REE 307Fluid Mechanics II

Lecture 1Sep 27, 2017

Dr./ Ahmed Mohamed Nagib Elmekawy

Zewail City for Science and Technology

Course Materials

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drahmednagib.com

COURSE OUTLINE

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• Fundamental of Flow in pipes• Losses in valves and connections.• Analysis of pipe networks (Pipes in Series -Pipes in Parallel -

Branching Pipes -Networks of Pipes)• The Boundary layer • The Differential and Integral Equations of the Boundary Layer• The Displacement and Momentum Thickness• Approximate Solutions of The Incompressible Laminar and

Boundary Layers• Unsteady Flow in Conduits (Oscillation of Liquid in a U-Tube,

Water Hammer Phenomena, Surge tanks).• The Navier-Stokes equations, Stokes' hypothesis

References

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• Munson, “Fundamental of Fluid Mechanics”, 7th Edition• White F. M., “Fluid Mechanics”, 8th Edition• Cengel Y., “Fluid Mechanics Fundamentals and Applications”, 3rd

Edition• Menon, Gas Pipeline Hydraulic• Gary Z. Waters, "Analysis and control of unsteady flow in

pipelines, 1979

Prerequisite Course:

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• Fluid Mechanics - ENGR 207

Classification of fluids - Definition of viscosity – surface tension -Hydrostatic pressure- Buoyancy - Bernoulli’s equation and its applicationfor ideal fluid - stream lines- velocity and acceleration in twodimensional flow – Differential Analysis of fluid flow (continuityequation – Navier-Stokes equations) - Moody diagram - IncompressibleFlow through Networks of Pipes – Unsteady Flow in Conduits

Fundamentals of Flow in Pipelines

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1. Incompressible flow through pipes2. Branching Pipe system3. Network pipe system4. Unsteady flow5. Compressible Flow in Pipes.

Revision

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Mechanics

Statics

Dynamics

Kinematics

Fluid

Compressible

Incompressible

Fluid Mechanics

Revision

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Mechanics

Statics: Concerned with the analysis of loads on physical system in static equilibrium

Dynamics: Concerned with the effect of forces on the motion of objects

Kinematics: Concerned with the space-time relationship of a given motion without considering the origins of forces

Revision

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Fluid

Liquids take the shape of the container and have a free surface

Gases: take the shape of the container but have no free surface

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• Fluids can sustain tension, compression, but can not withstand shear stresses, and therefore it is subjected to a continuous deformation.

• Solids bear tension, compression and shear stresses, and a deformation occurs in matter in case of failure:

1. Fracture 2. Yield

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Fluids

Gases

(compressible fluids)

𝑑𝜌

𝑑𝑝≠ 0.0

Liquids

(Incompressible fluids)

𝑑𝜌

𝑑𝑝= 0.0

• At lower Mach numbers (<0.3), Gases could be considered an incompressible fluid.

Revision

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Fluid Mechanics

Fluid Statics

Aerostatics

Hydrostatics

Fluid Kinematics

Fluid Dynamics

Aerodynamics (Gas Dynamics)

Hydrodynamics

Hydraulics

Revision

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Revision

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• Continuity Equation𝑚. = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝐴 × 𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

• Momentum Equation

𝑃

𝜔+ 𝑍 +

𝑉2

2𝑔= 𝑐𝑜𝑛𝑠𝑡

• Fluid Dynamics𝐹 = 𝑚𝑎

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• Fluid PropertiesDensity (𝜌), Viscosity (𝜐), Surface tension (𝜎)

• Flow PropertiesPressure (𝑃), Velocity (𝑉)

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• Assumptions1. Incompressible Flow2. 1D Flow3. Single Phase Flow4. Steady Flow

𝜕

𝜕𝑡𝐹𝑙𝑢𝑖𝑑 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 = 0

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• Pressure𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝐹𝑜𝑟𝑐𝑒/𝐴𝑟𝑒𝑎

psi = ponds per square inchPa = pascals = N/m2

bar = 105 pascalsatm = 1.013 × 105 pascals

0 psi gauge pressure= 14.7 psi absolute

Revision

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Pressure• Pressure ls also reported as height a liquid (water or

mercury) will rise in • A column with that pressure at the base of the column.

ft or m waterin or mm mercury

Friction Loss in pipes

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Head Loss Equations

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• Head loss equations are empirical relationships that predict head loss in pipes (or other conveyances).

• The four most common equations are:o Darcy-Weisbach: Most accurate and flexible but

relatively difficult to apply.o Hazen-Williams: Most commonly used in water

network modeling.o Colbrook: Most commonly used in network

modeling. o Manning: Widely used for wastewater, drainage and

open channel flow.

Darcy-Weisbach

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ℎ𝑙 =𝑓𝑙𝑉2

2𝑔𝑑ℎ𝑙 =

0.8 𝑓𝑙𝑄2

𝑔𝑑5

h = head loss f = friction factorL = length d = diameter V= velocity g = acceleratlon due to gravity

Friction factor depends on pipe roughness and Reynolds Number,𝑅𝑒 = 𝑉𝐷/𝜐

Friction factor can be estimated from a Moody diagram. However, the difficulty wlth the use of the Darcy Weisbach equation ls that the frlction factor ls not constant for a given pipe.

