Fluid Mechanics Flashcards
A solid is matter that…
retains it’s shape (lump of metal)
A liquid is matter that…
deforms to the shape of its container and forms a free surface (glass of water, cup of tea, ocean, etc)
A gas is matter that…
occupies the full extent of its container (i.e. air in a balloon etc.)
A fluid is a substance which is…
either a liquid or a gas
A fluid is a matter that deforms continuously under the action of…
Shear stress
Define a system
A quantity of matter or region of space chosen for study
What resides outside the system?
The surroundings
The system is enclosed by its ____?
Boundaries
Define real and solid boundaries
Real: Solid boundaries
Imaginary: Notional surfaces that define a control volume
A system can be…(3 things)
Open, close or isolated
What is not allowed and what is allowed to cross the boundaries in a closed system?
- Mass is not because mass is fixed
- Energy is
Boundaries are permitted to ____ in a closed system?
Move
What is allowed to cross in an open system?
Mass and energy
What can cross boundaries in an isolated system?
No mass or energy can cross the boundaries
Properties can either be…(2 things)
Intensive or extensive
Define intesive properties
Intensive properties are independent of the size of the system
Define Extensive properties
Extensive properties are dependent on the extent of the system
Give two examples of inentisive properties of a system and why
Pressure and temperature - they are independent of the size of the system
Give two example of extensive properties of a system and why
Mass and volume - they are dependent of the size of the system
Define density and it’s equation and units
The mass per unit volume of a substance
Define specific volume of a substance
- Specific volume defined as how much volume holds 1 kg of fluid mass
- The specific volume is the reciprocal of the density, i.e.
Define Continuum fluid
Homogeneous, continuous matter with no gaps
When is the assumption of continuum okay?
When the size of the system is large with respect to the spacing between molecules
What is the collision and rebound amongst molecules in the system represented by in the continuum?
Thermodynamic pressure, p (M/m2)
What is it called when the continuum assumption cal no longer hold and individual particles should be considered?
Rarified gas flow theory
Define viscosity
Internal resistance of fluid to motion
As fluid passes over a solid object, the interaction with the solid causes a ____ ____to form in the region near to the solid
As fluid passes over a solid object, the interaction with the solid causes a boundary layer to form in the region near to the solid
Define inviscid and viscous flows
- Inviscid flows are idealised and have zero viscosity
- All real flows are viscous
Define Internal and external flows
- Internal flows have an inlet and outlet (i.e. a pipe)
- External flows are unbounded (flow around an aeroplane)
Define Steady and unsteady flows
- Steady flows do not change with time
- Unsteady flows are time dependent
Define laminar and turbulent flows
- In laminar flows, fluid layers pass over each other smoothly
- Turbulent flows are chaotic and seemingly random
Define compressible and incompressible flows
- For compressible flows, density can change significantly and can affect the flow
- In incompressible flows, density is a constant
Define
- One-dimensional
- Two-dimensional
- Three-dimensional flows
- All flows are three-dimensional to some degree
- We can approximate if the flow does not vary significantly in certain dimensions
- The flow through a nozzle accelerates mainly along the nozzle - can be treated as a one-dimensional flow
What is the equation and symbol for Normal stress
What is the equation and symbol for axial strain
Equation for Young’s Modulus of Elasticity
In a control volume, the pressure inside the volume is ____ ____ as the external pressure
In a control volume, the pressure inside the volume is the same as the external pressure
Equation for change in pressure
Equation of the Modulus of elasticity
K = ?
and because
þ = 1/v
K = ?
K = -v(dp/dv)
and because
þ = 1/v
K = þ(dp/dþ)
Shear strain equation and symbol
Shear stress equation and symbol
G is the…?
