Chapter 6      Viscosity.

 

Introduction

I am in the process of describing an empirical science that is in existence and the next major thing to describe is the way that we deal with the flow of fluids in pipes. This cannot be done without the use of the property viscosity.

 

I have said that this is a personal text that lays out my view of this subject and here I am writing for any one who uses engineering science and is a bit more curious than most. You may not always agree with what I say but you are free to ignore it and I have found that what follows works for me. Viscosity has always troubled me because I learnt about it through Poiseuille and laminar flow as if this is the main application and then without more ado it was used in every conceivable type of flow. Subsequently, however I thought about it, there seemed to be no obvious justification for this extension of its use and yet it worked extremely well. With the rise of the internet I looked for explanations but I have never seen any satisfying explanation. The question that I now need to answer is, why is viscosity a property and therefore universal in its application and what does it quantify? In the event the question was in the front of my mind for many days before I could put together all that I know in some satisfactory way. Keep in mind that all we need as engineers is an explanation that lets us think about engineering applications without making silly mistakes of principle.

 

I think that people do not spend enough time just looking at water on the move. Water is the liquid that we see most often although no one else that I know ever bothers to look at it carefully even when they have an opportunity. When I first started looking at flowing water I could make little sense of it, mostly because my degree course was not a suitable preparation. Sometimes I could see no sense because I did not know the shape of the surfaces that affected the flow because they were obscured by the murky water. Year on year I learnt more and more observations fitted together and now I can often understand what is happening and why.

 

How liquids flow

Look at picture 6-1. It was taken in the River Medway in Kent in the UK. The flow in the river was greater than normal and the water flowed past bridge piers that have these pointed ends[1]. The water breaks away from the sharply angled corner and a swirling wake forms between the main flow and the side of the pier. Those who seek to make diagrams of this flow will usually just draw a lot of squiggly lines[2] to indicate no orderly flow. But, if you have the opportunity to look long enough, you can find some order and, in my view it is important to look for that order. In this flow there is a cyclic change in the flow pattern and what you need to know is that the foot of the pier has a large apron stretching over, and standing proud of, the natural bed of the river to protect it from scour. This affects the flow at the foot and these two flow patterns get out of phase. If you look at the wake you cannot avoid seeing eddies that form in the wake and get swept downstream. There is one in figure 6-2. These eddies are everywhere in the natural flow of water and air.

 

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Picture 6-3
Picture 6-3 is from Prandtl’s book where there are several pictures of eddies made using paint on glass and aluminium powder sprinkled on the surface of eddying water. Here he is showing us how a two-dimensional flow that has become detached can become re-attached as the speed of flow increases. I am interested in these eddies. Look at them, they are not accidental or random, they are an important part of the overall flow pattern. Note that the eddies all have the same direction of rotation and having three or more eddies in a line like these is quite common. Inevitably where two eddies exist side-by-side there is contra flow between them. Eddies are not necessarily circular nor are they the same shape instant-by-instant but over time they have a well-defined shape and appear to be an essential and predictable part of the flow pattern. Eddies do not rotate like wheels, instead the angular velocity increases with decreasing radius and, throughout the eddies, there is shearing and this shearing takes place continuously both within the eddies and between them. It seems to me that Nature has designed a flow pattern in which to lose mechanical energy to internal energy very effectively.

 

You can see from the horizontal stone course in picture 6-1 that the surface of the river Medway suffered a drop in level as it flowed past this bridge. This means that potential energy has been given up and it is now contained partially in an increased velocity and partly in the wakes from the several piers. Beyond the bridge the velocity will return to normal and the energy given up in the drop in level will be contained in the flow. If you try to think of ways to contain energy without an increase in level or average velocity you will find that the options are very limited and there is only one way for this energy to be contained and that is in eddies rotating with all sorts of different axes.[3] Go and look at your local river and see the way that it eddies after it has passed an obstruction. The well-defined eddies will have disappeared from the surface leaving a flow with much finer grained disturbance in it  and the kinetic energy will be dispersing into the water. The mechanism doing the dispersion is the shearing and it is reasonable to suppose that some liquids resist this shearing more than others. This means that, say, crude oil flowing in the natural bed of a river would present us with a new and exciting flow pattern to study. We, in ordinary parlance, talk of thin and thick liquids and we think of them as being runny or slow to pour. It is all about shearing. But ultimately it is not the shearing that we can see that matters, it is the shearing that goes on at molecular level that is the basic mechanism by which mechanical energy is dispersed into the fluid as random internal energy. That motion is going on all the time. It is the character of this motion that we quantify when we measure viscosity.

