Mechanical Engineering Systems 2008 Part 7 pptx

This is the equation of a parabola so the average or mean velocityequals half the maximum velocity, which is on the central axis.vmean = 1 2 Vaxis Now volume flow rate is Q = Vmean× area

Viscous shear stress = viscosity × velocity gradient

In its simplest form this can be applied to two flat plates, one movingand one stationary, in the following equation:

a difference in velocity of u The viscosity is correctly known as the

dynamic viscosity and has the units of Pascal seconds (Pa s) These are

identical to N s/m2and kg/m/s

For the case where these two plates are adjacent lamina or layers:

F = A × velocity gradient

We have replaced the term u/h with something called the velocity

gradient which will allow us later to apply Newton’s equation to

situations where the velocity does not change evenly across a gap

We can apply this to pipe flow if we now wrap the lamina intocylinders

Laminar flow in pipes (Figure 3.2.10)

We said earlier that most fluid flow applications of interest to mechanicalengineers involve turbulent flow, but there are some increasinglyimportant examples of laminar flow where the pipe diameter is small andthe liquid is very viscous The field of medical engineering has many suchexamples, such as the flow of viscous blood along the small diametertubes of a kidney dialysis machine It is therefore necessary for us to studythis kind of pipe flow so that we may be able to calculate the pumpingpressure required to operate this type of device

Consider flow at a volume flow rate Q m3/s along a pipe of radius R and length L The liquid viscosity is  and a pressure drop of P is required

across the ends of the pipe to produce the flow Let us look at the forces on

the cylindrical core of the liquid in the pipe, up to a radius of r.

If we take the outlet pressure as 0 and the inlet pressure as P then the

force pushing the core to the right, the pressure force, is given by

Fp = P × A

= r2P

Figure 3.2.10 Laminar flow

along a pipe

The force resisting this movement is the viscous drag around thecylindrical surface of the core Using Newton’s defining equation forviscosity, Equation (3.2.4),

Fdrag =  Acore surface× velocity gradient

We need to relate the pressure to the flow rate Q and the first step is

to find the velocity v at any radius r by integration.

This is the equation of a parabola so the average or mean velocityequals half the maximum velocity, which is on the central axis.

vmean = 1

2 Vaxis

Now volume flow rate is Q = Vmean× area

This is the continuity equation and up to now we have only applied

it to turbulent flow where all the liquid flows at the same velocity and

we do not have to think of a mean So

This is called Poiseuille’s law after the French scientist and engineer who

first described it and this type of flow is known as Poiseuille flow

Note that this is analogous to Ohm’s law with flow rate Q equivalent

to current I, pressure drop P equivalent to voltage E and the term 8L

R4

representing fluid resistance , equivalent to resistance R

Hence Ohm’s law E = IR becomes P = Q when applied to viscousflow along pipes This can be very useful when analysing laminar flowthrough networks of pipes For example, the combined fluid resistance

of two different diameter pipes in parallel and both fed with the sameliquid at the same pressure can be found just like finding the resistance

of two electrical resistors in parallel

Example 3.2.3

A pipe of length 10 m and diameter 5 mm is connected inseries to a pipe of length 8 m and diameter 3 mm A pressuredrop of 120 kPa is recorded across the pipe combinationwhen an oil of viscosity 0.15 Pa s flows through it Calculatethe flow rate

First we must calculate the two fluid resistances, one foreach section of the pipe

A1

V1 1

Examples of laminar flow in engineering

We have already touched on one example of laminar flow in pipeswhich is highly relevant to engineering but it is worthwhile looking atsome others just to emphasize that, although turbulent flow is the moreimportant, there are many instances where knowledge of laminar flow isnecessary

One example from mainstream mechanical engineering is thedashpot This is a device which is used to damp out any mechanicalvibration or to cushion an impact A piston is pushed into a close-fittingcylinder containing oil, causing the oil to flow back along the gapbetween the piston and the cylinder wall As the gap is small and the oilhas a high viscosity, the flow is laminar and the pressure drop can bepredicted using an adaptation of Poiseuille’s law