Shear modulus
In solids, the shear stress is ____ to the ____ ____
In solids, the shear stress is proportional** to the **shear strain
In fluids, viscous stress is ____ to the ____ of ____
In fluids, viscous stress is proportional to the rate of strain
When shear stress, Jxy(backwards J), acts on a fluid element, the deformation rate is…(define u)
da/dt = u
u is the upper surface fluid velocity
Equation for the shear strain rate
Equation given that vicous stress is proportional to the rate of strain and in the limit of L tending to zero
Vicous stress is proportional to the rate of strain
(Picture)
Limit of L tending to zero
limL→0Jxy(backwards) = u(du/dy)
In this equation;
what is the constant of proportionality,
what is it,
and what is it’s units
u
The dynamic viscosity
kg/ms
Define a Newtonian Fluid
A Newtonian fluid is defined as a fluid in which the dynamic viscosity is independent of du/dy
In a Newtonian fluid, the ____ is dependent on the shear rate
In a Newtonian fluid, the viscosity is dependent on the shear rate
- Dilatant or shear thickening fluid
- Viscosity ____ with shear rate
- Pseudoplastic or shear thinning fluids
- Viscosity ____ with shear rate
- Dilatant or shear thickening fluid -
- Viscosity increases with shear rate
- Pseudoplastic or shear thinning fluids
- Viscosity decreases with shear rate
The viscosity of a fluid varies with…?
Temperature
What’s it called to determine the value of viscosity for a gas and what is the equation?
Sutherland’s Law
This is Sutherland’s law used to determine the value of viscosity for a gas.
Define each symbol and it’s values
- μ0 =reference viscosity (kg/ms) at reference temperature T0
- T = input temperature (K)
- T0 = reference temperature (K)
- C = Sutherland’s constant for material in question
- For air,
- C = 110.56K
- T0 = 273.11K
- μ0 = 1.7894x10-5kg/ms
The viscosity of a gas ____ with temperature
The viscosity of a gas increases with temperature
Equation used determine the value of viscosity for liquids?
- For water,
- a =2.414x10-5 kg/ms,
- b = 247.8K,
- c = 140K
The viscosity of a liquid ____ with temperature
decreases
Define Kinematic viscosity
Kinematic viscosity is defined as the ratio of dynamic viscosity to density
What are the equation and units for Kinematic viscosity
m2/s
What is often used in an incompressible flow and why?
Kinematic viscosity is as density is constant in an incompressible flow
If the radius of the shaft, R, is much greater than the spacing l then…
On rotation in radians is…
2π radians
Angular velocity is…? (equation)
Where n dot is the number of rotations per unit time
What is pressure and the units for it?
Pressure is the force per unit area exerted by the fluid on the container
Pascals (N/m2)
1 bar = ?
1 atm = ?
1 psi = ?
You don’t have to remember these it’s just good too
Define Absolute pressure P<em>abs</em>
The pressure at a point relative to absolute vacuum (i.e. zero pressure)
Define Atmospheric pressure Patm
The atmospheric pressure at a point
Most pressure-measuring devices are calibrated to read ____ at atmospheric pressure
Zero
What is the difference called between absolute pressure and atmospheric pressure?
Gauge pressure, Pgauge
Pabs = ? (Two equations)
Pabs = Patm + Pgauge
Pabs = Patm - Pvac
What are pressures below atmospheric pressures called?
Vacuum pressure, Pvac
Pressure is infact a…?
Scalar - it has a magnitude but no specific volume
The pressure at a point in a fluid has the ____ ____ in all directions
The pressure at a point in a fluid has the same magnitude in all directions
P2 = ? (Manometer)
Pressure increases ____ with depth in a fluid
Linearly
Change in z or h are commonly referred to as…?
Pressure head
Why are all pressures the same here even though the weight of the water is different?
Because they are all the same height
What is Pascal’s Principle?
Pascal’s Principle:
- The pressure applied to a confined fluid increases the pressure throughout by the same amount
In an incompressible fluid this change is instantaneous
In an incompressible fluid when the Patm changes the pressure everywhere else in the fluid will change…
Instantaneous
F2/F1 = ?
F2/F1 = A2/A1
P2 = ?
Change in P = ? (Two equations)
- Pressure at point 1 must ____ the pressure at point 2
- The column of fluid with height h indicates that P2 is ____ than P1
- Pressure at point 1 must equal the pressure at point 2
- The column of fluid with height h indicates that P2 is greater than P1
The use of a manometer is based on the assumption that…
The use of a manometer is based on the assumption that the density of the manometer fluid is much higher than that of the fluid in the reservoirs
In an inclined manometer, P1 = ?
How do you use a Column Barometer and what are they commonly used for?
