) and its density (in kg/m3). The

density of water is about 1,000 kg/m3 – a litre weighs a kilogram. So

where g is the acceleration due to gravity - about 9.8 m/sec/sec. (Kilograms are

about mass, not force, so g has to be included to do the conversion.)

The pressure depends on the length of vertical pipe.

In fact, the length that matters is the height difference between the bottom of the pipe and the water surface at

the top – in the diagram on the left, the height h. This is the head. It’s measured in metres, or feet of water, or

mm of mercury, or pounds per square inch (psi), or Newtons per square metre (N/sq.m). Whatever units are

used, it’s still a pressure.

What’s a Newton?

This might be a good point to talk briefly about units. I'm used to thinking in

proper engineering units like Newtons and watts, not pounds and BTUs, whatever

they may be. British Thermal Underpants?

Isaac Newton was one of the first to say clearly that

Force = mass x acceleration

- for example, the down-force on your hand when you're holding an apple is the

mass of the apple (0.1 kg, say) times the acceleration due to gravity (about 10 m/

sec/sec). This force is (0.1 kg x 10 m/sec/sec) = 1 Newton. Force these days is

measured in Newtons. A Newton is the weight of a small apple - but then, the

apples were smaller in Newton's time.

Proper forces, like the weight you can lift with one hand, are lots of Newtons. A bucket holds 9 litres of water

(2 gallons) and weighs 8 kg (18 lb.). This weight is a force of 80 Newtons. The pressure on the bottom of the

bucket is about 3,000 Newtons per square metre (N/sq.m), though the pressure on your fingers when you’re

carrying the bucket by its handle is much higher – maybe 50,000 N/sq.m, or 7 psi (lb/sq.in).

A vertical 15mm pipe 1 metre long holds about 0.15 litres of water, which weighs 0.15 kg, giving a down-forceon the bottom end of about 1.5 N (Newtons). This force exerts a pressure of 10,000 N/sq.m at the end of the

pipe, which sounds huge but actually isn’t very big at all. You could poke your finger in the bottom of the pipe

to prevent the water flowing out.

By the way, normal atmospheric pressure is ten times greater than this, but we don't notice it because it

surrounds us. Atmospheric pressure is the same at both ends of the pipe, so it can't influence the flow. What

matters is pressure difference.

Hot water from gravity

There are lots of ways of heating the water in the cylinder, and one of the oldest methods can be the simplest

and cheapest. It depends on the apparently odd fact that hot water weighs less than cold water.

Here is a boiler heatin the water in a c linder. All the boiler does is heat











Here is a boiler heating the water in a cylinder. All the boiler does is heat

water in the pipe on the right. The water itself does the rest. To see why,

think about the pressures in the pipes on each side of the boiler. The right-

hand pipe holds hot water at, say, 82oC. The pressure at the bottom is then:

Similarly, the pressure at the bottom of the left-hand pipe, which holds

water at, say, 65oC, is:

If these two pressures are different, the water will be forced to move. And as the graph shows, water at 82oC is

about 1% less dense than the same water at 65oC. This may not sound very much, but it’s enough.

The difference in the density of the water in

the flow and return pipes is about 10 kg/m3. A

litre of the cold water weighs just 10 grams

more than a litre of the hot water. This tiny

difference - about half the weight of an

English robin - provides the force that makes

the water move.

The circulating pressure then seems to be:

but there is one more detail to factor in.

The water being moved is hot, so it weighs less than cold water, so the circulating pressure would make it

move slightly faster than if it were cold water. The correction is easy to add:

It’s more usual to express the pressure in a more general way that doesn’t directly involve h, because then you

can calculate how fast the water will flow round even a complicated circuit. But this is getting ahead of the

story. For now, assume that the boiler is on the ground floor and the cylinder is on the floor above, so that h is

(say) 3 metres. Then, putting in the numbers for this example, the pressure is

This is a real pressure difference, and the cold water will push the hot water around. The only major drawback

is that the pressure difference is so small, which means that the pipes have to be fat for it to work well. But it

does work, and it doesn't need a separate pump.

