Pipeline_Lec01

FLUID FLOW IN PIPES

Dr. M. Osama El-Samadony

Assistant Professor of Mechanical Power, Faculty of Engineering, KFS

University,Kafrelsheikh, Egypt.

Chapter 8: FLOW IN PIPESFundamentals of Fluid Mechanics 2

Objectives

Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow.

Calculate the major and minor losses associated with pipe flow in pipeline with a review both EGL and HGL, practice some cases and determine the pumping power. requirements

Understand the simplification and analysis of network having both series, parallel, branching(more than two tanks systems)


Our Plan

Review energy relationships in single pipes Review both EGL and HGL, practice some cases Extend analysis to progressively more complex systems

Pipes in parallel or seriesInterconnected pipe loops and reservoirs where flow direction is not obvious

Consider key factors in selection of pumps to add energy to fluid in a system


Overview of Pipe Networks

‘Pipe flow’ generally refers to fluid in pipes and appurtenances flowing full and under pressureExamples: Water distribution in homes, industry, cities; irrigationSystem components

PipesValves and bendsPumps and turbinesStorage (often unpressurized elevated tanks)


Energy Relationships in Pipe Systems

Energy equation between any two points:

Analysis involves writing expressions for hL in each pipe and for each link between pipes (valves, expansions, contractions), relating velocities based on continuity equation, and solving subject to system constraints (Q, p, or V at specific points).

2 22 2 1 1

2 1 ,2 2 pump turb L f

p V p Vz z h h h

g g

2 1 ,pump turb L fE E h h h


Energy Grade Line (EGL) and Hydraulic Grade Line (HGL)

Graphical interpretations of the energy along a pipeline may be obtained through the EGL and HGL:

EGLp V

gz

2

2

HGLp

z

EGL and HGL may be obtained via a pitot tube and a piezometer tube,respectively

In our discussion we will be taking atmospheric pressure equal to zero, thuswe will be working with gage pressures


Energy Grade Line (EGL) and Hydraulic Grade Line (HGL)

EGLp V

gz

2

2HGL

pz

h hL f - head loss, say, due to friction

z1


Energy Losses in Piping Systems

Darcy-Weisbach equation for headlosses in pipes (major headlosses):

2

2L

l Vh f

D g

For estimating friction factor f the type of fluid flow in the pipe has to be studied.


LAMINAR AND TURBULENT FLOWS

Laminar flow: characterized by smooth streamlines and highly ordered motion.

Turbulent flow: characterized by velocity fluctuations and highly disordered motion.

The transition from laminar to turbulent flow does not occur suddenly; rather, it occurs over some region in which the flow fluctuates between laminar and turbulent flows before it becomes fully turbulent.


Reynolds Number

The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid, among other things.

British engineer Osborne Reynolds (1842–1912) discovered that the flow regime depends mainly on the ratio of inertial forces to viscous forces in the fluid.

The ratio is called the Reynolds number and is expressed for internal flow in a circular pipe as


Reynolds Number

At large Reynolds numbers, the inertial forces are large relative to the viscous forces Turbulent Flow

At small or moderate Reynolds numbers, the viscous forces are large enough to suppress these fluctuations Laminar Flow

The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number, Recr.

The value of the critical Reynolds number is different for different geometries and flow conditions. For example, Recr = 2300 for internal flow in a circular pipe.


Reynolds Number for Non-circular Cross-sections

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

Ac = cross-section areaP = wetted perimeter

The transition from laminar to turbulent flow also depends on the degree of disturbance of the flow by surface roughness, pipe vibrations, and fluctuations in the flow.


Reynolds Number

Under most practical conditions, the flow in a circular pipe is

In transitional flow, the flow switches between laminar and turbulent randomly.


LAMINAR FLOW IN PIPES

We consider steady, laminar, incompressible flow of a fluid with constant properties in the fully developed region of a straight circular pipe.

In fully developed laminar flow, each fluid particle moves at a constant axial velocity along a streamline and the velocity profile u(r) remains unchanged in the flow direction. There is no motion in the radial direction, and thus the velocity component in the direction normal to the pipe axis is everywhere zero.There is no acceleration since the flow is steady and fully developed.

Free-body diagram of a ring-shaped differential fluid element of radius r, thickness dr, and length dx oriented coaxially with a horizontal pipe in fully developed laminar flow.


Free-body diagram of a fluid disk element of radius R and length dx in fully developed laminar flow in a horizontal pipe.

Boundary conditions

Maximim velocity at centerline

Velocity profile

Average velocity


Pressure Drop and Head Loss

A pressure drop due to viscous effects represents an irreversible pressure loss, and it is called pressure loss PL.

pressure loss for all types of fully developed internal flows

dynamic pressure

Darcy friction factor

Circular pipe, laminar

Head loss

In laminar flow, the friction factor is a function of the Reynolds number only and is independent of the roughness of the pipe surface.

