Chapter 1 Introduction, the structure of science, basic premises and systems of units.

 

SCIENCE—“Knowledge obtained by observation and experiment, critically tested systematised and brought under general principles.”-Chambers dictionary

 

Introduction

I am writing this book for the benefit of mechanical engineers. Unhappily as every man and his dog claim to be engineers I have to be more specific. I am writing for those who choose to follow a course of study in an attempt to be able to design a particular group of mechanical devices. Let me explain. I once went to visit a factory where cable was being made to be laid under the sea. There were two similar machines at work both producing cable and each was about 100 metres long. Initially copper wires were being extruded and spun into the conductor and then continuously covered with polythene. The capacitance of a short length of the cable was continuously measured and then the polythene automatically pared to keep the capacitance uniform. The wire was then covered with several layers of waterproof material and armoured. No single person had designed these machines from scratch. Indeed almost none of it could have come from any knowledge that depended on the general principles of science. I asked how it had been designed and was told that generations of designer-draughtsmen had each made a contribution. It was built by cut and try methods. Do not for one moment think that I am denigrating this method. It is the only way to design some things. Examples might be a light bulb, a washing machine, a pedal cycle, a machine for compacting rubbish and so on. In my view these are the output of technology. However some devices can be designed using the general principles of science. A pipeline to carry crude oil under the sea would not be designed just from technology but would be the subject of extensive calculation to predict the interaction between the throughput, the diameter of the pipe, the thickness of insulation and the power required for pumping. There is a large group of such applications of the general principles of science. This book is aimed at those who want to study those applications that arise in the flow of fluids and understand the underlying science.

 

If engineers use the general principles of science we must decide where they came from.

 

The evolution of science.

I am not a historian but I have lived long enough to have witnessed the evolution of some science. For example almost the whole of modern control theory has emerged since World War II. But where did science start? Certainly the Ancient Greeks had ideas that one might regard as the forerunners of science in their ideas of logic. However they had curious ideas about who should actually “do” logic. Typically they regarded arithmetic as wholly abstract and thought that its actual use was a task suitable only for slaves. They adopted a totally illogical position in saying dogmatically that when there was a conflict between logic and observation the observation must be wrong. That false position of the superiority of logic still echoes through the minds of men. There are claims for the evolution of science in the middle-east, the Indian subcontinent and so on. They may be correct but they came too early to have any parallel technological expertise to let science develop as it did subsequently in Europe.

 

It was not the men of science who designed and built steam engines for draining mines and for railways but practical men who by great good fortune had found an engine that would run and do useful work even though it was not understood nor well made. There can be no doubt that Newton and Bacon sowed the seeds of science and indeed took enormous strides to getting it started on a sound footing. Subsequently lots of gentlemen-scientists competed to explore what we now call physics and many of their names will never be forgotten in the English-speaking world. Unhappily this parallel development of technology by practical men and physics by theoreticians created a gulf of distrust that is in place to this day. It is sad but real.

 

When I was lecturing I set an examination paper in which I solved an equation by trial in three lines. When the external examiner returned the paper he had added four pages of detailed mathematics to find the same figure. He noted that it was not necessary to use a trial method. He must have had self-imposed rules that made him shy away from a trial solution yet we now know that it is the most powerful tool that we have when the trial is made instantly by a computer. Engineers find nothing wrong with a trial solution, mathematicians regard it as a failure of their mathematics. Yet Richard Feynman who started us along the road of quantum physics and played a major part during the Los Alamos nuclear bomb programme actually maintained and repaired the manual calculators that were worked to destruction solving differential equations by trial. But then he was also an engineer.