How fast can water flow?

The next obvious question is, what controls the speed of the water that flows out of the tap?

This is where things start getting a little more complicated. Pipes have been in use for quite a long time now,

and many clever people have tried to understand exactly what happens when you turn on a tap, but believe it or

not, the physics of water flowing in pipes is still more described than explained.

Sticky cylinders

When the water is moving slowly, it’s easy. Think of the water in the pipe as a series of thin-walled concentric

cylinders, one inside the other, each sliding relative to its neighbours. This is more than just a convenient

image. It gives a good picture of what’s really going on. The interesting stuff happens where the cylinders rub

past each other. The cylinder nearest the pipe wall doesn’t really move at all – it seems to be stuck to the pipe.

The next one in does move a bit, and as they slide past each other, the outer cylinder exerts a frictional drag on

the inner one and slows it down. Similarly, the next one in and the one inside that are slowed down too. The

cylinder right in the centre of the pipe moves the fastest.

If you don’t believe this, it’s possible (but not easy) to set up an experiment to prove it. It does happen.

Some liquids flow more freely than others, and the concept of viscosity was invented to describe the effect.

Viscosity is really a definition of how well a liquid resists shear stress – that is, the force making layers of the











Some liquids flow more freely than others, and the concept of viscosity was invented to describe the effect.

Viscosity is really a definition of how well a liquid resists shear stress – that is, the force making layers of the

liquid slide past each other.

Viscosity is measured like this. With a layer of

liquid trapped between two parallel plates, the top

one is pulled so that it slides steadily over the

liquid. The viscosity of the liquid is defined as

The units of viscosity are evidently Newton-seconds per square metre - that is, [Pressure x Time] - though somepeople prefer “poises” or (my favourite) “feet-slug-seconds”. That one really conjures up a picture. But it’s

useful to have the idea of viscosity, because it says explicitly that a liquid resists being made to move. A force

is needed, and the force increases with speed.

Engine oil has a viscosity of about 0.5 N.sec/sq.m., which means that plates 1 metre square separated by an oil

film 1mm thick could be slid slowly apart at 1 metre/sec by a strong man applying a force of 500 N (Newtons)

– about 100 lb, or 50 kg. A millimetre of cold water, by contrast, needs a force of only 1 N – just a few ounces.

A strong Newt could do it, if the Slugs didn’t get under its Feet.

If the viscosity of water is so low, why does it matter?

Think back to turning on the tap. As the water in the vertical pipe begins to move, the cylinders of water slide

reluctantly past each other. The viscous drag appears as a force acting upwards, opposing the down-force due

to the weight of water. The water accelerates, and since the opposing force depends on speed, this viscous dragincreases too. Eventually, when the tap is running freely, the forces exactly balance – there is just enough

down-force to overcome the friction force at this speed of flow. Viscous friction is the force that controls how

fast the water flows. Viscosity explains why the water leaving the tap doesn’t just carry on accelerating without

limit, as it would if there were no opposing force.

Now, the force pushing downwards is the pressure. More exactly, the net down-force on the water in a length

of pipe is the pressure difference between its ends. This down-force is exactly balanced by the opposing up-

force due to viscosity. So there must be a simple relationship between the pressure drop in a pipe and the

corresponding flow rate. It’s not all that difficult to derive it from first principles, if you enjoy integration (and I

know some people do). But if you don’t feel like doing that right now, the answer is

where µ is the viscosity of the fluid flowing in a pipe of radius R and length L. Unfortunately, although this

equation is nice and simple, it comes with a warning - it's only true for slow-moving fluid. But how slow is

slow?

Chaos rules

About 120 years ago, in Manchester, a man called Osborne Reynolds was trying to understand – or at least

describe – the flow of fluids in pipes. After many careful experiments, he decided that what happened to the

flow depended on four things - the viscosity and density of the fluid, the diameter of the pipe, and the speed

the fluid was moving. He put these four quantities together like this to make a dimensionless number:

Speed x Diameter x (Density / Viscosity)

The number has been known ever since as the Reynolds number, Re. Respect!

