Chap6-Behaviour of Real Fluids(1).pdf

8/17/2019 Chap6-Behaviour of Real Fluids(1).pdf

1/24

NDEJJE UNIVERSITY

FACULTY OF ENGINEERING

CIV 1201: FLUID MECHANICS I

Chapt er 6: Behaviour of Real Fluids

Tutor:

Mr. Solomon Mutebi


2/24

6.1 IntroductionIn the previous chapters, the basic equations of continuity, energy andmomentum, were introduced and applied to fluid flow cases where theassumption of frictionless flow was made (i.e. fluid being treated as being ideal).

This chapter introduces the concept of real fluid flow in which viscosity will beaccepted, leading to situations where frictional effects can not be ignored.

Two cases are considered; Bounded flow and Flow around a solid body(External flow). In this particular section, only ‘ Bounded flow’ , will be considered,where the fluid moves inside a pipe or duct or in a channel so that it is guidedby a boundary surrounding the fluid. Examples of ‘ External flow ’ include flow ofwind around a house, or an object moving through a stationary or moving fluide.g. an aeroplane in flight, or a sailing ship.

Note: In all the above cases, there is a velocity gradient and, thus shearstresses in the fluid. In order to maintain flow, the shear stress must bemaintained and this can only be achieved by additional force doing work on thefluid; in other words, there must be a continuous supply of energy for the flowto exist.

This energy supplied, solely to maintain flow in a bounded system, is usuallyexpressed per unit weight of the fluid flowing and, thus is in units of fluid head.

=gQ

pQgQ

pavgQ

t s pa

= hg

p

This head (or energy) is considered as ‘ lost ’ because it cannot be used for anyother purpose than to maintain flow, and hence is called ‘ Head Loss’ .

Energy supplied per unit time

Weight of the fluid flowing=

(Force) x (Distance)/time

(Specific weight) x (Discharge)


3/24

6.2 Incompressible steady and uniform Turbulent flow in boundedconduits

Consider a small element of fluid with in a conduit (Fig 6.1). The flow isassumed to be uniform and steady, so that the fluid acceleration in the flowdirection is zero.

Applying the momentum equation to the fluid element, in the flow direction yields

0sin021 wlp A p A p

where ‘ p’ is the wetted perimeter of the element defined as “that part of theconduit circumference in contact with the fluid”.

Note: Including the area under which the shear stress ‘ 0 ’ acts in the form of

‘ pl ’ effectively renders the derivation applicable to both open and closedconduits.

Substituting for Algw ,

andldz sin (negative because ‘ z’ reduces as ‘ l’ increases), gives

0021 z Aglp p p A where ‘ p1’ and ‘ p2’ are the static pressures in the flow at sections 1 and 2

Fig. 6.1: Turbulent flow in bounded conduits

Datum


4/24

thus, 01 021 A p

zg p pl

…………………………………….(6.1)

Note: The first term represent a drop in piezometric head over a length ‘ l’ of the

conduit, and the ratio ‘ p A

’ is known as ‘ Hydraulic mean depth’ , normally denoted

by ‘ m’

Thus, 01

0 mdx

dp …………………………….(6.2)

In order to express ‘ 0 ’ in equation (6.2), the concept of ‘ flow friction factor f ’ is

introduced, which is a non-dimensional experimentally measured factornormally introduced in the form

2

2

0v f

…………………………………………………………..(6.3)

where ‘ v’ is the mean flow velocity

thus, from equation (6.2),mv f

dxdp

2

2 …………………..……………(6.4)

if the friction head loss down a length ‘ l’ of the conduit is denoted by ‘ h f ’, thenthe rate of loss of piezometric pressure may be expressed as

mv f

dxdp

2

2 =

l

gh f

Orgm

l fvh f 2

2

………………………………………………………(6.5)

But, gz pdxd

dxdp

; where ‘z’ is the elevation of the conduit above some

datum.

For open channels, the static pressure ‘p’ may be assumed to remain constantalong the channel. Thus, it follows that

singdxdz

gdx

dp…………………………(6.6)


5/24

And for uniform flow,

the hydraulic gradient,l

h f the slope of the channel.

i.e. il

h f sin (the slope)

Therefore, from equations (6.4) and (6.5), it follows that

glm

l fv

2

2

, such that mi f g

v 2

If f g2

= C ………………………………………………………………………..(6.7)

Then, miC v ……………………………………………….….(6.8)

Equation (6.8) above is known as ‘ Chezy formula’ , where ‘ C ’ is the Chezyconstant.