Moody Chart

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Hazen-Williams

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ℎ =𝑘𝑙

𝑑1.16𝑉

𝐶

1.85

h = head loss d = diameter ( ft or m)k= 6.79 for V in m/s, D in mk= 3.02 for V in ft/s, D in ftV= velocityC= Hazen-Williams factorL = length

Hazen-Williams

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ℎ =𝑘𝑙

𝑑1.16𝑉

𝐶

1.85

C can be estimated from field measurements. The table on the next page provides initial estimates for C for pipes of different material, age and diameter. These estimates should be used with care and field checked when possible.C-factors range from 150 for very smooth pipes to 20 for very rough pipes. For rough pipes at high velocity, the C-factor can vary significantly and should be field tested.

Hazen-Williams

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Manning Equation

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𝑉 = 𝐶𝑜𝑅Τ2 3 ℎ

𝐿

0.5/𝑛

Co = 1.49 for English units and 1.0 for metric unitsV= velocity (ft/s or m/s)R = Hydraulic Radius = Cross sectional area / wetted

perimeter (ft or meters)h = head loss ( ft or m)L = lengthn = Manning’s roughness coefficient as follows

Manning Equation

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n = Manning’s roughness coefficient as follows

Comparison of friction equations

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Darcy – Weisbach Manning Hazen-Williams

All fluids Water only Water only

Difficult to get f Easy to get n Easy to get C

Good for all Roughness

Rough flow Smooth flow

Not commonly used Commonly used Commonly used

Minor Losses

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Minor losses caused by fittings, bends, valvesDescribed by coefficient K In h = KV2/2g Where,K = minor loss coefficienth = head loss due to minor loss

See following table for K representative values

Minor Losses Coefficient Table [K]

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Minor Losses Coefficient Table [K]

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Minor Losses Coefficient [K] fr valves

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For valves, a flow coefficient Cv is frequently given which defines the flow (gpm) that will pass through a valve at a pressure drop of 1 psi.

Cv can be converted to K, the minor loss coefficient:

𝐾 =888𝐷4

𝐶𝑣2

D is diameter in inches. Cv is a function of D, while K is independent of D.

Minor Losses

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Minor loss can also be given in terms of equivalent length of pipe that would give same head loss.(L/D) = K/fWhere,L = length added to account for minor loss D = pipe diameterf = Darcy Weisbach friction factor

The Energy and Hydraulic Gradient Lines

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• The Energy Line is a line that represent the total head available to the fluid• The Hydraulic Grade Line is a line that represent the total head available to the fluid minus the velocity head

Pipeline Design

Why do we need to study pipelines?

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Pipelines affect daily lives in most parts of the world. Modern people's lives are based on an environment in which energy plays a predominant role. Oil and gas are major participants in the supply oJ ener9y , and pipelines are the primary means by which they are transported.

Major factors that affect pipeline system design:

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• Fluid properties• Design conditions• Supply and demand magnitude/locations• Codes and standards• Route , topography, and access• Environmental impact• Economics• Hydrological impact• Seismic and volcanic impacts• Materiai• Construction• Operation• Protection• Long-term integrity

Classification of Pipelines:

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Classification of Pipelines:

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How to design a pipeline?

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1. Select pipe material2. Select/Design pipe diameter3. Select pipe thickness4. Select pumping/ compressor unit5. Select primover (Electric motor/ diesel engine/ gas turbine)

1. Selecting pipe material

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• When we select the material we must not that:1. No chemical reaction between pipe material and fluid material

(erosion, corrosion)2. Low roughness3. Low cost

2. Selecting pipe diameter

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𝑉 =𝑄

𝐴

Where V ranges between 1 and 3 m/s, because:1. At high velocities (high pressure drop, high friction loss)2. At low velocities (Deposition of suspending material in the

pipe)

3. Selecting pipe thickness

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𝑡 =𝑃𝐷

2𝑆𝑡

Wheret = Maximum required thickness, mmP= Maximum allowable working pressure, MpaD= Outside diameter of cylinder, mmSt= Maximum allowable stress value at the operating temperature of the metal

4. Selecting pipe thickness

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Power required by the pump determined by:1. Power loss2. Starting pressure head3. Flow propertiesWhere

𝑃𝑜𝑤𝑒𝑟 =𝜔𝑄𝐻

𝜂

5. Selecting Primover

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• Electric motor, Gas turbine, Steam turbine, Diesel engine, .. etc• Selecting the primover depends on:1. Speed2. Used source of energy3. Size

Branched Pipe System

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• Pipe in Series• Pipe in Parallel

Branched Pipe System

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Supply at several points

Branched Pipe System

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• Three Tank Problem

Branched Pipe System

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• Three Tank Problem