Unsteady Flow• Fundamental of unsteady flow
• Practical approach
• Transitional wave principle
• Water hammer
• Water hammer effect
• Water hammer protection
• Unsteady flow equations
• Rigid water column theory
• Elastic Theory
Fundamentals of unsteady flow
• Steady flow: the value of all fluid properties, as well asflow properties at any fixed point, are independent of time.
• Unsteady flow: a fluid or flow property – or more - at a given point in space (locally) varies with time.
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Fundamentals of unsteady flow
Unsteady flow may be:
1. Non-Periodic flow: shut down, start up,closure processes for hydraulic components.
2. Periodic flow: periodic injection of the Air-Gasoline mixture in spark ignition engines.
3. Random flow: occurs in turbulent flow and is absent from laminar flow.
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Fundamentals of unsteady flow
Mathematically
For steady flow
For Quasi-steady flow
For unsteady flow
• Quasi-steady flow: the change of any propertyat any point with respect to time is so small orbehaves in slow rate and could be neglected.
• For simplicity of flow analysis, it could beassumed as steady (or quasi-steady) flow insome applications.
0.0t
0.0t
0.0t
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Fundamentals of unsteady flow
• Application in practice
1. Start up operations
(transition from zero velocity to steady state operation)
2. Shut down operations
(transition from steady state velocity to static equilibrium
state (v=0.0))
3. Hydraulic component closure (valves)
(transition from steady state operation to zero velocity in
small periods)
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Calm Buoy System
• Calm Buoy
• In a typical SPM buoy / terminal application, the loading buoy(s)is anchored offshore and serves as a mooring point for tankers toload / offload their gas or fluid product for transfer between theonshore facility and the moored tanker. Although the SPM buoy isclearly a key component, a number of other components areintegral to such a system
Pipe Line End
Manifold
the terminal under• Installed on the seabed and which interfaces buoy hoses with the sea line .
CALM Buoy Body
➢ A Buoy body➢ Turn table➢ The buoy fluid product circuit➢ The anchoring system➢ The floating hoses➢ A Pipeline End Manifold (PLEM)➢ Onshore/offshore pipeline➢ Main bearing➢ Central pipe swivel
CALM Buoy Body
Water Hammer
• is a pressure surge orwave resulting when afluid in motion is forcedto stop or changedirection suddenly(Momentum Change).
• Watercommonly
hammeroccurs when
a valve is closedsuddenly at an end of apipeline system, and apressure wavepropagates in the pipe.
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Practical approach
Aircraft tank valve
Aircraft tank valve closes in 2-5 seconds.
Ventbox
High level shut-off
If aircraft tank valve fails to close, vent line is designed to handle 50 psi
Vent Tank
Aircraft manifoldOverflow
Aircraft coupling maximum 120 psi
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Practical approach
• Necessary to protect aircraft fuel systems fromexcessive fuelling pressure and surge pressures.
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►When the generator of a water turbine is disconnected from a power network, the turbine speed starts to increase. Consequently, the turbine controller closes the inflow to the turbine, thus creating water hammer in the penstock.Water hammer effects in thepenstock are created by any changes in discharge through the turbine, caused by changes in the connected power network, by the operators, or by breakdowns.
Sometimes, the entire system
becomes unstable due to themutual influence of a turbine equipped with a controller and to an unsteady
flow in the penstock. In such case, even small variations in pressure in the
penstock may increase steadily and perilously.51Monday, December 5, 2016 Flow in pipelines
Accident at Russia's Biggest Hydroelectric - Rev
00.pps
• Accident at Russia's Biggest Hydroelectric - Rev
00.pps
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Blood Hammer phenomenon
• The Blood hammer phenomenon is a suddenincrease of the upstream blood pressure in ablood vessel (especially artery or arteriole) whenthe bloodstream is abruptly blocked by vesselobstruction. The term "blood-hammer" wasintroduced in cerebral hemodynamics by analogywith the hydraulic expression "water hammer",already used in vascular physiology to designatean arterial pulse variety, the "water-hammerpulse".
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Watson's water hammer pulse is a characteristic medical signfirst described by Thomas Watson, M.D. in 1844. The waterhammer pulse is a pulse that is powerfully pulsating, similar innature to the pounding of a water hammer. This hyperdynamicpulse occurs when an increased amount of blood is pumpedwith each stroke of the left ventricle, the largest chamber ofthe heart. There is also a decreased resistance to outflow ofthe blood, leading to a widening of the range between thehighest and lowest numbers of a blood pressure reading, calledthe pulse pressure. The Corrigan's pulse, named for SirDominic Corrigan, M.D., refers to a water hammer pulse that isdetected in the carotid artery, whereas a Watson's waterhammer pulse pertains to a water hammer pulse detectedperipherally in an arm or leg.
