Chapter 8.The loss of head in turbulent flow through pipe fittings.
Introduction
In chapter 3 I concluded with the statement that, for steady, one-dimensional flow, the total energy/unit weight at point 1 in the flow equals the total energy/unit weight at point 2 down stream of 1 plus the energy/unit weight that has been lost between 1 and 2. This can be written in symbols:-
+ the loss between 1 and 2.
Now I have to extend this idea to the point where the loss between 1 and 2 can be evaluated for a pipe system.
I find that it is difficult to know how to start explaining the behaviour of pipes so that it is a logical progression because, of course, it did not evolve logically. It seems to me that I have to start by anticipating the outcome and observe that this empirical science has developed by evaluating each loss in the pipe as some fraction of the kinetic energy per unit weight in the flow. Then I can look at various pipe-fittings and see how the loss in incurred and put a magnitude to it.
We already have, from chapter 7, a way of finding the loss of head in steady flow in a pipe. Now we want ways of taking into account other losses due to bends, changes in section, valves and so on.
In chapter 3 we showed how the total head equation for a flow line could be adapted to the steady flow of a fluid through a pipe. We first put the total energy per unit weight at point 1 equal to the total energy head at point 2 plus the loss between 1 and 2. We then put the potential energy equal to its value on the axis of the pipe, regarded the pressure as uniform across every section, and let the kinetic energy head at any section equal to . Then we had to find a way to evaluate the loss of head to friction between points in the pipe.
In chapter 3 we also noted that pipes were normally constructed of lengths of straight pipe coupled by various fittings. We noted that this meant that we would require expressions for head loss in straight pipes in forms that can be used directly with the energy equation. We said nothing about the loss of head in the fittings. We must now adapt the total head equation still further to allow for these fittings. In order to do so we must consider the reasons for the loss of energy to friction in pipe fittings, find ways of measuring the losses and of storing the resulting data. We shall find that there is little justification for attempting to find expressions for the loss of head in pipe fittings when the flow is laminar so this chapter will be limited to the case of turbulent flow.
Losses in fittings.
Pipe work is commonly made up from lengths of straight pipe. Only when some special requirement has to be met are pipes bent. Clearly some system of making joints is required and these joints have to serve two functions. The first is to make a leak proof joint and the second is to provide some mechanical strength between lengths of pipe.
Suitable joints may be made by welding for steel and plastic. Steel pipe may also be joined by cutting screw threads on the ends of the pipe and screwing the pipe into a screwed sleeve. Copper and brass pipes are frequently joined by sliding the ends of the pipes into standard fittings and running solder into the gap between pipe and fitting so providing a seal and mechanical strength. Copper and brass pipes are also joined with compression fittings in which a soft metal ring is squeezed against the wall of the pipe by tightening a suitably shaped screwed nut on to the fitting. There are special joints for plastic plumbing pipes that are push-to-fit, need no jointing compound and can be dismantled very easily Flanged joints are very commonly used on middle and large size pipes. They may be cast with the pipe or welded to it if it is steel. There are other systems of making joints in pipes for use underground, for pipes carrying fluid at high pressure and for very large pipes. Cast pipes are often joined with spigot and socket ends rather like rainwater fittings on houses. Often underground pipes are lined with bitumen.
Here we are interested in the joints that cause a loss that is greater than that which would occur if the pipe were to be continuous and without a joint. Clearly straight joints that produce an effectively continuous surface do not add to the loss and it should be the aim of the designer of the joint and of the person who makes the joint to make a smooth joint. In practice, unless special care is taken, the many joints in a pipe system vary widely in standard and there is little point in trying to quantify the loss in a typical straight connector. The fittings that cause a loss that can be quantified, are those in which a change in direction or a change in area of cross-section of the flow occurs. These are in bends, elbows and tees and in reducers (for changes in diameter up or down), sharp entries, fully open valves and partially open valves.
The causes of losses in fittings.
The common feature in the flow through these devices is that, at some point, the flow breaks down to eddy in a way that causes a greater loss than would have occurred in the ordinary laminar or turbulent flow. In order to see what leads to the breakdown in the flow we must consider each device separately.