A second example which is more forward looking is from the field ofmicro-fluidics Silicon chip technology has advanced to the point wherescientists can build a small patch which could be stuck on a diabetic’sarm to provide just the right amount of insulin throughout the day Itworks by drawing a tiny amount of blood from the arm with a miniaturepump, analysing it to determine what dose of insulin is required andthen pumping the dosed blood back into the arm The size of the flowchannels is so small, a few tens of microns across, that the flow is verylaminar and therefore so smooth that engineers have had to go to greatlengths to ensure effective mixing of the insulin with the blood

Conservation of energy

Probably the most important aspect of engineering is the energyassociated with any application We are all painfully aware of the cost

of energy, in environmental terms as well as in simple economic terms

We therefore now need to consider how to keep account of the energyassociated with a flowing liquid The principle that applies here is thelaw of conservation of energy which states that energy can neither becreated nor destroyed, only transferred from one form to another Youhave probably met this before, and so we have a fairly straightforwardtask in applying it to the flow of liquids along pipes If we can calculatethe energy of a flowing liquid at the start of a pipe system, then we knowthat the same total of energy must apply at the end of the system eventhough the values for each form of energy may have altered The onlyproblem is that we do not know at the moment how to calculate theenergy associated with a flowing liquid or even how many types ofenergy we need to consider We must begin this calculation therefore byexamining the different forms of energy that a flowing liquid can have(Figure 3.2.12)

Figure 3.2.12 Energy of a

flowing liquid

If we ignore chemical energy and thermal energy for the purposes offlow calculations, then we are left with potential energy due to height,potential energy due to pressure and kinetic energy due to the motion.

In Figure 3.2.12 adding up all three forms of energy at point 1 for a

small volume of liquid of mass m:

Potential energy due to height is calculated with reference to some

datum level, such as the ground, in the same way as for a solid

PEheight = mgz1

Potential energy due to pressure is a calculation of the fact that the

mass m could rise even higher if the pipe were to spring a leak at point

1 It would rise by a height of h1equal to the height of the liquid in a

manometer tube placed at point 1, where h1 is given by h1 = p1/g.Therefore the energy due to the pressure is again calculated like theheight energy of a solid

PEpressure = mgh1

Note that the height h1is best referred to as a pressure head in order to

distinguish it from the physical height z1of the pipe at this point

Kinetic energy is calculated in the same way as for a solid.

Therefore if we cancel the m and divide through by g, we produce the

following equation:

z1+ h1+ v1/2g = z2+ h2+ v2/2g (3.2.6)

This is known as Bernoulli’s equation after the French scientist who

developed it and is the fundamental equation of hydrodynamics The

dimensions of each of the three terms are length and therefore they all have units of metres For this reason the third term, representing kinetic energy, is often referred to as the velocity head, in order to use the

familiar concept of head which already appears as the second term onboth sides of the equation The three terms on each side of the equation,

added together, are sometimes known as the total head A second advantage to dividing by the mass m and eliminating it from the

equation is that we no longer have to face the problem that it would bevery difficult to keep track of that fixed mass of liquid as it flowed alongthe pipe Turbulent flow and laminar flow would both make the massspread out very rapidly after the starting point

Paint/air spray Air

When carrying out calculations on Bernoulli’s equation it is

sometimes useful to use the substitution h = p/g to change from head

to pressure, and it is often useful to use the substitution v = Q/A because

the volume flow rate is the most common way of describing the liquid’sspeed

An example of the use of Bernoulli’s equation is given later inExample 3.2.5

Venturi principle

Bernoulli’s equation can seem very daunting at first sight, but it isworthwhile remembering that it is simply the familiar conservation ofenergy principle Therefore it is not always necessary to put numbersinto the equation in order to predict what will happen in a given flowsituation One of the most useful applications in this respect is thebehaviour of the fluid pressure when the fluid, either liquid or gas, ismade to go through a constriction

