Introduction A fluid is a substance which cannot preserve its own shape.. A fluid is a material substance which cannot sustain a shearing stress when it is at rest. A fluid can be either a liquid or a gas. A liquid retains a set volume and has a surface of separation above which is a vapour and below which is a liquid. A gas, whatever its quantity, when placed in an empty closed vessel will fill the vessel completely. It is important to note that a gas is generally considered to be an compressible fluid and a liquid is generally considered to be incompressible. In practice a liquid is compressible but generally very large pressures are required to cause quite small changes of volume. Introduction.... Thermodynamic Properties Fluids can be either liquids or gases. A liquid is hard to compress and takes the shape of the vessel containing it. However it has a fixed volume and has an upper level surface. Gas is easy to compress, and expands to fill its container. There is thus no free surface. Liquids are generally assumed to be incompressible fluids and gases compressible fluids. Liquids ar only compressible when they are highly pressurised, and the compressibility of gases may be disregarded whenever the change in pressure is very small. Important characteristics of fluids from the viewpoint of fluid mechanics are density, pressure, viscosity, surface tension, and compressibility. This section includes brief notes on these important characteristics. Symbols / Units A = Area (m 2 ) a = Speed of sound (m/s) g = acceleration due to gravity (m/s 2 ) h = fluid head (m) K = Bulk modulus (MPa ) M = mach number u /a M = Molecular weight p = fluid pressure (N /m 2 ) p abs - absolute pressure (N /m 2 ) p gauge - gauge pressure (N /m 2 ) p atm - atmospheric pressure (N /m 2 ) p s = surface pressure (N /m 2 ) u = fluid velocity (m/s) v = fluid velocity (m/s) x = depth of centroid (m) β = Compressibility (1/MPa) γ = Surface Tension (N/m) θ =slope (radians) ρ = density (kg/m 3 ) ρ r = density (kg/m 3 ) τ = shear stress
Transcript
Introduction
A fluid is a substance which cannot preserve its own shape..A fluid is a material substance which cannot sustain a shearing stress when it is at rest.
A fluid can be either a liquid or a gas. A liquid retains a set volume and has a surface of separation above which is a vapour and below which is a liquid. A gas, whatever its quantity, when placed in an empty closed vessel will fill the vessel completely. It is important to note that a gas is generally considered to be an compressible fluid and a liquid is generally considered to be incompressible. In practice a liquid is compressible but generally very large pressures are required to cause quite small changes of volume.
Introduction.... Thermodynamic Properties
Fluids can be either liquids or gases. A liquid is hard to compress and takes the shape of the vessel containing it. However it has a fixed volume and has an upper level surface. Gas is easy to compress, and expands to fill its container. There is thus no free surface. Liquids are generally assumed to be incompressible fluids and gases compressible fluids. Liquids ar only compressible when they are highly pressurised, and the compressibility of gases may be disregarded whenever the change in pressure is very small.
Important characteristics of fluids from the viewpoint of fluid mechanics are density, pressure, viscosity, surface tension, and compressibility. This section includes brief notes on these important characteristics.
Symbols / Units A = Area (m2)a = Speed of sound (m/s)g = acceleration due to gravity (m/s2 )h = fluid head (m)K = Bulk modulus (MPa ) M = mach number u /aM = Molecular weightp = fluid pressure (N /m2 )pabs - absolute pressure (N /m2 )pgauge - gauge pressure (N /m2 )patm - atmospheric pressure (N /m2 )p s= surface pressure (N /m2 )Q = Volume flow rate (m3 /s)R = Gas Constant (J/(kg.K)Ro = Universal Gas Constant (J/(kg.mol.K) ρ = fluid density (kg /m2 )
u = fluid velocity (m/s) v = fluid velocity (m/s) x = depth of centroid (m)β = Compressibility (1/MPa)γ = Surface Tension (N/m)θ =slope (radians)ρ = density (kg/m3)ρ r = density (kg/m3)τ = shear stress (N /m2)μ = viscosity (Pa.s) ν kinematic viscosity (m2�s-
1) υ = Specific volume (m3 / kg)γ= Ration of Specific Heats
Density
The mass per unit volume of material is called the density, which is generally expressed by the symbol ρ. The density of a gas changes according to the pressure, but that of a liquid may be considered constant unless the relevant pressures are very high The units of density are kg/m3 (SI). The ratio of the density of a material ρ to the density of water ρw ( at 4o C ) , is called the relative density, which is expressed by the symbol ρr (This is often called the specific gravity a term which is sometimes confusing..)
ρr = ρ /ρw
The density of gases gases and vapours are greatly affected by the pressure . For so called perfect gases the density can be calculated from the formula .
Ro = the universal Gas constant = 8314 J/(kg.K) and M = Molecular weight. Therefore R = 8314/M [ J/(kg.K) ]
The reciprocal of density, i.e. the volume per unit mass, is called the specific volume, which is generally expressed by the symbol υ
υ = 1/ρ
The dimensional formula for density = ML-3 and the dimensional formula for specific volume = M-1L3
Pressure
A fluid is always subject to pressure. Pressure is the force per unit area at a point. The absence of pressure occurs in a complete vacuum. A complete vacuum is really a theoretical concept.
The normal pressure experienced on the surface of the earth is called the atmospheric pressure and, in general, pressures are measured relative to the local atmospheric pressure. These measured pressures are called gauge pressures. The absolute pressure is the pressure relative to that of a perfect vacuum .
The figure below shows the relationship between the gauge pressure and the absolute pressure for two measurements : a pressure less than atmospheric (A) and a pressure greater than atmospheric (B) are shown .
pabs = pgauge + patm
The SI unit of pressure is the Pascal (abrev.= Pa) (Newton /m2 ). The dimensional formula for pressure is ML-1T-2.
In considering fluid pressures it has been found convenient in hydrostatics and in fluid dynamics to use fluid head as a method of measuring pressure. Considering the figure below. A quantity of fluid in an open vessel is experiencing an atmospheric pressure on its surface. A tube is routed vertically to a sealed container held at a pressure of absolute zero.
The liquid will be forced up the tube until the gravity force resulting from the level of fluid in the tube balances the force due to the pressure at the bottom of the head of fluid. Assuming the area of the tube is A t, the density of the fluid = ρ,and the pressure at the top of the tube is zero. The force at x-x
Fxx = 0 + hAρg.
The pressure at x-x=
Pxx = 0 + hAρg. / A = hρg.
For a fluid with a known fixed density the height h can be conveniently used to identify the pressure. For water the atmospheric pressure is about 10,5m. In practice water vaporises into the vacuum at the top of the tube reducing the vacuum this reduces the column height by about 180mm.
Mercury is used for measuring pressure and the height of a column of mercury which can be supported by atmospheric pressure is about 0,760m. Mecurey has a low vapour pressure and the vacuum is only reduced by about 0,16 Pa, (very small compared to atmospheric pressure of 105 Pa ). It is clear that gauge pressures and vacuum pressures are easily obtained using this method. The barometer identifies pressure readings in mm Hg.
Viscosity...
Additional ref notes.. ViscosityTables of fluid viscosities Fluid Viscosities
Perfect fluids cannot in theory transmit shear stresses. All real fluids resist shear flow. The viscosity property of the fluid defines the degree of resistance to flow it possess. This is illustrated using the figure below. A cylinder is located on a shaft and the space between is filled with a fluid. The cylinder is rotated at an angular velocity ω. The velocity distribution in the fluid as shown. The torque required to rotate the cylinder is an indication of the viscosity of the fluid.
Consider an element of fluid STQR which is subject to a shear stress τ
In a short period of time dt the fluid element distorts to S'T'QR. The fluid will experience a strain φ in time dt.
μ ( dφ /dt ) = μ τ
Note: The rate of shear strain is also measured as the deflection dx divided by the distance dy i.e dx/dy occuring over a time intervel dt. It is is effectively the velocity gradient dv /dy .. (dv = dx/dt)
If the element where an elastic solid it would distort a fixed amount proportional to the shear stress and the proportionality constant is called the Modulus of Rigidity (G). The fluid element distorts at a rate based on the viscosity of the fluid.
The SI unit for viscosity is the Pa.s (Pascal Second). This is simply derived from the units pressure /( velocity/ length) = Pa / (m /s / m )= Pa.s. The dimensional formula = ML-1T -1. The centipoise , a cgs unit, is commonly used because water has a viscosity of 1,0020 cP (at 20 �C;). 1 cP = 10-2 Poise. 1 Poise = 1g.cm.s-1 = 0,1 Pa.s........ Therefore 1 Pa.s = 1000cP
Kinematic viscosity The viscosity μ and the density ρ are both properties of a fluid. The ratio μ/ρ is called the kinematic viscosity and is also a property. Kinematic viscosity .ν can be completely defined in terms of length and time and has a dimensional equation L2T-1. The SI units for kinematic viscosity is the (m2�s-1). The cgs physical unit for kinematic viscosity is the stokes (abbreviated S or St). It is sometimes expressed in terms of centistokes (cS or cSt).
Newtonion /Non-FluidsSolids which distort an amount which is proportional to the stress are called elastic solids. Fluids which deform at a rate which is proportional to the tangential stress are called Newtonion fluids. Fluid mechanics generally
relates to Newtonion fluids. Fluid with high viscosities are called thick or heavy fluids and include tar, treacle and grease. Fluids of low viscosity are called thin fluids and include water, paraffin and petrol. Gases have very low values of viscosity. Non Newtonion fluids are studied under the heading of rheology.
Typical Non-Newtonion fluids include.
Pseudo plastic fluids e.g. solutions including gelatine, clay, milk and blood often have reduced viscosity when the rate of shear is increased.
Some fluids experience increased viscosity when the rate of shear is increased. This group includes concentrated solutions of sugar, and aqueous suspensions of starch.
Some materials, including metals, deform continuously with little increase in stress when stessed above their yield point. These behave as plastically above the yield point.
Surface Tension
The surface of a liquid is the interface between the liquid volume and the fluid above the liquid. Generally the liquid is water and the fluid above the liquid is air. The molecules within the liquid attract each other and at the interface there are more attractive forces towards the bulk of the liquid than there are towards the adjacent gas molecules. The molecular forces tend to pull into the fluid bulk. The surface of a liquid is apt to shrink, and its free surface is in such a state that each section pulls another as if an elastic film is being stretched. The surface behaves like a flexible membrane. This property is evident when overfilling a cup with water. The level of water in the cup will be higher than the cup edge before it overflows.
If a double line is drawn on the surface of a liquid there is a force normal to the lines holding the lines together.
The tensile strength per unit length of assumed section on the free surface is called the surface tension (symbon γ).
Liquid Surface FluidSurface TensionN/m
Water Air 0,0728
Mercury Air 0,476
Mercury Water 0,373
Paraffin Air 0,027
Water Paraffin 0,027
Methyl alcohol Air 0,048>
For large volumes of liquid the forces due to gravity and inertia are large compared to the surface tension forces. Therefore the surface tension is not considered in most hydrostatic and hydrodynamic calculations.
For small volumes and areas of fluid the surface tension becomes important and results in spherical water droplets and the capillary effect.
Compressibility
The volume of a fluid changes from V to V + δV as a result of the applied pressure changing from p to p + δp. The compressibility (β) is basically (δV / V ) /δp i.e. the ratio of the proportional change of volume to change of pressure . This is the reciprocal of the bulk modulus K. The bulk modulus K is similar to the spring factor , that is K .(δV/V) = δp
The volume of the fluid clearly decreases if the pressure increases and is proportionate assumed that the fluid does not change state during the process (it remains a liquid, solid or gas.
For water of normal temperature/pressure K = 2,06 x 109 Pa, and for air K = 1.4 x l05 Pa assuming adiabatic change. In the case of water, 1/K = 4.85 x l0-10Pa-1.Water compresses by about 0.005% when the pressure is increased by 1 atm (105 Pa).
The product of density ( ρ) and volume is the mass i.e. ρ V = m = constant. , and therefore volume V = m /ρ. The bulk modulus can be expressed in terms of density explained below
For gases the bulk modulus is very much dependent on the conditions : if the compression takes place at constant temperature the bulk modulus is called the isothermal bulk modulus and if the compression takes place with no transfer of heat across the system boundary the bulk modulus is colled the isentropic bulk modulus. The ratio of isentropic/isothermal bulk modulii is γ which is the ratio of specific heats.
Speed of Sound
The propagation speeds of traveling waves are characteristic of the media in which they travel and are generally not dependent upon the other wave characteristics such as frequency, period, and amplitude. The speed of sound in air and other gases, liquids, and solids is predictable from their density and elastic properties(bulk modulus). In a fluid medium the wave speed takes the general form
Consider a fluid in which a sound wave is being transmitted at a velocity c. The fluid velocity is u. To simplify the assessment this has been resolves such that the wave is stationary and the fluid has a velocity u-c. See figure below.