 

I have been looking at flow with a random element like this water in a river and perhaps in the wind but liquids can also flow in a totally orderly way. Look at picture 6-3. It is from chapter 9 of my book on model yachts that is on this website. It was produced on a home-built Hele-Shaw apparatus to try to visualise the flow round sails. (See details in book.)[4]  Here I am interested in the blow-up of part of the flow pattern. It is the total orderliness that I find so astonishing. The water was flowing between glass plates spaced about 1 millimetre apart and the lines are of ordinary fountain pen ink. The water was flowing round a brass shape representing a sail attached to a mast. Water in contact with the plates is stationary and the velocity distribution between the plates is parabolic. When there is a change in direction of the lines this parabolic velocity distribution takes on a transverse component as well and the line widens because there is the same pressure difference to produce the essential centripetal acceleration and because the water moves at different tangential speeds the centripetal acceleration to make them stay together is not the same for all the flow. When you actually see the apparatus running you can see the detail of the lines widening and narrowing and those “shaded” areas are actually sheets of ink having this parabolic shape across the flow. It is all too evident that water can flow without any discernable mixing even though it must be subjected to continual shearing at molecular level. I like these images that come from Hele-Shaw, I suspect that the apparatus is more versatile and instructive than is generally supposed but it needs experimental skill to get results from it.

 

My examples above show two very different aspects of this shearing and I need to attempt to find a mechanism that accounts for all the effects that are attributed to viscosity. I think that I must start with Newton, who, at a time (1687) when there was no concept of a molecular structure for solids, liquids and gases, could offer no insight to the mechanism that could create this internal friction that he called viscosity but could cut through the maze of different effects of viscosity and suggest a way of quantifying it.

 

The concept of viscosity

Newton saw that viscosity produced an effect just like solid friction in that mechanical energy was just lost into the liquid as it might be into a solid. His concept has obvious similarities with the way we think about solid friction as figures 6-3 and 6-4 show. In solid friction we put  and call m the coefficient of friction.  In the system shown in figure 6-4 a liquid is imagined to fill the space between the flat and level plane and the flat plate, of surface area , which is distance  above it. A horizontal force  is imagined to act on the block and to cause it to move at a steady speed . Newton then used this system to define a coefficient of viscosity for the liquid. He put:-

                                       where m is the coefficient of viscosity.

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              Figure 6-5a                                          Figure 6-5b                                     Figure 6-5c
Clearly Newton has proposed a system, under which the fluid is made to move with continual shearing, that is both well defined and easily visualised. I interpret Newton’s system as behaving as I have shown in figures 6-5a, b and c where a straight line through the liquid layer remains straight as the layer is continually distorted and there is a uniform rate of .

 

We need to have some mental model of what goes on in that layer because this is where mechanical energy is ultimately converted to the random motion of internal energy and I want to look towards the molecular motion of water as an example that might also apply to other liquids.

 

However it is easiest for me to start with gases. The kinetic theory of gases gives a very good picture of the structure of a gas. It postulates that a gas is composed of separate molecules, (which may be single atoms). The molecules “fly” freely at high speed, colliding frequently with other molecules and with the walls of the container in which they are enclosed. The scale of the structure of a gas is indicated by the following figures. The common gases at room temperature and at a pressure of one atmosphere have about  molecules per cubic centimetre. Even with this concentration there is still space between the molecules for them to move freely at high speed (about 350 m/s) through a distance of about 7 molecular "diameters" between collisions and the number of collisions made by each molecule each second is about . These are large numbers that I can accept but cannot imagine.

 

The molecules of the gas have mass. They move with high linear speed, rotate, and where the molecules have two or more atoms they can vibrate in the inter-atomic bonding. Kinetic energy can be stored in the gas in these motions. (Measurements seem to suggest that little energy is stored in vibration in a gas, nevertheless it does have this degree of freedom.) Consequently the gas may be regarded as having a stock of kinetic energy that is stored in a random manner in its rectilinear motion and, in thermodynamics, this is called internal energy.

 

The temperature and the pressure of a gas are measurements of two different aspects of the concentration of kinetic energy in the structure of the gas. The pressure is the result of the very large number of collisions that occur between the molecules of the gas and the solid surfaces containing it[5]. Each collision involves a very small force acting for a very short time at some angle to the surface and the continuous uniform pressure is the aggregate of all these short-lived forces. Of course the pressure acts normally to the surface because all the tangential components of the impact forces cancel out. So, in some way, pressure depends on the mass of the molecule, the mean kinetic energy of the molecules, and the concentration of molecules in the space (which, of course, determines the frequency of collisions with the walls). If, for a given gas, the mean kinetic energy were to be kept constant, the pressure would simply depend on the concentration of molecules, that is, on the density.