Note: For pipes running full, the wetted perimeter becomes the internaldiameter ‘ d ’ of the pipeline.

Hence,44

2 d d

d m

p A

Therefore, eequation (6.5) becomes

gv

d l f

h f 24 2

……………………………………………..…….(6.9)

This is known as the ‘ Darcy equation’ , which gives the head loss in circular pipes


6/24

6.3 Separation Losses in pipe flow

Whenever the uniform cross-section of the pipeline is interrupted by theinclusion of a pipe fitting such as a valve, bend, junction or flow measurementdevice, then a pressure losses will occur. These losses are referred to as'Separation losses '.

Generally, the flow separates from the pipe walls as it passes through theobstructing pipe fitting, resulting in the generation of eddies in the flow, withconsequent pressure loss (Fig 6.2), for the case of sudden enlargement.

Note: For small complex pipe networks such as those found in some chemicalprocess plants, air crafts fuel and hydraulic systems, and in ventilationsystems, the total effect of separation losses may be the predominant factor inthe system pressure loss calculation, exceeding the contributions of the pipefriction at the design flow rate. Conversely, in large pipe systems, such as waterdistribution networks, the losses due to pipe fitting may be negligible comparedwith the frictional losses and may often be ignored.

Fi 6.2: Se aration losses in sudden enlar ement


7/24

6.3.1 Losses in Sudden expansion and contractionFig (6.3) illustrates a sudden enlargement of the pipeline flow.

Considering a control volume ABCDEF, if 1 p and 2 p are the pressures at

sections 1& 2 respectively, then

from continuity of flow,

2211 u Au A where 1u and 2u are the respective meanvelocities.

Applying the momentum equation between points 1 and 2,

i.e. (Resultant force flow direction) = (Rate of change of momentum)

12221211 ' uuQ A p A A p A p

Where, 22 AuQ and ' p is the pressure acting on the annulus represented by

AB and CD of the cross-section area 12 A A .

Note: Since the radial acceleration at entry to the larger-diameter duct at

section ABCD is small, then 1' p p , giving

12221 uuQ A p p = 1222 uuu A

12221 uuu p p ……………………………………………..(6.10)

Now, applying Bernoulli’s equation between points 1 and 2,

Fig 6.3 Suddenenlargement


8/24

h zg

ug

p z

gu

g p

2

222

1

211

22 ; where ‘ h ’ represents separation losses

Simplifying give,2

22

2121 uu

g p ph

………………………………..(6.11)

Combining equations (6.10) and (6.11), gives

guu

guu

uh2

22

2112

2

= 22121 221

uuuug

=

guu

2

22

21

Thus, the loss due to sudden enlargement is given by;

guu

h2

221

…………………………………………………..(6.12)

Alternatively, from the continuity equation,2

2

12

1 1

2

A

A

g

uh =

2

1

22

2 1

2

A

A

g

u…………………………(6.13)

The above expression is sometimes referred to as ‘ Borda-carnot ’ relationship, andis usually within a few percent of the experimental results for the separationloss incurred by sudden enlargement in coaxial pipelines.

Note: The losses into the reservoir may be obtained by considering equation

(6.13). As 2 A (i.e. the reservoir is large), so 02 u .

Therefore,g

uh

2

21

; i.e. the kinetic energy of the approaching flow


9/24

Sudden Contraction

It is not possible to apply the momentum equation between sections 1 and 2 forthe above figure owing to the uncertain pressure distribution across the face

ABCD.However, experiments have shown that pressure losses occur as a result ofeddies formed as flow area expands from the vena contracta up to the fullcross-section of the downstream pipe.

If the area of the vena contracta is ‘ c A ’, then the expression for sudden

enlargement may be applied between the vena contracta and section 2

Giving,2

22

2 12

c A

A

g

uh =

222 1

1

2

cC g

u…………………..…..(6.14)

Where ‘ cC ’ is the coefficient of contraction for the junction based on the

smaller-pipe entry diameter BC.

In general, expression (6.14) may be written in the form:

gu

K h2

22

; where2

11

cC K is known as ‘ loss coefficient’

The table below shows some experimental values of ‘ cC ’ and the corresponding

values of ‘ K ’ obtained with sharp pipe edges.