- Barometers are commonly used to measure ambient pressure
- Fill a tube up completely with the working fluid
- Place it upside down in an open vessel of the working fluid
- The top chamber is essenDally a vacuum
- The height of the column of working fluid above the top of the open vessel gives the ambient pressure
Equation for ambient pressure, Pa
1atm of mercury = ?
1atm = 760mmHg = 101,325Pa
What is meniscus?
When there is a bend in water - it is never quite flat
Does a liquid have surface tension?
Give an example.
Yes, pond skaters for example
What is the symbol for surface tension and units?
N/m
Define the Capillary effect
When surface tension affects fluid in small-diameter tubes
Define concave and convex meniscus
What is the equation for the weight of the fluid of the column?
Define the contact angle, Ø
The angle that the tangent of the liquid surface makes with the solid surface at the point of contact
For wetting liquids, Ø ? 90°
So the rise is…?
Ø < 90°
So the rise is positive
For non-wetting liquids, Ø ? 90°
So the rise is…?
Ø > 90°
So the rise is negative
Capillary effect is normally negligible for tubes of diameter ____ than 1 cm
Capillary effect is normally negligible for tubes of diameter greater than 1 cm
How do we minimise errors in manometry from the capillary effect?
- Use sufficiently wide tubes
- More dense liquids are less affected by capillary rise
Manometry is a branch of…?
Hydrostatics
Summarise the principles of hydrostatics
- There are no shearing forces within a body of static fluids
- Pressure is perpendicular to the surface on which it acts
- Pressure at any point in a fluid at rest is the same in all directions
- Pressure increases linearly with depth in a static fluid
- Pressure is proportional to the density of the fluid
- Pressure is the same at all points in the same horizontal plane in a fluid at rest
- External pressure applied at any point to an enclosed fluid is transmitted unchanged throughout the fluid (for an incompressible fluid this transmission is instantaneous)
Fluid statics is the study of…
Hydrostatics it the study of…
Aerostatics is the study of…
…fluids at rest
…liquids at rest
…gases at rest
Define centroid
The centre of the body
Force = ?
Pressure x Area
The pressure at the centroid of the surface, Pc, is the same as…?
…the average pressure acting on the surface
FR = ?
FR = PaveA
Pave = ?
Pave <span>= </span>P0 + þghc
hc is the…?
…vertical distance of the centroid from the liquid free surface
The resultant force does not act on the ____, it acts at the ____ __ ____
The resultant force does not act on the centroid, it acts at the centre of pressure
Equation for second moment of area about the x-axis
The moment from the pressure distribution must equal the moment from…?
…the resultant force
What is a pressure prism?
When,
- Area is the area of the base plate
- Height is the pressure distribution
???
Finding the hydrostastic force reduces to finding the…?
…volume of the prism and the position of its centroid
For a rectangular plate, the second moment of area passing through the centroid is given by…? (Equation)
yc =?
yc = s + b/2
If the plate is vertical, sinØ = ?
1
If the plate is horizontal, there is a…?
…uniform pressure distribution across the plate
For forces on a curved surface, how do we calculate the magnitude acting on the curved surface?
- Horizontal force - FH
- Vertical force - FV
The magnitude of the resultant force acting on the curved surface is… (picture)
If
- Horizontal force - FH
- Vertical force - FV
What is the equation for the force makes an angle with the horizontal?
FB = ÞfgV which is…?
…the weight of the liquid of the same volume as the body
What is Archimedes’ Principle?
The buoyant force acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body, and it acts upwards through the centroid of the displaced volume
Why do some bodies float and other bodies sink?
- Less dense than the surrounding fluid =
- Same density as the surrounding fluid =
- More dense than the surrounding fluid =
- Less dense than the surrounding fluid will float
- Same density as the surrounding fluid are neutrally buoyant
- More dense than the surrounding fluid will sink
How do we define the stability of a system?
- A small disturbance applied to the ball in (1) results in the ball returning to its original position - the system is stable
- A small disturbance applied to the ball in (2) results in the ball moving, but not running off or returning to its original place - the system is neutrally stable
- A small disturbance applied to the ball in (3) results in the ball rolling away and never returning to its original posiFon - the system is unstable
Rotational stability depends on the locations of…
…the centre of gravity, G, and the centre of buoyancy, B
If the centre of gravity is ____ the centre of ____ = ?