Consider what happens in Figures 3.2.13 and 3.2.14

In both cases the fluid, air, is pushed through a narrower diameterpipe by the high pressure in the large inlet pipe The velocity in the

narrow pipe is increased according to the relationship v = Q/A since the volume flow rate Q must stay constant Hence the kinetic energy term

v2/2g in Bernoulli’s equation is greatly increased, and so the pressure head term h or p/g must be much reduced if we can ignore the change

in physical height over such a small device The result is that a very lowpressure is observed at the narrow pipe, which can be used to suck paint

in through a side pipe in the case of the spray gun in Figure 3.2.13, orpetrol in the case of the carburettor in Figure 3.2.14 This effect isknown as the Venturi principle after the Italian scientist and engineerwho discovered it

Measurement of fluid flow

Fluid mechanics for the mechanical engineer is largely concernedwith transporting liquid from one place to another and therefore it isimportant that we have an understanding of some of the ways ofmeasuring flow There are many flow measurement methods, some ofwhich can be used for measuring volume flow rates, others whichcan be used for measuring flow velocity, and yet others which can beused to measure both We shall limit ourselves to analysis of oneexample of a simple flow rate device and one example of a velocitydevice

Figure 3.2.13 Paint sprayer

Figure 3.2.14 Carburettor

Q Q

in the pressure as the total head remains constant In the Venturi effect

the large increase in velocity through a constriction causes a markedreduction in pressure, with the size of this reduction depending on thesize of the velocity increase and therefore on the degree of constriction

In other words, if we made a device which forced liquid through aconstriction and we measured the pressure head reduction at theconstriction, then we could use this measurement to calculate thevelocity or the flow rate from Bernoulli’s equation

Such a device is called a Venturi meter since it relies on the Venturi

effect In principle the constriction could be an abrupt change ofcross-section, but it is better to use a more gradual constriction and aneven more gradual return to the full flow area following theconstriction This leads to the formation of fewer eddies and smallerareas of recirculating flow As we shall see later, this leads to less loss

of energy in the form of frictional heat and so the device creates less

of a load to the pump producing the flow A typical Venturi meter isshown in Figure 3.2.15

The inlet cone has a half angle of about 45° to produce a flow pattern

which is almost free of recirculation Making the cone shallower wouldproduce little extra benefit while making the device unnecessarilylong

The diffuser has a half angle of about 8° since any larger angle leads

to separation of the flow pattern from the walls, resulting in theformation of a jet of liquid along the centreline, surrounded byrecirculation zones

The throat is a carefully machined cylindrical section with a smaller

diameter giving an area reduction of about 60%

Measurement of the drop in pressure at the throat can be made usingany type of pressure sensing device but, for simplicity, we shall considerthe manometer tubes shown The holes for the manometer tubes – thepressure taps – must be drilled into the meter carefully so that they areaccurately perpendicular to the flow and free of any burrs

Figure 3.2.15 A typical Venturi

meter

Analysis of the Venturi meter

The starting point for the analysis is Bernoulli’s equation:

z1+ h1+ v1/2g = z2+ h2+ v2/2g

Since the meter is being used in a horizontal position, which is the usual

case, the two values of height z are identical and we can cancel them

from the equation

h1+ v1/2g = h2+ v2/2g

The constriction will cause some non-uniformity in the velocity of theliquid even though we have gone to the trouble of making the change incross-section gradual Therefore we cannot really hope to calculate thevelocity accurately Nevertheless we can think in terms of an average ormean velocity defined by the familiar expression:

v = Q/a

Therefore, remembering that the volume flow rate Q does not alter

along the pipe:

h1+ Q2/2a1g = h2+ Q2/2a2g

We are trying to get an expression for the flow rate Q since that is what

the instrument is used to measure, so we need to gather all the termswith it onto the left-hand side:

(Q2/2g) × (1/a1– 1/a2) = h2– h1

Q2 = 2g(h2– h1)/(1/a1– 1/a2)

Since h1 > h2 it makes sense to reverse the order in the first set ofbrackets, compensating by reversing the order in the second set aswell:

of energy, and head, no matter how well we have guided the flowthrough the constriction We could experimentally measure a head lossand use this in a modified form of Bernoulli’s equation, but it iscustomary to stick with the analysis carried out above and make a finalcorrection at the end

Any loss of head will lead to the drop in heights of manometer levels

across the meter being bigger than it should be ideally Therefore the

calculated flow rate will be too large and so it must be corrected by

Q Q

Orifice

applying some factor which makes it smaller This factor is known as the discharge coefficient CDand is defined by:

Qreal = CD × Qideal

In a well-manufactured Venturi meter the energy losses are very small

and so CDis very close to 1 (usually about 0.97)

In some situations it would not matter if the energy losses caused bythe flow measuring device were considerably higher For example, ifyou wanted to measure the flow rate of water entering a factory theneven a considerable energy loss caused by the measurement would onlyresult in the pumps at the local water pumping station having to work alittle harder; there would be no disadvantage as far as the factory wasconcerned In these circumstances it is not necessary to go to the trouble

of having a carefully machined, highly polished Venturi meter,particularly since they are complicated to install Instead it is sufficientjust to insert an orifice plate at any convenient joint in the pipe,

producing a device known as an Orifice meter (Figure 3.2.16) The

orifice plate is rather like a large washer with the central hole, or orifice,having the same sort of area as the throat in the Venturi meter

The liquid flow takes up a very similar pattern to that in the Venturimeter, but with the addition of large areas of recirculation In particularthe flow emerging from the orifice continues to occupy a small cross-section for quite some distance downstream, leading to a kind of throat.The analysis is therefore identical to that for the Venturi meter, but the

value of the discharge coefficient CD is much smaller (typically about0.6) Since there is not really a throat, it is difficult to specify exactlywhere the downstream pressure tapping should be located to get themost reliable reading, but guidelines for this are given in a BritishStandard

Measurement of velocity – the Pitot-static tube

Generally it is the volume flow rate which is the most importantquantity to be measured and from this it is possible to calculate a meanflow velocity across the full flow area, but in some cases it is alsoimportant to know the velocity at a point A good example of this is in

a river where it is essential for the captain of a boat to know what

Figure 3.2.16 An orifice meter

Inner Pitot tube

Outer static tube

strength of current to expect at any given distance from the bank;calculating a mean velocity from the volume flow rate would not bemuch help even if it were possible to measure the exact flow area over

an uneven river bed

It was exactly this problem which led to one of the most commonvelocity measurement devices A French engineer called Pitot was giventhe task of measuring the flow of the River Seine around Paris andfound that a quick and reliable method could be developed from some

of the principles we have already met in the treatment of Bernoulli’sequation Figure 3.2.17 shows the early form of Pitot’s device.The horizontal part of the glass tube is pointed upstream to face theoncoming liquid The liquid is therefore forced into the tube by thecurrent so that the level rises above the river level (if the glass tube wassimply a straight, vertical tube then the water would enter and rise until

it reached the same level as the surrounding river) Once the water hasreached this higher level it comes to rest

What is happening here is that the velocity head (kinetic energy) ofthe flowing water is being converted to height (potential energy) insidethe tube as the water comes to rest The excess height of the column ofwater above the river level is therefore equal to the velocity head of theflowing water:

The Pitot-static tube is still widely used today, most notably as thespeed measurement device on aircraft (Figure 3.2.19)

The two tubes are now combined to make them co-axial for thepurposes of ‘streamlining’, and the pressure difference would be

Figure 3.2.17 The Pitot tube

Figure 3.2.18 An early

Pitot-static tube

Figure 3.2.19 A modern

Pitot-static tube

measured by an electronic transducer, but essentially the device is thesame as Pitot’s original invention Because the Pitot tube and the static

tube are united, the device is called a Pitot-static tube.