 

It becomes pertinent to ask whether there is a deep-rooted difference in the mind-sets of engineers and scientists. I think that there is. The gentlemen-scientists of the 19^th century experimented to find simple mathematical laws to describe the observed behaviour of the natural things that were all around them. They produced simple laws relating temperature, pressure and volume for gases and for the elongation of metals under loads and a host of other similar examples. In the main they made little attempt to apply these laws in any practical way although William Thompson became wealthy from his work on telegraph cables and Davey is said to have invented the miner’s lamp. Horny-handed engineers made things and it is doubtful whether much of the output of the scientists impinged on the technological world. (Slaves did arithmetic!) Engineers could not wait for science to catch up, they had railways, bridges and ships to build and empirical experience to guide them.

 

A long time ago I read an account of an American scientist from space and aviation who wanted to design a sailing boat to travel at 40 knots. He noted that the designers of sailing craft seemed to be totally unaware of the developments in aeronautics during the first 60 years of the 20^th century. Nothing has changed. Search the web for theory of sailing to see what I mean. The much-hyped warplanes designed between say 1935 and 1945 seem to have owed much more to the “can-do” men of industry than the aviation scientists employed by the governments. The can-do men could not wait because they were engineers who did not have the leisure to experiment systematically. Aeroplanes were needed immediately to fight a war not at some time in the future when some interesting theoretical problem had been solved. 

 

The fundamental difference between the approach of scientists and engineers is that scientists have no goal because they are on a voyage of discovery whereas engineers do have a well-defined goal in the form of a bridge or a boiler or an aeroplane to design and build. They can use mathematics and physics just as a scientist does but they also have a goal that tells them that when the contribution from mathematics is exhausted or too time consuming or just not worth any more effort it is time to get designing.

 

One might argue for Euclid as a “scientist in the making” but his work gave us an fascinating tool to use and admire but not anything that might be regarded as a general principle. Galileo without doubt used deductions from observation to upset the dogmatic views of the established church on the structure of a planetary system. Francis Bacon made a major step forward when he said that the right way to build a science was to make observations and test them by experiment and to formulate the “laws” of nature as we go. He has been proved to be correct. Newton appears to have followed this idea and to have used abstract ideas to describe and explain the behaviour of real things in the natural world. The classic example of this is Newton’s statement of what are now known as “Newton’s laws of motion”.[1] The first and third of these laws can only be stated in words and they are complete as Newton expressed them. The second law is capable of being expressed in symbols, ie  and then a system of mathematics is needed to apply this apparently simple statement to the motion of bodies in specified circumstances, eg when travelling in a circular path. If one cares to think of Newton as exploring the motion of bodies it becomes evident that he would not have made much progress without calculus. Of course he had to “invent” calculus, if that is the right word for discovering in the literal sense, something as useful as calculus that might also be regarded as innate to mathematics.[2]

 

Newton’s many contributions to the emergence of science are well known and it is clear that he belonged to a new school of thought where deductive reasoning about the physical world was tested against accurate observation. Perhaps Newton was one of the first to find out that if you look and observe and then attempt some rational analysis the next time you look you see more with a better understanding.

 

These ideas took root in European philosophy if nowhere else but science did not then emerge as a structured whole. It was like the six blind men and the elephant. Lots of people dabbled with bits of science and from time to time new unifying ideas came along each giving some coherence to a part of the whole. We now have a fairly coherent body of knowledge that usually works very well if used by someone who understands it. Unhappily when this knowledge is cherry picked it can be reconstructed to produce the most outrageous “theories” and the internet is full of them.

 

It is pertinent to ask where we are in the evolution of science. It seems to me that at the normal levels of science for everyday use by engineers we are finished except for tidying up some loose ends. We have explanations for most of the things that have sufficient order in their behaviour for us to quantify the behaviour. Even the weather is reasonably predictable for several days ahead.

 

However there seems to me to be one impediment to the application of science and that is the way that the internet is used. Often the outcome of science is a mass of useful data, e.g. the NACA data on aerofoils that is so good that it will never be superseded and the physical data for liquids gases and solids that has been gathered over many years. It is a very large data-base and it is an obvious next step to put all this data on the internet for use by anyone with a computer.[3] Unfortunately this data has cost money to gather and has become intellectual property and is available only if purchased. This means that a would-be user must weigh the direct cost of buying from the internet against the cost and effort needed for the acquisition of data by traditional methods. It makes the methods of science and engineering not as useful as it could be.