 

Temperature is a concept that springs from our natural concept of hot and cold and this appears to be essential to human survival because we are more vulnerable to temperature changes than animals, birds and reptiles. In order to measure the temperature of a gas we bring some thermometric device into contact with it. The atoms of the material of the device are held together by atomic bonds, that appear to be perfectly elastic, and they vibrate in a random manner within the limits imposed by the bonding. The molecules of the gas, when they collide with the surface of the device, do not meet a rigid, flat, stationary surface but one that is irregular in shape, and at the molecular scale, is in violent motion. The molecules of the gas and the atoms of the surface continually exchange energy. We wait until equilibrium is established between the average rate at which energy is given to the surface of the device by the molecules of the gas and the average rate at which energy is given to the gas by the atoms in the surface of the device. Thermodynamicists call this thermal equilibrium and say that the gas and the solid are at the same temperature. Thus temperature is an independent variable, to which we can relate other properties, and which is a measure, in the case of a gas, of the mean total kinetic energy of each molecule, which physicists observe to be the same for most gases at the same temperature.

 

The thermometric device will have been chosen so that some easily observed feature of it changes during the process of reaching thermal equilibrium and reaches a steady value. The device is calibrated to read temperature in an arbitrary system of units. The important thing to note, is that the reading of the device, that is the temperature, is dependent on the mean kinetic energy of any single molecule, whereas pressure is a measure of the concentration in many molecules of random kinetic energy in the space occupied by the gas.

 

We now have a picture of the gas in which each molecule is involved in many millions of collisions every second. The image brought to mind is of a continuous exchange of kinetic energy, both between the molecules of the gas, and between the molecules of the gas and the atoms of the material of the container. Such a process tends to produce uniform values for the mean velocity etc of the molecules. If there were to be some way in which the kinetic energy of the molecules of the gas and the atoms of the isolated container could dissipate their kinetic energy there would be a steady fall in temperature and pressure. Neither of these things happens. We can only conclude that the energy of the molecules and atoms is not dissipated and therefore the molecular interactions are perfectly elastic.

 

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Graph 6-1
 Now I can look at water as a liquid that behaves in a way that is typical of many other liquids. In chapter 2, I mentioned that the molecule of water has a T shape but one might prefer to see it as three atoms joined in a triangle. I have no mental image of an atom, except as a nucleus with electrons whizzing round it, but I know that atoms vary in physical size so that the atom of oxygen is larger than the atom of hydrogen and that the molecule of water will have an odd shape because it has three atoms. Water is an agglomeration of densely packed molecules of . I have said that, in the commonly used, simple image of a molecular “structure”, molecules are regarded as being perfectly elastic. This means that, for millions of millions of molecules all moving in a seething mass, collisions and other exchanges of kinetic energy do not lead to a change in the sum total of kinetic energy. Only an external effect can produce a change. I need some model of the system in which this kinetic energy is stored by the molecules. As I understand it (I may be totally wrong.) the molecules seem to experience an intermolecular attraction when they are close together and a repulsion when they are too close together. As both the attraction and the repulsion increase as the distance between molecules decreases I have supposed, in order to make progress, that it might be possible to represent these two forces by the graph 6-1. For the system of two molecules to have my claimed character these two plots must cross as I have shown them. At some point P the attraction will equal the repulsion and when the two molecules get closer the net force will act to push them apart and when they are further apart than P they will be attracted.

 

The repulsion will obviously go on rising rapidly as the separation decreases and this must mean that a large number of molecules forming a continuum will be effectively incompressible. The implication of having an attraction is that the same continuum will be capable of withstanding tension, which is, of course, the case. This tension has a significant value compared with the forces ordinarily impressed on water in engineering applications and the tensile forces created during shearing are easily resisted by water without rupture at the molecular level.

 

The molecules have mass and if they also have motion the molecules could oscillate between the two positions that I have shown as  and  and the combination of forces of attraction and repulsion produce a motion like that of a weight suspended from a non-linear spring. The amplitude of the oscillation will vary with the total kinetic energy of the two molecules. It looks to me as though the kinetic energy can have any value between zero and some value when the attraction is too weak to prevent the molecules flying apart if that is, in fact, physically possible.[6]

 

Molecules of water that are part of a continuum of molecules do not join to form pairs. They are very closely packed so it follows that any molecule will also be in engaging with several other molecules and have some complex system of oscillation with all of them simultaneously. Given that the molecules are not symmetrical this makes one wonder how many molecules can be in contact simultaneously. I looked at tennis balls and they do not fit snugly round a central ball. I suppose that 8 or 9 can be in loose contact at once. New Scientist says that water molecules form clusters of ever-changing numbers with a probable size of 5 molecules and that fits with my figure.