Fig 6.4 : Suddencontraction


10/24

1

2 A

A 0.1 0.3 0.5 0.7 1.0

cC 0.61 0.632 0.673 0.73 1.0

K 0.41 0.34 0.24 0.14 0

Table 6.1: Loss coefficients for sudden contraction

Questions

1. A pipe of 0.09m 2 area is suddenly enlarged to an area of 0.36m 2 . Thedischarge through the pipe is 0.27m 3/sec, and the pressure at the smallerpipe is 83.3kN/m 2 . Determine the following:

i. the head loss due to change of section

ii. pressure at the larger part of the pipe iii. work done in forcing the water through the enlargement

2. The diameter of a pipe is suddenly reduced from 15cm to 10cm, with acorresponding change in pressure from 1.2bar to 1bar. Assuming acoefficient of contraction to be 0.62, find the discharge through the pipelinein litres/sec.

6.4 Losses in Pipe fittings, Bends and Pipe entry

Losses in pipe fittings are usually expressed in the form,g

uK h

2

2

; where ‘ K ’ is

the fitting loss coefficient. It is a non-dimensional constant whose value isobtained experimentally for any pipe fitting. Table 6.2 show some typical values.

Note: The major advantage of expressing losses due to separation in the aboveform is that it can easily be incorporated into the steady flow energy equation.


11/24

Fitting Loss coefficient K

90 0 elbow 0.945 0 elbow 0.4Return bend 2.2Large-radius 90 0 bend 0.6

Tee junction 1.8Sharp pipe entry 0.5Sharp pipe exit 0.5Gate valve (open to 75% shut) 0.25 25Globe valve 10

Table 6.2: Head loss coefficients for a range of pipe fittings

A Globe valve is used for regulating flow in a pipeline and consists of a movabledisk-type element and a stationary ring seat in a generally spherical body

A Gate Valve , or Sluice Valve, as it is sometimes known, opens by lifting around or rectangular gate/wedge out of the path of the fluid


12/24

Figure (6.5) illustrates the flow in a pipe bend, demonstrating the area of flowseparation which results in the loss coefficients for bens listed in Table 6.2. Asthe bend becomes sharper, so the areas of separation become extensive and theloss coefficient increases.

Fig 6.5. Separation at pipebends


13/24

Losses at entry to a pipe from a reservoir are a special case of suddencontraction, in which the velocity in the reservoir is considered to be zero.Owing to the fact that the fluid enters the pipe from all directions, a venacontracta is formed downstream of the pipe inlet and, consequently, the loss isassociated with enlargement from the vena contracta to the full-bore pipe.

Note: considering the above illustration for pipe entry losses, it can be seenthat the sharper the entry Conner, the smaller is the vena contracta, and,hence, the greater the flow separation and the higher the value of K

6.5 Incompressible, steady and uniform turbulent flow in circular cross-section pipes

The head loss in turbulent flow in a closed section pipe is given by the Darcyequation (6.9),

i.e.g

vd

l f h f 2

4 2

From the equation, all other parameters are measurable apart from the friction

factor ‘ f ’. Thus the following are noted:1. ;lh f

2. ;2vh f

3. ;1 d h f

4. f h depends on the surface roughness of the pipe walls;

Fig 6.6. Pipe entry losses


14/24

5 f h depends on the fluid density and viscosity;

6 f h is independent of pressure.

Thus, the value of ‘ f ’ which depends on the above factors listed, must be

selected to determine the correct value of ‘ f h ’. Expressed in a form suitable for

dimensional analysis, implies that

k d v f ,,,, …………………………………………….(6.15)

where ‘k’ is a measure of the size of wall roughness

In general rough pipe case, dimensional analysis yields an expression

d k vd f ,2 Or in terms of Reynolds Number, d k R f e ,2

Hence ‘ f ’ is a function of ‘ Re’ and ‘ k/d ’. This relationship has been foundexperimentally for turbulent flow conditions, and the values of ‘f’ have beenplotted on a chart for different values of ‘ R e’ and ‘ k/d ’. This chart is known as the‘ Moody Chart ’ (Fig 6.7


15/24

This chart allows the calculation of ‘ f ’ provided the values for ‘ R e’ and ‘ k/d ’ areknown.