Taking a small area normal to the wave front ΔA continuity requires that
.....(equation a)
For the volume enclosed by ΔA
the force to the right = (p + δ p)ΔA -pΔA
= The rate of increase of momentum towards the right
= ρ(u-c)ΔA (- δ u)
Therefore ...δp = ρ(c - u)δ u.....(equation B)
Elimination of δ u from (A) and (B) above
For a weak pressure wave with δp and δρ --> zero
This equation states that a sound wave which is a weak pressure surge of value √(∂ p /∂ ρ ) move through a fluid at a velocity of ( c-u ) =a (the speed of sound ) relative to the fluid ahead of it moving with a velocity u. The assumption is that the friction is low and the resulting temperature difference across the wave is small. The movement of the wave is considered to be isentropic. (not heat transfer and no friction).
Now the bulk modulus is defined (see above) in terms of density by K = ρ ( ∂ p /∂ ρ )and therefore
Considering gases subject to isentropic processes. The law pvγ = constant (k) applies. Therefore
The bulk modulus as defined above K = - v.(dp/dv) and therefore
K = γp and therefore for a perfect isentropic gas
Mach Number
The mach number M is the ratio of the velocity of gaseous flow in relation to the sonic velocity
Fluids velocities less than the speed of sound are called sub-sonic (M < 1) and fluid velocities greater than the speed of sound are called supersonic (M >1 )
Table showing approximate sonic velocities for various mediums
Solid Velocityof
Liquid Velocityof
Gas Velocityof
Sound(a)bar /bulk
Sound(a) Sound(a)
(m/s) (m/s) (m/s)
Aluminium5100 / 6300
Water-Fresh
1430 Air 331
Copper3700 / 5000
Water-Sea 1510 Oxygen 315
Iron 3850 Alcohol 1440 Hydrogen 1263
Steel 5050-6100 Mercury 1460Carbon Monoxide
336
Lead 1200Carbond Dioxide
258
Glass5100 / 5600
Rubber 30
Wood 04-5000
Introduction
This webpage includes various notes relating to fluid flow and flowpaths and flow patterns at a very basic level.
Symbols a = Acceleration (m/s2 A = Area (m2)) a = Speed of sound (m/s)F = Force (N)g = acceleration due to gravity (m/s2 )h = fluid head (m)K = Bulk modulus (MPa )m = mass (kg)M = mach number u /aM = Molecular weightp = fluid pressure (N /m2 )pabs - absolute pressure (N /m2 )pgauge - gauge pressure (N /m2 )patm - atmospheric pressure (N /m2 )p s= surface pressure (N /m2 )Q = Volume flow rate (m3 /s)q = Heat transfer /unit mass (J/kg)
R = Gas Constant (J/(kg.K)Ro = Universal Gas Constant (J/(kg.mol.K) ρ = fluid density (kg /m2 )s = specific volume (m3 /kg)u = fluid velocity (m/s) v = fluid velocity (m/s) x = depth of centroid (m)β = Compressibility (1/MPa)θ =slope (radians)ρ = density (kg/m3)ρ r = density (kg/m3)τ = shear stress (N /m2)μ = viscosity (Pa.s) ν kinematic viscosity (m2�s-1) υ = Specific volume (m3 / kg)γ= Ratio of Specific Heats
Patterns of Flow
When considering the flow of a fluid it is often convenient to consider the flow as a number of imaginary curves along which individual particles of fluid flow. These lines are called streamlines or flowlines. fluid particles only flow along the streamlines and no particles cross the lines. A number of streamlines bundled together is termed a streamtube. he boundary of a streamtube is composed of streamlines and by definition of a streamline fluid can only enter and leave a streamtube at its ends.
The flow of fluid is always clearly three dimensional. However when considering flow problems it is generally more convenient and practical to consider flow as two dimensional or one dimensional. When considering flow along a pipe it is convenient, and practical to consider the flow along the centreline of the pipe run, that is as one dimensional flow. The pressure loss, and variations of velocity etc are along the pipe centre line. Macro pipe flow is more nearly two dimensional because the fluid velocity varies across the diameter with zero velocity at the pipe wall and the maximum velocity at the pipe centre. This problem is overcome in pipe flow calculations by assuming the flow velocity as the mean velocity. Water flow over a long weir is in reality three dimensional at the ends but can be considered as two dimensional flow with corrections for the ends.
Types of Flow
Internal flow is flow within the boundary walls. Types of internal flow include pipe flow, channel flow, airflow in ducts. This type of flow is controlled using valves, fans , pumps.
External flow is flow outside of a boundary or body. Examples of this type of flow include flow over aircraft wings, flow around immersed bodies, air flow around buildings and airflow around cars.
A very important separation of flow types relates to the velocity of the flowing fluid. Fluid flow at low velocities is smooth with the fluid particles moving in straight lines along the direction of flow. This type of flow is called laminar flow. The majority of flows in practice are turbulent with no uniform motion at the local level but an average velocity in the direction of flow. By using experimentally derived results Osborne Reynold (Manchester UK ) determined flow ranges separating the flow types for the majority of fluids. A dimensionless quantity was identified ( vρl/μ ) called Reynolds number. If the conditions and properties of flowing fluid are such the relevant Reynolds number is less than 2000 the flow will be laminar. If the relevant Reynolds number is greater than 4000 then the flow will be turbulent. Flows in the transition region between are termed critical and may be laminar or turbulent or a bit of each. For pipe flow v is the mean velocity, l is the pipe diameter ρ is the fluid density and μ is the viscosity. This differentiation of flow occurs if the fluid is a liquid or a gas.
Another classification of flow is steady flow which is defined as the type of flow in which the various parameters at any point do not change with time. Flow which changes with time is unsteady or non-steady flow. Real flows are generally the latter type but in completing flow assessments it is often more practical to assume steady flow conditions.
Viscous Flows are flows whose flow patterns are dominated by the viscous properties of the fluid. This occurs in fluids where the velocity gradients are large e.g. within pipes close to the walls of the pipe. When the viscous
properties are not dominant the flow is defined as inviscid flow. This type of flow is prevelant in the centre region of flowing pipes and in gas flows.
The majority of liquids are virtually incompressible under the majority of operating conditions from open channels to high pressure hydraulic systems. Flow involving liquids are generally considered to be incompressible flows. It is only during exceptional flow events such as water hammer when liquids do not behave as incompressible fluids. Gases are compressible fluids and flow involving gases are often compressible flows . The effects of variation of density along the flow paths require special consideration when determining the operationing parameters such as the head loss along the flow path. For many gaseous flows the velocities are low and the pressure drops are also low such that they can be considered as compressible flows.
It is often important to consider the flow velocity relevant to the speed of sound and the Mach number (Fluid velocity/ Fluid sonic velocity) is important. Fluid velocities are generally limited such that the mach number does not exceed 1.
Continuity Equation
The continuity equation is really a mathematical version of the principle of the conservation of mass as applied to fluid flow. If a region is defined in a fluid and steady flow conditions apply ..then.
The rate at which mass enters the region = The rate at which mass leaves the region
Considering a stream tube as shown below whose section is so small that the velocity u and the density ρ do not vary across it.The flow across the section = udA and the mass flow across the section is uρdA.
There is not mass accumulation within the stream tube under steady flow conditions therefore
u 1ρdA 1 = u 2ρdA 2 = u 3ρdA 3 = Constant
Considering a cross section of flow i.e. a pipe length with cross section A consisting entirely of stream tubes the equation above can be integrated as follows
The local velocity u is assumed to be normal to the local cross section and the density and velocity are constant over the entire local section then.
uρA = Constant
For fluids of constant density (incompressible fluids -liquids) with average flow u this reduces to
uA = Constant = Q
That is, under the conditions specified, when the area increases the velocity reduces and vice versa
Bernoulli's Equation..Perfect Fluids
The equation is an expression of the conservation of energy. Initially the only forces considered are gravity, pressure and inertial forces. The viscosity forces are assumed to be neglible. The fluid is assumed to be a perfect inviscid fluid under steady flow conditions.
The velocity of a fluid varies and this is associated with forces which may be linked using Newtons first law
Newtons First law ...Every body continues in a state of rest or of uniform rectilinear motion unless acted upon by a force
The relationships between the changes can be analysed using Newtons second law..
Newtons Second law ...The time rate of change of linear momentum of a body is proportional to the unbalanced force acting on the body and occurs in the direction in which the force acts. i.e Force = Mass.Acceleration.
Applying Newtons first law to a small element of fluid within a single streamline of small cross section as shown below. The ends of the streamline are assumed normal to the centre line of the streamline . Considering the forces of pressure and gravity around the element and ignoring all other forces (viscosity , surface tension, magnetism, nuclear etc).
Upstream the pressure is p and downstream the pressure is (p + δp). δp may be negative. The forces at the side of the element vary but it will be assumed that the mean pressure is (p +kδp ) with k being some value less than
one. Taking forces in the direction of flow and noting that the hydrostatic forces on the sides mean = (p +kδp ) are acting on the area δA - all other side forces are perpendicular to the axis and are ignored. The weight of the element = ρgAδs
Resultant Force = pA - (p + δp)(A+ δA ) + (p + kδp)δA - ρgAδs cos θ
ignoring second order of small quantities...
Resultant Force = Aδp - ρgAδs cos θ
The mass of the element is constant and therefore the resultant force must be equal to the mass times acceleration of the element in the direction of the force du/dt .
ρAδs (du/dt) = Aδp - ρgAδs cos θ
Now ( δs cos θ) = δz where z represents height above a horizonatal datum level.Dividing throughout by ρAδs and taking the limit δs -> 0
Consider a particle moving along a steady flow streamline for which the velocity changes as the particle moves along the streamline e.g the velocity increases as the area of the streamline reduces. δu = [ du/ds ]δs. Therefore in the limit δt -> 0 the local acceleration with respect to time is
Therefore if (du/dt) is replaced by u(du/ds) the following equation results...
For fluids of constant density the equation can be integrated with respect to s as follows.
The resulting equation (in both forms) is known as Bernouli's equation and is probably the most widely used in fluids and hydraulics and explains many of the phenomena encountered in these areas of engineering. In the second form each term has the energy per unit weight ρg and has units [ML2 /T2] / [ML /T2] = [L]. The terms are identified as pressure head, velocity head, and gravity head.
Proof of Bernoulli's equation using the principle of energy
Upstream the pressure is p1 and downstream the pressure is p2. The area upstream is A 1 and the area downstream is A 2. Assume the fluid is incompressible the volume of fluid moved ( δ V ) in a short period of time is δs 1 A 1 upstream which is equal to the volume moved downstream δs 2 A1 The work done in moving the fluid upstream is F 1δs1 and the work done in moving the fluid downstream is- F 2δs2. The net work done by the two elements of fluid in the period of time under consideration is..
Also as a result of the small flow the change in potential energy of the stream tube as a result of the mass of fluid being effectively moved from the downsteam part of the streamtube to the upstream end is
The increase in kinetic energy as a result of the fluid motion is
The work done by the pressure results in the increase in potential and kinetic energy
This is Bernoull'is equation proved using the principle of energy
The figure below shows a hypothetical fluid system taking no account of friction losses in the pipe line. The gauge points show the pressure head at the connection point. The differences in level between the gauge level (red line) and the tank level represents the kinetic head.
The figure below illustrates the condition of a perfect fluid flowing out of reservoir through a sharp edged orifice. The fluid flows out as a free jet. The datum z is assumed to be at the centre line of the orifice.
At the inlet the total energy is equal to the hydrostatic head of the fluid h1 = p /( ρg ). At the exit from the orifice at the point of maximum velocity (vena contracta) the head is virtually = to the velocity head h2 = u2 /(2g). As h1 = h2 it is clear that the maximum velocity of the fluid flowing out of the tank is
Stagnation Point
Considering bluff object located in a flowing fluid.
The off line flowstreams generally divert round the object as shown but the centre flow stream is such that at the surface of the object the velocity in the direction of the flowsteam = zero. This is called a stagnation point.
Now by bernoulli's equation the (pressure + velocity + head) energy is constant along the flowstream and consequently at the stagnation point the pressure is increased from p to p + (1/2) ρ u2.as the velocity energy is converted to pressure energy. For a constant density fluid the value of ( p + (1/2) ρ u2 ) is known as the stagnation pressure of the streamline.
A manometer connected to point S would indicate the stagnation pressure (p /ρg +u 2 /2g) and therefore if the static head (p /ρg) was known then by subtraction the velocity head and hence the velocity could easily be calculated. The is how the pitot tube meter works ref. Pitot tube meter .