 

Somehow we have to form a mental picture of innumerable closely-packed molecules oscillating in complex ways in ever-changing groups to form an extremely active continuum of molecules. Clearly kinetic energy will be contained at some mean level in every molecule. The kinetic energy has no mean direction so it cannot give the mass centre of the water a direction and so there is no way that the sum of all this kinetic energy can be extracted and stored in the gravitational field. It is totally random energy.

 

There is a further piece of information that is common experience is that water is a poor conductor of heat. The sum total of the random kinetic energy can be increased by heating the water and this proves to be a slow process if it takes place by dispersion through the molecular structure. This is consistent with this model.

 

Now I need to look at this model of molecular behaviour in the layer of water envisaged by Newton. He applies a force to the layer of liquid. Presumably, if an external force tending to compress the system of molecules were to be applied, it would occur through the whole system instantly. A depression would also affect the system instantly. So it would be reasonable to expect a shearing force to be exerted instantly throughout the molecular structure and not depend on a slow dispersion process.

 

This seems to me to be a plausible model for the molecular structure and its behaviour and that it is adequate to continue this study of viscosity.

 

Quantifying viscosity

I think that it is worth examining Newton’s expression. Newton, like everyone else, must have been aware that the there are plenty of liquids that flow more easily when they are heated and have expected that his coefficient of viscosity would decrease as the temperature rises. However I do not remember whether he said anything about this or about the fact that the coefficient of viscosity might be dependent of the velocity gradient  as well.

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Figure 6-4
Anyone thinking of using this system to actually quantify the coefficient of friction for the liquid in the space finds out very quickly that it is not easy because any mechanical system that is devised causes effects other than those due to viscosity. No doubt many people tried to devise ways of realising Newton’s idea in practical hardware but the hardware that was needed did not appear until Hagen, who was an engineer and ignored by physicists, gave a basis for it in 1839 only for it to be rediscovered by Poiseuille in 1840. It is interesting to contemplate what went on in the years between 1687 and 1840. We know that Robert Hooke, who was contemporary with Newton and often entertained the fellows of the new Royal Society with demonstrations much as the Royal Society entertains the public now, appears to have had reasonably well-developed engineering skill at his disposal. This must mean that various devices must have been invented that were of a mechanical nature e.g. time of discharge to measure liquid mobility, rotating discs to measure shear, falling balls and so on. This suggests that the failure to exploit Newton’s proposal to produce a method that could give an accurate measurement of viscosity did not lie with the technology of the time. The one system that did work was dependent on the use of calculus. Newton “invented” calculus but he wrote his Principia in Latin and, even though it was translated into English in 1729 it is still very difficult to read and understand. This must have delayed the dissemination of his ideas and, as the methods used by both Poiseuille and Hagen depended on being able to integrate, the delay may have been inevitable. I suppose that both Hagen and Poiseuille showed that when a liquid flows in a tube of small diameter the pressure drop is proportional to the flow. Perhaps they thought that this was an unlikely result because it is not true for larger pipes but they both pursued it. They found a method of quantifying viscosity that has proved to be useful well beyond the direct result. It made possible the non-dimensional group that we call Reynolds’ number and the use of Reynolds’ number opened the way to storage and retrieval systems for immense amounts of valuable experimental data. It was a big step on the way to constructing an empirical science.

 

Hagen and Poiseuille knew that, for a small diameter pipe, the pressure drop was proportional to the flow and they must also have known that the actual pressure drop varied with temperature. They required something that took account of both temperature and the liquid that was flowing in the pipe. Newton’s definition of the coefficient of viscosity was there to use and we can reconstruct their method. It is implicit in Newton’s work that the liquid in contact with a fixed surface behaves as if it is stationary and this means that in flow through a pipe the liquid in contact with the pipe has zero velocity. Obviously there must be velocity elsewhere in the pipe and the most reasonable supposition is that across any section the velocity at a given radius is uniform and that this velocity changes from zero to at the outer radius to some maximum on the axis. Poiseuille imagined the flow to take place as if a series of coaxial cylindrical layers moving in an orderly way at different speeds. This clearly reflects Newton’s method of defining a coefficient of viscosity but does not provide uniform velocity in the layers nor a constant area.[7]