Reynolds found that when the fluid moves slowly, the flow stays smooth and even, but as the speed is

increased it eventually becomes rough and turbulent - chaotic, we would say now. The transition to turbulencealways happens at a Reynolds number between 2,000 and 3,000, no matter what fluid is used. (The reason for

the uncertainty is probably to do with small variations in initial conditions – chaos is like that.)

In other words, the simple equation above relating flow rate to pressure drop is only valid when the moving

water has a Reynolds number of 2,000 or less. If Re is higher than this, the water starts bouncing around

unpredictably, and it takes more energy to shift it along.

As well as speed and pipe size, the Reynolds number Re depends on the ratio of the water's density to its

viscosity, so to save having to work out (ρ / µ) each time you need to calculate Re I've included a table so that

you can simply look it up.

Table 1: Viscosity and density of w ater

Temo

C Viscosit Densit ( / )











40 0.000567 995.1502 1,760,000

50 0.000427 993.6288 2,330,000

60 0.000321 992.1097 3,090,000

70 0.000242 990.5929 4,100,000

80 0.000182 989.0785 5,440,000

90 0.000137 987.5663 7,210,000

The ratio ( ρ / µ ) for water varies from about 106 at 20oC to 4x106 at 70oC. This means that cold water flowing

in 15mm pipe starts to become turbulent at speeds above about 0.2 metres/sec, and for hot water the criticalspeed is even lower. Does this matter?

The question is, how long are you prepared to wait for the sink or the bath to fill? A flow rate of 0.2 metres/

second means that just 20cm of the pipe's contents come out of the tap in one second. Now, 22mm pipe (the

size normally used to plumb in a bath) has an internal cross-sectional area of 320 sq.mm, so the volume

occupied by 20cm of water is just:

Volume = Length x Area = (320 x10-6) x (20 x10-2) = 64 x10-6 cubic metres = 0.064 litres.

A bath holds typically 100 litres. It would take nearly half an hour to fill at this rate. Clearly 0.2 metres/second

is far too low a speed to be useful. If a higher speed means turbulence, then so be it.

The trouble is that allowing turbulence is really not a good idea. For one thing, the particles of water are

bashing into each other all the time, and that takes energy, and that means a much bigger force has to be

applied to move it. For another, the force you need can’t be calculated. It has to be inferred from other peoples’experiments.

Shake, rattle and moan

But the worst part of turbulence in pipes is the noise. In turbulent flow, particles of water move in random

directions at random speeds. Well, so what? One particle of water is much like another. The problems begin

when a particle decides, all by itself, to change into steam.

Large pressure differences can appear across very small volumes of

turbulent water, especially when the water flows round a bend, or

through a constriction like a valve or a tap.

In fact, bends can cause the flow to separate from the wall of the pipe,

like this. Because the water is suddenly forced to move sideways acrossthe pipe, whilst at the same time it is moving along the pipe, vortex

eddies appear. They spiral off down the pipe, wasting energy as they go,

until they are damped out by viscous friction. Bends should be avoided. If

they are inevitable, then the more gradual they are, the better.

Sudden changes of velocity (that is, speed in a particular direction) cause equally sudden – and dramatic – loca

changes in pressure.

This creates problems because water at a lower pressure boils at a lower temperature. At the normal

atmospheric pressure of 1 bar (14.5 psi, or 100,000 N/sq.m), water boils at 100oC of course, but if the pressure

drops to 0.1 bar it will boil at only 47oC. The water in a central heating system is hotter than this.

When the local pressure somewhere in the water drops low enough, a particle of water turns immediately into a

bubble of steam. The bubble soon moves back into a region of higher pressure and collapses, and the resulting

shock wave zips through the water, bouncing off the pipe walls. The more turbulent the flow, the more often

this happens. The process is known as cavitation, and it can corrode the pipework as well as a making a

disturbing amount of noise. If you think about it, bubbles do form spontaneously in turbulent water. A

waterfall, or the wake of a ship, or rocks in a stream all cause the water to foam. So does flushing the loo!