Flow of real fluids..... reference First Law Of Thermodynamics Steady Flow
Bernoulli's equation is fundamental to fluid flow analysis but it is subject to certain important simplifications. It assumes the fluid is inviscid and incompressible and that the flows are steady and relationships have been derived using newtons laws based on flows alone a single streamline. When considering the flow of real fluids it is necessary to include for energy losses and energy gains. The first law of thermodynamics is reviewed in outline on webpage Laws of Thermodynamics and basically identifies the transfer of energy over a complete cycle for a system as follows
Δ Q = ΔE + ΔW
Δ Q is the increase of energy supplied to a system, Δ E is the increase in the total energy of a system and Δ W is the mechanical work done by the system. In fluids the processes are flow processes and the first law is expressed in the form of the "steady flow energy equation as shown below..
q = net heat transferred to fluid per unit mass = the net work done by the fluid [p2 / ρ2 - p1 / ρ1 + w ] + the net increase in the kinetic energy (u 2 2 /2 - u 1 2 /2 ) + the net increase in the gravitational energy (z2 g - z1 g ) + the net increase in the internal energy (e1 - e2).
This law applies to fluids with steady and continuous flows and the conditions at the inlet and outlet points (section 1) and (section 2), heat and work energy transfer rates are constant.
This very general law applies to all flowing fluids. When applied to liquids under normal flow conditions e.g in pipes and channels, with no heat transfer and no mechanical work being performed the steady flow energy equation can be reduced to.
h f is the energy lost as a result of friction and is dissipated in increasing the internal energy of the fluid (e2 - e1 ) and as heat lost to the environment (q)
Pressure in a moving fluid
Bernoulli's theorem indicates that if the hydrostatic head (p/ρg +z) decreases then the fluid accelerates. Consider a streamtube (or pipe) as shown below.
Bernoulli's formula can be rewritten as
From the basic laws of motion ref. Dynamics
u1 is the initial velocity and u1 is the final velocity, s = distance , and a = acceleration.
Therefore
The left hand side of this equation is the drop in the piezometric head - ( the head measured by a manometer type gauge). If the slope of this decrease in piezometric head is i then the head drop is iδs ..s being the distance along the streamtube or pipe.. Therefore....
According to Newtons second law acceleration = force/unit mass.
Form the intitial notes on bernoulli's principle above ref Bernoulli's Equation
This equation relates to forces in the direction of the axis of motion. For an incompressible fluid the piezometric head gradient ( i ) in any direction produces a corresponding force per unit mass in that direction equal to ( ig ). Assuming negligible viscous friction this is the only force acting on the fluid and can be equated to the corresponding fluid acceleration in that direction.
Introduction
The following notes should enable a mechanical engineer to establish basic flow conditions and head losses along pipe routes in which fluids are flowing. The equations are most relevant to liquids although approximate sizing for gases can be carried out if appropriate correction factors are used,where necessary, and low gas velocities are considered.
Symbols A = Pipe Cross Section Area (m2)a = Velocity of sound ( m /s) c p = Specific Heat Capacity at Constant pressure (kJ/(kg K))c v = Specific Heat Capacity at Constant Volume (kJ/(kg K))ε = Pipe roughness (m)ε mm = Pipe roughness (mm)D = diameter (m)f = friction factor fT = friction factor (flow in zone of complete turbulence). h = Specific Enthalpy (kJ/kg )k = Thermal Conductivity (W/(m K))r = radius of pipe bend (m)K = f (L/D )
p = Absolute Pressure N / m2
Pr = Prantl Number =c p. mu / k (Dimensionless) Q = Volume flow Rate (m3 /s )q = Heat Input per unit mass ( kJ /kg ) R = Gas Constant = R o / M (kJ /(kg.K) Re = Reynolds Number = v.ρD/μ t = Temperature (C )T = Absolute Temperature (K) u = Specific Internal Energy (kJ/kg)v =Fluid Velocity (m/s)w = Work Output per unit mass (kJ/kg)ρ = Density ( kg /m3 )
L = Pipe Length (m)
μ =Fluid Viscosity = (Ns/m2 = Pa s)z = Elevation (m ) g = gravitational acceleration ( 9.81 m /s2)
Fluid Flow
Fluid flowing in pipes has two primary flow patterns. It can be either laminar when all of the fluid particles flow in parallel lines at even velocities and it can be turbulent when the fluid particles have a random motion interposed on an average flow in the general direction of flow. There is also a critical zone when the flow can be either laminar or turbulent or a mixture. It has been proved experimentally by Osborne Reynolds that the nature of flow depends on the mean flow velocity (v), the pipe diameter (D), the density (ρ) and the fluid viscosity Fluid Viscosity( μ). A dimensionless variable for the called the Reynolds number which is simply a ratio of the fluid dynamic forces and the fluid viscous forces , is used to determine what flow pattern will occur. The equation for the Reynold Number is
For normal engineering calculations , the flow in pipes is considered laminar if the relevant Reynolds number is less than 2000, and it is turbulent if the Reynolds number is greater than 4000. Between these two values there is the critical zone in which the flow can be either laminar or turbulent or the flow can change between the patterns...
It is important to know the type of flow in the pipe when assessing friction losses when determining the relevant friction factors
Steady Flow Equation....
Reference :
The steady flow equation steady flow equation (energy per unit mass ) for a system is identified below...Reference... Steady Flow
If q = w = 0 and the fluid is incompressible and frictionless and if the variables are converted to measured heads of the fluid , that is the units are per unit weight (ρg) - then the Bernoulli's equation results ..Reference .. Bernoulli's Equation ideal fluids..
In real flow systems there are losses due to internal and wall friction which result in increase in the internal energy of the fluid. (q > 0). Reference Bernoulli's Equation real Fluids . The bernoulli equation is modified to reflect
these losses by adding a term h f = Head loss due to friction.= (u2 -u1 - q) The modified bernoullis equation is therefore ..
The object of most pipe flow head loss calculations is to determine the friction head loss and allow estimation of the pump /compressor power required to pump the fluid along the piping. In most fluid transfer cases the fluid is a incompressible (a liquid) and the flow rate (Q) is constant along the pipe run and therefore the velocity at any point can easily be calculated. The head (z) can also be easily obtained from the pipeline geometry. The system pressure and head loss are therefore the variables generally subject to the detailed pipeline calculations....
Pipe Flow Calculations
In determining the head loss (pressure drop) along a pipe as a result of friction losses it is first necessary to determine the following:Diameter (m), Length (m), Fluid Viscosity( μ), Fluid density (ρ) and the fluid velocity (v). It is then necessary to obtain the relevant Reynolds number..
The equation for the Reynold Number
Consistent units to be used i.e Typically ρ = kg/m3, v = m/s, D= m, μ = Ns/m2 ( 1 Ns/m2 = 103cP)
The value for the Reynold number is to be used to evaluate if the flow is laminar or turbulent and can be used to obtain the friction factor " f " from a moody chart. The moody chart plots the friction factor (f) against the Reynold number with a number of different plotted lines for different values of absolute roughness/Diameter .
The head loss along the pipe can now be calculated using the Darcy-Weisbach equation
The result of the calculation is in units of head of the fluid. . It is based on the pipe being all one dia and the fluid is incompressible
For a single pipe line with a number of fittings the total head loss is calculated as
K p = f (L/D) for the length of pipe. ( this may be made up of ∑ f(L/D). for a number of different pipe lengths of different diameters ) K 1..n = fT(L/D) equivalent for each fitting
A Moody chart (see below) is used to determine the turbulent flow friction factor from the Reynolds number and the relative roughness of the pipe. If the flow is laminar then the fricton factor is 64/Re.
Note: it is suggested that for laminar flow in pipe at Re number approaching 2000 the above K values are used for bends and fitting with reasonable accuracy
A moody chart and tables for roughness values and (L/D) factors for various fittings are provided below
Moody Chart
Various typical values of hydraulic roughness (ε)
Note: In the moody chart above (ε /D ) is identified with both numerator and denominator in metres (for consistency with all other equations on this page. It is probably more convenient to express both in (mm).i.e a 50mm cast iron pipe (ε mm; /Dmm ) would simply be (0,203 /50 ).
Type of Pipe ε .103..( = εmm )
Cast Iron 0,203
Galvanised Steel 0,152
Steel/Wrought Iron 0,051
Rivetted Steel 0,91 - 9,1
Asphalted Cast Iron 0,12
Wood-Stave 0,18 - 0,91
Concrete 3,0
Spun Concrete 0,203
Drawn Copper, Brass Steel,Glass Smooth
Typical Values of L/D for Fittings
The losses through fittings are generally evaluated by obtaining K = fT(L/D)
Table of pipe friction values for clean pipe in region of complete turbulence
The K180 value for a 180o bend may be derived from the equivalent K90 which is calculated from the above tables using the equation
The K180 = 0,25.π.fT . r/D + 1,5.K90
For laminar fluids with low Re numbers ( "<" 500) the K values obtained using the above are probably very innaccurate. The table below illustrates how this affects the K values
FittingK values for low Reynolds Number fluids
Re = 1000 500 100 50
90 deg Elbow Short Radius 0,9 1 7,5 16
Gate Valve 1,2 1,7 9,9 24
Globe Valve 11 12 20 30
Plug Valve 12 14 19 27
Angle Valve 8 8.5 11 19
Swing check Valve 4 4,5 17 55
K values for Sudden Expansion-Contraction & Orifice
The losses through these fitting are generally evaluated by first obtaining β = d2 / d1
Important Note: the resulting K values as tabled below are based on the flow velocity in the larger pipe if the flow velocity in the small pipe is used to evaluate the head loss then the K values tabled below should be multiplied by ( β ) 4 = (d2 / d1) 4
Table of Ke,Kc & KO against β = d2 / d1
β K e K c K o β K e K c K o
0.15 1887.42 965.43 2852.85 0.6 3.16 2.47 5.63
0.2 576 300 876 0.65 1.87 1.62 3.49
0.25 225 120 345 0.7 1.08 1.06 2.14
0.3 102.23 56.17 158.4 0.75 0.6 0.69 1.29
0.35 51.31 29.24 80.55 0.8 0.32 0.44 0.76
0.4 27.56 16.41 43.97 0.85 0.15 0.27 0.42
0.45 15.51 9.72 25.23 0.9 0.06 0.14 0.2
0.5 9 6 15 0.95 0.01 0.06 0.07
0.55 5.32 3.81 9.13 1 0 0 0
Reasonable Velocities of fluid in Pipes
Medium Pressure (bar) Service Velocity (m/s) Notes
Steam (sat) 0 - 1.7 Heating 20 to 30 + 100mm dia
Steam (sat) over 1.7 Process 30 to 50 +150mm dia
Steam (sup) over 14 Process 30 to 100 +150mm dia
Air Forced Air Flow 5 to 8 e.g. AC Reheat
Water - General 1 to 3
Water Concrete Pipe 4,7
Water Pump Suction 1,2
Water Horizontal Sewer 0,75 Minimum
Water Pump discharge 1,2 to 2,5 Minimum
Water Boiler Feed 2,4 to 4,6 Minimum
Oil Hydraulic Systems 2,1 to 4,6 Minimum
Ammonia Compressor Suction 25 Max. Permissable
Ammonia Compressor Discharge 30 Max. Permissible
Introduction
Definition (Hydrostatics)...That part of fluid mechanics restricted to fluids in which the velocity (linear or angular) of mass motion does not vary from point to point. The term hydro comes from a Greek word meaning water. This term is generally used for water but it also applies to other fluids both liquid and gaseous.
Symbols A = Area (m2)Ixx = Second moment of area about horizontal axis (m4) g = acceleration due to gravity (m/s2 )h = fluid head (m)kG = radius of gyration of surface about centroid (m)kO = radius of gyration of surface about axis O-O'(m)
The buoyancy of a body wholly or partly immersed in a fluid at rest , situated in a gravitational field or other field of force is defined as the upward thrust of the fluid on the body. Generally all problems relating to buoyancy can be resolved by applying the principles of Archimedes.
Put simply The buoyancy of any body is vectorially equal and opposite to the weight of the fluid displaced by the body and has the same line of action.
The upward thrust which the surrounding fluid exerts on an object is referred to as the force of buoyancy. This thrust acts through the centroid of the displaced volume, referred to as the centre of buoyancy. The centre of buoyancy is not the same as the centre of gravity which relates to the distribution of weight within the object. If the object is a solid with a uniform density exactly the same as water and the body is immersed in water the force of buoyancy will be exactly equal to the weight and the centre of buoyancy will be the same as the centre of gravity. The object will be in equilibrium with the surrounding fluid.
This principle also applied to gases as well as liquids and explains why balloons filled with gases which have lower density compared to air rise to such a height that the weight of the air displaced is equal to the weight of the gas in the balloon.
A body which hovers in a fluid and is in equilibrium is said to have neutral buoyancy.