 

In the science of fluids both at rest and in motion we have sorted out good ways of predicting the things we want to know in many cases. We have two major groups of problems that resist our efforts. They are those for which the mechanical properties of the fluid cannot be quantified eg slurry, and those for which the boundaries cannot be quantified, eg the flow on watercourses. It is unlikely that these will ever yield.

 

Basic premises

It is hard to know how Newton came by his ideas because it was the fashion of the times to publish one’s ideas in as concise a form as possible and let other people try to follow and see the consequences. It almost looks like a game of “ this is what I think now prove me wrong”.

 

Newton’s laws of motion are nothing to do with gravity as such, they tell us how a body behaves when it is subjected to a force or a system of forces.

 

Newton tells us first that a force cannot exist by itself, there must be something resisting the force.[4] That something might be a force or a combination of forces but it might also be the inertia of a body that is accelerating under the action of the force. A force might be a push or a pull but it might also be a friction force. Clearly we shall find all sorts of forces and bodies undergoing all sorts of accelerations.

 

Newton then makes a bold statement that relates force and acceleration. He asserts that bodies have mass and that force = mass times acceleration. We still do not know any more about mass than Newton knew but his assertion has enabled us to create the whole science of dynamics and use it very successfully indeed.

 

Finally Newton tells us what happens when a body is moving in the absence of any net force on it. He says that it will move in a straight line at constant speed or, of course, be at rest. This raises the question of a reference frame for measuring speed. The reference frame is that of the distant stars that, in human terms, are fixed. Engineers work on a much, much smaller frame and they work relative to Earth. Usually they are interested in the corollary to this law. They note that if a flow of fluid is moving in a path other than in a straight line there will be a force or system of forces acting on it somewhere.

 

It may not have astonished Newton when he recognised that action at a distance without any apparent means was possible but the more recent extension of the idea to say that it acts at the astronomical distances between galaxies is unimaginable even if you can get used to the idea. Fortunately we have only a need for gravity on a terrestrial scale.

 

It is our daily experience that a vertical force is exerted on our bodies continually and without letting up for just an instant. That force can easily be measured on a set of bathroom scales. We do not need to question that it exists but when we try to think of a “mechanism” by which it might be seen to operate then it is very troublesome. It is fun to think about it but we do not need anything complicated. We just need Newton’s laws. Newton postulated that the force of attraction between two masses would depend directly on the product of the masses and inversely as the distance between them. He wrote this in mathematical shorthand as:-

                                            the force  where m₁ and m₂ are the masses, d is the distance between their centres of mass and G is a constant[5].

 

For most purposes this simple model of the gravitational attraction between two bodies can be used exactly in this form. Only in astrophysics do we require a more refined model.

 

Newton’s model of the relationship between the gravitational force exerted by the Earth on a body that is not in contact with the Earth seems to stand up if the mass of the Earth is regarded as being at a point at the centre of the Earth. According to the model the force on a body should decrease with distance from the centre. This has been checked by experiment and appears to be correct.

 

The earth is very large when compared with any device we might make in engineering. It is a large, ever-present body that gives every other body a weight that does not change much wherever it may be.

 

Clearly we shall need to be able to quantify the force due to the gravitational attraction of the earth. We use the statement that force = mass times acceleration. The gravitational force on any body can be measured. Its acceleration in free fall can be measured and it is usually taken to be 9.81 m/s/s. This gives a relationship between mass and weight and all we need is some system of units.

 

Before we look at units there is an important feature of a gravitational field to be noted. Work can be stored in it without loss. A mass can be raised and in doing so work is done equal to the weight times the increase in height. This work can be recovered in its entirety by lowering the mass without acceleration. The gravitational field seems to be perfectly elastic. (I cannot think of any other system for which this is true.)

