# For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then:

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1 Hydraulic Coefficient & Flow Measurements ELEMENTARY HYDRAULICS National Certificate in Technology (Civil Engineering) Chapter 3 1. Mass flow rate If we want to measure the rate at which water is flowing along a pipe. A very simple way of doing this is to catch all the water coming out of the pipe in a bucket over a fixed time period. Measuring the weight of the water in the bucket and dividing this by the time taken to collect this water gives a rate of accumulation of mass. This is know as the mass flow rate. For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then: Performing a similar calculation, if we know the mass flow is 1.7kg/s, how long will it take to fill a container with 8kg of fluid? 2. Volume flow rate - Discharge. More commonly we need to know the volume flow rate - this is more commonly know as discharge. (It is also commonly, but inaccurately, simply called flow rate). The symbol normally used for discharge is Q. The discharge is the volume of fluid flowing per unit time. Multiplying this by the density of the fluid gives us the mass flow rate. Consequently, if the density of the fluid in the above example is 850 then: 1

2 An important aside about units should be made here: As has already been stressed, we must always use a consistent set of units when applying values to equations. It would make sense therefore to always quote the values in this consistent set. This set of units will be the SI units. Unfortunately, and this is the case above, these actual practical values are very small or very large ( m 3 /s is very small). These numbers are difficult to imagine physically. In these cases it is useful to use derived units, and in the case above the useful derived unit is the litre. (1 litre = m 3 ). So the solution becomes. It is far easier to imagine 1 litre than m 3. Units must always be checked, and converted if necessary to a consistent set before using in an equation. 3. Discharge and mean velocity If we know the size of a pipe, and we know the discharge, we can deduce the mean velocity Discharge in a pipe If the area of cross section of the pipe at point X is A, and the mean velocity here is. During a time t, a cylinder of fluid will pass point X with a volume A t. The volume per unit time (the discharge) will thus be 2

3 So if the cross-section area, A, is and the discharge, Q is, then the mean velocity,, of the fluid is Note how carefully we have called this the mean velocity. This is because the velocity in the pipe is not constant across the cross section. Crossing the centerline of the pipe, the velocity is zero at the walls increasing to a maximum at the center then decreasing symmetrically to the other wall. This variation across the section is known as the velocity profile or distribution. A typical one is shown in the figure below. A typical velocity profile across a pipe This idea, that mean velocity multiplied by the area gives the discharge, applies to all situations - not just pipe flow. 4. Continuity Matter cannot be created or destroyed - (it is simply changed in to a different form of matter). This principle is known as the conservation of mass and we use it in the analysis of flowing fluids. The principle is applied to fixed volumes, known as control volumes (or surfaces), like that in the figure below: 3

4 An arbitrarily shaped control volume. For any control volume the principle of conservation of mass says Mass entering per unit time = Mass leaving per unit time + Increase of mass in the control volume per unit time For steady flow there is no increase in the mass within the control volume, so For steady flow Mass entering per unit time = Mass leaving per unit time This can be applied to a stream tube such as that shown below. No fluid flows across the boundary made by the streamlines so mass only enters and leaves through the two ends of this stream tube section. We can then write A stream tube 4

5 Or for steady flow, This is the equation of continuity. The flow of fluid through a real pipe (or any other vessel) will vary due to the presence of a wall - in this case we can use the mean velocity and write When the fluid can be considered incompressible, i.e. the density does not change, r 1 = r 2 = r so (dropping the m subscript) This is the form of the continuity equation most often used. This equation is a very powerful tool in fluid mechanics and will be used repeatedly throughout the rest of this course. Some example applications We can apply the principle of continuity to pipes with cross sections which change along their length. Consider the diagram below of a pipe with a contraction: A liquid is flowing from left to right and the pipe is narrowing in the same direction. By the continuity principle, the mass flow rate must be the same at each section - the mass going into the pipe is equal to the mass going out of the pipe. So we can write: 5

6 (with the sub-scripts 1 and 2 indicating the values at the two sections) As we are considering a liquid, usually water, which is not very compressible, the density changes very little so we can say also says that the volume flow rate is constant or that. This For example if the area and and the upstream mean velocity,, then the downstream mean velocity can be calculated by Notice how the downstream velocity only changes from the upstream by the ratio of the two areas of the pipe. As the area of the circular pipe is a function of the diameter we can reduce the calculation further, Now try this on a diffuser, a pipe which expands or diverges as in the figure below, If the diameter at section 1 is and at section 2 and the mean velocity at section 2 is. The velocity entering the diffuser is given by, 6