The head loss represents the additional height that the fluid needs to be raised by a pump in order to overcome the frictional losses in the pipe.


The relation for pressure loss (and head loss) is one of the most general relations in fluid mechanics, and it is valid for laminar or turbulent flows, circular or noncircular pipes, and pipes with smooth or rough surfaces.

The pumping power requirement for a laminar flow piping system can be reduced by a factor of 16 by doubling the pipe diameter.

Horizontal pipe

Poiseuille’s 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 diameter of the pipe.


Pressure Drop and Head Loss

In the above cases, the pressure drop equals to the head

loss, but this is not the case for inclined pipes or pipes with variable cross-sectional area.

Let’s 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?


Laminar Flow in Noncircular Pipes

Friction factor for fully developed laminar flow in pipes of various cross sections


TURBULENT FLOW IN PIPES

Most flows encountered in engineering practice are turbulent, and thus it is important to understand how turbulence affects wall shear stress.

However, turbulent flow is a complex mechanism. The theory of turbulent flow remains largely undeveloped.

Therefore, we must rely on experiments and the empirical or semi-empirical correlations developed for various situations.


TURBULENT FLOW IN PIPES

The intense mixing in turbulent flow brings fluid particles at different momentums into close contact and thus enhances momentum transfer.

Most flows encountered in engineering practice are turbulent, and thus it is important to understand how turbulence affects wall shear stress.

Turbulent flow is a complex mechanism dominated by fluctuations, and it is still not fully understood.

We must rely on experiments and the empirical or semi-empirical correlations developed for various situations.

Turbulent flow is characterized by disorderly and rapid fluctuations of swirling regions of fluid, called eddies, throughout the flow.

These fluctuations provide an additional mechanism for momentum and energy transfer.

In turbulent flow, the swirling eddies transport mass, momentum, and energy to other regions of flow much more rapidly than molecular diffusion, greatly enhancing mass, momentum, and heat transfer.

As a result, turbulent flow is associated with much higher values of friction, heat transfer, and mass transfer coefficients


TURBULENT FLOW IN PIPES (Skipped)

Turbulent flow is characterized by random and rapid fluctuations of swirling regions of fluid, called eddies, throughout the flow.

These fluctuations provide an additional mechanism for momentum and energy transfer.

In laminar flow, momentum and energy are transferred across streamlines by molecular diffusion.

In turbulent flow, the swirling eddies transport mass, momentum, and energy to other regions of flow much more rapidly than molecular diffusion, such that associated with much higher values of friction, heat transfer, and mass transfer coefficients.


Turbulent Shear Stress

Eddy motion and thus eddy diffusivities are much larger than their molecular counterparts in the core region of a turbulent boundary layer.

The velocity profiles are shown in the figures. So it is no surprise that the wall shear stress is much larger in turbulent flow than it is in laminar flow.

Molecular viscosity is a fluid property; however, eddy viscosity is a flow property.


Turbulent Velocity Profile

Typical velocity profiles for fully developed laminar and turbulent flows are given in Figures.

Note that the velocity profile is parabolic in laminar flow but is much fuller in turbulent flow, with a sharp drop near the pipe wall.


Turbulent Velocity Profile

Turbulent flow along a wall can be considered to consist of four regions, characterized by the distance from the wall.

Viscous (or laminar or linear or wall) sublayer: where viscous effects are dominant and the velocity profile in this layer is very nearly linear, and the flow is streamlined.

Buffer layer: Next to the viscous sublayer, viscous effects are still dominant: however, turbulent effects are becoming significant.

Overlap (or transition) layer (or the inertial sublayer): the turbulent effects are much more significant, but still not dominant.

Outer (or turbulent) layer: turbulent effects dominate over molecular diffusion (viscous) effects.


Turbulent Velocity Profile (Skipped)

The Viscous sublayer (next to the wall):The thickness of this sublayer is very small (typically, much less than 1 % of the pipe diameter), but this thin layer plays a dominant role on flow characteristics because of the large velocity gradients it involves.

The wall dampens any eddy motion, and thus the flow in this layer is essentially laminar and the shear stress consists of laminar shear stress which is proportional to the fluid viscosity.

The velocity profile in this layer to be very nearly linear, and experiments confirm that.


Effect of Viscous Sub-layer on flow resistance


The Moody Chart

The friction factor in fully developed turbulent pipe flow depends on the Reynolds number and the relative roughness /D.

Colebrook equation (for smooth and rough pipes)

Explicit Haaland equation

The friction factor is minimum for a smooth pipe and increases with roughness.