If it is top heavy = ?
If the centre of gravity is below the centre of buoyancy = More stable
If it is top heavy = bad
An object in which the centre of gravity and centre of buoyancy coincide is…?
…neutrally stable
How can a floating body can be stable even if the centre of gravity is above the centre of buoyancy?
- A disturbance causes the floating body to rotate
- The centre of buoyancy moves as the shape of the submerged body changes
- B’ is the new centre of buoyancy, and B’ and G act to create a restoring moment
What is the distance GM?
And the larger it is…?
Distance GM is the metacentric height - the larger it is, the more stable the floating body
Visulsation techniques that are commonly used include…(4 things)
- Smoke/Dye visualisation
- Schlieren and shadowgraph imagery
- Laser Induced Fluorescence
- Particle Image Velocimetry
(Don’t worry too much about this though)
What is the smoke/Dye visulisation method?
- Inject smoke into air flow, or dye into a liquid flow
- The injected material acts as a marker in the flow
Who was the first person to use the visulaisation technique?
Osborne Reynolds
What is now known as the Reynolds number?
Reynolds number is to recognise the dependence of the state of the flow on certain parameters
With an increasing Re, what happens to flows?
Reynolds number is a….?
…dimensionless number
Re = ?
Re = ? For an incrompressible flow
Define each symbol in these equations
- ρ is the fluid density
- V is an appropriate velocity scale in the flow
- L is an appropriate length scale in the flow
- Choice between V and L is arbitory, be careful with them
- μ is the dynamic viscosity of the fluid
- ν is the kinematic viscosity of the fluid
What would it look like when Re = 3000?
What would Re = 16,500 look like?
Schlieren imagery requires a…
…change in refractive index in the medium
How does Schileren Imagery work?
- Light source is collimated by a mirror and passed through the field of interest
- A second mirror refocuses the light, re- imaging the point where it is intercepted by an adjustable knife edge
- The knife edge removes bent rays of light
How does the Planar Laser Induced Fluorescence work?
- This visualisation technique uses a laser sheet to illuminate the fluid
- The fluid contains a fluorescent substance
- The laser sheet excites electrons in the medium, which then emit light waves as the electrons return to their ground state
- This technique is commonly used to investigate chemical reactions and mixing in turbulent flows
How does Particle Image Velocimetry work?
- In PIV a Laser sheet is used to illuminate the fluid
- The fluid is seeded with small particles which can be tracked by taking
- repeated photographs of the flow
- Velocity information can be obtained by tracking the particles in the flow
- This technique is a non-intrusive method of obtaining velocity information from a flow
Define streamline
A streamline is a curve whose tangent at any point is in the direction of the velocity vector at that point
Define streakline
A streakline is the path formed by the release of a series of particles at different times from a fixed location in the flow
Define pathline
A pathline is the path formed by one fluid particle as it moves in the flow
In a steady flow…
…streamlines, streaklines, and pathlines are equivalent
Can fluid cross a streamline?
Why?
No fluid can cross a streamline as the tangent of the streamline is the velocity vector at any point
The 3-D extension of a streamline is called a…?
…streamtube
All fluid entering a streamtube at one end must exit it at the other end
Total change of velocity = ?
Difference of velocity between points at the given instant + change of velocity occurring at the target point in time δt
For fluid particle motion, v = ?
v = ds/dt
Fluid problems can be treated in…
Lagrangian or Eulerian frames of reference
What is Lagrangian frame of reference?
- Apply physical principles to fluid particles
- Frame of reference moves along with the particles
What is Eulerian frames of reference
- Apply physical principles to a control volume
- Control volume is fixed in space - flow travels through it
Mass flowing per unit time = ?
Rate of increase of momentum from AB to CD = ?
Pressure force in direction of motion = ?
Pressure force opposing motion = ?
Side pressure force producing a component in direction of motion = ?
Gravitational force opposing motion = ?
Resultant force in direction of motion = ?
Weight of fluid element Angle = ?
Pressure always acts…?
Inwards
What is Bernoulli’s Equation?
Try to remember the other 3 Bernoulli equations when you intergrate
To what is Bernoulli’s equation limited?