7 Another example of the use of the continuity principle is to determine the velocities in pipes coming from a junction. Total mass flow into the junction = Total mass flow out of the junction r 1 Q 1 = r 2 Q 2 + r 3 Q 3 When the flow is incompressible (e.g. if it is water) r 1 = r 2 = r If pipe 1 diameter = 50mm, mean velocity 2m/s, pipe 2 diameter 40mm takes 30% of total discharge and pipe 3 diameter 60mm. What are the values of discharge and mean velocity in each pipe? The Bernoulli equation 7

8 1. Work and energy We know that if we drop a ball it accelerates downward with an acceleration (neglecting the frictional resistance due to air). We can calculate the speed of the ball after falling a distance h by the formula (a = g and s = h). The equation could be applied to a falling droplet of water as the same laws of motion apply A more general approach to obtaining the parameters of motion (of both solids and fluids) is to apply the principle of conservation of energy. When friction is negligible the Sum of kinetic energy and gravitational potential energy is constant. Kinetic energy Gravitational potential energy (m is the mass, v is the velocity and h is the height above the datum). To apply this to a falling droplet we have an initial velocity of zero, and it falls through a height of h. Initial kinetic energy Initial potential energy Final kinetic energy Final potential energy We know that kinetic energy + potential energy = constant so Initial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energy 8

9 so Although this is applied to a drop of liquid, a similar method can be applied to a continuous jet of liquid. The Trajectory of a jet of water We can consider the situation as in the figure above - a continuous jet of water coming from a pipe with velocity. One particle of the liquid with mass travels with the jet and falls from height to. The velocity also changes from to. The jet is travelling in air where the pressure is everywhere atmospheric so there is no force due to pressure acting on the fluid. The only force which is acting is that due to gravity. The sum of the kinetic and potential energies remains constant (as we neglect energy losses due to friction) so As is constant this becomes This will give a reasonably accurate result as long as the weight of the jet is large compared to the frictional forces. It is only applicable while the jet is whole - before it breaks up into droplets. Flow from a reservoir We can use a very similar application of the energy conservation concept to determine the velocity of flow along a pipe from a reservoir. Consider the 'idealised reservoir' in the figure below. 9

10 An idealised reservoir The level of the water in the reservoir is. Considering the energy situation - there is no movement of water so kinetic energy is zero but the gravitational potential energy is. If a pipe is attached at the bottom water flows along this pipe out of the tank to a level. A mass has flowed from the top of the reservoir to the nozzle and it has gained a velocity. The kinetic energy is now and the potential energy Initial kinetic energy Initial potential energy. Summarising Final kinetic energy Final potential energy We know that kinetic energy + potential energy = constant so so 10

11 We now have a expression for the velocity of the water as it flows from of a pipe nozzle at a height below the surface of the reservoir. (Neglecting friction losses in the pipe and the nozzle). Now apply this to this example: A reservoir of water has the surface at 310m above the outlet nozzle of a pipe with diameter 15mm. What is the a) velocity, b) the discharge out of the nozzle and c) mass flow rate. (Neglect all friction in the nozzle and the pipe). Volume flow rate is equal to the area of the nozzle multiplied by the velocity The density of water is so the mass flow rate is In the above examples the resultant pressure force was always zero as the pressure surrounding the fluid was the everywhere the same - atmospheric. If the pressures had been different there would have been an extra force acting and we would have to take into account the work done by this force when calculating the final velocity. We have already seen in the hydrostatics section an example of pressure difference where the velocities are zero. 11

12 The pipe is filled with stationary fluid of density pressures and at levels and respectively. What is the pressure difference in terms of these levels? has or This applies when the pressure varies but the fluid is stationary. Compare this to the equation derived for a moving fluid but constant pressure: You can see that these are similar form. What would happen if both pressure and velocity varies? 2. Bernoulli's Equation Bernoulli's equation is one of the most important/useful equations in fluid mechanics. It may be written, 12

13 We see that from applying equal pressure or zero velocities we get the two equations from the section above. They are both just special cases of Bernoulli's equation. Bernoulli's equation has some restrictions in its applicability, they are: Flow is steady; Density is constant (which also means the fluid is incompressible); Friction losses are negligible. The equation relates the states at two points along a single streamline, (not conditions on two different streamlines). All these conditions are impossible to satisfy at any instant in time! Fortunately for many real situations where the conditions are approximately satisfied, the equation gives very good results. The derivation of Bernoulli's Equation: An element of fluid, as that in the figure above, has potential energy due to its height z above a datum and kinetic energy due to its velocity u. If the element has weight mg then Potential energy = Potential energy per unit weight = Kinetic energy = 13