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A pulse is the rhythmic throb of blood flow due tothe heartbeat. The pulse can be felt in many siteson the human body. Common sites for checking apulse include in the neck, at the wrist, on the insideof the elbow, behind the knee, and near the anklejoint. It can also be ascertained by assessing theheartbeats directly using a stethoscope. Both pulserate and quality reveal the underlying status of theheart and blood vessels.
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Systolic and diastolic readings constitute the numerical boundaries of blood pressure. Theyrepresent opposite ends of the cardiac cycle andthe highest and lowest levels of blood pressure fora given individual. The pulse pressure is anindicator of the force that the heart generates eachtime it contracts. In healthy adults, the pulsepressure in a seated position is approximately 40,but can rise to 100 during exercise. Some studiesindicate that the pulse pressure may be a betterprognostic indicator of clinical outcome than eitherthe systolic or the diastolic blood pressure alone.
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There are many symptoms associated with water hammer pulse, the most common of which aremuscle weakness and fatigue. Other associatedsymptoms include shortness of breath, lowerextremity swelling, and headache. Patient mayexperience chest pains and palpitations. Cardiacarrhythmia, irregular heartbeat, may occur due toimpaired electrical conduction in the heartchambers.
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•A water hammer pulse is most often associated with a
leaking aortic valve. The aortic valve is the valve that
normally keeps blood that has been pumped out of the
heart from flowing backward into the heart again. Aortic
regurgitation or leakage occurs when the valve does not
close properly, allowing blood to leak backward through it.
As a result, the left ventricle has to pump more blood than
usual, with progressive expansion due to the extra
workload. The symptoms of aortic regurgitation can range
from mild to severe, with some patients having no
symptoms for years.
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Water hammer effect
Assume:
• The difference between the energy gradeline (EL) and the hydraulicgradeline (HGL) will be neglected
• Horizontal, constant-diameter pipe.
• Neglect friction.
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Sudden Closure of gate valve, visualized by a
heavy steel spring
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Transitional wave principle
• Wave: is the substance reaction against anychange in an exerted phenomena and differsaccording to the substance itself.
Pressure wave → Substance reaction due to pressure change
Thermal wave → Substance reaction due to Thermal change
Surface wave → Substance reaction due to Height change
• Shock wave: is a produced wave due to verylarge change in the exerted phenomena in thesubstance.
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Transitional wave principle
the effect of
Pressure waves
1. Pressure waves: Transmits pressure rise/increase.
the effect of2. Expansion waves: Transmitspressure drop/decrease.
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Water hammer effect
• Where
t : time sincevalve was closed
a : velocity of pressure wave
L : Length of the pipe
ΔH : increasingin pressure head
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What is water hammer?
Water hammer (or hydraulic shock) is themomentary increase in pressure, which occurs in awater system when there is a sudden change ofdirection or velocity of the water.
When a rapidly closed valve suddenly stops waterflowing in a pipeline, pressure energy is transferredto the valve and pipe wall. Shock waves are set upwithin the system. Pressure waves travel backwarduntil encountering the next solid obstacle (orchange in density), then forward, then back again.The pressure wave’s velocity is equal to the speedof sound; therefore it “bangs” as it travels back andforth, until dissipated by friction losses.
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Anyone who has lived in an older house is familiarwith the “bang” that resounds through the pipeswhen a faucet is suddenly closed. This is an effectof water hammer.
A less severe form of hammer is called surge, aslow motion mass oscillation of water caused byinternal pressure fluctuations in the system. Thiscan be pictured as a slower “wave” of pressurebuilding within the system. Both water hammer andsurge are referred to as transient pressures.
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If not controlled, they both yield the same results:damage to pipes, fittings, and valves, causing leaksand shortening the life of the system. Neither thepipe nor the water will compress to absorb theshock.
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Investigating the Causes of Water Hammer
A water transport system’s operating conditions arealmost never at a steady state. Pressures and flowschange continually as pumps start and stop,demand fluctuates, and tank levels change. In
normal events, unforeseenaddition to these events, such as power outages and equipmentmalfunctions, can sharply change the operatingconditions of a system. Any change in liquid flowrate, regardless of the rate or magnitude of change,requires that the liquiddecelerated from its initial
be accelerated orflow velocity. Rapid
changes in flow rate require large forces that areseen as large pressures, which cause waterhammer.
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Entrained air or temperature changes of the wateralso can cause excess pressure in the water lines.Air trapped in the line will compress and will exertextra pressure on the water. Temperature changeswill actually cause the water to expand or contract,also affecting pressure. The maximum pressuresexperienced in a piping system are frequently theresult of vapor column separation, which is causedby the formation of void packets of vapor whenpressure drops so low that the liquid boils orvaporizes. Damaging pressures can occur whenthese cavities collapse.