If the centre of gravity (G) is not in the same location as the centroid (centre of buoyancy-B). The body will orient itself such that the centre of Gravity is below the centre of buoyancy. (See diagram below). The diagram below shows a hollow vessel with a heavy weight occupying a small segment. The diagram below shows the object in a fully stable equilibrium position. In theory if the G was vertically above B then there is no force (moment) tending to rotate the object and it is still in a position of equilibrium. In this position however it is considered to be unstable.
.
The figures below show two positions of a similar submerged object which represent positions of stable and unstable equilibrium. The definition of stable and unstable equilibrium are stated thus.
If an immersed body initially at rest is displaced so that the force of buoyancy and the force of the centre of gravity are not in the same vertical line :...The body is stable if the resulting couple tends to bring the body back to its original position and ...The body is unstable if the resulting couple tends to move the body away from its original position.
Metacentre and Metacentric Height
Consider a rectangular vessel immersed as shown below in the first figure the centre of buoyancy at B and the centre of gravity is at G. with the water line at S-S Now if the vessel is heeled such that the water line is at S'=S'. The centre of buoyancy now moves to B' as shown in the second figure below. There is now an upthrust (W) due to buoyancy at B' and the weight of the vessel(W) is acting down at G and there is a couple W.a acting to restore
the vessel to its original position. The locus of each position of B' as the vessel heels to different angles is called the buoyancy curve. Also the curve joining the tangents of each line of thrust, drawn relative to the vessel, is known as the curve of metacentres. The cusp of this curve is known as the initial metacentre . This is shown on the third figure which combines the first and second figures
The initial metacentre M is the point where the line of action of the upthrust intesects the original vertical line through the centre of buoyancy B and the centre of gravity G for an infinitesimal angle of heel.
The righting moment is calculated as W.GM.sinθ..The angle of heel being θ. For small values of heel up to about 15o GM is fairly constant and is the value generally accepted as ..the traverse metacentric height of the vessel
A floating vessel is stable if the metacentre lies above the centre of gravity G.A floating vessel is in neutral equilibrium if the metacentre lies on the centre of gravity G.A floating vessel is unstable if the metacentre lies below the centre of gravity G.
Pressure in liquids
A perfect fluid cannot resist or exert any shear force and is defined as non viscous or inviscid under all conditions. The intensity of normal forces is called the pressure and is positive if compressive. Considering a small element of fluid of uniform thickness which is subject to pressures p, px, and py as shown the element is assumed to be so small that the pressures are assumed to be uniform (the effect of gravity is ignored). Equating forces in the x and y directions results in the equations
p A sin θ = A sinθ py p A cos θ = A cosθ px
p = py = px
This simple example illustrates that for perfect fluids and, to some extent, for real fluids the pressure at a point is the same in all directions. Thus in static fluids it is reasonable to identify the pressure at a point in any direction of direction.
****** To determine the pressure p at any depth h below a free surface in is necessary to examine the vertical equilibrium of an imaginary vertical cylinder of a fluid. The fluid column must be supported by the pressure difference across its ends. If the atmospheric pressure is ps. The weight of the fluid = the density multiplied by the volume ρ.A.h Assume the cylinder has unit Area A= 1
p - ps = W = ρA.h
The pressure in a liquid under the influence of gravity increases uniformly with depth is proportional to the density and is in addition to the surface pressure.
p = ps + ρgh
Liquids are asssumed to be virtually incompressible and ρ is therefore assumed to be constant. If the pressure is measured above atmospheric pressure then the pressure is called the gauge pressure pg. Ordinary dial pressure gauges measure gauge pressure. The liquid pressure at different depths based on gauge pressure is rewritten as
pg = ρgh
The figure below illustrates the hydrostatic paradox .. It implies that using the relevant formula the force on the inside base of the vessel can be many times the weight of the fluid contained. This is explained by the fact that most of the downward pressure is balanced by the upward pressure on the downward facing surfaces of the vessel.
Using the pressure at depths to establish the buoyancy consider the figure below. Assume an immersed body is composed of and infinite number of vertical cylinders each of area δA. and length h. The upward thrust on each cylinder = ρghδA. This is equal to ρgv when v = hδA
.
Adding the upthrust for all the cylinders making up the volume (V) of the immersed body.V = Σ v and F = the total upthrust (buoyancy).
F = ρg Σ v. = ρg V = The weight of the displaced fluid
If the object is grounded such that the area in contact with the ground is A l there is a loss of buoyancy = ρg hA l This can have very serious consequences for ships grounded on sandbanks.
.
Force on Submerged surfaces
Consider a submerged plane surface of area A -see figure below. The surface is subject to a pressure which varies linearly from R to S from pR to pS.
The force on each elementary strip =
δF = p.δA = ρgd.δA = ρgx sin θ.δA
The total force =
F = ρg sin θ ∑x .δA
∑x .δA is the first moment of area of the plane about the line of intersection 0-O' of the immersed surface projected to intersect the liquid surface. This first moment is equal to A xG where xG is the slant depth of the centroid G. Now sin θ xG is simply equal to dG which is the depth of the centroid and ρg dG = pG The liquid force acting on the surface is therefore.
F = ρg dGA = pG A
Centre of Pressure on Submerged surfaces
The point at which the resultant fluid force is considered to act on a plane area is called its centre of pressure. This is shown on the above figure at point P. This point is found by summing the moments of the elementary forces about the imaginary axis. O - O'.
M = ∑ δ M = ∑ p x δ A = ρ g sin θ ∑ x 2 δ A
This is equivalent to the moment exerted by the resultant force F acting through the centre of pressure P. Thus
M = F xP = [ ρg sin θ ∑ x δA. ] x P
And from above the force (F) on the plate is
F = ρg sin θ ∑ x .δA
Therefore
The second moment of area of the plane figure about its centroid G is IG The first moment of area of a plane figure about O - O= A.xG.Using the parallel axis theorem IO = IG + A. xG
2 . This can be expressed in terms of radii of gyration as A ko
2 = A [ kG2 + xG
2 ] thereofore
Therefore the centre of pressure of a plane area lies below the centroid G of the area by a distance P - G = x P - x G = k G
2 / x G measured along the slope of the plane. As the radius of gyration of the surface about it's centroid k G is fixed the difference reduces as the depth of the surface increases.
Introduction
The notes below relate primarily to compressible fluids flowing in pipes. The notes are of a basic level sufficient for a mechanical engineer to be able to estimate operating conditions in a pipeline transferring vapours or gases.
It is much more difficult to determine the operating characteristics of compressible fluids (vapours and gases) as the density is not constant under flowing conditions.
The extremes conditions encountered are adiabetic flow (PVγ = constant ) or isothermal flow (PV = constant). Adiabetic conditions occur when no heat is transferred across system boundaries. Isothermal conditions occur when the system changes occur at constant temperatures. For short insulated pipes adiabetic conditions can be assumed. For long pipes with reasonable levels of insulation isothermal conditions provide good approximations to real conditions... Operating conditions occuring at some point between the extremes can often be related to the polytropic process (PVn = constant)
Compressible fluid flows also have a maximum velocity which is limited by the speed of propogation of a pressure wave travelling at the speed of sound for the fluid under consideration. If the differential pressure along a pipe is such that the fluid velocity approaches sonic speed then any further increase in differential pressure will not be accompanied by an increase in fluid velocity.
For many real life pipe flow conditions it is possible to use the Darcy equations and factors as provided on webpage Pipe Flow Calcs subject to the following restrictions.
1).. If the calculated (estimated ) pressure drop (upstream (P1) - downstream (P2 ) is less than 10% of the inlet pressure (P1) then reasonable accuracy is achieved if the specific volume used is based on the known conditions. (Upstream or downstream).
2)..If the calculated (estimated ) pressure drop is greater than 10% but less than about 40% of the inlet pressure P1 then reasonable accuracy is achieved by using a specific volume based on the average of the downstream and upstream pressures.
3)..If the calculated (estimated) pressure drop is greater than 40% of P1 then methods as identified on this page should be used.
Symbols a = Acceleration (m/s2 A = Area (m2)a = Speed of sound (m/s)F = Force (N)g = acceleration due to gravity (m/s2 )h = fluid head (m)K = Bulk modulus (MPa )L = Pipe length (m )m = mass (kg)m = mass flow rate (kg/s)M = mach number u /aM = Molecular weightP1 = Inlet fluid pressure (gauge) (N /m2 )P2 = Outlet fluid pressure (gauge) (N /m2 )P1 = Inlet fluid pressure (abs) (N /m2 )P2 = Outlet fluid pressure (abs) (N /m2 )P - Absolute pressure (N /m2 )pgauge - gauge pressure (N /m2 )patm - atmospheric pressure (N /m2 )p s= surface pressure (N /m2 )
Q = Volume flow rate (m3 /s)q = Heat transfer /unit mass (J/kg)R = Gas Constant (J/(kg.K)Ro = Universal Gas Constant (J/(kg.mol.K) ρ = fluid density (kg /m2 )s = specific volume (m3 /kg)u = fluid velocity (m/s) v = specific volume (m3/kg) v1 = specific volume at inlet conditions(m3/kg) x = depth of centroid (m)β = Compressibility (1/MPa)θ =slope (radians)ρ = density (kg/m3)ρ r = density (kg/m3)τ = shear stress (N /m2)μ = viscosity (Pa.s) ν kinematic viscosity (m2�s-1) υ = Specific volume (m3 / kg)γ= Ratio of Specific Heats
Compressible Fluid Flow equations.
The flow of compressible fluids in long lines approximates isothermal conditions. The flow rate in a pipe under isothermal conditions is provided by the equation below...
This equation has been developed on the basis of a number of assumptions including: Isothermal flow, steady flow, perfect gas laws apply, constant friction value, straight and horizontal pipe. The equation below is a simplified version and assumes no acceleration along streamlines.
Limiting Flows.
The equations above do not take into account the fact that , for a particular fluid, there is a maximum speed which cannot be exceeded in the compressible fluid flowing in a pipe.
The maximum velocity of a compressible fluid is limited by the velocity of a pressure wave travelling at the speed of sound in the fluid. Reference Sonic velocity. Clearly the maximum velocity will be at the downstream end of the pipes as the velocity will progressively rise as the pressure falls resulting in a increase in the specific volume. Now if the pressure drop is sufficiently high such that sonic velocities is about to be exceeded the
resulting pressure decrease and hence driving force will not be transmitted upstream and consequently there will be no increase in flow rate.
The sonic velocity which cannot be exceeded is expressed as
vs = √ (γRT ) = √ (γP v )
More Notes to follow -March -2007
Table of Air Flows through sched 40 Piping.
Notes : Factors for other conditions..
1) For inlet pressures (p o) other than 7 bar gauge... [ Multiply table pressure drop value by 8,013/(p o + 1,013) ]2) For inlet temperatures (t o) other that 15 deg C... [ Multiply table pressure drop values by (273 + t o ) / 288 ] 3) For Pipe sizes (d o ) other than sched 40 ( d 40 )... [ Multiply table pressure drop values by (d 40 /do ) 5 ]4) Pressure drop is proportional to length. For pipe lengths l o other than 100m.... [ multiply table presuure drop by l o /100 ]It is important to note that this table should only be used for crude estimates. For serious work then detailed calculations should be used.
Pipe Sizes 1/8" to 2"
Pressure drop of air in bars per 100m of schedule 40 commercial pipe
The study of forces resulting from the impact of fluid jets and when fluids are diverted round pipe bends involves the application of newtons second law in the form of F = m.a. The forces are determined by calculating the change of momentum of the flowing fluids. In nature these forces manifest themselves in the form of wind forces, and the impact forces of the sea on the harbour walls. The operation of hydro-kinetic machines such as turbines depends on forces developed through changing the momentum of flowing fluids.
Symbols α = jet angle (radian)a = Acceleration (m/s 2) ρ = density (kg/m 3)ρ = density (kg/m 3)F = Force (N)m = mass (kg)V = fluid velocity(m/s)
Newtons Second Law can be stated as: The force acting on a body in a fixed direction is equal to rate of increase of momentum of the body in that direction. Force and momentum are vector quantities so the direction is important. A fluid is essentially a collection of particles and the net force, in a fixed direction, on a defined quantity of fluid equals the total rate of momentum of that fluid quantity in that direction.
Consider a mass m which has an initial velocity u and is brought to rest. Its loss of momentum is m.u and if it stopped in a time interval t then the rate of change of momentum is m.u /t. The force F required to stop the moving mass is therefore F = m.u / t . Now if this is applied to a jet of fluid with a mass flow rate ( m / t ) which is equivalent to the volumetric flow rate times the density ( Qρ ) the equivalent force on a flowing fluid is F = Qρ u. Also in accordance with Newtons third law the resulting force of the fluid by a flowing fluid on its surroundings is (-F). Newtons third law states that for every force there is an equal and opposite force.