I drew
the diagrams in figures 8-1a to 8-1f 20 years ago and I am not really satisfied
with them because
they do not show the region of eddying sufficiently accurately but I cannot, with any confidence improve them. Fortunately for me we are not really interested in the flow pattern beyond knowing qualitatively how the loss is caused.
Figures 8-1a, b, c, d, e, and f represent the typical fittings. The diagrams have been drawn with square corners for two reasons. The first is that these represent the worst case so that any data for such fittings would overstate the loss. The second is that these shapes can be described in the fewest dimensions and words. Typically the sudden enlargement shown in figure 8-1d can be described in terms of the diameters of the two pipes and perhaps the word "square". A real enlargement would in fact be a transition piece and be profiled with two radii if only to facilitate manufacture. These radii improve the flow pattern in the transition and so the loss caused by the real fitting would be less than that caused by the sudden enlargement.
In looking at the diagrams of figures 8-1 it must be remembered that the approach flow would normally contain fine grain turbulence and the flow lines depicted would not be shown up by injecting dye. However the lines do represent the mean paths of the liquid and in each case show how the shapes of the solid boundaries lead to the breakdown of flow.
In the cases of the sharp-edged entry to a pipe, the sudden contraction and the mitre bend, the flow lines separate at the corners, over-contract, and then, breakdown occurs when the liquid expands again to fill the pipe. The random eddies decay into fine grain turbulence as the liquid continues along the pipe until the former turbulence level is re-established.
When liquid flows from a pipe into a reservoir a secondary flow is entrained by the emerging stream and the two flows mix as indicated in figure 8-1b. Ultimately the mixing flows spread out into the main body of liquid and the kinetic energy that the liquid has as it leaves the pipe is dissipated to increase the stock of internal energy. The flow through a sudden enlargement is clearly allied to the flow into a reservoir with the larger pipe suppressing the secondary flow but some of the eddying flow moves back upstream into the corner to circulate back into the flow. The function of a partially open valve is the control of the rate of flow through a pipe. This is done by creating a controllable loss and the dissipation of energy into the structure of the fluid. The valve is really a sudden contraction followed by a sudden enlargement with the throat area being variable. Figure 8-1f shows the section through a partially open gate valve. The main loss occurs downstream of the valve. This can be so intense that valves have to be designed to control the flow during the divergent phase just in order for the valve to have a reasonable working life.
Measurement of loss in fittings.
It will
be evident from the diagrams that the main loss occurs downstream of the
fitting and may affect the flow for a length of pipe equal to several pipe
diameters but some loss will occur upstream of the fitting as well. This means
that the loss in a fitting takes place in the pipe and not in the fitting. Many
have been tempted to investigate the character of these various flow patterns
but that is not the best route. All it does is complicate things unnecessarily.
The quality of the data available for evaluating the loss in the pipe caused by
friction is not good enough to justify having any data of quality for the
losses in fittings.
The method used to create this part of our empirical science is to regard the loss at any given fitting as occurring at the fitting and not in the adjacent pipe-work. Then the loss is found by experiment. The methods used are quite straightforward. Let us consider a length of pipe that is set up horizontally and through which fluid flows at a steady rate. We would expect from the work of chapter 7 that the total head would fall uniformly along the pipe as is shown in figure 8-2a. Now suppose that a fitting is introduced at the mid-point of the pipe and the rate of flow kept the same. As I have explained, we should now expect the total head to fall uniformly as before up to the vicinity of the fitting, for there to be some departure from this uniform loss as the fluid approaches and leaves the fitting, and then, a return to the uniform loss in the second part of the pipe. Figure 8-2b shows the pipe with its fitting and the lines of total head for the lengths of the two sections of the pipe for which the loss of head is uniform. No attempt has been made to show the change of total head as the fluid flows up to, through the fitting, and on into the second part of the pipe because, in this context, it is of no interest.
It will be evident from figure 8-2b that the two total head lines could be extended to the plane of the fitting to give a step change as shown in figure 8-2c. This step change is clearly the additional loss in head caused by the fitting however it may in fact have occurred. If we choose to call this step change the loss of head in the fitting we could experiment to find its value. Then we could find values for the whole range of fittings which we might use, and store this data in some storage and retrieval system. Then, in a real pipe system, we could use Darcy and Moody to find the loss in the pipes and use our data on fittings to find the loss in the fittings and simply sum these losses. It is a simple strategy and, in my experience, works well.