 

This is how the rest of the argument goes. Now we can look at figure 6-4. It shows, at some instant, a section of the flow of a liquid in a small tube of radius [8] between two plane surfaces  apart. The pressure on the upstream section is  and on the downstream section is . In figure 6-4 I have drawn a section of a cylindrical surface of radius  that is coaxial with the tube. There is a net force acting on this element equal to:-

 

                                                  = 

If the flow pattern is taken to be symmetrical about the axis, we can replace the  of Newton’s definition by , use Newton’s relationship, and put the force exerted on the liquid in the small tube equal to:-

                                                               

Then :- , where the minus sign allows for the fact that  will be negative.

Then we can rearrange and put:-

                                                       which can be integrated to give :-

                                                       where  is a constant.

Using Newton’s simplifying decision that the fluid behaves as if the solid boundary is the same as a stationary layer of liquid we get  where  and then   and :-

                                                        , which is the equation to a parabola.

This is not yet in the form of a rational expression because c is not a measurable quantity. We need an expression in terms of the mean velocity   and:-

                                                        

                                                =

                                                =  or, changing to diameter, = 

If this is now integrated along the pipe we get:-

                                                         and this has been named after Poiseuille and sometimes also called the Hagen-Poiseuille expression.

 

It is not in the form that is best for engineering. We need:-

                                                                

This is an expression that contains only measurable quantities and it is a rational expression on which the coefficient has been based for experiment. When it was first produced no one knew whether liquid could flow with this parabolic velocity distribution. Its success would depend on whether it turned out to be useful in practice. It was remarkably successful for liquids like water and oil. Look in any science data book and you will find values of  quoted at 20°C for a string of liquids. Presumably these liquids share the fact that they are all agglomerations of one type of molecule and not of curious mixtures of liquids, solids and semi-solids and, as a result, the coefficient does not depend on the rate of shearing. We do not find paint, milk or confectionary cream on the list.

 

The units of viscosity.

There is a group of liquids and gases for which the viscosity is independent of velocity gradient and, for practical purposes, dependent only on temperature. The group includes most of the common mobile liquids and most of the common gases, which from the point of view of engineers, is very convenient. Liquids and gases in this group are called Newtonian fluids. By implication there are non-Newtonian substances typified by milk, paint, toothpaste and egg white.

 

Viscosity has units and these can be deduced from Newton's original definition in any given system of units. Unfortunately for engineers, systems of units have cycles of acceptability like fashion and as a result useful data is stored in at least three consistent systems and in many inconsistent systems as well.

 

If we start with the current fashion for S.I. we can put each term in kilograms, metres and seconds :-

 

 can be rearranged to give  which, when written in basic units, gives m the units of

                                                   .

which rearranges to give m the units of  in the S.I system. It can also be expressed in . Neither unit has a name.

 

In the c.g.s. system the units are gram/cm.sec and 1 gram/cm.sec is called a poise and frequently viscosities are quoted in centipoise to give easily remembered numbers like 1 centipoise as the viscosity of water. The two Imperial systems give units of pound/foot.sec and slug/foot.sec neither of which have names.

 

Measuring viscosity

The Hagen-Poiseuille expression is used as the basis for measuring the viscosity of liquids and gases. If a straight pipe of uniform diameter can be set up horizontally, and a liquid or gas made to flow through it, the pressure drop and the rate of flow can be measured and, from this, the value of the coefficient of viscosity can be deduced. Any attempt to do this will show that the coefficient of viscosity varies with temperature and a further constraint must be imposed on the method of measuring the coefficient of viscosity; the measurement must be made at uniform temperature. As laminar flow necessarily  involves internal heating, the flow can only be at constant temperature if it is slow with a small pressure drop and undertaken in an environment that is temperature controlled. Then, for liquids and gases for which m is independent of the velocity gradient it is possible to measure m. We, as engineers, must accept that physicists are quite capable of doing this and of producing accurate values of viscosity of all the common liquids and gases. We should note that laminar flow at constant temperature is a very special case and normally, especially in oil hydraulics, the heating is significant and, if we choose to use the Poiseuille expression to deal with such a flow, we should expect it to become inaccurate in such cases and be prepared to create a computer model that allows for the heating.

 

The method outlined above is regarded as a primary method of measuring viscosity and, without doubt, it is a method for use in a laboratory.  One must contemplate the position of engineers in respect of viscosity.

 

Viscometers.