Measurements show that the cavitation

noise from fittings (that is, elbows and

tees) goes up with the speed the water is

moving. Each increase of 1 m/s raises

the noise level by a factor of about 4,

and it’s generally agreed that a water

speed above about 3 metres per second











makes cavitation noise unacceptably loud.

Cavitation isn’t the only source of noise. Turbulence causes eddies to appear in the flow, and besides wasting

energy they cause noise and vibration in the pipe network. Large eddies can be moving at up to 10% of the

average speed of flow, and contain energy at frequencies from a minimum defined by

fmin

= (Average water speed) / (Pipe diameter)

on upwards. For a 15mm pipe carrying water at 1.5 metres/sec, f min

turns out to be 100 Hz. Frequencies above

100 Hz contain progressively less energy, because viscous friction damps them more quickly.

The energy is coupled to the pipe network and may cause some part of it to resonate. The moving water acts

rather like a white noise generator, seeking out any resonances in the pipe network. That’s why pipes should be

clamped firmly to the wall at intervals of no more than a metre or so. The speed of sound in water is about 1450

m/sec (at 15oC) – about 30% faster than in air – so a 1 metre length of pipe can’t resonate at a frequency below

about 700 Hz, and there shouldn’t be enough stray energy there to worry about.

All this boils down to an engineering trade-off between cost and convenience. Slow-moving water implies

large-diameter pipes, which would cost more to install. The key question is, how much noise will people

accept?

Most sources recommend that the speed of water in pipes should be kept to less than 2 m/sec, and some

specify a maximum speed of 1.5 or even 1 m/sec. Remember the Slug. The Reynolds numbers corresponding to

usable water speeds for each size of pipe are summarised here.

Table 2: Maximum Reynolds numbers for standard pipes

Re at… 6 mm 10 mm 15 mm 22 mm 28 mm 35 mm 42 mm 54 mm

0.3 m/s 2,600 4,800 7,300 11,000 14,000 18,000 21,000 28,000

1.0 m/s 8,600 16,000 24,000 36,000 47,000 59,000 71,000 93,000

2.0 m/s 17,000 32,000 49,000 73,000 94,000 120,000 140,000 190,000

The table shows clearly that the Reynolds number for water moving at 2 metres/sec is way larger than the

2,000-3,000 maximum that would guarantee non-turbulent flow. Real plumbing in real houses is designed on

the basis that the water flow will be chaotic and turbulent. Unfortunately, there is as yet no proper theory to

describe turbulent flow, so systems have to be designed on the basis of experience rather than physics. The

simple theory I've been investigating just doesn't apply.

What happens in the real world?So if there is no simple theory, is there a complicated one? Pipeline systems do get built, after all, and the

engineers who design them must know what they're doing. How do they manage it?

One key tool seems to be an expression called the Darcy-Weisbach equation, which predicts how much

pressure would be needed to push a given fluid along a pipe at a particular speed. What makes it tricky to use is

that it includes a "friction factor" (f) which depends not only on the smoothness of the pipe - since copper is

smoother than, say, concrete, you'd expect it to have a smaller friction loss - but also the Reynolds number of

the flow. But that in turn depends on the speed of flow. In other words, you can only calculate the speed if you

already know the speed!

It's not quite as daft as it sounds, but it's certainly too complicated if you only want to design the plumbing in a

house. Real-world engineers prefer a simpler approach. They use the Hazen-Williams equation. The equation

is strictly only valid for water at below about 25oC, but that's OK. It's much simpler than the alternatives. To

use it, we need to know the head h, the pipe's length L and the 'hydraulic radius' (Rh

- half the pipe's internal

radius). And because the material the pipe is made from can also make a difference, there's a friction

coefficient C which for ordinary copper or plastic pipe can be taken as 150. The graph below illustrates what

the equation predicts will happen when the head is 3m, as it might be for a bath or shower. It shows that a

small increase in the length of a short pipe makes a big difference to the flow-rate. On the other hand, you

could safely add another 10m length to the garden hose without it making much difference at all.