 

How then shall we visualise the gravitational field of the Earth? The answer is that it depends what you plan to use your field model for. If you want to fly satellites in orbits round the Earth then it may well be necessary to let the field take into account the concentrations of mass that occur in various points in and on the Earth. (This data is readily available.) For the purpose of engineering we can manage with Newton’s model and visualise the field as a series of concentric spherical surfaces. The force exerted on a given mass by the Earth would be the same for any point on a given spherical surface. The value of g decreases with height and it is easy to show that  where g_h is the value of g at height h and is the value of g at the mean radius of the Earth. As the radius of the Earth is about 6500 km at a height h equal to 1 kilometre . For our purposes g is constant at 9.80665 m/s² and for most purposes in this book this can be rounded to 9.81 m/s².

 

The gravitational force is thought of as being towards the centre of the Earth and this completes our model of the field. This raises another important consequence.

 

Fluids exist like the rest of the material in a gravitational field and the thing that is special is that, because they can flow, gravity determines where they end up. A fluid will just go on flowing perhaps over thousands of years until it can get no lower because there are solid impermeable surfaces under it. Water might flow through a watercourse and so to the sea but it gets into the watercourse either over the surface of the land or by percolation through the ground. This could not happen to a solid. Water and lots of other liquids can flow in large quantities and in very small quantities because the ability to flow comes from its molecular structure. It flows under the influence of forces that are created by gravity and rigid but not necessarily impervious surfaces.

 

The atmosphere surrounds the solid surface of the Earth. Air is compressible but it too gets as low as it can but, being compressible, it has greater density at the surface simply because of the weight of air above it. It is about 10 km thick.[6]

 

Systems of units

I am sure that somewhere there is a definitive history of systems of units. We are still in a dreadful muddle with units and the muddle shows no sign of disappearing. We need to see why.

 

First we must decide what we mean by units. Perhaps we should start by looking at weights and measures. Probably there was not any problem with weights and measures at one time. They were needed for commerce. Initially when the number of sheep that could be traded for a given herd of goats only a system of numbers was needed and that might only be a tally stick. But once a commodity like grain or salt is to be traded for money a system of packing commodities in known quantities was needed. All sorts of “standard” quantities were used and often these quantities were peculiar to the trade involved. There was nothing wrong with this although one can imagine that there were opportunities for sharp practice in every transaction[7]

 

There was a common feature to these measures. They were chosen to suit the commodity. Grain was sold by the sack-full but diamonds weren’t. The proliferation of weights and measures no doubt led to sharp practice and some limit could be placed on this if some legal system of weights and of length could be established.[8] Arbitrary systems were established in various countries and colonialism ensured that some were promulgated across the world. It was inevitable that commerce would have to proceed with this mixed bag of weights and measures. To the practitioners of trade this was probably no problem and, if busybodies in politics and science had kept out of it there would be no problem.

 

Then came the rise of science. Now there were people who had to think about units for a different reason to those engaged in commerce. In the 17th century Newton pointed out that mass and weight are not the same thing and the problems started. Scientist wanted not a system of weights and measures but a system of units. They observed that, if you started from the unit of length, now measured to some staggering accuracy, all the measures of volume could be expressed in cubic feet or whatever the unit was. But cooks knew in their bones that a pinch of salt could never be measured in cubic feet and retain any meaning.

 

Scientists went one better. They could see that you could have a system that allowed every physical quantity that could be imagined to be related to each other by one set of specified units for length, mass, time and temperature and a few others. They called this a consistent system of units. The still went shopping in the ordinary measures of course but some, who liked to be in control, could not but feel that it would be nice if everybody was forced to used a consistent system of units regardless of the snags.