The flow must be…(5 things)
- Steady: du/dt =0
- Frictionless: μ =0
- No work or heat transfer to system
- Incompressible: ρ = constant
- Irrotational - no vorticity
Bernoulli’s equation states the conversation of…
…total pressure
Define stagnation pressure
The stagnation pressure is the sum of the static and the dynamic pressures
What is the equation for stagnation pressure?
In a horizontal flow, or in an air flow of velocity greater than 5 m/s, changes in ρgz are ____, and stagnation pressure is ____
negligible, and stagnation pressure is constant
For aerodynamic applications, the Bernoulli equation simplifies to…?
What device uses different pressures to measure the velocity in a flow?
A Pitot Tube
What are the dimensions of this equation and what do the seperate parts respresent?
- The dimensions of this equation are of L
- p/ρg = pressure head
- v2/2g = velocity head
- z = elevation head
- H = total head - the height to which fluid rises in a pitot tube open to the atmosphere
What is Einstein’s theory of Special Relativity?
E = mc2
Mass cannot be ____ __ ____ in a fluid flow with no ____ reactions
Mass cannot be created or destroyed with no nuclear reactions
Mass of fluid entering the system per unit time = ?
Mass of fluid entering the system per unit time =
Mass of fluid leaving per unit time + increase of mass in the control volume per unit time
Define this symbol and its units
Mass flow rate (kg/s)
For a steady flow, the mass inside the control volume is…?
…constant
Mass of fluid entering the system per unit time = ?
= Mass of fluid leaving per unit time
What is the continuity equation?
For an incompressible flow, the ____ is constant, so the continuity equation is…?
For an incompressible flow, the density is constant so the continuity equation is…
What is Q in this equation and it’s units?
Q is the volumetric flow rate (m3/s)
Q1 = ? + ?
So,
A1v1 = ? + ?
Q1 = Q2 + Q3
So,
A1v1 = A2v2 + A3v3
What equation and device can be used to measure the velocity of a flow?
The continuity equation and a venturi flow meter
Simplier continuity equation ? = ?
Where
? > ?
A1V1 = A2V2
Where
A1 > A2
The value of H can be found from the…?
…Manometer reading
Manometer contains a fluid of density…?
ρman
The volumetric flow rate is Q = ?
Q = A1V1
For a Venturi meter, the discharge coefficient typically has a value of…?
0.96
Some loss of energy will occur in a real system, so a coefficient of discharge has to be introduced
Qactual = ?
Some loss of energy will occur in a real system, so a coefficient of discharge has to be introduced
Qactual = CdQtheoretical
What is it when you insert a sudden contraction into a pipe
An orifice plate
Define the vena contracta point
The point of maximum convergence which occurs slightly downstream of the orifice plate
Define the momentum of a particle or object
momentum = mv
If a fluid stream undergoes a change in velocity, there will also be a…?
…change in momentum
Newton’s 2nd Law dictates that a force is required to produce a change in momentum, so F = ?
Or F = (mv - mu) / t
Force and velocities are vector quantities, so they have a…?
…magnitude and a direction
Aplying continuity to the control volume…(equation)
Force = Rate of…
Force = Rate of change of momentum
Newton’s 3rd Law dictates that the fluid will…
…exert an equal and opposite reaction force on the surroundings
What do you do if the fluid is not travelling in a straight line?
Split the flow up into its x and y components
In general, the total force F acting on a control volume in a given direction will be made up of three components
Force exerted in the given direction on the fluid in the control volume
- by any solid body within the control volume, or coinciding with the boundaries of the control volume
- by body forces such as gravity
- by the fluid outside the control volume
Imagine a jet of fluid impacting on a fixed, symmetrical smooth vane
With no fricton or losses, the tangential velocity of the flow leaving the plate has the…
…same velocity as the initial jet
From continuity, the mass flow out of the vane must equal…
…the mass flow into the vane
For the impact of a jet on a cup, θ = ?
θ = 0°
How do you calculate the resultant force for a deflection of a jet?
Reaction force R is equal and opposite to the…?
…resultant force
What forces act on this system?