Real pipes for real houses

In the real world the range of pipe sizes you can actually buy is quite restricted. Builders’ merchants usually

stock the standard sizes listed in the table below. The size given (eg. 15 mm) is the outside diameter of the

pipe.

Table 3: Standard plumbing pipe sizes

Plumbers' merchants call it ... 10 mm 15 mm 22 mm 28 mm 35 mm 42 mm 54 mm

Some plumbers know it as ... 3 /8 in

1 /2 in

3 /4 in 1 in 11

/4 in 11 /2 in 2 in

Internal diameter (mm) 8.8 13.6 20.2 26.2 32.6 39.6 51.6

Cross-sectional area (in mm2) 61 145 320 539 835 1,232 2,091

In passing, it’s worth pointing out that the standard sizes changed slightly when metrication was introduced.

This might matter if the half-inch pipe that was installed in your kitchen thirty years ago is not exactly the

same size as the new 15mm pipe you've just bought to plumb in the new dishwasher. The fittings may not

quite, er, fit.

Smaller pipe must be cheaper and easier to install, so why isn't 10mm pipe used for everything? (In France, it

often is!)

Small isn’t always beautiful

A bath should fill in five minutes or so. Suppose in the interests of economy you decide to plumb it in with

10mm pipe. The bath holds about 100 litres, so to fill it in 300 seconds will require a flow rate of (100 / 300) =

0.33 litres/second. Now, the internal cross-sectional area of 10mm pipe is about 61 sq.mm, so a 1 metre length

of it holds about 61 cubic millimetres, or 61 milli-litres (ml). A flow rate of 0.33 litres/sec therefore means a

water speed of about (0.33 / 61 x10-3) = 5.4 metres/second – about 12 mph. Given enough pressure you

probably could do it, but the roaring noise would frighten children and small animals, and cavitation damage

would mean you’d have to replace the pipes and fittings after a few years anyway. It's probably a bad idea.










Each size of pipe is intended to carry a specific flow rate, quietly. Cross-sectional area is what matters here. A

simple calculation of volumes shows that, at the maximum recommended water speed of 2 metres/second, the

maximum flow rates are:

Table 4: Quiet flow rates (in litres/sec) of standard pipes

10 mm 15 mm 22 mm 28 mm 35 mm 42 mm 54 mm

1.5

m/sec0.08 0.22 0.45 0.82 1.3 1.9 3.1

2.0

m/sec0.1 0.3 0.6 1.1 1.7 2.5 4.2

To check the actual flow rate, get a bucket and a watch. A bucket usually holds 2 gallons ( = 16 pints, = 9.1

litres). At a 15mm pipe's maximum flow rate of 0.3 litres/sec, the bucket will fill in 30 seconds. At a 22mm

pipe's maximum flow rate of 0.6 litres/sec, it will take 15 seconds.

Bath taps and sink taps

If 10mm pipe is not an option for the bath, then what is?

The idea is to choose the pipe sizes so that the water flows fast enough to fill the bath or the sink in a sensible

time without making too much noise. A kitchen sink holds 10 or 12 litres of water. So to fill a 10 litre sink with

water moving at 2 m/sec would take 85 seconds using 10mm pipe, 34 seconds with 15mm, 15 seconds with

22mm ... and just 2 seconds with 54mm pipe.

Now, if the sink filled in 2 seconds it wouldn’t save much time on the washing up, and besides, water wouldsplash all over the kitchen. Most people don’t mind waiting half a minute or so for the sink to fill, and that’s

why kitchen sink taps are designed to be connected to 15mm pipe.

A bath holds about ten times as much as a sink: 100 to 120 litres. If both the hot and cold pipes are 22mm, and

you run both at once, the flow rate is over 1.2 litres/second, so the bath fills in less than a minute and a half.

Bath taps are designed for 22mm pipe.

If you want to know how to calculate real flow rates in a real system, the method is given here in Part 2.

Copyright © John Hearfield 2007, 2009







domestic water-supply system - theory - john hear field

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