The figure below illustrates this principle at two locations.
The fluid flowing into the tank is brought to rest from a velocity u to zero velocity and the force on the jet is F = Q.ρ. u. . The reaction force on the tank contents is -F.
The fluid flowing in the pipe in the horizontal direction is forced to change direction at the bend such that its velocity in the orginal direction is zero. Therefore the force on the flowing fluid is F = Q. ρ u. The reaction force on the pipe is -F in the horizontal direction as shown.
In its simplest form, with steady flow conditions, the force on a fluid flow in a set direction is equal to its mass flow rate times by the change in velocity in the set direction. The fluid flow also exerts an equal and opposite reaction force as a result of this change in momentum.
F = Qρ (u 1 - u 2). ..( F and u are vector quantities)
The resultant force on a fluid in a particular direction is equal to the rate of increase of momentum in that direction.
Jet Forces on Stationary Plates
Jet force on a flat plate
Considering only forces in a horizontal direction u 1 = V and u 2 = 0 therefore F = QρV = ρAV 2
Jet force on a flat plate at an angle θ
Considering only forces Normal to plate surface u 1 = V sinθ and u 2 = 0 therefore F = QρV sin θ = ρAV 2 sin θ
when θ =90o then F = ρAV 2 as above
Jet force on an angled plate. (θ < 90 o)
Considering only forces in a horizontal direction u 1 = V and u 2 = V cos θ therefore F = QρV (1 - cosθ ) = ρAV 2(1 - cosθ )
Jet Force on an angled plate (θ > 90 o)
Considering only forces in a horizontal direction u 1 = V and u 2 = V cos θ therefore F = QρV (1 - cosθ ) = ρAV 2(1 - cosθ )
Jet Force on an angled plate (θ = 180 o)
u 1 = V and u 2 = V cos θ therefore F = QρV (1 - cos180 o ) = 2QρV = 2ρAV 2
Jet Forces on Moving Plates......Fluid Machines - Pelton Wheel
Jet force on a moving flat plate
Considering only forces in a horizontal direction u 1 = V and u 2 = V p and let r = V p / V therefore F = Qρ( V - V p ) = ρAV(V -V p) and F = ρAV 2( 1 - r
The power (P) generated by the force on the moving plate = P = F. V p
Jet force on an angled moving plate
Considering only forces in a horizontal direction u 1 = V and u 2 = V p + (V - V p) cos θ and let r = V p / V therefore F = ρA V( V - V p ) ( 1 - cosθ ) = ρA V 2 ( 1 - r ) ( 1 - cosθ ) The power (P) generated by the force on the moving plate P = F. V p
Note: The moving plate with an angle θ = 180o is the typical of the rotating plate of the pelton wheel. The ideal value for r resulting in the maximum power output is clearly 0,5
Jet Forces on Vanes......
Additional notes can be found on webpagesFluid Machines - Francis WheelSteam Turbines - Impulse blades
Force on fixed Vane.
In the x Direction: u 1x = V cos θ 1 , u 2x = -V cos θ 2
F x = QρV(cos θ 1 + cos θ 2 ) = ρAV 2(cos θ 1 + cos θ 2 )
In the y Direction u 1y = V sin θ 1 u 2y = V sin θ 2
F y = QρV(sin θ 1 - sin θ 2 ) = ρAV 2(sin θ 1 - sin θ 2 )
Force on Moving Vane.
The notes below related to vanes as used in impulse turbines. These turbines derive the mechanical energy mainly from the change in momentum as the fluid passes through the vanes. The conditions as shown when the vectorial sum of V v + V r1 = V1 results in smooth entry with efficient transfer of energy of the fluid to the vane. When this does not occur there will be turbulent flow over the vane with significant losses.
In the x Direction u 1x = V 1 cos αu 2x = V v - V r2 cos θ 2 .... ( V r2 = V r1 = V 1 sin α /sin θ 1 )F x = QρV 1 (cos α + [sin α /sin θ 1 ] cos θ 2 - r > )r = V v / V 1
In the y Direction u 1y = V 1 sin αu 2y = V r2 sin θ 2 .... ( V r2 = V r1 = V 1 sin α /sin θ 1 )In the y direction F y = QρV 1sin α (1 - sin θ 2 / sin θ 1 )
If the vane is moving in the x direction the power developed by the vane P = F x.V V
Force on Pipe Wall
The notes below related to the force on a pipe wall resulting from the changes in fluid pressure and fluid momentum as the fluid flows round a pipe bend . Gravity and friction effects are not considered. The fluid is assumed to be flowing under steady state conditions.
In the x Direction
u 1 = V 1 u 2 = V 2
F x = p1.A1 - p2.A2 + ρ (A1V12 - A2V2)2
In the x Direction
u 1x = V 1 u 2x = V 2cos θF x = p1.A1 - p2.A2cos θ + ρ (A1V1
2 - A2V22 cos θ )
In the y Direction
u 1y = 0 u 2y = V 2sin θF y = - p2.A2sin θ - ρA2V2
2 sin θ
The resultant reaction force on the pipe = Fr =√ (Fx2 + Fx
2 )The angle α of the resultant force to the x axis = tan-1 ((Fy /(Fx)
Introduction
Many physical relationships in engineering and especially in fluid mechanics are, by nature, extremely complex. Often a phenomenon is too complicated to, theoretically, derive a formula describing it e.g the forces experienced when an object moves through a fluid. Dimensional analysis is then used to identify variables which can be combined in groups which are definitely related. Experiments can then be completed to formulate this relationship and allow determination of the actual performance characteristics of real world systems..
This method derives from the principle that each term in an equation depicting a physical relationship must have the same dimension. Non-dimensional quantities expressing the relationship among the variables are constructed e.g. [Length / (Velocity.Time)], or [ Force / (Mass /Acceleration)]. These are equated and then experiments are complete to determine their functional relationship.
The principles of dimensional analysis are developed from the principle of dimensional homogeneity which is self evident.
It is characteristic of physical equations that only like quantities, that is those systems having the same dimensions, are added or equated.
It is OK to to equate forces. ( 5 newtons = 2 newtons + 3 newtons.)It is clearly not OK to equate forces with lengths ( 5 newtons = 2 newtons + 3 m)
Quantities Symbols and Dimensions
Quantity Symbol Dimensions Quantity Symbol Dimensions
Mass m M Mass /Unit Area m/A 2 ML -2
Length l L Mass moment ml ML
Time t T Moment of Inertia I ML 2
Temperature T θ - - -
Velocity u LT -1 Pressure /Stress p /σ ML -1T -2
Acceleration a LT -2 Strain τ M 0L 0T 0
Momentum/Impulse mv MLT -1 Elastic Modulus E ML -1T -2
Force F MLT -2 Flexural Rigidity EI ML 3T -2
Energy - Work W ML 2T -2 Shear Modulus G ML -1T -2
Power P ML 2T -3 Torsional rigidity GJ ML 3T -2
Moment of Force M ML 2T -2 Stiffness k MT -2
Angular momentum - ML 2T -1 Angular stiffness T/η ML 2T -2
Angle η M 0L 0T 0 Flexibiity 1/k M -1T 2
Angular Velocity ω T -1 Vorticity - T -1
Angular acceleration α T -2 Circulation - L 2T -1
Area A L 2 Viscosity μ ML -1T -1
Volume V L 3 Kinematic Viscosity τ L 3T -1
First Moment of Area Ar L 3 Diffusivity - L 2T -1
Second Moment of Area I L 4 Friction coefficient f /μ M 0L 0T 0
Density ρ ML -3 Restitution coefficient M 0L 0T 0
Specific heat-Constant Pressure
C p L 2 T -2 θ -1 Specific heat-Constant volume
C v L 2 T -2 θ -1
Note: a is identified as the local sonic velocity, with dimensions L .T -1
Types of Similarity.
In order that the relationships determined for a model can be applied to a real life application (prototype) there has to be a physical similarity between the parameters involved in each one. The two systems are said to be physically similar in respect to specified physical quantities when the ratio of the corresponding magnitudes of these quantities between the two systems is everywhere the same. Within the general term physical similarity there are a number of types of similarity some of which are listed below.
Geometric similarity... This is basically the similarity of shape. Any length of one system is related to that of another system by a ratio which is normally called the scale. All parts of the scale model of a car should be in direct scale to the full scale item if it is truly geometrically similar. This should ideally include such features as the surface roughness. This does not include non dimensional features e.g. weight...
Kinematic Similarity... This is basically the similarity of motion and implies that the geometric similarity and similarity of time intervals. i.e ratios of length are fixed (r l) and ratios of time intervals (r t) are fixed. The velocities (ds/dt) of corresponding parts should also be in fixed ratios ( r l / r t ) and the ratios of acceleration (dv/dt) are in ratios ( r l / r t 2 ).
Dynamic Similarity.. This is the similarity of forces. The magnitude of forces at two similarly located points are in a fixed ratio. For systems involving fluids the forces may be due to viscosity, gravitation, pressure, inertia, surface tension, elasticity etc etc... It is generally accepted in fluid mechanics that the ratio of inertia forces is the most useful ratio.
Dynamic similarity involving flow with viscous forces...The are numerous instances of fluid flow affected only by viscous pressure and inertia forces. A fluid flowing in a full pipe is such a case. For dynamic similarity the ratio of magnitude of any two forces must be the same at corresponding points (in a steady flow situation) . The ratio of inertia force to net viscous force is chosen for review. The inertia force is the mass x acceleration. [density (ρ ) x volume ( l 3 ) x acc'n ( u 2 / l )]. Note: The acceleration is chosen to be the characteristic velocity ( u ) divide by a particular time interval ( l/ u ) = u 2 / l . The magnitude of the inertia forces are therefore proportional ( ρ.l 3 )( u 2 / l ) = ρ l 2 u 2
The magnitude of the shear stress resulting from viscosity is the product of the viscosity (μ )and the rate of shear ( u / l ) acting over an area proportional an area l 2 . This is therefore proportional to ( μ ) ( u / l ) x ( l 2 ) = ( μ u l )
The ratio of inertia forces to viscous forces is therefore as follows:
This ratio is very important in fluid mechanics, mainly for problems involving flowing fluids, and it is called Reynolds number. The ratio for dynamic similarity between two flows past geometrically similar boundaries and affected by only viscous and inertia forces is the same if the fluids have the same reynolds number. In the UK for pipe flow studies the characteristic length( l ) is the diameter ( D ) and the characteristic velocity u is chosen as the mean velocity.
Dynamic similarity involving flow with gravity forces... When considering forces with free surfaces e.g. flows over weirs, channel flows, or surface motion around ships, the most significant relationships is the ratio between the gravity forces and the inertia forces. These are summerised below..
This ratio u /( lg ) 1/2 is called the Froude number . Dynamic similarity exists between two flows which involve fluids subject to only gravity and inertial forces if the Froude number , based on corresponding velocities and lengths, is the same for both fluids...
The primary dimensionless groups in Fluid mechanics are listed below.
Group NameRepresents Ratio ofForces
Symbol
ρ l u / μ Reynolds Number Inertia / Viscous Reu / ( lg ) 1/2 Froude Number Inertia / Gravity Fru / ( lρ / γ )1/2 Weber Number Inertia / Surface Tension Weu / a Mach Number Inertia / Elastic M
Note: a is identified as the local sonic velocity, with dimensions L .T -1
Simple Example
Consider a body moving with constant acceleration. The relationship is expressed as ..... s = ut + at 2 /2 Expressing this in terms of dimensions..
Note....[ z ] is used to say the dimensions of z
The above examples simply illustrates that the equation is dimensionally correct. This exercise can be continued to produce a non-dimensional equation.
The terms within the brackets are non dimensional groups which can be considered a single variables or groups. These are generally called denoted using the symbol Π the above equation can be expressed as
Π 1 = 1 + Π 2..... or.....Π 1 = F [ Π 2]
There is no real advantage in using the principle for this simple example but for more complex relationships the benefits can be significant.
Buckinghams Π theorem
Consider a physical phenomenon with an unknown defining equation.
First define what relationship is require. e.g The wind force experienced by a sphere
List the number of dependent variables and all relevant variables. eg F = f (d,u,ρ,μ )
Using base dimension (say L,M,T,η..F), set down the dimensions of all the variables. e.g. F->[MLT-2] , d->[L] , u-> [LT-1] , ρ ->; [ML-3] , μ ->[ML-1 T-1]
Count the number of variables (n = 5) count the number of base dimensions used to dimension the variables ( j = M,L,T).note: For fluids j will generally = 3
Select j variables which include in their dimension which collectively include all the base dimensions (in this case M,L & T]. e.g. Choose d->[L], u-> [LT-1], and ρ ->; [ML-3]
Form k dimensionless groups [k = n - j = 2]
Use the resulting dimensionless groups to establish a relationship in which one group which includes the dependent variable (F) as a function of the other groups..