The proper place for making measurements of viscosity that are to be used in evaluating non-dimensional groups is in a physics laboratory using primary methods. This would not be done in the normal day-to-day business of engineering.

 

The main fluids that are moved about by engineers are probably water, oil, beer and milk in that order and water and oil dominate. The movement of oil is the one that causes the engineer to become involved with viscosity. The oil that is extracted from the Earth’s crust varies in its composition. It is made up of all sorts of inflammable compounds of hydrogen and carbon and, when they are separated, some are heavy and thick and some are light and runny and everything between. For use in motor-cars, whether petrol or diesel, crude oil is refined and the light fractions separated to become petrol or light diesel[9]. What is left after distillation is inflammable and therefore a fuel. It will still contain a mix of different compounds some of which are not very desirable in flue gas. It is also thick stodgy stuff that needs careful management if it is to be burnt in power stations or ships. It is also typical of tar oils.

 

The main problems for engineers can be illustrated by that of burning heavy oil in power stations. There the oil has to be kept hot enough to flow into pumps to shift it into and out of tankers whether they are road tankers or sea tankers and also for the rest of its journey to the boilers. Once delivered to storage tanks on site it must be kept warm usually by steam heating and then pumped through burners in the boilers. These burners atomise the oil, that is, they produce a continuous spray of finely divided oil that must evaporate before it can burn. The flame is so hot that, it can only be viewed through welding goggles, but even so it can be stratified by the different temperatures at which the several fractions evaporate. Burning this oil so that the combustion products are least harmful and the combustion most efficient needs careful control of the flame by controlling the inlet conditions to the burner. Atomisation depends on the high pressure at the burner, the mobility of the oil and I suppose other properties of the oil like film strength and surface tension. It is not just about viscosity and even if it were to be there is still the problem of its composition of many fractions. I think that the flame is mainly monitored by just looking at it.

Text Box:  
Figure 6-5

From the above it is clear that engineers want to know about the state of the oil as it goes through its various stages before burning and, whilst it might be interesting to have an accurate value for the variation of the viscosity of the oil with temperature, some hands-on method of finding a working value of viscosity is much more desirable at the final point of injection at the burner. There are various viscometers available and I can make a assessment of their suitability for engineering use.

 

The Ostwald viscometer

I must start with the Ostwald viscometer because it is the one that might just escape from the laboratory. The following (In blue.) is what I wrote for a laboratory exercise for students in 1984. I make no apologies about including it because it tells us how empirical sciences grow. I have no time to bring it up to date and, for my purpose, it would be pointless.

 

British Standard 188:1977

 

This standard is the latest (In 1984.) in a series that have been published in 1923, 1929, 1937, 1957, and now 1977. and the original reason for the publication was to encourage the use of CGS units of viscosity rather than the arbitrary units such as Redwood seconds and Saybolt seconds.

 

The standard introduced by the BSI is shown in figure 6-5 with its standardised dimensions.

 

In 1957 there was a range of 8 viscometers to measure the viscosities of most liquids. However between 1957 and 1977 manufacturers of viscometers produced their own ranges with detailed differences in design and, in 1977 BSI removed the length constraints and approved commercial designs.

 

In 1977 BSI changed to SI units of viscosity and also introduced a new definition for coefficient of viscosity and introduced ideas of shear stress and rate of shear to take the concept of viscosity as a property a further step away from Newton’s concept.

 

The standard lays down test procedures for measurement of viscosity and these are more suited to a physics laboratory than to a power station.

 

The standard says “ the principle of the test is the measurement of the time taken for a reproducible volume of liquid to flow through a capillary viscometer under an accurately reproducible head and at a closely controlled temperature. The kinematic viscosity (  ) is then calculated from the measured flow time and the calibration factor of the viscometer.”

 

In use a viscometer is filled with the test sample to just above level G. It is then mounted in a bath of water that is gently stirred and heated under the control of an accurate thermostat. When the temperature is steady at some desired value the level of the sample is adjusted to be at G by extracting some of the sample. The sample is then sucked up tube 2 until the level is above E. Then the time is taken for the level to fall from E to F.

 

Then the kinematic viscosity is found from an expression of the form:-

                          where  is the elapsed time and  and  are calibration constants.

 

Look on the internet and you will see that BSI have not displaced Redwood seconds and Saybolt seconds as practical measurements of viscosity for use by engineers. In my experience the BSI instrument is bound to fail. Their viscometer is much more a device for use in a laboratory and once it goes into a lab bureaucratic paperwork and procedural delays make it useless for the engineer who could well have a whole tank full of oil to burn efficiently in a boiler now, not in a month’s time. Redwood viscometers and Saybolt viscometers have been reworked to make them more “user-friendly” and are still very much in use.