14 kinetic energy per unit weight = At any cross-section the pressure generates a force, the fluid will flow, moving the cross-section, so work will be done. If the pressure at cross section AB is p and the area of the cross-section is a then Force on AB = When the mass mg of fluid has passed AB, cross-section AB will have moved to A'B' Volume passing AB = Therefore distance AA' = work done = force distance AA' = work done per unit weight = This term is known as the pressure energy of the flowing stream. Summing all of these energy terms gives or As all of these elements of the equation have units of length, they are often referred to as the following: 14

15 pressure head = velocity head = potential head = total head = By the principle of conservation of energy the total energy in the system does not change, Thus the total head does not change. So the Bernoulli equation can be written As stated above, the Bernoulli equation applies to conditions along a streamline. We can apply it between two points, 1 and 2, on the streamline in the figure below or or Two points joined by a streamline total energy per unit weight at 1 = total energy per unit weight at 2 total head at 1 = total head at 2 This equation assumes no energy losses (e.g. from friction) or energy gains (e.g. from a pump) along the streamline. It can be expanded to include these simply, by adding the appropriate energy terms: 15

16 3. An example of the use of the Bernoulli equation. When the Bernoulli equation is combined with the continuity equation the two can be used to find velocities and pressures at points in the flow connected by a streamline. Here is an example of using the Bernoulli equation to determine pressure and velocity at within a contracting and expanding pipe. A fluid of constant density = 960 A contracting expanding pipe above tube. The diameters at the sections are is flowing steadily through the. The gauge pressure at 1 is and the velocity here is. We want to know the gauge pressure at section 2. We shall of course use the Bernoulli equation to do this and we apply it along a streamline joining section 1 with section 2. The tube is horizontal, with z 1 = z 2 so Bernoulli gives us the following equation for pressure at section 2: But we do not know the value of. We can calculate this from the continuity equation: Discharge into the tube is equal to the discharge out i.e. 16

17 So we can now calculate the pressure at section 2 Notice how the velocity has increased while the pressure has decreased. The phenomenon - that pressure decreases as velocity increases - sometimes comes in very useful in engineering. (It is on this principle that carburettor in many car engines work - pressure reduces in a contraction allowing a small amount of fuel to enter). Here we have used both the Bernoulli equation and the Continuity principle together to solve the problem. Use of this combination is very common. We will be seeing this again frequently throughout the rest of the course. 4. Pressure Head, Velocity Head, Potential Head and Total Head. By looking again at the example of the reservoir with which feeds a pipe we will see how these different heads relate to each other. Consider the reservoir below feeding a pipe which changes diameter and rises (in reality it may have to pass over a hill) before falling to its final level. Reservoir feeding a pipe To analyses the flow in the pipe we apply the Bernoulli equation along a streamline from point 1 on the surface of the reservoir to point 2 at the outlet nozzle of the pipe. And we know that the total energy per unit weightor the total head does not change - it is constant - along a streamline. But what is this value of this constant? We have the Bernoulli equation 17

18 We can calculate the total head, H, at the reservoir, as this is atmospheric and atmospheric gauge pressure is zero, the surface is moving very slowly compared to that in the pipe so is the elevation of the reservoir., so all we are left with A useful method of analyzing the flow is to show the pressures graphically on the same diagram as the pipe and reservoir. In the figure above the total head line is shown. If we attached piezometers at points along the pipe, what would be their levels when the pipe nozzle was closed? (Piezometers, as you will remember, are simply open ended vertical tubes filled with the same liquid whose pressure they are measuring). Piezometer levels with zero velocity As you can see in the above figure, with zero velocity all of the levels in the piezometers are equal and the same as the total head line. At each point on the line, when u = 0 The level in the piezometer is the pressure head and its value is given by. What would happen to the levels in the piezometers (pressure heads) if the if water was flowing with velocity = u? We know from earlier examples that as velocity increases so pressure falls 18

19 Piezometer levels when fluid is flowing We see in this figure that the levels have reduced by an amount equal to the velocity head,. Now as the pipe is of constant diameter we know that the velocity is constant along the pipe so the velocity head is constant and represented graphically by the horizontal line shown. (this line is known as the hydraulic grade line). What would happen if the pipe were not of constant diameter? Look at the figure below where the pipe from the example above is replaced be a pipe of three sections with the middle section of larger diameter Piezometer levels and velocity heads with fluid flowing in varying diameter pipes The velocity head at each point is now different. This is because the velocity is different at each point. By considering continuity we know that the velocity is different because the diameter of the pipe is different. Which pipe has the greatest diameter? Pipe 2, because the velocity, and hence the velocity head, is the smallest. 19

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