Buckinghams's theorem simply states that if there is a relationship involving n variables and j base dimensions then k = n- j dimensionless groups ( Π groups ) can be created allowing physical relationships to be developed using experimental methods
Dimensional Analysis
Introduction
This page concerns fluid flows down channels and pipes which are not full. The fluid has a free surface which is subject to atmospheric pressure. This naturally occurs with rivers, and canals, and drainage ditches. The notes also include fluid flowing over weirs and notches.
SymbolsA = Area (m2)F1 = Force of fluid down channel (N)F2 = Force up fluid down channel (N)g = acceleration due to gravity (m/s2 )h = fluid head (m)i = incline l = length down slope (m)lh = length --- horizontal (m)m = wetted mean length (m)p = fluid pressure (N /m2 )p s= surface pressure (N /m2 )P = perimeter (m)ρ = fluid density (kg /m2 )
s wetted surface length (m)u = velocity (m/s) v = velocity (m/s) x = depth of centroid (m)θ =slope (radians)ρ = density (kg/m3)τ o = shear stress (N /m2)
Channel Flow
In an open channel, the flowing water has a free surface and flows by the action of gravity. See figure below. The water flows with a velocity v down a channel with an incline θ. The water depth is uniform and therefore the downward force F1 is balanced by the upward force F2. The only force causing motion is the weight component in the direction of motion ρgAl sin θ.
The fluid is not accelerating so the downward gravity force is balanced only by the friction force between the fluid and the wall. If the length of wetted perimeter = s and the shear stress at the wall = τ osl
ρgAl sin θ = τ osl
Now let the incline i be x / l h. For small angles i = sin θ
.
ρgAli = τ osl .. and therefore ..τ o = ρgAi / s
Now let m be the mean wetted depth (m = A/s) the resulting equation is
.
τ o = ρgmi
Note: The relationship between τ o and f is proved at the bottom of this page..
The quantites 2g/f are combined as a single constant ( C2 ) yielding the equation known as Chezy 's formula
The value of C can be obtained using the Ganguillet---Kutter equation: with the relevant n values provided in the table below
Mannings formula C = m1/6 /n also applies...Using the same tabled values of n
Table showing n coefficients for using in Mannings equation and Ganguillet---Kutter equation:
Descriptionn Normal
n Range
Glass 0,010 0,009---0,013
Concrete
Culvert straight and free of debris 0,011 0,010---0,013
Culvert with bends, connections and some debris 0,013 0,011---0,014
Sewer with manholes, inlet etc straight 0,015 0,013---0,017
Unfinished steel form 0,013 0,012---0,014
Unfinished smooth wood form 0,014 0,012 --- 0,016
Finished wood form 0,012 0,011---0,014
Clay
Drainage tile 0,013 0,011---0,017
Vitrified clay sewer 0,014 0,011---0,017
Vitrified clay sewer with manholes inlet etc 0,015 0,013---0,017
Vitrified sub drain with open joint 0,016 0,014---0,018
Brickwork
Glazed 0,013 0,011---0,015
Lined with cement mortar 0,015 0,012---0,017
Sewer coated with slimes , with bends 0,013 0,012---0,016
Rubble masonary 0,025 0,018---0,030
Cast Iron
Coated 0,013 0,010---0,014
Uncoated 0,014 0,011---0,016
Excavated or Drained Channels
Earth after weathering ---straight or uniform 0,022 0,018---0,025
Gravel straight uniform 0,025 0,022---0,030
Earth winding clean 0,025 0,023---0,030
Earth with some grass, weeds 0,030 0,025---0,033
Earthe bottom rubble sides 0,030 0,028---0,035
Dragline excavated, no vegetation 0,028 0,025---0,033
Clear straight, fullstage no rifts or deep pools 0,030 0,025---0,033
As above but with more stones and weeds 0,035 0,030---0,040
Clean, winding some pools and shoals 0,040 0,035---0,045
As above but some weeds and stones 0,045 0,035---0,050
Flood Plains
Pasture short grass 0,030 0,025---0,035
Pasture high grass 0,035 0,030---0,050
Cultivated Areas
No crop 0,030 0,020---0,040
Mature row crops 0,035 0,025---0,045
Mature field crops 0,040 0,030---0,050
Major Streams Width > 30m
Regular section with no boulders or bush 0,025---0,060
Irregular and rough 0,035---0,10
Thin Plate Weirs1) Full Width Weir
Flow Q = 0,66√(2g). Cd b he 1,5 CD = 0,602 + 0,083 h/p
he = h + 0,0012m (h = measured head)
2) Supressed Weir
Flow Q = 0,66√(2g). Cd b. he 1,5 CD = 0,616(1 --- 0,1h/b)
he = h + 0,001 (h = measured head ---m)
3) Vee Notch Weir
Flow Q = (8/15)√(2g).Cd tan (θ /2 ). he 2,5
he = h + hk (h = measured head ---m)
θ Degrees
Cd hk
20 0,592 0,0027
40 0,581 0,0018
60 0,576 0,0011
80 0,578 0,0010
90 0,579 0,0009
Notes showing relationship between τo and f...Provided in support of proof of Chezy Formula above..
Showing relationship between τo and f
Darcy conducted experiments and proved that for pipes of uniform cross section and roughness and fully developed flow the head loss due to friction (hf ) along a pipe is in accordance with the following formula.
The fluid shear stress (τ o ) at the boundary wall is related to the pressure differential along the pipe by the expression.
P = perimeter length, A = Area of section
The differential head along the pipe is related to the differential pressure as follows/
The equation for shear stress is modified as ..
Now for fully developed flow with no axial sudden changes the flow pattern along the pipe is constant and dh/dl is equal to h / l therefore ..
Drag on objects moving through fluids
IntroductionFluid Flow
Fluid flowing past an object tends to drag the object along in the direction of fluid flow. If an object is moving through a stationary fluid the drag tends to slow the object down. If the object is stationary in a flowing fluid the drag tends to move the object in the direction of flow. The drag comprises two components:
- Pressure drag or form drag which is based on the pressure difference between the upstream and downstream surfaces of the object
- Skin Friction which results from the viscous shear of the fluid flowing over the object surfaces.
The form drag is the resultant of resolved forces normal to the surface of the object and the skin-friction is the resultant of resolved forces tangential to the surface. The total drag on an object is called the profile drag and is the sum of the pressure and skin-friction drag When the drag is primarily viscous drag, the body is streamlined, and when the drag force is primarily pressure drag the body is called a bluff body.
A perfect fluid flowing past an infinitely long cylinder is represented as streamlines which are arranged such that the flow through each streamline is fixed at Q . The streamlines flow over the cylinder and all forces are balanced front/back and top /bottom and there is therefore no form drag. A perfect fluid cannot transfer shear stress so there is no viscous drag. In real fluids there is a pressure build up on the front surface as the fluid is slowed and the streamlines are re directed round the cylinder. As the fluid flows over the cylinder the fluid separates into a wake which is a lower pressure region. There is therefore significant form drag. There is also skin-friction drag as the fluid passes over the surface.
If the cylinder rotates, as shown below, the drag between the surface and the fluid results in the fluid flow as shown. The flow results in higher fluid velocities above the cylinder compared with the flow below the cylinder . Application of bernoulli's equation results a lower pressure above the cylinder and a consequent lift.
Symbols a = Acceleration (m/s2 A = Area (m2)a = Speed of sound (m/s)C D =Drag coefficientF = Force (N)g = acceleration due to gravity (m/s2 )h = fluid head (m)K = Bulk modulus (MPa )m = mass (kg)M = mach number u /aM = Molecular weightp = fluid pressure (N /m2 )p abs - absolute pressure (N /m2 )p gauge - gauge pressure (N /m2 )p atm - atmospheric pressure (N /m2 )p s= surface pressure (N /m2 )
Q = Volume flow rate (m3 /s)Re = Reynolds Number = u.ρD/μ Re x = Local Reynolds Number = u.ρD/μ ρ = fluid density (kg /m2 )s = specific volume (m3 /kg)u = fluid velocity (m/s) u n = Normal flow fluid velocity (m/s) v = fluid velocity (m/s) θ =slope (radians)ρ = density (kg/m3)ρ r = density (kg/m3)τ = shear stress (N /m2)μ = viscosity (Pa.s) ν kinematic viscosity (m2�s-1) υ = Specific volume (m3 / kg)
Boundary Layer and Wake
Flow past a slender body or flat plate arranged parallel to the flow results in a flow regime as shown below. The flows at the surface are brought to rest relative to the surface and flows close to the surface are slowed. This effect reduces rapidly as the distance from the surface increases until the flow is completely unaffected. The flow pattern may be split into two regions: A thin boundary layer in which friction (viscous) drag forces are important and a region in which bernoulli's equation primarily applies i.e. the total head is constant.
The boundary layer concept where the influence of viscosity is concentrated bridges the gap between classical hydrodynamics based on inviscid fluids and bernoulli's theorem and the behaviour of real fluids.
The boundary layer as formed by a fluid flowing along one side of the flat plate is initially laminar, as shown below. The velocity gradients are primarily due to the viscous drag near the surface of the plate. The velocity upstream of the plate and in the region outside the boundary layer is u n .
At the leading edge of the plate the fluid is retarded and the boundary layer is initiated. As more fluid is slowed because of the viscous forces the boundary layer thickens. Initially the boundary layer includes only laminar flow but as the boundary layer thickness increases the laminar layer loses stability and the flow becomes less even. The point at which the laminar flow starts to deteriorate is called the transition point and is at the start of a region called the transition region where the flow changes from laminar flow to turbulent flow. At the plate surface a thin laminar sub-layer remains below the turbulent boundary layer. The sketch shows the boundary layer thickness very much magnified relative to the length along the plate. The thickness of the boundary layer on an aircraft wing may only be a few mm thick.
Consider a fluid flow over a curved surface as shown below. The pressure of the fluid is initially higher at the approach (A) compared to that when the fluid has accelerated over the surface (B). This is because the pressure head falls as the velocity head rises. When the fluid starts to decelerate as is moves over the surface the pressure rises from B to C to D. The initial pressure change from A->B assists the flow over the surface and the pressure gradient is favourable. However the pressure gradient from B- > C -> D is not favourable and there is a tendency for the pressure to retard the boundary layer. The effect of this is greater at the solid -fluid surface because the local momentum of the fluid is least.
There is a point where the reverse pressure causes sufficient flow that the boundary layer velocity gradient becomes zero and there is a tendency for the fluid to separate. In the figure above this is shown at point C and is called the point of laminar separation. The result of this, so called,separation is to impair the process of conversion between kinetic energy and pressure energy with internal fricitonal losses to heat. A laminar boundary layer has less kinetic energy compared to a turbulent boundary layer and this effect is reduced if a turbulent boundary layer flow can be encouraged. A golf ball is covered with dimples to encourage a turbulent boundary layer and therefore delay the onset of separation
The region of constant reduced pressure behind the at the rear of an object, called the wake, lies beyond the separation point and is responsible for the pressure drag.
A streamline shape is such a form that the change in velocity head downstream of the maximum velocity is reduced such that the separation point is moved as far back as possible .see figure below.
Drag notes.
When calculating the total drag a very simple equation is used as shown below. This equation is not valid for relative fluid velocities approaching sonic velocity.
For bluff bodies A = The frontal area of the body facing the flow. For thin flat plates and similar shaped items subject to primarily skin-friction A = the total surface area swept by the fluid flow (both sides) For wings in aeronautical calculations A is the product of the wing span and the mean wing chord.
The denominator is simply the product of the dynamic head of the undisturbed fluid [(1/2) ρ u n2 ] and the
specified area.
The flow patterns in the wake are dependent on the Reynolds number as illustrated by considering an infinitely long cylinder in a fluid flow providing a two dimensional flow pattern.
At low reynolds numbers (Re < 0,5)the inertial forces are negligible compared to the viscous forces and the streamlines come together behind the cylinder. The drag is therefore roughly proportional to u n. Drag is inversely proportional to C D.
At reynolds number 2 to 30 a wake is formed behind the cylinder but the streamlines come together behind the cylinder. Eddies form which rotate in opposite direction
At Reynolds numbers 40 to 70 the eddies elongates and wake flow instability inititiates. At high Reynolds numbers eddies form and break off each side of the cylinder alternately.
At higher levels (Re > 90 ) the eddies form vortices downstream of the cylinder. This unsymetrical flow pattern
causes telephone wires to sing in the wind, cause venetian blinds to flutter and are a major design problem for tall chimneys. This arrangement of vortices is known as a Karman vortex street.