This simple method can only be implemented if we can find a way of measuring the total head at any section in a pipe. There are no ways of making a direct measurement that does not involve an unacceptable interference with the flow pattern. However we can measure the pressure at a given section of a pipe with good accuracy and, from such a measurement, we could derive a value for the pressure head. As we have already decided to put the kinetic head equal to , where , the kinetic head can be found from measurements of the rate of flow and the diameter. As the pipe is horizontal its axis can be used as the datum for potential energy. So a plot of total head is possible.
The
method can be illustrated with the system shown in Figure 8-3. In this case the
fitting is an enlargement from one pipe diameter to one of a larger diameter.
The enlargement is used to join two long pipes that are set up horizontally.
Pressure is measured, by pressure gauge or by piezometer tube as is
appropriate, at suitable positions along both pipes but not in the vicinity of
the enlargement. Sufficient tapping points must be provided for it to be
possible to pick out with confidence the gradient of the line of total head.
The variation of pressure head and of total head is shown in figure 8-3. For
some real test rig a plot could be made of the lines of total head and the
magnitude of the step giving the additional loss in head caused by the
enlargement could be determined.
A method of this sort could be used for any of the fittings shown in figure 8-1.
Storage of data on the loss of energy in fittings.
It has already been noted that, for practical pipes, the loss in head in the fittings is usually small compared with that in the pipe. We know that, by the nature of the Moody diagram, the order of accuracy of the figure for will not be sufficient to justify high accuracy for data on the loss in the fittings or for any unnecessary complexity in the form of any empirical expressions we might devise. It is more important that the data should be stored in a way that fits easily into the total head equation. The most simple system is to express the loss equal to a fraction of the kinetic head, that is to write :-
loss of head in the fitting = , where is a coefficient and is the velocity in the section of pipe in which the loss mainly occurs.
Using this system typical values are :-
|
the loss at entry to a pipe |
figure 8-1a |
k = 0.5 |
|
the loss at exit from a pipe |
figure 8-1b |
k = 1 |
|
the loss at a mitre bend |
figure 8-1e |
k = 0.5 - 0.75 |
|
the loss at a fully open gate valve |
figure 8-1f |
k = 0.25 |
and in each case the velocity is that in the pipe.
Other fittings are not treated in this simple way. The reducer fitting which is used to join pipes of different diameters may be used at an increase in diameter or at a reduction in diameter. The only difference is the direction of flow. The loss will depend on the two diameters as well as the direction of flow. The mitre bend produces a much greater loss than a bend with a large radius, that is, a swept bend and the loss will depend on the diameter of the pipe and the radius of the bend. Clearly expressions that take account of these dimensions will be desirable provided that they are not too complicated to use.
The sudden enlargement
The expression that is commonly used for loss in a fitting that is used as an enlargement is interesting because of the way in which the total head equation and the momentum equation are used to derive it in a form that involves measurable quantities and on which we can base a coefficient.
The
variety of shapes of the enlargement is too great to justify comprehensive
experiment so the sudden enlargement shown in figure 8-1d is taken to be the
worst case. The transition is abrupt
from one diameter to the other so that a flat annulus is created. An expression
for the loss is required which must involve the two diameters and a
coefficient. Figure 8-4 shows an enlargement joining two pipes A and B of
substantial length and two sections 1 and 2 in the pipes that are remote
from the enlargement. If we regard the flow as being one dimensional we can
apply the total head equation to 1 and 2 and write :-
+ the loss between 1 and 2. If we follow our previous decision on how to deal with losses in fittings we can act as if the loss occurred at the enlargement and put :-
the loss between 1 and 2 = the loss in pipe A + the loss in enlargement + the loss in pipe B and find the pipe losses using Darcy.
Then if we put the loss in pipe A = and the loss in pipe B = the total head equation can be rearranged to give:-
the loss in the enlargement =
Now we can apply the momentum equation to the liquid between 1 and 2 and say that the net force exerted on the liquid as it flows from 1 to 2 equals the increase in momentum per second between 1 and 2. The net force will be made up of the pressure forces acting on areas and , the force exerted by the annulus and the friction drag exerted by the walls of the two lengths of pipe.