 

The snags were mainly that these new consistent units are often extraordinarily small compared with the size of thing to be measured or, of course, extraordinarily large. We have enough trouble with imagining size because people live most of the time in a people-sized world. We do not often sit and contemplate the flow over the Niagara Falls or the size of a molecule of gas nor yet the fact that spiders have working knee points. Should we do so we find that ordinary units do not convey any idea of the magnitudes involved. What we do not want is to find that the ordinary units do not convey an idea of magnitude when they are used for ordinary things. The metric system was based on an arbitrary length derived from the circumference of the Earth. Users of the system devised all sorts of non-preferred units to make the units fit the size of the thing to be measured which is of course our starting point. I suppose they were comfortable with them. I lived in a world of feet and inches and pounds and ounces. Doing physics and electrical calculations in grams and centimetres caused no great problem to those trained in science, one simply got used to two systems. All these units were people sized.

 

However calculations for electrical engineering gave rise to ordinary quantities measured in millions or millionths and they fancied the metre, kilogram, second system because it fitted their calculations better. Then, when Britain joined the European Union, the question of units got tangled up with the decimal system versus the duodecimal systems. Everyone forgot how easy it is to work in two systems and persuaded Britain to move to the SI system of units that was a derivative of the MKS system. Suddenly lorries appeared with their weights painted on the back in kilograms instead of tons or tonnes and in Newtons. No doubt the French lorry drivers were just as mystified by weights in Newtons as everyone else. It was a great triumph for the scientific lobby over good sense. No one puts weights in Newtons now because the process of common sense is at work devising new non-preferred units to make the numbers intelligible to ordinary people.

 

Unfortunately the SI system does not fit with mechanical engineering and we are lumbered with it. If you want to buy petrol you are charged by the litre but you cannot buy a single litre of petrol. The next preferred unit is the cubic metre! When I asked a group of engineering students from overseas who had all been brought up using the metric system to say how many cubic metres of space there is in a mini car they simply had no idea. They had never thought in cubic metres before.

 

Our current position is that an engineer trained in the SI system will go to work in organisations using all sorts of hybrid systems of units. They will meet new designs with metric and Imperial dimensions in the same component because the firm is taking advantage of cost and availability. Standardisation is a far off dream and we have to work in one system now and expect to adapt to circumstances as they arise. One unforeseen consequence is that if a customer is paying for design calculations and these calculations are in some incomprehensible units like Newtons per square metre he cannot check to see whether they make sense in the ordinary concept of common sense. It is a paradise for rogues.

 

So what systems of units have been used? The most commonly used systems are the Imperial system and the Metric system.

 

In the Imperial system one pound force (denoted lbf) will give a mass of one pound an acceleration of 32.2 feet/second. In the system in common use in Europe 1 kilogram force (denoted kgf or kilopond) will give a mass of 1 kilogram an acceleration of 9.81 metres/second

 

Other systems have been used. The ones I know about are:-

the dyne-gram-centimetre-second system (C.G.S.) which was certainly in use in school physics. The gram is a very small mass to be a unit and the dyne a very tiny force. But it was used for small scale applications and seemed to be workable for these.

the poundal-pound-foot-second system which had its adherents but was not very popular because the poundal, being equal to 1/32.2 lb was too small a force to be an acceptable unit.

the pound force-slug-foot-second system much used in aeronautics. The slug has a mass equal to 32 2 pounds but the name for the unit of mass was somehow repulsive and it was really too large.

 

Before the arrival of the SI system most people worked in either the Imperial system or the metric system. They accepted the pound force and the pound mass or the kilogram force and the kilogram mass and used the appropriate value of g to make calculations. Then forces and masses were measured in people-sized units.

 

In Great Britain the driving force behind the change to the SI system was the National Physical Laboratory. They published the rationale for it in a pamphlet called “Changing to the metric system”(published in June 1965), a title which was misleading (probably deliberately) because the change was to the new SI system. That rationale looks to be just as dubious now as it was then and the introduction of a consistent system of units with all its problems was hidden under the argument that working in tens is better than working in a mixed collection of factors even if these factors were more suitable for the task in hand. One is forced to wonder what system of units the NPL used when they claim that the Imperial system was the yard-pound-second system. I have never seen g quoted in yards/sec/sec.