The Moody Chart

Estimating f Graphically


The Moody Chart

The Moody chart presents the Darcy friction factor for pipe flow as a function of the Reynolds number and /D over a wide range. It is probably one of the most widely accepted and 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 to ±15% due to roughness size, experimental error, curve fitting of data, etc


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

• The friction factor is a minimum for a smooth pipe and increases with roughness. The Colebrook equation in this case ( = 0) reduces to the Prandtl equation.

• The transition region from the laminar to turbulent regime is indicated by the shaded area in the Moody chart. At small relative roughnesses, the friction factor increases and approaches the value for smooth pipes.

• At very large Reynolds numbers (to the right of the dashed line on the Moody chart) the friction factor curves corresponding to specified relative roughness curves are nearly horizontal, and thus the friction factors are independent of the Reynolds number. The flow in that region is called fully turbulent flow or just fully rough flow because the thickness of the viscous sublayer decreases with increasing Reynolds number, andit becomes so thin that it is negligibly small compared to the surface roughness height.The Colebrook equation in the fully rough zone reduces to the von Kármán equation.

Observations from the Moody chart

4/1Re/316.0fBlasius formula:


Types of Fluid Flow Problems

1. Determining the pressure drop (or head loss) when the pipe length and diameter are given for a specified flow rate (or velocity)

2. Determining the flow rate when the pipe length and diameter are given for a specified pressure drop (or head loss)

3. Determining the pipe diameter when the pipe length and flow rate are given for a specified pressure drop (or head loss)

The three types of problems encountered in pipe flow.

To avoid tedious iterations in head loss, flow rate, and diameter calculations,these explicit relations are used


Types of Fluid Flow Problems (Again)

Explicit relations have been developed which eliminate iteration. They are useful for quick, direct calculation, but introduce an additional 2% error (Swamee- Jain Eqns.)


Alternative Equations for Flow - Headloss Relationships in Turbulent Pipe Flow

Hazen-Williams equation – widely used for hL as function of flow parameters for turbulent flow at typical velocities in water pipes:

0.540.630.849 LhHW

hV C R

l

1.85

4.87 1.85

110.7L

HW

Qh l

D C

2 4

4 2flow

hwetted

D D R

D

AR

P

Coefficients shown are for SI units; for BG units, replace 0.849 by 1.318 and 10.7 by 4.73.


Comparison of Equations for Transitional and Turbulent Curves on the Moody Diagram

Darcy - Weisbach Hazen-Williams* Manning*

V

Q

hL (=S*l)

12 fhgD

l f

0.50 0.50 0.502 g D S f

0.63 0.540.849 HW hC R S

0.63 0.540.354 HWD S C

0.67 0.501hR S

n0.67 0.50 1

0.397D Sn

2.50 0.50 0.502

4

gD S f 2.63 0.540.278 HWD S C 2.67 0.50 1

0.312D Sn

22 5

8

f l

Qg D

1.854.87 1.85

110.7

HW

lQ

D C2

5.33 2

110.3

lQ

D n

*Coefficients shown are for SI units (V in m/s, and D and Rh in m); for BG units (ft/s and ft), replace 0.849 by 1.318; 0.354 by 0.550; 0.278 by 0.432; 10.7 by 4.73;1/n by 1.49/n; 0.397 by 0.592; 0.312 by 0.465; and 10.3 by 4.66.


1.85fh KQ

2fh k Q

Summary


Minor Losses

Piping systems include fittings, valves, bends, elbows, tees, inlets, exits, enlargements, and contractions.

These components interrupt the smooth flow of fluid and cause additional losses because of flow separation and mixing.

The head loss introduced by a completely open valve may be negligible. But a partially closed valve may cause the largest head loss in the system which is evidenced by the drop in the flow rate.

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


Minor Losses

• KL is the loss coefficient (also called the 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.


Minor Losses

The minor loss occurs locally across the minor loss component, but keep in mind that the component influences the flow for several pipe diameters downstream.

This is the reason why most flow meter manufacturers recommend installing their flow meter at least 10 to 20 pipe diameters downstream of any elbows or valves.

Minor losses are also expressed in terms of the equivalent length Lequiv, defined as


Nomogram of fitting equivalent length


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 diameteri pipe sections j components


Head loss at the inlet of a pipe

The head loss at the inlet of a pipe is a strong function of geometry. It is almost negligible for well-rounded inlets (KL = 0.03 for r/D = 0.2), but increases to about 0.50 for sharp-edged inlets (because the fluid cannot make sharp 90° turns easily, especially at high velocities; therefore, the flow separates at the corners).

The flow is constricted into the vena contracta region formed in the midsection of the pipe.