The friction drag on the walls has to be overcome by a pressure difference which, in pipe A is equal to , and in B is equal to . Then :
,
We have to decide what to do about the force on the annulus. There will be a pressure on the annulus but there is no reason to suppose that the pressure is uniform. Nevertheless the best result comes from supposing the pressure on the annulus to be uniform and equal to and then the force on the annulus is equal to and,
which, on dividing by and simplifying, gives :
But , and and it follows that
Putting this into the expression for the loss gives :-
Loss at enlargement = , which reduces to :
The loss at a sudden enlargement =
This is an attractively simple expression which can be rewritten in the form :-
The loss at a sudden enlargement = which is, of course, all in measurable quantities.
It is worth reflecting on this expression. No one can pretend that it is the result of a rigorous analysis. The decision that was taken about the force on the annulus is obviously questionable but the expression that follows from that decision is simple and turns out to be useful. Had this not been the case the expression would have disappeared and the decision about the annulus along with it. The expression follows from the same decisions as we have already taken for dealing with pipe losses so we can expect it to have the same validity as the rest of our expressions.
Then we can put :
the loss at a sudden enlargement = , where is a coefficient. For most purposes it is adequate to put = 1. On such methods is an empirical science constructed.
Real transition fittings are shaped for ease of manufacture and do not have a sudden change in diameter and the loss in such fittings when used as enlargers is less than that in a sudden enlargement. In order to take account of the better flow pattern in the real fitting, a reduced value of k is used. [1]
Sudden contraction
Where a liquid flows through a sudden contraction as shown in figure 8-1c the loss occurs almost wholly in the small pipe. The liquid in the large pipe converges smoothly as it approaches the sudden change in diameter and continues to converge for a distance of about before it starts to diverge again. The flow pattern breaks down into intense eddying and the main loss occurs in this eddying. As the loss occurs in the small pipe the expression for the loss will be in terms of and, as the contraction depends on the two diameters, will depend in some way on the ratio of the diameters or the ratio of the areas. Then, if the loss is given by , useful values of k are given in the table below.
|
0.3 |
0.4 |
0.5 |
0.6 |
0.7 |
0.8 |
0.9 |
0.95 |
|
|
0.09 |
0.16 |
0.25 |
0.36 |
0.49 |
0.64 |
0.81 |
0.9 |
|
|
0.36 |
0.35 |
0.32 |
0.27 |
0.22 |
0.14 |
0.05 |
0.015 |
A real fitting will have a shape that will cause less loss than the sudden contraction and so the figures above should be regarded as worst case values.
In figure 8-1e I showed that the loss at a mitre bend results from the contraction and subsequent divergence. When the bend is swept ie has a radius and the radius of a swept bend is greater than about 2.5 times the pipe diameter the contraction does not occur. The flow then exhibits the characteristics of free vortex flow in that the velocity on the inside of the bend is greater than that on the outside. The transition to this flow starts upstream of the bend and the transition back to normal flow takes place downstream of the bend. The loss is attributed to this double change in flow pattern.
The flow through such a bend is unpredictable even if the effects of the connections are suppressed in some way and generally it is found to be adequate to put the loss in a swept bend where the radius of the bend is greater than equal to .
Allowances for losses in other fittings can be made by choosing figures that are based on similar geometry to the figures given above. For example the loss in a tee used as a bend could be put equal to that for a mitre bend.
Valves
There are valves for every conceivable purpose in fluid flow. Ultimately they all depend on creating a restriction in which kinetic energy is created and subsequently lost into the molecular structure of the flowing fluid. Generally no special arrangements are needed to “lose” the kinetic energy but, for some hydroelectric applications, the energy to be dissipated is so great that valves have to be designed to dissipate it in a controlled manner. Where a valve controls the flow in a long pipe the inertia of the long column of fluid flowing along the pipe can lead to destructive pressures if the flow is stopped suddenly and great care has to be taken to control these pressures.
Figures
8-5, 6 and 7 seven show the principles of the three most commonly used valves.