The drag drag coefficient C D reaches a minimum of about 0,9 at Re = 2000 and then increases a little because the turbulance of the wake increases and the position of separation gradually moves upstream. At this stage the profile drag is nearly all due to pressure (form) drag.
At an Re value of about 20 x 105 the laminar boundary layer becomes turbulant before separation. The turbulant boundary layer has higher kinetic energy than the laminar based layer and is better able to withstand the adverse pressure gradiant. There is therefore a sudden drop in the drag down to about 0,3. As the Reynold number rises there is a conequent increase in the drag coefficient to about 0,7. For increases in Reynold number above 4 x 106 the drag is independent of the Re number.
Drag for Infinitely long PLATE and infinitely long CYLINDER
Drag calculated is better calculated using the variation of the drag equation
The drag coefficient for a shape which is finitely long is affected by fluid flow around the ends. This is illustrated by considering the plate for reynolds number above 1000. The drag coefficient varies as follows for different Length(L) to Breadth (d) ratios.
L /d 1 2 4 10 18 InfiniteC D 1,1 1,15 1,19 1,29 1,4 2,01
Drag for long plate parallel to flow and streamlined strut
The drag coefficient is mainly friction drag (C f ). The area is the length x width (l.d )Drag calculated is better calculated using the variation of the drag equation
Drag factor for various long objects
Drag for DISC and SPHERE<
Drag factor for various objects
Drag for a human body = C D = 1 to 1,3
Drag for a normal car = C D = 0,37
Fluids -Curved Motion- Circulation & Vorticity
Introduction
The notes on this page relate to fluid motion with curved streamlines and rotary fluid motion. The notes are of a very basic nature such that a mechanical engineer can obtain an understanding of the principles involved. The notes include sections on curved motion, fluid rotation, free vorticity, forced vorticity, and circulation.
Symbols
a = acceleration (m/s2)an = normal acceleration (m/s2)in = Hydraulic Gradient (m) towards the centre of curvatureu = velocity (m/s)p = pressure (N/m2) pi = pressure at infinite radius (N/m2)
un = normal velocity (m/s)R = Radius (m)s = distance moved along path (m)t = time (s)ρ = density (kg/m3)ζ = vorticity (rad/s)Γ = Circulation (m3 /s )
Motion in a curved path
Consider the motion of a fluid element moving in a curved path moving from point 1 to point 2 see below.
Now if δs and δθ are small the following equations apply
The change in velocity normal to the path results from these expressions
The corresponding normal acceleration is therefore
The equation becomes exact at δs = 0
The normal acceleration may be equated to the corresponding force per unit mass. For an incompressible fluid the force per unit mass is equal to the downward slope of the hydraulic gradient (piezometric head) toward the centre of rotation times g. ref Pressure in a moving fluid
Hence the slope of the piezometric head across a curved stream-line is.
Ignoring the force due to gravity the force per unit mass in the n direction is ref Pressure in a moving fluid
The resulting equation of motion is therefore
As an example consider an incompressible fluid circulating steadily about a fixed axis. The value of i n at radius r is
The positive direction of the normal is opposite to that of the radius from the centre. Therefore the slope of the transverse Piezometric head is
Bernoulli's equation (H = z + (p /ρg ) +u 2/2g). The equivalent variations in total head (H) across curved stream-tubes may be evaluated . Its gradient in the radial direction from the centre of rotation is
Substituting u 2 /rg for the traverse piezometric head gradient d( p/ρg +z )/dr the following results
..
And thus
Fluids -Curved Motion- Circulation & Vorticity
Fluid Rotation
Consider a fluid element ABCD with sides dx and dy located at O which moves to location O' in time period dt and deforming in the process
AB in the x direction moves to A'B' in the process rotating dθ1 and AD in the y direction moves to A'D' in the process rotating dθ2
The angular velocities of AB and AD being ω1 and ω 2 respectively
The centre O has an angular velocity which is the average of (ω1 and ω 2 ) / 2
The fluid vorticity for the z axis is ζ is defined as
When the fluid vorticity ζ= 0 the fluid movement is called irrotational flow..
Free Vorticity
Circular fluid motions frequently occur in regions which are sensibly of constant energy throughout. A typical example of this type of motion occurs when fluid in a vessel empties through a small hole . In this case the liquid makes a rotary motion but its the water elements always face in the same direction. The flow is irrotional
In assessing this movement it is necessary to examine the the motion corresponding to the rule H = constant both along and across the fluid streamlines in all locations in the regions at all times. This type of motion is termed a free, potential, or irrotional vortex. This type of motion is such that dH/dr = 0 and hence the equation derived in the the notes on curved fluid flow above applies.
Integrating the the final expression between two radii results in the following :
That is
The corresponding pressure distribution across such a free vortex may be obtained from Bernoulli's equation since DH/dr = 9 for this motion, that is, H has the same value across each stream tube.
The constant c depends on the strength of the vortex... therefore
If the axis of rotation is vertical with z being constant at any radius then ρg (H-z) is simply the pressure at infinite radius say (pi )
That is
The figure below shows a section through a free vortex and a real vortex which includes real losses due to fluid friction. It is clear that at small radii for a ideal free vortex the velocity would approach infinity. In practice at small radii the fluid velocities are more proportional to the radius and not its reciprocal. At larger radii the agreement between the theory and practice is quite good
Forced Vorticity
Fluid entrained in the rotating impeller of a centrifugal pump before the pump discharge valve is opened undergoes virtual forced vortex motion. A fluid contained in a rotating vessel, as shown below, also over time forms into a forced vortex. Such a vortex results whenever a fluid is whirled bodily about an axis with a constant angular speed.
In conditions of forced vorticity the linear velocity of the fluid is proportional to the radius of rotation, ... the fluid is being rotated as if it were a solid body i.e.
u = r ω
It has been shown that hydraulic gradient (piezometric head slope) in fluid i = a g ref Pressure in a moving fluid...A fluid in a circular motion has an accelaration of (u2 / r ) towards the centre of rotation. Therefore the piezometric slope towards the centre of rotation is in = an /g and an = u2 /r.. therefore
A forced vortex is such that the if the relevant liquid has a free surface (p/ρg = 0) then the radial slope of this surface (in) increases linearly with r..... the cross section is a parabola and the free surfaces forms a paraboloid..(see above figure).
The positive direction for the radius is outwards from the centre while that for the normal n is inwards towards the centre of rotation. Thus
Integration of this expression between two radii (r1 and r2 ) results in the following equations
Bernoulli's equation (H = z + p /ρg +u 2/2g). The equation below can thereofre be written in terms using bernoulli's equation as follow .
If H has a value H0 ar the rotation axis (when r = 0 and u = 0), then the value of H at any other radius is given by...
The surface profile results if (p/ρg)= 0. The equation corresponding to the surface profile is therefore ...
And z at any radius results from...
Now since the pressure variation across any horizontal plane is simply ρg time the head of liquid above it ...
For a forced vortex both the pressure and total head increase parabolically with the radius.
Most real life circulatory motions which occur in nature are approximations to a forced vortex at the core surrounded by a free vortex as shown in the figure on the section on free vorticity.
Circulation
Consider a line AP of unit thickness in a flowing fluid.
The volume flow rate across the line = Qn and also the flow along the line can be expressed as Qs. These flows are obtained as follows
Now if the line is a fixed closed circuit the flow around the circuit is called the circulation ( Γ ) . . Convention is that positive circulations are ani-clockwise (ACW ) flows.
The circulation round a large circuit equals the sum of the circulations round components small circuits contained within the large circuit ( provided that the boundaries of all circuits are wholly in the fluid). This is illustrated by the figure below. The large circuit is subdivided into smaller ones. M and N are typical sub-circuits. The contribution of flow QsM along AB from circuit M along the common boundary is positive (ACW) while the contribution of flow from circuit N is QsN is negative and cancels out the contribution from M. Therefore all flows along common boundaries cancel each other and circulation then only consists of the flow round the periphery
Now considering a small element in a flowing fluid as shown below..
The circulation is calculated as follows
Therefore
Fluid properties of Water
Table of Properties of water at varying temperatures and Saturation Pressure
Note :
Isothermal compressibility β (1/MPa) = 1/K ...(K=Bulk Modulus).For liquid this is virtually the same as the isentropic compressibility.For vapours it is not the same as isentropic compressibility ref Compressibility
1000 �Pa-s = 1 cP
Temp.
SaturationPressure
Density
Density
Specific Volume
Specific Volume
Compr'ty.Isothermal
Compr'ty.Isothermal.
Speedof sound
Speedof sound
Viscosity
Viscosity
Surf. Tension
t psat ρ liq ρ vap υ liq υ vap β liq β vap a liq a vap μ liq μ vap γ
Table of Properties of water at 20 deg C Pressures 1 to 20 Mpa (10-200 bar A)
Notes :
Isothermal compressibility β (1/MPa) = (1/K (Bulk Modulus).For liquid this is virtually the same as the isentropic compressibility.
1000 �Pa-s = 1 cP
Pressure
DensitySpcificVolume
Compressibility Isothermal
Speed of Sound
Viscosity
Surf. Tension
p ρliq υ liq βliq a μ liq γliq
[MPa][kg/m^3]
[m^3/kg] [1/MPa] [m/s][�Pa-s]
[N/m]
1 998,6 0,001001 0,0004579 1484 10010,07274
2 999,1 0,001001 0,0004567 1485 10010,07274
3 999,5 0,001 0,0004555 1487 10000,07274
4 1000 0,001 0,0004543 1489 10000,07274
5 10000,0009996
0,0004532 1490 999,60,07274
6 10010,0009991
0,000452 1492 999,20,07274
7 10010,0009987
0,0004509 1494 998,80,07274
8 10020,0009982
0,0004497 1495 998,40,07274
9 10020,0009978
0,0004486 1497 998,10,07274
10 10030,0009973
0,0004474 1499 997,70,07274
11 10030,0009969
0,0004463 1500 997,30,07274
12 10040,0009964
0,0004452 1502 9970,07274
13 1004 0,000996 0,0004441 1504 996,60,07274
14 10040,0009955
0,0004429 1505 996,30,07274
15 10050,0009951
0,0004418 1507 9960,07274
16 10050,0009947
0,0004407 1509 995,60,07274
17 10060,0009942
0,0004396 1510 995,30,07274
18 10060,0009938
0,0004385 1512 9950,07274
19 10070,0009933
0,0004374 1514 994,70,07274
20 10070,0009929
0,0004363 1515 994,40,07274
Table of Properties of water at 100 deg C Pressures 1 to 20 Mpa (10-200 bar A)
Notes :
Isothermal compressibility β (1/MPa) = (1/K (Bulk Modulus).For liquid this is virtually the same as the isentropic compressibility.
1000 �Pa-s = 1 cP
Pressure
DensitySpecific Volume
Compressibility Isothermal (β)
Speed of Sound
Viscosity
Surf. Tension
p ρliq υ liq βliq a μ liq γ
[MPa][kg/m^3]
[m^3/kg] [1/MPa] [m/s][�Pa-s]
[N/m]
1 998,6 0,001001 0,0004579 1484 10010,07274
2 999,1 0,001001 0,0004567 1485 10010,07274
3 999,5 0,001 0,0004555 1487 10000,07274
4 1000 0,001 0,0004543 1489 10000,07274
5 10000,0009996
0,0004532 1490 999,60,07274
6 10010,0009991
0,000452 1492 999,20,07274
7 10010,0009987
0,0004509 1494 998,80,07274
8 10020,0009982
0,0004497 1495 998,40,07274
9 10020,0009978
0,0004486 1497 998,10,07274
10 10030,0009973
0,0004474 1499 997,70,07274
11 10030,0009969
0,0004463 1500 997,30,07274
12 10040,0009964
0,0004452 1502 9970,07274
13 1004 0,000996 0,0004441 1504 996,60,07274
14 10040,0009955
0,0004429 1505 996,30,07274
15 10050,0009951
0,0004418 1507 9960,07274
16 10050,0009947
0,0004407 1509 995,60,07274
17 10060,0009942
0,0004396 1510 995,30,07274
18 10060,0009938
0,0004385 1512 9950,07274
19 10070,0009933
0,0004374 1514 994,70,07274
20 10070,0009929
0,0004363 1515 994,40,07274
Viscosities of Various Fluids
Notes:The values identified below are for dynamic viscosity as measured using centiPoise (cP) and SI units N.s m -2. = Pa. s 1 Centipoise = 1 mPa s (milliPascal Second)= 10 -3 Pa.s1 Pa.s = 1000 centiPoise
The information has been obtained from reference texts and the internet. A number of values have been obtained using nonograms provided in Perrys and by taking values from graphs of fluid properties. The values are therefore only to be used for initial estimates. Please use quality information sources for detail design work.