Head loss at the inlet of a pipe


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


Gradual Expansion and Contraction (based on the velocity in the smaller-diameter pipe)

Typos in the text


Some Examples on Energy

and Energy Losses


Total energy gradient (TEL – EGL - EL) line is equal to sum of all heads: pressure head ,velocity head, and datum head

For a fluid flow without any losses due to either friction or minor losses - the energy line would be at a constant level.

In a practical, the energy line decreases along the flow due to losses (except for pump).

A turbine in the flow reduces the energy line and a pump in the line increases the energy line.

Energy gradient lines

velocityhead

2

2

p VEL EGL TEL z

g

elevationhead (w.r.t.

datum)pressure

head


Hydraulic Grade Line (HGL )

Hydraulic gradient line is the sum of pressure head and datum head (piezometric head)

where

The hydraulic grade line lies one velocity head below the energy line and parallel to it.

zγp

HGL


Case 1: pipe flow connecting two tanks

55


Case 1: pipe flow connecting two tanks


Case 2: pipe discharge to atmosphere

57


Case 3: pipe flow connecting two tanks with cross crack

58


Case 4: pipe flow connecting two tanks with a fitted valve

c) Fully Closed.

Valve may be either: a) Fully Opened (hv assumed negligible ). b) Partially Opened (partially closed).


Case 5: Pump System


Pump Terminology

Pump head (dynamic head) – Hp

Pump discharge – Q

Pump speed – n

Pump power – P


Centrifugal Pump

62


Pump Terminology

Power P

Motor efficiency m

Pump Efficiency p

Q, H


Pump Terminology

Pump Output (Water) Power (Q in m3, H in m, γ is specific gravity and dimensionless, and P in horsepower)

Pump Input (Brake) Power

pwQ HP

p

p

Q HBP


Pump Terminology

Electric Motor Power

Typically motor efficiency is approximately 98%

p

p m

Q HMP


Pump Performance

Variable-Speed pumps – may be desirable when different operating modes require different pump head or flow

Similarity lawsQ1/Q2 = n1/n2

H1/H2 = (n1/n2)2

P1/P2 = (n1/n2)3


Pump Performance Curves

67


Pump Terminology

Static Lift (Suction head - Hs) elevation difference between pump centerline

and the suction water surface. If the pump is higher, static lift is positive. If pump is lower, static lift is negative.

Static Discharge (Discharge head - hd) elevation difference between the pump centerline and the end discharge point. If pump is higher, static discharge is negative.

Total Static Head (Hst) – sum of static lift and static discharge.

st s dH H H


Pump Terminology

Shutoff Head – head at 0 flow

Operating point – the point where the pump curve and the system curve intersect.

A system curve is a curve describing the head-flow relationship of the pipeline system.

A pump performance curve is a curve describing the head-flow relationship of the pump 2

pH a bQ cQ

sys st dynH H H


System Curve

friction losses

Hst

Hdyn

Hsy

s


Operating Point


Preventing Cavitation


Multiple-Pump combination


Pipe systems in Series


For parallel pipes, perform CV analysis between points A and B

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


Pipe systems in Parallel


EXAMPLE : Pumping Water through Two Parallel Pipes

Water at 20°C is to be pumped from a reservoir (zA = 5 m) to another reservoir at a higher elevation (zB = 13 m) through two 36-m-long pipes connected in parallel.


EXAMPLE Pumping Water through 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 in pipes that connect the parallel pipes to the two reservoirs are considered to be negligible. Determine the total flow rate between the reservoirs and the flow rate through each of the parallel pipes.

Solution:

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

developed.


EXAMPLE 8–7 Pumping Water through Two Parallel Pipes

Solution: 3 The elevations of the reservoirs remain constant. 4 The minor losses and the head loss in pipes other than

the 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 is determined from


EXAMPLE soln: Pumping Water through 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 friction factor, and the head loss in each pipe are expressed as



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



Note that Re > 4000 for both pipes, and thus the assumption of turbulent flow is verified.

Discussion The two parallel pipes are identical, except the diameter of the first pipe is half the diameter of the second one. But only 14 percent of the water flows through the first pipe. This shows the strong dependence of the flow rate (and the head loss) on diameter.


Branching Pipe systems


= 24 = 8


Graphical solution for 3 reservoir prob.

1) Intially, guess hD value (avg. of all heads is good guess).

2)find Q1, Q2, Q3 from:

3)Calculate Q,

4) recalculate Q with another hD

4) Plot hD verus Q


Pipe Equivalence Simplifcation

Two general types of networks connections are

Pipes in seriesVolume flow rate is constantHead loss is the summation of parts

Pipes in parallelVolume flow rate is the sum of the componentsPressure loss across all branches is the same

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