chapter 8: flow in pipes. 2 objectives 1.have a deeper understanding of laminar and turbulent flow...

Chapter 8: Flow in Pipes

Chapter 8: Flow in Pipes 2

OBJECTIVES

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

2. Calculate the major and minor losses associated with pipe flow in piping networks and determine the pumping power requirements

3. Understand the different velocity and flow rate measurement techniques and learn their advantages and disadvantages


INTRODUCTION

Flowing real fluids exhibit viscous effects, they:Stick to solid surfacesHave stresses within their body

From earlier we saw relationship between shear stress and velocity gradient:

τ α du/dyThe shear,τ in a fluid is proportional to velocity gradient the rate of change of velocity across the flowFor a ‘Newtonian’ fluid we can write:

τ = μ du/dyWhere μ is coefficient of velocity (or simply viscosity)


LAMINAR & TURBULENT FLOW

Injecting a dye into the middle of flow in pipe, what would we expect to happen.

All this three would happen with different flow rates.


LAMINAR & TURBULENT FLOW

Laminar flow:Motion of the fluid particles is very orderly, all particles moving in straight lines parallel to the pipe wallsEncountered when highly viscous fluid such as oil flows in small pipes or narrow passage.

Turbulent flowMotion is, locally, completely random but the overall direction of flow is one wayIn practice, most flows are turbulentThe intense mixing fluid in turbulent flow as the result of rapid fluctuation enhance momentum transfer between fluid particles which increase the friction force on the surface and thus required pumping powerThe friction factor reached maximum when flow become fully turbulent


REYNOLDS NUMBER, Re

Critical Reynolds number (Recr) for flow in a round pipe

Re < 2300 laminar2300 ≤ Re ≤ 4000 transitional Re > 4000 turbulent

Note that these values are approximate.

For a given application, Recr depends upon

Pipe roughnessVibrationsUpstream fluctuations, disturbances (valves, elbows, etc. that may disturb the flow)

Definition of Reynolds number

Reynold number is a non-dimensional number.

Has no unit!


Laminar and Turbulent Flows


Laminar and Turbulent Flows

For non-round pipes, define the hydraulic diameter Dh = 4Ac/PAc = cross-section areaP = wetted perimeter

Example: open channelAc = 0.15 * 0.4 = 0.06m2

P = 0.15 + 0.15 + 0.5 = 0.8mDon’t count free surface, since it does not

contribute to friction along pipe walls!

Dh = 4Ac/P = 4*0.06/0.8 = 0.3mWhat does it mean? This channel flow is

equivalent to a round pipe of diameter 0.3m (approximately).


The Boundary Layer


Fully Developed Pipe Flow

Comparison of laminar and turbulent flowThere are some major differences between

laminar and turbulent fully developed pipe flows

LaminarCan solve exactly

Flow is steady

Velocity profile is parabolic

Pipe roughness not important

It turns out that Vavg = 1/2Umax and u(r)= 2Vavg(1 - r2/R2)


Fully Developed Pipe Flow

TurbulentCannot solve exactly (too complex)Flow is unsteady (3D swirling eddies), but it is steady in the meanMean velocity profile is fuller (shape more like a top-hat profile, with very sharp slope at the wall) Pipe roughness is very important

Vavg 85% of Umax (depends on Re a bit)No analytical solution, but there are some good semi-empirical expressions that approximate the velocity profile shape.

Instantaneousprofiles


Types of Fluid Flow Problems

In design and analysis of piping systems, 3 problem types are encountered

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

2. Determine V, given L, D, p3. Determine D, given L, p, V (or flow rate)

1. Types 2 and 3 are common engineering design problems, i.e., selection of pipe diameters to minimize construction and pumping costs

2. However, iterative approach required since both V and D are in the Reynolds number.


Major and Minor losses

Total Head Loss( hLT) = Major Loss (hL)+ Minor Loss (hLM)

g

V

D

LfhEquationsDarcy l 2

'2

Due to wall friction Due to sudden expansion, contraction, fittings etc

g

VKhlm 2

2

K is loss coefficient must be determined for each situation

For Short pipes with multiple fittings, the minor losses are no longer “minor”!!


Major loss

Physical problem is to relate pressure drop to fluid parameters and pipe geometry

Differential Pressure Gauge- measure ΔP

PipeD

V

L

ρ μ ε

),,,,,( DLVP

Using dimensional analysis we can show that

DD

LVD

V

P

,,

2

1 2


Friction factor

g

V

D

Lfh

g

P

VD

LfPor

VfD

LPie

D

VDffactorfrictiondefine

VD

VD

D

LP

D

VD

D

L

V

P

LL

L

2

2

1

2

1

,

2

1,

,

2

1

2

2

2

2

2

2

2

2

11

2

VL

DPf

g

V

D

Lfh

L

L


Friction Factor

For Laminar flow ( Re<2300) inside a horizontal pipe, friction factor is independent of the surface roughness.

Df

D

VDf

Re,,

asiprelationshfunctionalthederivecanwellyTheoretica

onlyfie .Re

• For Turbulent flow ( Re>4000) it is not possible to derive analytical expressions.

• Empirical expressions relating friction factor, Reynolds number and relative roughness are available in literature

Re

64f For Laminar flow


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

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

• KL is the loss coefficient.

• Is different for each component.

• Is assumed to be independent of Re.

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


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


Flow at pipe inlets and losses from fittings

Rounded inlet Sharp-edged inlet

Head loss for inlets, outlets, and fittings:

where K is a parameter that depends on the geometry.For a well-rounded inlet, K = 0.1, for abrupt inlet K = 0.5(much less resistance for rounded inlet).


Bends in pipes:

Sharp bends result in separation downstream of the bend.

The turbulence in the separation zone causes flow resistance.

Greater radius of bend reduces flow resistance.


Transition losses and grade lines

Head loss due to transitions (inlets, etc.) is distributed over some distance.Details are often quite complicated.

Approximation: Abrupt losses at a point.


Piping Networks and Pump Selection

Two general types of networks

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



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



Head loss relationship between branches allows the following ratios to be developed

Real pipe systems result in a system of non-linear equations. Very easy to solve with EES!Note: the analogy with electrical circuits should be obvious

Flow flow rate (VA) : current (I)Pressure gradient (p) : electrical potential (V)

Head loss (hL): resistance (R), however hL is very nonlinear



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

The useful head of the pump (hpump,u) or the head extracted by the turbine (hturbine,e), are functions of volume flow rate, i.e., they are not constants.Operating point of system is where the system is in balance, e.g., where pump head is equal to the head losses.


Pump and systems curves

Supply curve for hpump,u: determine experimentally by manufacturer. When using EES, it is easy to build in functional relationship for hpump,u.

System curve determined from analysis of fluid dynamics equations

Operating point is the intersection of supply and demand curves

If peak efficiency is far from operating point, pump is wrong for that application.

chapter 8: flow in pipes. 2 objectives 1.have a deeper understanding of laminar and turbulent flow...

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