I have omitted all the mechanical details that are required to make these into
practical devices. Figure 8-5 is a gate valve and is really the adaptation to a
circular pipe of the square paddle used in lock gates. The gate is circular,
has a small taper from top to bottom to fit snugly into faced matching seats.
The hole through which the fluid flows is a crescent and there is a complex
relationship between the area of the hole and the position of the disc.
However, when it is fully open it offers little resistance to flow. It is often
used in as an isolating valve in water mains.
Figure 8-6 is the common globe valve used where the rate of flow is to be controlled. The restriction is between the disc-shaped jumper and the seat. Clearly the path through the valve is never unrestricted and it will always offer a resistance to flow even when fully open. The valve can be adapted by reducing the size of the hole in the seat and changing the disc for a tapered needle to give a very sensitive valve.
The plug valve is used very extensively for control of the flow of gas. It is simple and has a good seal. The blending of the circular pipe into the rectangular hole in the plug reduces the loss when the valve is open and is necessary anyway. It can be closed very quickly and needs to be used with care.
The variations on these basic designs are endless.
The valve is not a device that
works in isolation and often it is either somewhere along or at the end of a
pipe. An important pipe system is illustrated in figure 8-6. It is just a pipe
supplied with water from a 
tank
and fitted with a valve at its end. That valve is there to control the rate of
flow from the pipe. Usually the valve is screw operated but it may be a plug
valve. As explained above whatever type of valve is used it works by creating a
hole of adjustable size through which the water flows. As the area at the
restriction is reduced the speed and hence the kinetic energy of the water
leaving the restriction, increases from that in the supply pipe and this
kinetic energy comes from the energy head on the system. As a result the energy
available to overcome friction falls and the speed of the flow in the pipe
falls, As the valve is progressively closed there is a switch from all the
energy going to overcome friction in the pipe to all the energy going to
produce a small jet of water moving at high speed with almost no speed in the
pipe. There must be a relationship between the area of the restriction and the
flow. A glance at the three valves shows that the flows are complicated and
different for each one. This is too complex for analysis. But, if minor losses
are ignored, and we say that, at any valve position, the total head on the
system is used either in overcoming friction or in creating the kinetic energy
of the water flowing through the restriction a useful relationship is possible.
It seemed to me that the most likely way to make progress is to define an area
ratio A for the partly open valve. A is the ratio of the area at the
restriction divided by the area of the pipe. Then the velocity at the
restriction where is the velocity in the pipe.
Then
From this and
This can now be converted to a graph of flow versus area ratio using Mathcad.
I had never seen graph 8-4 before I plotted it for this text and at first I was surprised by its shape. I plotted initially first for one length of pipe actually 70 metres and then realised that I needed a family of graphs for several lengths. Clearly all this family of graphs are tangential to one line and that line is for no pipe at all.
The line in black shows how the flow changes with valve opening when there is no pipe and it is what one might expect for a variable orifice attached to the tank. But, once a pipe exists between the tank and the nozzle, the growth of the friction loss with flow starts to reduce the energy available to create kinetic energy to give this characteristic shape.
The valve has no characteristic behaviour as such, its behaviour is closely linked to the pipe system in which it is used. You can check this graph by using your own outside tap. All that matters takes place in the first turn.
The diffuser
There are applications in which it is very desirable to recover kinetic energy as a rise in pressure. Probably the classic example is the draught tube that is used on low-head, water turbines. In those turbines there is an incentive to keep the size of the turbine down and this can only be achieved at the expense of having a higher than desirable velocity at exit. A diffuser that starts off circular in section and changes to square or elliptical and turns through 90° as well can recover the inevitable kinetic energy as a rise in pressure that converts to a reduced pressure at exit from the turbine. Another application is the transition that must act as a diffuser between the square working section of a wind tunnel and the inlet to the fan. It is often a make or break component of the tunnel. Get it wrong and the performance is impaired.