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Summary
Water hammer refers to fluctuations caused by a suddenincrease or decrease in flow velocity. These pressurefluctuations can be severe enough to rupture a watermain. Potential water hammer problems should beconsidered when pipeline design is evaluated, and athorough surge analysis should be undertaken, in manyinstances, to avoid costly malfunctions in a distributionsystem. Every major system design change or operationchange—such as the demand for higher flow rates—should include consideration of potential water hammerproblems. This phenomenon and its significance to boththe design and operation of water systems is not widelyunderstood, as evidenced by the number and frequencyof failures caused by water hammer.
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Pressure and velocity waves in a single-conduit
frictionless pipeline following its sudden closure.
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Water hammer effect
● Pressure head at the valve
● Pressure head at the midpoint
● Pressure head at thereservoir
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The unsteady flow equations
Considering only the streamline direction, Newton’s second law gives
Where m = fluid particle mass, and s signifies the streamline direction.
Substituting the force components and mass from the figure into thisequation results in
dts s F ma m
dv
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s g dt
pA
p p
sAW sin sd
W dv
The unsteady flow equations
After some manipulation, we end up with the one-dimensional Euler equation
1 p
z
4
1 dv
s s d g dt
Expanding the particle diameter to the size of the pipe cross-section and introduction the average velocity gives a more useful equation
1 p
z
4 0
1 dv
s s D g dt
Where D is the pipe diameter and τ0 is not directly useful, we will substitute a
relation between τ0 and the Darcy-Weiscach friction factor ƒ. The result of
this substitution is
1 p
z
f V 2
1 dv
s s D 2g g dt
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The unsteady flow equations
Recognizing that z is a function only of s and represents the elevation abovesome datum of the pipe centerline, we can change the partial derivative to
a total derivation. Finally, the equation has the form
1 p
dz
f V 2
1 dv
s ds D 2g g dt
This equation is valid for:
1- Compressible/ incompressible flow,
2- Steady/ unsteady flow,
3 Real/ Ideal flow,
4 Rigid and elastic pipe.
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The unsteady flow equations
• Assuming
1. Ideal flow
Neglect viscous force
2. Steady flow
1 p
dz
1 dv
s ds g dt
D 2g
2
f V
0.0
dv 0.0
dt
V fn(s,t)
dt s dt t
dv v
ds v
dts t
dv
v ds
v
t
v 0.0
dv
Vv
dt s
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The unsteady flow equations
0.01 p
dz
1 dv2
s ds g 2ds
p Z E
2g
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• For incompressible flow (γ = constant)
v2
Euler equation forsteady ideal flow
Bernoulli’s equation
Rigid Water Column Theory
• 1- Neglect compressibility of the fluid i.e.,
ρ = Constant
• 2- neglect Elasticity of the pipe,
D = Constant
0.0
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1 p
dz
1 dv2
s ds g 2ds
Flow establishment in horizontal pipe
Because the pressure head p1/ω = constant = H0 and because p2/ ω = 0 for t>0
ds 1 dV
dsL ds L 2gD L g dt
1 pds
dzds
fV2
L s
g dt
(dz/ds) = 0, and V is a function of time only, assume the ƒ-value in unsteady
flow is the same as for a steady flow at a velocity equal to the instantaneous
value.
p1 p2
fL V 2
LdV
2gD
H 0
fL V 2
L dV
2gD g dt
0
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2gD
fL V 2
H g
Integration is performed by separating the variables to form
dt L
dV
Flow establishment in horizontal pipe
The integration gives the following equation for the time necessary toaccelerate the flow to given velocity V
Where log denotes natural logarithm. Recognizing thatsteady state velocity, the equation for t becomes
fL
LD fL
V
V
t 2gDH0
2gDH0
log2gfH0
t LV0 log
V0 V
2gH0 V0 V
V0 2gH0 D / fL , the
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Flow establishment in horizontal pipe
gH0
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• It is important to note that as steady flow isapproached, V → V0 and as a consequence t → ∞.
Of course this answer is unacceptable so wepropose that when V = 0.99 V0, we haveessentially steady flow. With this interpretation,
t 2.65LV0
Flow establishment in horizontal pipe
• When the valve isclosed pressure iseverywhere equalto H0.
• When the valve issuddenly opened.The pressure atthe valve dropsinstantly to zeroand the fluidbegins toaccelerate.
100
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Example.1
A horizontal pipe 24” inside diameter, 10,000 ftlong leaves a reservoir 100 ft below surface andterminates with a closed valve. If the valve openssuddenly. How long would the velocity takes toreach 99% of its final value, neglecting minorlosses valve. Friction factor of 0.018 is to beassumed constant during the acceleration phase
• Given:
d = 24” = 2 ft L = 104 ft Ho = 100 ft F = 0.018= const
• Required:
t for a velocity of 0.99Vo
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Example.1
• Solution
flo
2gdHo
0.018104
232.22100 8.45 ft /s
V
t lVo ln
Vo V
2gHo Vo V
ln232.2100 8.46 (10.99)
104 8.46 8.46 (10.99)t
t 70 second
102
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Water Hummer Protection
Using Accumulators and Air Chambers
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