 

The Systeme Internationale d'Unites (S.I.) is the Newton-kilogram-metre-second system. This is really a metric version of the old poundal-pound-foot-second system with the same old problems. The unit of force is small. However journalists, politicians and academics joined forces to give the SI system the backing of the law and we are stuck with it. I suppose that the consequences of the introduction of the SI system can be typified by one example. A cubic centimetre that could be imagined by anyone was suddenly the millilitre a descriptive word that simply defies the imagination. Units derived by some arcane formula and a hierarchy simply create confusion for the vast majority of ordinary people who are not technically trained. The man digging a hole in the road claims that it is one and a half metres deep and he knows this because he has been told that his shovel is a bit less than a metre long. What is his unit of length, the metre or the shovel? I say to students of engineering that your task has been made more difficult by the SI system and it is essential for you to understand it and be able to understand other systems so that you can switch around as the need arises. If you set about it with a will it is really quite easy.

 

The S.I. unit of force is the Newton, a large hen's egg weighs about 50 grams and that is about a half a Newton, and, as the consistent unit of area is the square metre, the unit of pressure is 1 Newton/square metre. The diagrams for this book were drawn on good quality note-paper weighing 100grams/square metre so a single sheet of that paper would exert a pressure of nearly Newton/square metre (one Pascal) on any horizontal surface on which it was placed. It is clearly not a tangible pressure. An insect can crawl from under a sheet of paper. In these units the atmospheric pressure is about 100,000 N/square metre. Power engineers did not fancy swapping pounds/square inch or kilogram force per square centimetre for Newtons/square metre and the galactic numbers that would be entailed so they put 100,000 Newtons/square metre equal to 1 bar and got on with their work. What is the man who wants to pump up the tyres of his car to do when he finds the tyre pressure quoted in Pascals, bar, and pounds/square inch?

 

Nor is it easy to comprehend the magnitude of the unit of pressure. Clearly the S.I. system is not ideal, it may fit science but it does not fit people.

 

The fact is that the S.I. system was introduced at a time when two other systems, the foot-pound-second system and the metric system (the kgf-cm-second system), were well established. They are systems which make sense to ordinary people when used to measure ordinary things and are used quite easily in science and engineering. It is 40 years since S.I. was introduced yet it is still not in general use. In practice many non-consistent secondary units are still used and others evolved to suit particular applications. Governments may legislate to enforce the use of some preferred system but, in time, as its deficiencies become evident and the offending politicians find different things to worry about, the will to enforce the system wanes and modifications are introduced or, the former systems are revived.

 

The science of fluids

This is the knowledge on which this book depends and it is not just a few simple principles. Fluids are particularly troublesome to deal with because they have no shape other than that imposed on them from outside and when they move they do not move like a solid. They move with internal motion at three levels, the motion at molecular level, the motion of small scale turbulence and the larger scale motion of eddying. Just looking at water flowing in a small river leads to the feeling that this is just the same as the last time you looked at it yet the detail of the flow is simply not the same. It is the same old flow with different detail. In a sense this gives us a clue to the possibility of success because it suggests that if we try to avoid the detail we might find an overall order to things that is adequate for engineering purposes. This last proviso is dominant. Once science and experiment can provide us with numbers that are adequate for our purposes there is no incentive for further refinement and it pays to remember that those numbers might be changed for political, financial or commercial expediency.

 

In this book I want to attempt to assess precisely what we are doing and to spell out how it fits into the overall order of things.

 

I wrote it originally in 1990 and since then computer programmes like Mathcad have made the exploration of the many equations of fluids very easy to do. This exploration gives insights that were formerly not available because it was too laborious to make the exploration. I hope to re-draft this text to include some of these explanations.