Example 3.2.4

A Pitot-static tube is being used to measure the flow velocity

of liquid along a pipe Calculate this velocity when the heights

of the liquid in the Pitot tube and the static tube are 450 mmand 321 mm respectively

The first thing to do is calculate the manometric headdifference, i.e the difference in reading between the twotubes

Head difference = 450 mm – 321 mm = 129 mm

= 0.129 mThen use Equation (3.2.7),Velocity = (2 × 9.81 × 0.129) = 1.59 m/s

Losses of energy in real fluids

So far we have looked at the application of the familiar ‘conservation ofenergy’ principle to liquids flowing along pipes and developedBernoulli’s equation for an ideal liquid flowing along an ideal pipe.Since energy can neither be created nor destroyed, it follows that thethree forms of energy associated with flowing liquids – height energy,pressure energy and kinetic energy – must add up to a constant amounteven though individually they may vary We have used this concept tounderstand the working of a Venturi meter and recognized that apractical device must somehow take into account the loss of energyfrom the fluid in the form of heat due to friction This was quite simplefor the Venturi meter as the loss of energy is small but we must nowconsider how we can take into account any losses in energy, in the form

of heat, caused by friction in a more general way These losses can arise

in many ways but they are all caused by friction within the liquid orfriction between the liquid and the components of the piping system.The big problem is how to include what is essentially a thermal effectinto a picture of liquid energy which deliberately sets out to exclude anymention of thermal energy

Modified Bernoulli’s equation

Bernoulli’s equation, as developed previously, may be stated in thefollowing form:

z1+ h1+ v1/2 g = z2+ h2+ v2/2 g

All the three terms on each side of Bernoulli’s equation have dimensions

of length and are therefore expressed in metres For this reason the total

value of the three terms on the left-hand side of the equation is known

as the initial total head in just the same way as we used the word head

to describe the height h associated with any pressure p through the

expression

p = gh

Similarly the right-hand side of the equation is known as the final total

head Bernoulli’s equation for an ideal situation may also be expressed

in words as:

Initial total head = final total head

What happens when there is a loss of energy due to friction with a realfluid flowing along a real pipe is that the final total head is smaller thanthe initial total head The loss of energy, as heat generated by the frictionand dissipated through the liquid and the pipe wall to the surroundings,can therefore be expressed as a loss of head Note that we are notdestroying this energy, it is just being transformed into thermal energythat cannot be recovered into a useful form again As far as the engineer

in charge of the installation is concerned this represents a definite losswhich needs to be calculated even if it cannot be reduced any further.What happens in practice is that manufacturers of pipe systemcomponents, such as valves or couplings, will measure this loss of headfor all their products over a wide range of sizes and flow rates They willthen publish this data and make it available to the major users of thecomponents Provided that the sum of the head losses of all thecomponents in a proposed pipe system remains small compared to thetotal initial head (say about 10%) then it can be incorporated into amodified Bernoulli’s equation as follows:

Initial total head – head losses = final total head

With this equation it is now possible to calculate the outlet velocity orpressure in a pipe, based on the entry conditions and knowledge of theenergy losses expressed as a head loss in metres Once again we see theusefulness of working in metres since engineers can quickly develop afeel for what head loss might be expected for any type of fitting and how

it could be compensated This would be extremely difficult to do ifworking in conventional energy units

Example 3.2.5

Water is flowing downwards along a pipe at a rate of 0.8 m3/sfrom point A, where the pipe has a diameter of 1.2 m, to point

B, where the diameter is 0.6 m Point B is lower than point A

by 3.3 m The pipe and fittings give rise to a head loss of0.8 m Calculate the pressure at point B if the pressure atpoint A is 75 kPa (Figure 3.2.20)

Since the information in the question gives the flow rate Q

rather than the velocities, we shall use the substitution

Q = a v = a v