 

It is hard to see where BSI was going when it sought to alter the definition of viscosity. But the mindset of BSI is that of the physicist and not that of the engineer just as the NPL was when SI was introduced. They seem not to be interested in how engineers, or anyone else for that matter, go about their work.

 

The Redwood viscometer.

In the late nineteenth century three secondary viscometers came to prominence, the Redwood viscometer in Great Britain, the Saybolt viscometer in America and the Engler viscometer in Europe. They all depended on the same method of performing a “time-of-discharge” test on a sample where the time depended on laminar flow through a short pipe.

 

The sample of oil to be tested is held in a pot that is filled to a set point and, when the temperature of the sample is at some desired value, the sample is allowed to drain from the pot through the short pipe until a set volume has drained away. The kinematic viscosity, if that is what is measured, is calculated from the elapsed time for the draining.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The Redwood viscometer was made[10] by Sir Boverton Redwood in about 1880. I looked for an authentic drawing of a Redwood viscometer but the important details seem to have changed in with successive re-drafting for textbooks. Lewitt’s book of 1943 shows in figure 6-6 the design of the viscometer as it was somewhere between 1923 and 1943 and I am inclined to think that this is correct. I give a copy of Lewitt’s diagram in figure 6-6 and I have drawn the essential central pot to a larger scale in figure 6-7. The brass pot that has a height of about 90mm or 3.5² and an internal diameter of  46.5mm, that is, the inside diameter of 2² tubing having a 14 swg wall thickness. The bottom of the pot is clearly a turned insert. The all-important feature is the agate insert. It has a bore of 1/16² or 1.62mm and is 1/2² or 12.5 mm long. This is an Imperial size but it was a volume of 50 cc that was allowed to drain out.[11] The unlikely detail is the creation of a spherical seat at entry to the pipe so that it might be sealed with a ball on the end of a wire. I think that we would avoid this shape of entry to the pipe nowadays. The pot is fitted with a hook gauge to give an accurate position for the upper surface of the sample before carrying out the test. The level drops by 30mm during the test to drain away 50cc of the sample. This is measured with the small graduated flask under the hole.

 

The pot is surrounded by a water bath that is electrically heated by coils round the bath. The water is manually stirred with the paddles round the pot. Thermometers in the sample and in the water permit the adjustment of temperature. The test is started by lifting the ball off its seat but I do not know where it is placed during the test, perhaps it just rests in the groove in the base plate. Wherever it goes its position is part of the test procedure. The Redwood viscometer is still in regular use as are the others and I need to offer reasons for this to be the case.

 

There can be no dispute about the fact that if you carry out the same test with the same oil in identical equipment you will get the same result.[12] This means that, if you want to buy fuel for a power station and a supplier offers the 10,000 tons you need with a Redwood number, when tested at a set temperature, of say 1,200 seconds and you know that your equipment can burn fuel with this Redwood number, there is no physical reason not to buy if the price is agreeable to you both. The seller knows that when the fuel arrives you can test a sample long before very much of it has entered your storage tanks. Here we have the legal and practical basis for a transaction. Nowhere in this is viscosity even mentioned, only the Redwood seconds because, in the end, the final adjustment that ensures that the fuel is burnt as well as possible is all about looking at the flame just as a plumber does to adjust his gas torch. The Redwood viscometer that makes this transaction possible is a simple piece of equipment that is very suited to the task for which it was designed and this is why it, and the other two viscometers, will go on being used. There are two versions of this viscometer and the second is designed for heavy oils.

 

The Redwood viscometer is capable of being reproduced very accurately and however much one might question the use of a ball with its spherical seat that produces such an unlikely entry to the short pipe it cannot be changed now and there is no undisputed case to do so.

 

So what does the Redwood viscometer actually measure?

 

If the energy equation is applied to the viscometer at some instant during a test when the difference in level between the free surface and the exit from the short pipe is  we get:-

                .As  and  are both atmospheric pressure,  is zero and  = , this reduces to

                   .

The loss is the friction lost in laminar flow in the short pipe. There is little option but to use the Pouseuille expression and put:-

                           The loss between 1 and 2  where  and  are the diameter and length of the short pipe,  and  are the viscosity and density of the sample. Then:-

                                                                

This expression tells us that the Redwood viscometer will only work if the kinetic energy is very small when compared with the loss in the pipe. So, if the expression is reduced to

                                                         we can examine it.