Laminar flow

When laminar flow is present, the friction factor may be computed analyticallyfor both smooth and rough walls as

2

16 R

f …………………………………………………….(6.16)

Turbulent flow

For turbulent flow in a smooth pipe, the equation for friction factor developedby ‘Blasius’ is given by

Fig 6.7 The Moody Chart


16/24

25.02

079.0

R f ………………………………….(6.17)

Otherwise the Moody chart should always be used to determine the value of ‘ f ’given ‘ R e’ and ‘ k/d ’. Colebrook expressed the information provided by the Moody

chart for the turbulent regime in form of an equation known as Colebrook-Whiteequation , i.e.

f Rd k

f e

3.92ln74.148.3

1………………………………..(6.18)

Note: This equation is transcendental, and therefore iteration is always neededto calculate ‘ f ’

Question

1 Calculate the loss of head due to friction and the power required tomaintain flow in a horizontal circular pipe of 40mm diameter and 750mlong when water (coefficient of dynamic viscosity 1.14 x 10 -3 Nsm -2 ) flowsat a rate of: (a) 4.0 litres/min, (b) 30 litres/min. Assume that for thepipe the absolute roughness is 0.00008m.

2 A hydraulic plant consists of a reservoir which supplies a turbine via a1.5m diameter, commercial steel pipe which is 7km long. The turbine issituated 260m below the water level in the reservoir and flow iscontrolled by a butterfly valve just upstream of the turbine. With thevalve half open, its loss coefficient is 8.5 and the volumetric flow rate is4.5m 3 /s. If the turbine is 88% efficient, determine the output powergenerated.

(Assume that flow, after passing through the turbine, discharges to atmosphereand that all other specific loses may be neglected. Take the kinematic viscosityof water as 1.3 x 10 -6 m 2 /s )

3. A pump is to transport 0.02 m 3 /s of water from a tank to another tank 30mabove. The commercial steel pipe with a diameter of 100mm, and 62m longis used. If the specific losses amount to 8.3 times the kinetic energy head ofwater in the pipe, calculate

a. The pipe friction factorb. The specific work input to the waterc. The input power to the pump which has an efficiency of 68%


17/24

Take mskg /10*01.1 3

4. A 300 mm diameter pipe connects a water reservoir to a turbine 275m belowthe reservoir. The head loss (frictional + specific) in the pipeline amounts to56 times the kinetic energy head of water in the pipe. The combined

mechanical and electrical efficiency of the turbine/alternator is 78%. For awater velocity of 4.9 m/s in the pipe, calculate

a. The mass flow rate of waterb. The efficiency of transmission of hydraulic power to the turbinec. The electrical power developed

6.6 STEADY INCOMPRESSIBLE FLOW IN PIPE AND DUCT SYSTEMS

This section is concerned with the analysis of the steady flow of a liquid inclosed or open conduits.

A C losed conduit ; is a pipe or duct through which the fluid flows while completely

filling the x-section. Since the fluid has no free surface, it can be either a liquid

or a gas, its pressure may be above or below atmospheric pressure, and this

pressure may vary from x-section to x-section along its length.

An Open conduit is a duct or open channel along which a liquid flows with the

free surface. At all points along its length, the pressure at the free surface will

be the same, usually atmospheric. An open conduit may be covered provided

that it is not running full and the liquid retains a free surface; a partly filled

pipe would, for example, be treated as an open channel.

In either case as the fluid flows over a solid boundary, a shear stress will be

developed at the surface of contact and will oppose fluid motion. This is called

frictional resistance which results in energy transfer with in the systems,

experienced as a “ loss ” measurable in a fluid flow by changes in the fluid

pressure or head. In addition to these losses attributable to friction, separation

losses due to the flow disruption at changes in section, direction or around


18/24

values and other flow obstructions also contributes to the overall energy

transfers to be accounted for.

Consider the fig below showing energy changes in a flowing fluid.

Applying the energy equation yields the steady flow energy equation as. (i.e.

between A and B)

222

21

21

21 kugzv p pgzv p B B B pump A A A

defined in terms of pressure, e.g. for steady flow in air duct systems

The Pressure losses due to friction and separation being represented as;2

1 2ku u = local flow velocity

k = Constant dependent upon the pipe or conduit parameters

i.e. length, diameter, Roughness or fitting type.

Expressed in energy per unit weight (energy head), gives

gku zg

vhh zgvh B B B pump A A A 222

222

the head term being measured in meters of flowing fluid.


19/24

The continuity of flow equation is given by

Volume per unit time = Volume per unit time

Entering a control volume at A leaving the control volume

At B

Analysis of all steady flow problems in pipes and channels with based on the

application of these equations (stead flow energy eqn.) and the continuity of

volumetric flow equation applied between suitable points in the system.