- Pressure from the pipe inlet and outlet
- Gravity
- Forces exerted by the walls of the pipe
Mechanical energy is a form of energy that can be converted to…
…work by an ideal turbine
Mechanical energy is the sum of…
…flow energy plus kinetic energy plus potential energy
The mechanical energy per unit mass of a flow is…(equation)
The change in flow state from point 1 to point 2 produces…?
…work output
What is the inflow energy per unit mass (J/kg)?
What do each seperate part of these equations represent?
When is the maximum theoretical work output obtained?
When emech2 = 0
The efficiency of the idealised system can be defined as…
Turbine efficiency can be defined as…
Pump efficiency can be defined as…
Friction again produces heat in the system which…
…reduces the useful work output
Motor efficiency can be defined as…
Generator efficiency can be defined as…
What is the combined efficiency when a motor and pump are connected together?
What is the combined efficiency when a turbine and generator are connected together?
Define these symbols
Each term in the steady flow energy equation is a measure of ____, and is a useful means to evaluate ____ ____ in systems
Each term in the steady flow energy equation is a measure of distance, and is a useful means to evaluate head loss in systems
With laminar flow, flow is…
…ordered, layers of fluid pass each other smoothly
With turbulent flow, flow is
…seemingly random, chaotic, and irregular
The transition of a flow is determined by…?
…the Reynolds number
For a pipe flow, the Reynolds number is given by…
- vm is the mean flow velocity (m/s)
- Dh is the hydraulic diameter of the pipe (m)
For a smooth walled pipe, flows of the same Reynolds number will have the same…
…characteristics, i.e. be in the same flow regime
Define the critical Reynolds number
The critical Reynolds number is the value at which the flow will become turbulent
Re < Recrit = ?
Laminar flow
Recrit ≤ Re ≤ 4,000 = ?
Transtional flow
Re > 4,000 = ?
Turbulent flow
Reynolds number for smooth walled pipes is…
The mean velocity, vm, is…
The mean velocity, vm, is the velocity at which an
idealised, inviscid fluid has the same mass flow rate
as the real flow
Mean velocity, vm = ?
Define the hydraulic diameter
The hydraulic diameter is the diameter of an equivalent circular pipe
Equation for Hydraulic diameter
And does the symbol p stand for?
P is the perimeter of the original conduit
What does the no-slip condition do and what does it form?
The no-slip condition gives the fluid next to the pipe wall a zero velocity - this forms a boundary layer
The boundary layer grows until it reaches all parts of the pipe - the flow is then said to be…?
…fully-developed
The distance from the entrance to the point where the boundary layers grow to the centreline is called the…?
…entrance length, Lh
For laminar flow, Lh/D = ?
For turbulent flow, Lh/D = ?
The wall shear stress is at its highest at…?
…the entrance of the pipe
The value of the wall shear stress reaches a constant at…?
…the end of the entrance length, Lh
What is Reynolds averaging?
A time-averaging procedure when applied to turbulent flow
Adding friction to the flow results in a pressure drop along the pipe caused by…?
…the shear stress
If there is no shear stress, there is no pressure drop and the flow becomes…?
…idealised
For fully-developed pipe flow we know that…(equation)
What is ∆PL ?
ΔPL is the pressure loss along the pipe due to viscous effects
What is the Hagen-Poiseuille equation written in terms of head loss?
What equation is used to show head loss for a laminar flow?
What is the the Darcy-Weisbach equation for a circular pipe?
f is called the friction factor
Define the friction factor
Friction factor is the ratio of the pressure drop to the dynamic pressure
Darcy’s friction factor for a laminar flow = ?
Define H (total head)
H = total head - the height to which fluid rises in a pitot tube open to the atmosphere
What gives the Energy Grade Line a negative gradient along the flow?
Frictional losses give the Energy Grade Line a negative gradient along the flow
What is the head loss, hf?
The head loss, hf, is the hydrostatic height difference between two horizontally connected reservoirs
What is the hydraulic head?
The hydraulic head is the sum of the pressure head and the elevation head
Hhyd = ? (equation and definition)
Hhyd is the height to which fluid rises in a piezometer
What is a piezometer?
A piezometer is a static pressure tap with the vertical tube open to the atmosphere
What do EGL and HGL stand for?
EGL = Energy grade line
HGL = Hydraulic grade line
What does the hydraulic grade line do?