We have
already met a diffuser in connection with the Venturi meter. There the diffuser
is straight and just a cone with an included angle of about 6°. In a diffuser the object is
to convert kinetic energy to pressure energy with the minimum loss. If the flow
were to be like that in a parallel pipe the loss would be attributable to friction
and could be calculated using Darcy and integration. Clearly the longer the
length of divergent pipe that contains high speed flow the greater the loss
will be over that that which would occur in the same length of parallel pipe
carrying the same flow. But the flow in the diffuser is not like that in a
parallel pipe. We know that when a stream of water issues from a pipe into a
reservoir the stream flows straight on and entrains water from around it. If we
fitted a conical divergence to the outlet to the pipe the entrainment would be
stopped and what then happens depends on the angle of the cone. Figures 8-7 and
8-8 show in principle what happens. In 8-7 the stream of water retains its
basic form but creates eddies around it as shown. These eddies extend into the
down-stream pipe where most of the kinetic energy is lost. At a smaller
included angle the pattern changes and the stream changes to remain attached to
the wall on one side and to create a large eddy on the other[2].
Again the loss of kinetic energy takes place in the large pipe. It is hard to
see any reason to regard either of these transitions as diffusers. We need to
concentrate on diffusers in which the flow is likely to be attached all round
and over the whole length.
I started looking at this diffuser several days ago and it seems to get more and more complicated as the days go on. For a start this simple frustum of a cone does not produce a constant retardation of the flow. In the end I chose to avoid the complications and turned to Mathcad to have look at a few significant numbers. The programme and its outcome is shown in calculation 8-1. I chose an included angle of 6°, a flow of 0.022 m3/s because it produced velocities in the range 0 to 3 m/s. This is a frustum and it is necessary to calculate from the notional apex. I worked with lengths from 0.4 m and 1 m from the apex. This gave realistic figures. The first graph is of the diameter and that changes from about 42 mm to about 100 mm in a length of 600 mm.
The next graph is of velocity on the basis on one-dimensional flow and the range of velocity is from about 16 m/s to about 2.5 m/s and these are practical figures. Then the third graph gives rate of change of velocity with distance along the diffuser.
The graph of rate of change of velocity with
distance along the diffuser shows that the velocity is changing most quickly at
inlet. For this retardation to exist there must be a pressure gradient acting
against the flow and this may not come into existence. However it is clear that
diffusers do have attached turbulent flow and we must conclude that if
detachment is to occur it will be near the inlet. It may be that a trumpet
shaped diffuser would be best. It also tells us that this is a complicated flow
pattern and that the divergence is nothing like as predictable as a convergence
where the loss is usually very small.
From an engineering point of view this diffuser is going to be difficult to design and it would be prudent to look at existing designs if you are ever called upon to design one. Be prepared for rules of thumb like “use sweeping curves that are mathematically continuous”.
There have been experiments on the performance of simple conical diffusers and the maximum value for included angle for efficient performance is about 16° and there is a well-defined minimum that stretches from about 4° to about 10° with its turning point at about 5°. However one must remember that the factors that influence engineering design may well make it unnecessary to do anything more than use a diffuser with an effective included angle of 6°.
As an aside, if you want to design a transition from square to circular find the diameter of a circle of area equal to that of the square and work out a length having a notional taper of 6°. It can be constructed from triangles and parts of a cone.
The relative magnitudes of pipe losses and fittings loss in practical pipe lines.
We now
have the Darcy expression to be used with the Moody diagram, and the
expressions for losses in fittings. We know that the Moody diagram must be used
on the basis that the figures are all good approximations. Now we have
expressions for losses that may well involve similar approximations. We need to
have some idea of the use of these expressions in practice. This involves making
some calculations for a typical pipeline by way of illustrative example.
Let us take as an example a simple system of a new copper pipe of 32 mm diameter connecting two tanks filled with water with a valve in the pipe to control the rate of flow between the tanks. The system is shown in figure 8-5 and it can be seen that there are four bends in the pipe and these can be taken to be swept bends. Let us suppose that the pipe has a total length of 70 m, that the velocity of water in the pipe is to be 3 m/s, and attempt to find the minimum difference in level between the tanks that would be required to maintain this velocity. We can apply the total head equation to points 1 and 2 in the free surfaces and write :
+ the loss between 1 and 2.
As and this reduces to :
= the loss between 1 and 2
The minimum loss between 1 and 2 is the sum of the loss in the pipe, the loss at entry to the pipe, the loss in four bends, the loss in the open valve and the loss at exit from the pipe. We have expressions for all these losses, so we can write :
The loss between 1 and 2 = .
