chapter 08_flow in pipes

7/28/2019 Chapter 08_Flow in Pipes

1/92

Chapter 8: FLOW IN PIPES

Department of Hydraulic EngineeringSchool of Civil Engineering

Shandong University2007

Fundamentals of Fluid Mechanics


2/92

Chapter 8: FLOW IN PIPESFundamentals of Fluid Mechanics 2

Objectives

Have a deeper understanding of laminar andturbulent flow in pipes and the analysis of fullydeveloped flowCalculate the major and minor lossesassociated with pipe flow in piping networksand determine the pumping power requirements

Understand the different velocity and flow ratemeasurement techniques and learn their advantages and disadvantages


3/92


Introduction

Average velocity in a pipeRecall - because of the no-slipcondition, the velocity at thewalls of a pipe or duct flow iszero

We are often interested only inV avg , which we usually call justV (drop the subscript for convenience)Keep in mind that the no-slip

condition causes shear stressand friction along the pipewalls

Friction force of wall on fluid


4/92


Introduction

For pipes of constantdiameter andincompressible flow

V avg stays the samedown the pipe, even if the velocity profilechanges

Why? Conservation of

Mass

Vavg Vavg

samesame

same


5/92


Introduction

For pipes with variable diameter, m is still thesame due to conservation of mass, but V 1 V 2

D2

V2

2

1

V1

D1

m m


6/92


LAMINAR AND TURBULENT FLOWS

Laminar flow: characterized bysmooth streamlines and highly ordered motion. Turbulent flow: characterized by

velocity fluctuations and highly disordered motion.The transition from laminar toturbulent flow does not occur suddenly; rather, it occurs over some region in which the flowfluctuates between laminar andturbulent flows before it becomesfully turbulent.


7/92


Reynolds Number

The transition from laminar to turbulent flow depends onthe geometry, surface roughness, flow velocity, surfacetemperature, and type of fluid, among other things.British engineer Osborne Reynolds (1842 1912)

discovered that the flow regime depends mainly on theratio of inertial forces to viscous forces in the fluid.The ratio is called the Reynolds number and isexpressed for internal flow in a circular pipe as


8/92


Reynolds Number

At large Reynolds numbers, the inertial forces are largerelative to the viscous forces Turbulent Flow At small or moderate Reynolds numbers, the viscousforces are large enough to suppress these fluctuations

Laminar FlowThe Reynolds number at which the flow becomesturbulent is called the critical Reynolds number, Re cr .The value of the critical Reynolds number is different for

different geometries and flow conditions. For example,Re cr = 2300 for internal flow in a circular pipe.


9/92


Reynolds Number

For flow through noncircular pipes, the Reynolds number is based on the hydraulicdiameter Dh defined as

Ac = cross-section areaP = wetted perimeter

The transition from laminar toturbulent flow also dependson the degree of disturbanceof the flow by surfaceroughness, pipe vibrations,and fluctuations in the flow.


10/92


Reynolds Number

Under most practicalconditions, the flow in acircular pipe is

In transitional flow, the

flow switches betweenlaminar and turbulentrandomly.


11/92


THE ENTRANCE REGION

Consider a fluid entering a circular pipe at a uniform velocity.


12/92


THE ENTRANCE REGION

The velocity profile in the fully developed region is parabolic in laminar flow and somewhat flatter (or fuller )in turbulent flow.The time-averaged velocity profile remains unchanged

when the flow is fully developed, and thus u = u(r ) only.The velocity profile remains unchanged in the fullydeveloped region, so does the wall shear stress.The wall shear stress is the highest at the pipe inlet

where the thickness of the boundary layer is smallest,and decreases gradually to the fully developed value.Therefore, the pressure drop is higher in the entranceregions of a pipe.


13/92


THE ENTRANCE REGION


14/92


Entry Lengths

The hydrodynamic entry length is usually taken to be thedistance from the pipe entrance to where the wall shear stress (and thus the friction factor) reaches within about 2percent of the fully developed value.In laminar flow, the hydrodynamic entry length is givenapproximately as

In turbulent flow, the hydrodynamic entry length for turbulent flow can be approximated as

The entry length is much shorter in turbulent flow, asexpected, and its dependence on the Reynolds number isweaker.