The Hydraulic Grade Line joins the hydraulic head height through a flow
The Hydraulic Grade Line is below the Energy Grade Line by…?
v2/2g
What is the formula used to calculate the mass flow rate through a pipe with friction? (A law not a formula)
Poiseuille’s Law
For laminar flow, Poiseuille’s Law = ?
Vm = ? (Poiseuille’s law)
m dot = ? (Poiseuill’s Law) so m dot = ?
m ̇ = þ vmA so =
For a circular pipe, A = ?
A = πD2/4
For hydraulic pumping power, W =?
For uphill flow, θ ? 0
θ > 0
For downhill flow, θ ? 0
θ < 0
For Poiseuilles law on inclined pipes, what do you add into these equations?
You add sinθ
If the pipe is not circularm we must use the hydraulic diameter, so Poiseuilles Law equations become?
Notice the D2h and D4h
For the same free-stream velocity, a turbulent flow will have a much thicker boundary layer than a…?
…laminar flow
The velocity gradient next to the wall is much higher in a…?
The wall shear stress is much higher in a…?
…turbulent flow
The velocity profile for a laminar flow is…?
…parabolic in nature
The velocity profile for a turbulent flow is…?
…much fuller, with a sharp drop near the pipe wall
The turbulent profile consists of four regions…(layers)
- The viscous sublayer
- The buffer layer
- The overlap layer
- The turbulent layer
What is the viscous sublayer?
Viscous sublayer - the thin layer next to the wall where viscous effects are dominant
- The velocity profile in this layer is very nearly linear
What is the Buffer layer?
Buffer layer - turbulent effects start to become significant but the flow is still dominated by viscous effects
What is the overlap layer?
Overlap layer - turbulent effects are much more significant, but still not yet dominant
What is the Turbulent layer
Turbulent layer - turbulent effects dominate over viscous effects
How do you approximate the outer part of the turbulent velocity profile? (Equation)
The viscous sublayer is usually less than…?
…1% of the pipe diameter
How to you determine the friction factor for a turbulent pipe flow?
By using the Moody chart
How do we use the Moody chart? (5 things)
- Determine the mean velocity of the pipe flow
- Determine the Reynolds number
- Determine the pipe relative roughness ε/Dh
- Read the friction factor from the Moody Chart
- Evaluate the pressure loss
The Fanning Friction Factor is a factor of _ smaller than the Darcy Friction Factor.
So if a question comes up requiring the determination of a friction factor be sure to multiply the number obtained from the data book by…?
The Fanning Friction Factor is a factor of 4 smaller than the Darcy Friction Factor.
x4!
Losses in pipe systems fall into two catergories…?
Major and Minor losses
What are the major losses in engineering pipelines?
Frictional losses
What are the minor losses in long pipelines?
The inlet, outlet, bends and valves
In small pipe networks, separation losses due to bends, contractions, expansions may…?
…dominate the pressure loss in the system
Head losses for fittings in pipe systems can be determined by…? (Equations)
K in this equation is the…?
…loss coefficient
A sudden increase in pipe cross-section will produce loss in the system, what is the equation for this?
For a pipe flowing into a reservoir, A2 in this equation tends too…?
And so K = ?
A2 tends to infinity ∞
Losses in pipe inlets are driven by the curvature of the flow through the inlet
- For a protruding inlet, K = ?
- For a sharp-edged inlet, K = ?
- For a rounded inlet, K tends to ____ as the radius of curvature of the inlet approaches ____
- The value of K ____ as the inlet curvature ____
- K = 1.0
- K = 0.5
- K tends to zero** as the radius of the curvature of the inlet approaches **0.14D
- The value of K changes** as the inlet curvature **varies
The smaller the vena contracta, the…?
…greater the flow separation and the higher the value of K
As with the pipe inlet, the loss coefficient of the bend depends on its…?
… radius of curvature
The sharper the bend, the ____ the vena contracta and the ____ the loss coefficient becomes
The sharper the bend, the smaller the vena contracta and the higher the loss coefficient becomes
For a mitre bend,
r = ?
K ≈ ?
r = 0
K ≈ 1.1
For a flanged elbow bend,
K ≈ ?
K ≈ 0.3
InvesFgated the stability of submerged and floaFng bodies
InvesFgated the buoyant force