Incompressible Flow through Ducts and Pipes

The pressure loss p or energy lost per unit volume due to friction may be

expressed via the Darcy equation.

Dv fl p 2

4 2

And for circular x-section conduit flowing full, terms of head lost

gD flvh 2

4 2

For laminar flowe R

f 16

and hence depends only on flow velocity.

Separation losses may be expressed as pressure term 221. vk or head term

gvk 2.

2

Where ‘ k’ depends on the type of fitting encountered


20/24

Question

Water discharges from a reservoir (fig below) through a 100mm pipe 15m long,

which rises to its highest point at B, 1.5m above the free surface of the

reservoir, and discharges direct to the atmosphere at C, 4m below the free

surface at A. the length of pipe L1 from A to B is 5m and the length of pipe L2

from B and C is 10m. Both the entrance and exit of the pipe are sharp and the

value of f = 0.08.

Calculate,

a) the mean velocity of water leaving the pipe at C

b) the pressure in the pipe B.

Incompressible flow through pipes in series

When pipes of different diameters are connected end-to-end to form a

pipeline, so that the fluid flows through each pipe in turn, the pipes are said

to be in series. The total loss of energy, or pressure loss, over the whole

pipeline will be the sum of the losses for each pipe together with any

separation losses such as might occur at the junctions, entrance or exit.


21/24

The losses encouraged are:

i. Loss at entrance to pipe AC . This is a separation loss and is given as

gv

h2

5.02

11

ii. Friction loss in AC . Given by the Darcy formula, as loss of head in friction in

1

21

2

41

1 gd

v flh AC f f

iii. Loss at change of section at C . There will be a separation loss at the sudden

change of section given as

Loss of head at sudden enlargement,

gvv

h2

221

2

iv. Friction loss in CB. Given by Darcy formula as loss of head in friction in CB,

2

222

24

2 gd v fl

h f

v. Loss of head at exit . Which is a separation loss given as

gv

h2

22

3


22/24

Incompressible flow through pipes in parallel

When two reservoirs are connected by two or more pipes in parallel (refer to fig

below), the fluid can flow from one to the other by a number of alternative

routes. The difference of head h available to produce flow will be the same for

each pipe. Thus each pipe can be considered separately, entirely independent of

any other pipes running in parallel.

For incompressible flow, the steady flow by each route and the total volume rate

of flow will be the sum of the volume rates of flow in each pipe.

Incompressible flow through branching pipes; the three-reservoir problem

If the flow from the upper reservoir passes through a single pipe which then

divides and the two branch pipes lead to two separate reservoirs with different

surface levels (refer to below), sometimes it is difficult to decide the direction of

flow in one of the pipes

However, if the hydraulic gradient lines are drawn as shown, flow will be from

‘D’ to ‘B’ if the level of the hydraulic gradient at ‘D’ is above the level of the free

surface at ‘B’. If below the level of B, then flow will be in the reverse direction

i.e. from B to D.

Unfortunately, the hydraulic gradient cannot be drawn until the problem has

been solved and so its value ( g p z D D ), at D cannot be determined initially.


23/24

In many cases, the direction of flow is reasonably obvious, but if it is doubtful,

e.g. in DB, imagine that this branch is closed and calculate the value of

( g p z D D ) when there is flow from A to C only. If ( g

p z D D ) is greater than

ZB for this condition, flow will initially be from D to B when the branch DB is

opened.

At point D, it follows from continuity of flow that

321 QQQ

Question

Two sharp-ended pipes of diameters d1 = 50mm and d2 = 100mm, each of

length 100m, are connected in parallel between two reservoirs which have a

difference of level h = 10m. If the Darcy coefficient f = 0.008 for each pipe,

calculate,i. The rate of flow in each pipe

ii. The diameter D of a single pipe 100m long that would give the same

flow if it was substituted for the original two pipes


24/24

Question

Water flows from reservoir A through a pipe of diameter d1 = 120mm and

length l1 = 120m to a junction at D, from which a pipe of diameter d2 = 75mm

and length l2 = 60m leads to reservoir B, in which the water level 16m below

that in reservoir A. A third pipe of diameter d3 = 60mm and length l3 = 40m

leads from D to reservoir C, in which the water level is 24m below that in

reservoir A. Taking f = 0.01 for all the pipes and neglecting all losses other than

those due to friction, determine the volume rate of flow in each pipe

Documents

Chap6-Behaviour of Real Fluids(1).pdf