15/92


Entry Lengths

In the limiting laminar case of Re 2300, the hydrodynamicentry length is 115 D. In many pipe flows of practical engineering interest, theentrance effects for turbulent flow become insignificantbeyond a pipe length of 10 diameters, and thehydrodynamic entry length is approximated as

In turbulent flow, it is reasonable to assume the flow isfully developed for a pipe whose length is several timeslonger than the length of its entrance region.


16/92


LAMINAR FLOW IN PIPES

In this section we consider the steady laminar flow of anincompressible fluid withconstant properties in thefully developed region of a

straight circular pipe.In fully developed laminar flow, each fluid particlemoves at a constant axialvelocity along a streamlineand no motion in the radialdirection such that noacceleration (since flow issteady and fully-developed).


17/92



Now consider a ring-shapeddifferential volume element of radius r, thickness dr, and length dx oriented coaxially with the pipe. Aforce balance on the volumeelement in the flow direction gives

Dividing by 2 pdrdx and rearranging,


18/92



Taking the limit as dr, dx 0 gives

Substituting t = - m(du /dr ) gives the desiredequation,

The left side of the equation is a function of r , andthe right side is a function of x . The equality musthold for any value of r and x ; therefore, f (r ) = g ( x )= constant.


19/92



Thus we conclude that dP /dx = constant and we can verify that

Here t w is constant since theviscosity and the velocity profileare constants in the fullydeveloped region. Then wesolve the u(r) eq. by rearrangingand integrating it twice to give

r 2


20/92



Since u/ r = 0 at r = 0 (because of symmetry about thecenterline) and u = 0 at r = R, then we can get u (r)

Therefore, the velocity profile in fully developed laminar flowin a pipe is parabolic. Since u is positive for any r , and thusthe dP /dx must be negative (i.e., pressure must decrease inthe flow direction because of viscous effects).The average velocity is determined from


21/92



The velocity profile is rewritten as

Thus we can get

Therefore, the average velocity in fully developed laminar pipe flow is one half of the maximum velocity .


22/92


Pressure Drop and Head Loss

The pressure drop

P of pipe flow is related to the power requirements of the fan or pump to maintain flow. SincedP /dx = constant, and integrating from x = x 1 where thepressure is P 1 to x = x 1 + L where the pressure is P 2 gives

The pressure drop for laminar flow can be expressed as

P due to viscous effects represents an irreversiblepressure loss, and it is called pressure loss P L toemphasize that it is a loss.


23/92



The pressure drop represents the pressure loss P L (Noviscosity No loss)In practice, it is found convenient to express thepressure loss for all types of fully developed internalflows as

It is also called the Darcy Weisbach friction factor, named after theFrenchman Henry Darcy (1803 1858) and the German Julius Weisbach(1806 1871)


24/92



It should not be confused with the friction coefficient C f ,Fanning friction factor, which is defined asC f = 2 t w / (r V 2avg ) = f /4.

The friction factor for fully developed laminar flow in a

circular pipe

In the analysis of piping systems, pressure losses are

commonly expressed in terms of the equivalent fluid column height , called the head loss hL.

(independent of the roughness )

(Frictional losses due to viscosity)


25/92



Once the pressure loss (or head loss) is known, therequired pumping power to overcome the pressure lossis determined from

The average velocity for laminar flow in a horizontal pipeis

The volume flow rate for laminar flow through ahorizontal pipe becomes

This equation is known as Poiseuilles law, and this flow is calledHagen Poiseuille flow.


26/92


Pressure Drop and Head Loss Poiseuilles law

For a specified flow rate,the pressure drop and thus the required pumping

power is proportional to

the length of the pipe and the viscosity of the fluid,but it is inversely

proportional to the fourth power of the radius (or diameter ) of the pipe .Since

)128

( 4 D L

V V pV W p

m


27/92


Pressure Drop and Head Loss (Skipped)

In the above cases, the pressure drop equals to the headloss , but this is not the case for inclined pipes or pipeswith variable cross-sectional area.Lets examine the energy equation for steady,incompressible one-dimensional flow in terms of heads as

Or

From the above eq., when the pressure drop = the head loss?