First it can be rearranged to give             where  is a constant that can be calculated from the dimensions of the viscometer and .

Now, in a short time  a volume of the sample  will flow from the viscometer and cause a drop in level of  where  are the diameters of the pipe and the pot.

Substituting for  and re-arranging we get:-

                                                         which can be integrated between 0 and T and the start and finish values of h. The result of this is that the time for 50 cc of sample to drain from the pot is given by T = a constant .

 

So the Redwood viscometer does not give us a value for the viscosity but for  which crops up so often that it is given a special name, the kinematic viscosity and denoted . It now has units of metres2/second where it used to have units of centistokes with units of centimetres2 /second. To change from centistokes to SI units divide by 10,000.

 

So the Redwood viscometer has the potential to measure kinematic viscosity but, of course, it all depends on whether we were justified in using the Pouseuille expression to relate the time of discharge and the kinematic viscosity. We cannot know from argument but it is evident that the pipe is too short for the shape at entry to have no effect so we must turn to the actual performance of the viscometer.

Text Box:  
Graph 6-1

In graph 6-1 I have plotted the performance of a Redwood number 1 viscometer when tested using fluids of known viscosity at 21°C. I think that most engineers would be delighted to have any basic data with this sort of accuracy. The graphs for the viscometer at 60°C and 93°C show little more error.

 

I was trained to denigrate these secondary viscometers as being very poor compared with U tube viscometers. I regret passing this view on to others when lecturing. This text gives me a chance to offer a more reasoned view.

 

 

 

Other viscometers

Stokes showed that the force exerted on a sphere moving with laminar flow of fluid round it can be given by                      where .

 

If ball, e.g. a steel bearing ball, falls freely through a liquid that is expansive compared with the size of the ball it will reach its terminal velocity and if that velocity is measured the value of the viscosity can be deduced from it. This is the principle of the falling ball viscometer.

 

There is a case for continuously monitoring the viscosity of fuel oil just before it enters the burners of a boiler. It could then become part of an automatic control system. Then a cylinder rotating inside a sleeve would give a suitable element to form part of a transducer. It could be a rotating disc.

 

There used to be a falling ball viscometer that was really portable. It was in two parts, a steel ball about 1² in diameter and a steel cup attached to a handle. The cup had the same radius as the ball but it had three high spots that created a small clearance between the ball and cup when one was fitted inside the other. In use the ball was immersed in a sample of oil and the cup, on its handle, was placed over the ball under the oil and left for the ball and cup to come into contact when sufficient oil had been squeezed out.  The clearance was then full of oil. The viscometer was then lifted and the time taken for the ball to fall out of the cup was measured to give a useful estimate of the viscosity.

 

 

Endnote

The fact that viscosity is measured from laminar flow in a pipe does not mean that it has no significance in other modes of flow. The fact is that the use of laminar flow is just a means to quantify a property of the molecular structure of the liquid and is not dependent on the way that it is measured. Of course the viscosity will be important in many practical applications in engineering like bearings and oil hydraulics but it also has an important role in creating storage and retrieval systems in our empirical science and a pivotal role in computer fluid dynamics.

 

Scroll down for exercises

 

 

 



[1] Not much has changed. Those bridge builders of long ago used an intuitive approach to their designs. It took a long time for the pier to become rounded. Still, when someone attempts to design a leading edge for a keel or a bow for a boat the first thought is to make it sharp. It is sad.

[2]I have in the past and will again in this text.

[3] Sailors know this. They call it gusting, veering and backing because that is what it feels like to them.

[4] I cannot now remember when I first saw a Hele-Shaw apparatus but I did have one made when I was teaching. I learnt a lot from it but, in retrospect, not as much as I might have done. No other lecturer took any interest in it partly I suppose because not all lecturers in engineering turn out to be good with their hands.

[5] The gases surrounding the Earth are located by the surface of the Earth and by gravity. If a gas under pressure is suddenly free from its constraining surfaces it cannot sustain its pressure except during the short period when it is accelerating as part of a general mass of gas.

[6] This corresponds to evaporation.

[7]Of course no one then knew whether this was possible and it was not until 1896 that Osborne Reynolds demonstrated the existence of this mode of flow

[8] Engineers do not generally work in radii but it is convenient here.

[9] Generally slow speed engines can burn heavy diesel oil and ships use very thick oil and lorries something in between.

[10] It is not clear whether he designed it.

[11] Engineers still use a mix of units. I found a model engine that used ball races of metric and imperial sizes on the crankshaft. It is nor a problem.

[12] Science depends on it.