Clearly we need a value for but we do not know Re or the relative roughness . However, for water, = 1000 kg/m3 and can be taken to be 0.001 kg m/s2 and then :
Re = . For new copper pipe is 0.0015 and then the equivalent sand grain roughness for the pipe = .
From these figures = using the Moody diagram. Then :
the loss between 1 and 2 = = =
In this calculation the suspect figure is that for . Had we used a higher value of of, say 0.002, the calculated value of the loss in the pipe would be 21.07m. The difference of 0.8m is more than a half of the calculated value of the other losses. There is nothing unusual in the outcome of these calculations and it is not surprising that engineers decide to ignore the loss caused by fittings when they see that the context of the calculation makes this justifiable. The loss in the fittings is then frequently referred to as a minor loss.
The characteristics of the flow in pipe systems.
The main characteristics of the flow in pipe systems can be illustrated if we choose to ignore the losses in the pipe fittings. When this decision is taken a different form of the Darcy expression may be more useful.
This is the form of the Darcy expression which involves instead of . We have chosen to put where . From this it follows that . If g is put equal to 9.81 to suit the S.I system this reduces to :-
Joules/Newton or just metres.
In practice this is used in the form below because it is easier to remember and because the available data is not of the same order of accuracy as the number 3.025.
Joules/Newton or metres
[It is pertinent to note that the original form of the Darcy expression is unusual in that the 4 is not cancelled with the 2. The obvious reason for the decision not to cancel the 4 and the 2 is in order to retain the kinetic energy/unit weight as an identifiable element of the expression. Unfortunately there is another form of this expression that is widely used in the United States of America. There, energy loss/unit weight, is put equal to . So we have to watch for values of friction factor, both denoted , with one being 4 times the other.]
If this pipe system is to be treated as a one-dimensional flow, liquid flowing in a pipe has kinetic energy that can be evaluated as . I think that it is worth plotting this kinetic energy term against velocity.
It is given in graph 8-1. It is an obvious sort of graph, just the graph of but it does give us the magnitudes of the velocity and the kinetic energy head for any liquid.
In fact it is not kinetic energy that concerns us but, if the losses to be inserted into the energy equation are to be evaluated as a fraction of the kinetic energy per unit weight of the liquid flowing in the pipe, it is useful to have some idea of the magnitude of the kinetic energy per unit weight. After all we already know that this kinetic energy is often lost at exit from a pipe simply because no provision is made to recover it and the obvious next step is to draw a graph of energy flow through a pipe of a given diameter in order to put a magnitude to this potential loss. The kinetic energy has units of Nm/N and if we multiply by weight flow we get power in watts, or if it were a loss, the rate of loss of energy in watts.
Potential
loss = where is the diameter of the pipe.
In graph 8-2 I have plotted this loss for a pipe of 25, 50 and 100 mm diameter.
However,
we have chosen to express all our losses in pipes as fractions of the kinetic
energy of the flow through the pipe. It follows that an interesting graph is
that of flow of kinetic energy per unit area of the pipe which is just the
graph of .
I think that this is one of the many graphs that crop up in this subject that
effectively set ranges to go and no go areas on graphs. I know that it is
obvious but I found that students often do not see graphs as anything other
than y versus x as if they were still plotting their first graph. This is a
graph of two real quantities and engineers want x to be as large as possible
and y to be as small as possible so that the losses are small. This graph says
that this is what you can have and you must choose. A balance has to be found
between velocity and possible power loss and the graph says that the loss at 1
m/s is about 1/8 of the loss at 2 m/s and 1/27 of that at 3 m/s and 1/64 of
that at 4 m/s. The balance is struck at about 3 m/s as a working maximum.
In my view it is important to get a “feel” for energy exchanges and the energy exchanges that take place on the free surfaces of water are the only ones that are visible. They need some idea of speed and as a guide small boats (25 feet long) travel at about 3 m/sec = 10 km/hr or 6 mph or the speed of water in pipes. Ducks make up to 2 mph or 1 m/s. Keep looking at them to find the true character of the flow.
See supplementary examples.