28/92


Pressure Drop and Head Loss Inclined Pipes

Similar to the horizontalpipe flow, except there isan additional force which isthe weight component inthe flow direction whosemagnitude is

Analogous to horizontal pipe. Read by yourself


29/92



The force balance now becomes

which results in the differential equation

The velocity profile can be shown to be


30/92



The average velocity and the volume flow rate relationsfor laminar flow through inclined pipes are, respectively,

Note that > 0 and thus sin > 0 for uphill flow, and 30,but neither velocity

profile is accurate inthe buffer layer, i.e., theregion 5 < y + < 30. Also,the viscous sublayer

appears much larger inthe figure.

Turbulent Velocity Profile ( Turbulent layer )


50/92


Turbulent Velocity Profile ( Turbulent layer )(Skipped) A good approximation for the outer turbulent layer of pipeflow can be obtained by evaluating the constant B bysetting y = R r = R and u = umax , an substituting it backinto Eq. 8 46 together with k = 0.4 gives

The deviation of velocity from the centerline value umax - u iscalled the velocity defect , and Eq. 8 48 is called thevelocity defect law. It shows that the normalized velocityprofile in the turbulent layer for a pipe is independent of theviscosity of the fluid. This is not surprising since the eddymotion is dominant in this region, and the effect of fluidviscosity is negligible.

(8-48)

Turbulent Velocity Profile


51/92


Turbulent Velocity Profile(Skipped)Numerous other empirical velocity profiles exist for turbulent pipe flow. Among those, the simplest and the bestknown is the power-law velocity profile expressed as

where the exponent n is a constant whose value dependson the Reynolds number. The value of n increases withincreasing Reynolds number. The value n = 7 generallyapproximates many flows in practice, giving rise to the termone-seventh power-law velocity profile .



52/92


Turbulent Velocity Profile(Skipped)Note that the power-lawprofile cannot be used tocalculate wall shear stresssince it gives a velocitygradient of infinity there, and

it fails to give zero slope atthe centerline. But theseregions of discrepancyconstitute a small portion of

flow, and the power-lawprofile gives highly accurateresults for turbulent flowthrough a pipe.



53/92


Turbulent Velocity Profile(Skipped)

The characteristics of the flow in viscous sublayer are very

important since they set the stage for flow in the rest of thepipe. Any irregularity or roughness on the surface disturbsthis layer and affects the flow. Therefore, unlike laminar flow,the friction factor in turbulent flow is a strong function of surface roughness.The roughness is a relative concept, and it has significancewhen its height e is comparable to the thickness of thelaminar sublayer (which is a function of the Reynoldsnumber). All materials appear rough under a microscopewith sufficient magnification. In fluid mechanics, a surface ischaracterized as being rough when e > dsublayer and is said tobe smooth when e < dsublayer . Glass and plastic surfaces aregenerally considered to be hydrodynamically smooth.


54/92


The Moody Chart

The friction factor in fully developed turbulent pipe flowdepends on the Reynolds number and the relativeroughness e/D, which is the ratio of the mean height of roughness of the pipe to the pipe diameter . It is no way to find a mathematical closed form for frictionfactor by theoretical analysis; therefore, all the availableresults are obtained from painstaking experiments.Most such experiments were conducted by Prandtls studentJ. Nikuradse in 1933, followed by the works of others. The

friction factor was calculated from the measurements of theflow rate and the pressure drop .Functional forms were obtained by curve-fitting experimentaldata.

h d h


55/92


The Moody Chart

In 1939, Cyril F. Colebrook combined the available data for transition and turbulent flow in smooth as well as roughpipes into the Colebrook equation :

In 1942, the American engineer Hunter Rouse verifiedColebrooks equation and produced a graphical plot of f.In 1944, Lewis F. Moody redrew Rouses diagram into theform commonly used today, called Moody chart given in theappendix as Fig. A 12.


56/92


Th M d Ch


57/92


The Moody Chart

The Moody chart presents the Darcy friction factor for pipeflow as a function of the Reynolds number and e/D over awide range. It is probably one of the most widely acceptedand used charts in engineering. Although it is developed for circular pipes, it can also be used for noncircular pipes by

replacing the diameter by the hydraulic diameter.

Both Moody chart and Colebrook equation are accurate to15% due to roughness size, experimental error, curve fitting

of data , etc


58/92

Ob i f h M d h


59/92


Observations from the Moody chart

For laminar flow, the friction factor decreases withincreasing Reynolds number, and it is independent of surface roughness.

The friction factor isa minimum for asmooth pipe andincreases withroughnessThe data in thetransition regionare the leastreliable.


60/92

T f Fl id Fl P bl


61/92


Types of Fluid Flow Problems

In design and analysis of piping systems, 3 problemtypes are encountered

Determine p (or h L) given L, D, V (or flow rate)Can be solved directly using Moody chart and Colebrook

equation

Determine V, given L, D, pDetermine D, given L, p, V (or flow rate)

Types 2 and 3 are common engineering designproblems, i.e., selection of pipe diameters to minimizeconstruction and pumping costs

However, iterative approach required since both V andD are in the Reynolds number.


62/92


63/92

EXA MPLE 8 4 Determining the Diameter


64/92


gof an Air Duct

Heated air at 1 atm and35 C is to be transported ina 150-m-long circular plasticduct at a rate of 0.35 m 3/s, If the head loss in the pipe isnot to exceed 20 m,determine the minimumdiameter of the duct.

Solution:

This is a problem of the third type. We can solve this problem bythree different approaches:(1) An iterative approach by assuming a pipe diameter, calculating

the head loss, comparing the result to the specified head loss,and repeating calculations until the calculated head loss matchesthe specified value;

(2)Writing all the relevant equations (leaving the diameter as anunknown) and solving them simultaneously using an equationsolver;

(3)Using the third Swamee Jain formula. We will demonstrate theuse of the last two approaches.

Method (2)

The roughness is approximately zero for a plastic pipe (Table 8 2).Solving for the four equations, then we can get

Note that Re > 4000, and thus the turbulent flow assumption is verified.The diameter can also be determined directly from the third Swamee Jain formula to be

EXA MPLE 8 5 Determining the Flow Rate


65/92


gof Air in a Duct

Reconsider Example 8 4.Now the duct length isdoubled while its diameter ismaintained constant. If thetotal head loss is to remainconstant, determine the dropin the flow rate through theduct.

Solution:

Method (2)

Solving them

Then the drop in the flow rate becomes

The new flow rate can also be determined directly from the secondSwamee Jain formula to be

Minor Losses


66/92


Minor Losses

Piping systems include fittings, valves, bends, elbows,tees, inlets, exits, enlargements, and contractions.These components interrupt the smooth flow of fluid andcause additional losses because of flow separation andmixing.

The head loss introduced by a completely open valvemay be negligible. But a partially closed valve maycause the largest head loss in the system which isevidenced by the drop in the flow rate.

We introduce a relation for the minor losses associatedwith these components as follows.

Minor Losses


67/92


Minor Losses

KL is the loss coefficient (also calledthe resistance coefficient ).

Is different for each component.

Is assumed to be independent of Re(Since Re is very large).

Typically provided by manufacturer or generic table (e.g., Table 8-4 intext).


68/92

Minor Losses


69/92


Minor Losses

Total head loss in a system is comprised of major losses (in the pipe sections) and the minor losses (in the components)

If the piping system has constant diameter

i pipe sections j components

Head loss at the inlet of a pipe


70/92


Head loss at the inlet of a pipe

The head loss at the inlet of a pipe isa strong function of geometry. It isalmost negligible for well-roundedinlets ( K L = 0.03 for r /D = 0.2), butincreases to about 0.50 for sharp-

edged inlets (because the fluidcannot make sharp 90 turns easily,especially at high velocities;therefore, the flow separates at thecorners).The flow is constricted into the venacontracta region formed in themidsection of the pipe.


71/92


72/92


Whether laminar or turbulent, the fluid leaving the pipe loses all of its kinetic energy as it mixes with thereservoir fluid and eventually comes to rest


73/92



74/92


75/92

Piping Networks and Pump Selection


76/92



Two general types of networksPipes in series

Volume flow rate is

constantHead loss is thesummation of parts

Pipes in parallel

Volume flow rate is thesum of the componentsPressure loss across allbranches is the same



77/92



For parallel pipes, perform CV analysis between points Aand B

Since p is the same for all branches, head loss in allbranches is the same


78/92



79/92



The analysis of piping networks, no matter howcomplex they are, is based on two simpleprinciples:

Conservation of mass throughout the system must besatisfied.Pressure drop (and thus head loss) between two

junctions must be the same for all paths between the

two junctions.



80/92



When a piping system involves pumps and/or turbines,pump and turbine head must be included in the energyequation

The useful head of the pump (h pump,u ) or the headextracted by the turbine (h turbine,e ), are functions of volume flow rate, i.e., they are not constants.Operating point of system is where the system is inbalance, e.g., where pump head is equal to the headlosses.

Pump and systems curves


81/92


Supply curve (or characteristicor performance curves) for hpump,u : determineexperimentally by manufacturer.System (or demand) curvedetermined from analysis of fluid dynamics equationsOperating point is theintersection of supply anddemand curvesIf peak efficiency is far fromoperating point, pump is wrongfor that application.

Pump and systems curves

Examples on pages from 358 to 364In the text


82/92

EXA MPLE 8 7 Pumping Water through


83/92


Two Parallel Pipes Water is to be pumped by a 70 percent efficient motor

pump combination that draws 8 kW of electric power during operation. The minor losses and the head loss inpipes that connect the parallel pipes to the two reservoirsare considered to be negligible. Determine the total flow

rate between the reservoirs and the flow rate througheach of the parallel pipes.

Solution:

Assumptions:1 The flow is steady and incompressible.2 The entrance effects are negligible, and the flow is fullydeveloped.

EXA MPLE 8 7 Pumping Water throughll l


84/92


Two Parallel Pipes Solution:

3 The elevations of the reservoirs remain constant.4 The minor losses and the head loss in pipes other thanthe parallel pipes are said to be negligible.5 Flows through both pipes are turbulent (to be verified).

The useful head supplied by the pump to the fluid isdetermined from

EXA MPLE 8 7 Pumping Water throughll l


85/92


Two Parallel Pipes The energy equation for a control volume between these

two points simplifies to

or

WhereWe designate the 4-cm-diameter pipe by 1 and the 8-cm-diameter pipe by 2. The average velocity, the Reynolds number, the frictionfactor, and the head loss in each pipe are expressed as


86/92

EXA MPLE 8 7 Pumping Water throughT P ll l Pi


87/92


Two Parallel Pipes

This is a system of 13 equations in 13 unknowns, andtheir simultaneous solution by an equation solver gives

EXA MPLE 8 7 Pumping Water throughT P ll l Pi


88/92


Two Parallel Pipes

Note that Re > 4000 for both pipes, and thus theassumption of turbulent flow is verified.Discuss ion The two parallel pipes are identical, exceptthe diameter of the first pipe is half the diameter of thesecond one. But only 14 percent of the water flowsthrough the first pipe. This shows the strong dependenceof the flow rate (and the head loss) on diameter.


89/92

Laser Doppler Velocimetry (LDV)


90/92


pp y ( )

LDV is an optical technique to measure flow velocity atany desired point without disturbing the flow.

Laser Doppler Velocimetry (LDV)


91/92


pp y ( )

When a particle traverses these

fringe lines at velocity V , thefrequency of the scattered fringelines is.

Particles with a diameter of 1 mmThe measurement volumeresembles an ellipsoid, typicallyof 0.1 mm diameter and 0.5 mmin length.


92/92

chapter 08_flow in pipes

Documents