It should be noted that the viscosity is not a nice convenient fixed property of a fluid but it is affected very much by the ambient conditions of temperature and pressure and material parameters such and density, and chemical make-up.
1) A table of fluid (gas) viscosities using cP at room temperatures is provided and also a table of Pa.s values at various temperatures is also provided.2) A table of fluid (liquid viscosities using cP at room temperatures is provided and also a table of Pa.s values at various temperatures is also provided.
Table of dynamic gas viscosity values in cP at room temperatures (20 deg C) and atmospheric pressure
Fluid
Dynamic viscosity
cP(Centipoise)
Acetylene 0,00935
Ammonia 0,00918
Argon 0,021
Carbon Dioxide 0,014
Carbon Monoxide 0,0166
Chlorine 0,0129
Cyanogen 0,00928
Ethylene 0,0097
Helium 0,0186
Hydrogen 0,00835
Hydrogen Chloride 0,0138
Hydrogen Sulphide 0,0117
Methane 0,0103
Nitric Oxide 0,0178
Nitrous Oxide 0,0135
Oxygen 0,0192
Sulphur Dioxide 0,0117
Dynamic viscosities (Pa.s) of various gases at different temperatures at atmospheric pressure
Example value Air (0 deg C) dynamic viscosity = 17 X 10 -6 Pa.s = 0,017 cPoise
GasSymbol
Temperature (Deg. C)
0 100 250 500
Viscosity Pa s ( Actual x 106 )
Oxygen O2 19 24 30 39
Helium He 18 22 28 37
Air - 17 21 27 35
Nitrogen N216,5
21 265 35
Carbon Monoxide
CO16,5
21 26 35
Carbon Dioxide
CO213,7
18 24 34
Sulphur Dioxide
SO211.8
15 20 28
Ammonia
NH3 9,5 13 18 27
Hydrogen
H2 8510,5
12,5
16,5
Ethane C2H6 8,511,5
15,5
22,5
Propane C3H8 7,7 1013,2
18
Butane C4H20 810,5
14,2
20
Dynamic Viscosities of Various liquids at room Temperature (20 degres C)
Fluid
Dynamic viscosity
cP(Centipoise)
Acetic Acid 1,219
Acetone 0,324
Benzene 0,647
Bromine 0.993
Carbon Disulphide 0,375
Carbon Tetrachloride 0.972
Chloroform 0,569
Glycerol 1495
Mercury Hg 1,552
Methyl Alcohol 0,594
Nitrobenzene 2,03
Olive Oil 84000
Lubricating Oil 799,4
Paraffin Oil 1000
Phenol 12,74
Toluene 0,585
Turpentine 1,49
Water 1
Dynamic Viscosities (Pa.s) of Various Liquids at different temperatures
Example value Water (20 deg C) dynamic viscosity = 0,001 Pa.s = 1 cPoise
Liquid..........Chemical symbol
Temperature (Deg. C)
-25 0 20 50 100 200
Viscosity centiPoise (cP) (val's x 10-
3 = Pa.s )
Water H20 -1,7918
1,0028
0,5471
0,2817
0,1346
Light Machinery Oil
- - - - 28 4,9 -
Spindle Oil - - - 13 5,49 2,0 -
EthanolC2H5OH
3,241
1,786
1,201
0,701
0,326
-
MethanolCH3OH
-0,817
0,584
0,396
- -
Benzene C6H6 - -0,649
0,436
0,261
0,113
Toluene C7H8 -0,773
0,586
0,419
0,269
0,133
Sulphur Dioxide SO2 -0,368
0,304
0,234
- -
Ammonia NH30,215
0,169
0,138
0,103
- -
Freon 113 - 1,4 0.92 0,75 0,5 0,29 -
Fuel Oil (sg=0.94)
- - - 1000 100 11 2,5
Crude Oil (sg = 0,855)
- - 16 7,5 4 2,5 1,5
Kerosene - - 3,7 2,5 1,4 0,6 -
Sulphuric Acid (100%)
90 36 25 10,5 3,4 -
Sulphuric Acid (60%)
16 9 6 3,5 1,5 -
Nitric Acid (95%)
2,1 1,5 1,2 0,85 0,50 -
Nitric Acid (60%)
6,0 3,5 2,5 1,4 0,7 -
Mercury 2 1,60 1,55 1,5 1,45 1,1
Fluids Orifices, Nozzles, Venturies
Introduction
The notes on this page related to the methods of measuring flow using devices which are based on bernoulli's equation. There are many other devices which are convenient to use and are very accurate which are based on other principles including vortex shedding, ultrasonics (doppler), turbines, and variable orifice. To obtain information on these devices please consult the linked sites at the bottom of this page.
Pitot tubes, Orifice Plates, Nozzles and Venturi meters are established methods of measuring flows in pipelines . These methods relate the pressure difference across the upstream and downstream sides of the units to the pipeline fluid velocities. In modern piping systems various high technology methods many of which are non-intrusive are replacing these systems. Orifice plates and nozzles are also used as flow balancing and/or limiting devices.
For information on flow resistance provided by an orifice plate refer Expansion/Contractions/Orifice Plates
Pitot tubes are specially designed probes inserted into pipes to establish the flow velocity at fixed points in the pipe bore. The flow rate is established using special techniques.
Orifice plates are low cost devices consisting of thin plates trapped between flanges. The orifice plate includes a sized hole with a downstream bevel, through which the fluid flows. The flanges include tapping points to measure the pressure upstream and downstream of the plates. The accuracy of the orifice plate method is about ± 2%
Nozzles are the same as orifice plates except that the thin plate is replaced by a contoured nozzle. The accuracy of the nozzle method is about ± 1%
The venturi is a converging length of pipe followed by a short parallel throat and then a divergence. The accuracy of the venturi method is better than ± 1%
The main differences in the devices are that the orifice plate results in signficant losses, the nozzle has relatively low losses and the venture meter and the pitot tubes are very efficient. The venturi meter is also the most accurate followed by the nozzle and the orifice plate. The orifice plate system is the most widely used because it is the cheapest and most convenient to install and maintain.
Symbols A = Area (m2)A 2= Area of Orifice(m2)a = Speed of sound (m/s)C d = Coefficient of DischargeC c = Coefficient of Contractionρ = density (kg/m3)ρ 1 = density at inlet condtions(kg/m3)g = acceleration due to gravity (m/s2 )ε = Expansion factor h = fluid head (m)L = Pipe length (m )m = mass (kg)m = mass flow rate (kg/s)P 1 = Inlet fluid pressure (gauge) (N /m2 )P 2 = Outlet fluid pressure (gauge) (N /m2 )P 1 = Inlet fluid pressure (abs) (N /m2 )P 2 = Outlet fluid pressure (abs) (N /m2 )P - Absolute pressure (N /m2 )
p gauge - gauge pressure (N /m2 )p atm - atmospheric pressure (N /m2 )p s= surface pressure (N /m2 )Q = Volume flow rate (m3 /s)Q m = Mass flow rate = Qρ (kg /s)Re = Reynolds Number = u.ρD/μ s = specific volume (m3 /kg)u = fluid velocity (m/s) v = specific volume (m3/kg) v 1 = specific volume at inlet conditions(m3/kg) x = depth of centroid (m)θ =slope (radians)β = Ratio of largest pipe dia to small diameterτ = shear stress (N /m2)μ = dynamic viscosity (Pa.s) ν = kinematic viscosity (m2�s-1) υ = Specific volume (m3 / kg)γ= Ratio of Specific Heats
Relevant Standards
The following standards provide detailed information on measuring fluid flow using venturis, orifice plates and nozzles.
Moff = Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full.
BS EN ISO 5167-1:2003 Moff: General principles and requirements
BS EN ISO 5167-2:2003 Moff: Orifice plates
BS EN ISO 5167-3:2003 Moff: Nozzles and Venturi nozzles
BS EN ISO 5167-4:2003 Moff: Venturi tubes
Note: BS 1042:Part 1(5 sections) and Part 3 :Standards withdrawn and replaced by above standards
MoFFiCC= Measurement of Fluid Flow in closed conduits
BS 1042-1.4:1992 BS 1042-2.1:1983, ISO 3966:1977 MoFFiCC. Velocity area methods. Method using Pitot static tubes
BS 1042-2.2:1983, ISO 7145:1982 MoFFiCC. Velocity area methods. Method of measurement of velocity at one point of a conduit of circular cross section
BS 1042-2.3:1984, ISO 7194:1983 MoFFiCC. Velocity area methods. Methods of flow measurement in swirling or asymmetric flow conditions in circular ducts by means of current-meters or Pitot static tubes
BS 1042-2.4:1989, ISO 3354:1988 MoFFiCC. Velocity area methods. Method of measurement of clean water flow using current meters in full conduits and under regular flow conditions
BS 1042:Part 1:Section 1.5:1997 MoFFiCC. Pressure differential devices. Guide to the effect of departure from the conditions specified in BS EN ISO 5167-1
BS 1042:Part 1:Section 1.5:1987 MoFFiCC. Pressure differential devices. Guide to the effect of departure from the conditions specified in Section 1.1
Pitot Meter
Consider three glass tubes positioned in a pipe which is carrying flowing fluid
Now the static head of the fluid (p /ρg ), that is the height that the fluid rises in the tube with the fluid velocity at zero, is indicated by the tube at position B. At the interface of a flowing liquid with a solid surface the fluid velocity is zero. The head at position A is a measure of the stagnation head (p /ρg +u 2 /2g) Reference ..Stagnation point.
Now if the flowing fluid was in an open stream that static pressure would be the atmospheric pressure and the stagnation head would be simply the level of fluid in the manometer above the surface level of the flowing fluid. For an enclosed stream the velocity head is the difference in level of the static head as measured by tube B and the stagnation head as indicated by Tube A. The level of fluid in the tube at C is not useful because the fluid is flowing past the end of the tube.
The figure below shows a typical design of pitot tube flow meter. The inner tube pressure is the stagnation pressure and the annulus surrounding the inner tube is at the static pressure i.e. it indicates the pressure at the surface of the pitot tube which is static.
If ρ m is the density of the manometer fluid and ρ is the density of the flowing fluid then the the fluid velocity results from the equation.
u = C √ (2 Δp /ρ)........ Δp = (ρ m - ρ)gx....... and.... Δh = [(ρ m / ρ ) - 1] x
The pitot tube meter is used to indicate the velocity of the fluid flow in an enclosed pipe or duct. It is very accurate and involves minimum energy losses in the flowing fluid. It requires good alignment with the flow direction to achieve best results. Pitot tube meters are able to achieve accuracy levels of better than 1% in velocity with alignment errors of up to 15o
Venturi Meter Reference ..Fluid Flow
A venturi meter includes a cylindrical length, a converging length with an included angle of 20o or more, and short parallel throat, and a diverging section with an included angle of about 6o. The internal finishes and proportions are such to enable the most accurate readings while ensuring minimum head losses.
Assuming an inviscid fluid with no losses due to viscocity. The velocities at section 1 and 2 are u 1 and u 2. The velocities are steady and uniform over areas A 1 and A 2
The contuity exquation applies therefore A 2 u 2 = A 2 u 2 = Q
Applying bernoulli's equation to a streamline passing along the axis between the two sections( 1 & 2 ).
Applying bernoulli's equation to the two sections.
Therefore the ideal discharge is given by
Now in practice there is a slight friction loss between 1 and 2 which would result in a high Δh reading and a consequent value of Q which is too high. For real fluids therefore a factor is introduced called the coefficient of discharge factor (C d ).For low viscosity fluids C d = 0,98. The actual discharge as measured by a venturi is therefore given by.
Design and performance parameters of venturi flow meters are provided in BS EN ISO 5167-4:2003
Nozzle Flow Meter
The nozzle as shown is practically a venturi with the diverging part removed. The basic equations used are the same as for the venturi meter. The friction losses are slightly larger than for the venturi but this is offset by the lower cost of the unit. The fact that the manometer connections cannot be located in the ideal positions for measuring the required piezometric pressures is allowed for in selection of the coefficient of discharge factor C d..
Design and performance parameters of nozzles flow meters are provided in BS EN ISO 5167-3:2003
Orifice Flow Meter
The simplest and cheapest method of measuring the flow using the bernoulli equation is the sharp edged orifice as shown below.
The fluid flow pattern in the region of and orifice is shown in the diagram below..
Application of Bernoulli's equation to the fluid flowing through the orifice.
Now u 1 = Q /A 1 and u 2 = Q /A c where A c = The area of the vena-contracta which is the reduced area of the fluid after leaving the orifice hole. (A c = C c A 2 ).A 2 = the area of the orifice and C c is the Coefficient of contraction.Finally C d = Coefficient of discharge = C c.C v
Using these factors a relationship for Q can be developed from the above equation
To arrive at a final equation a overall discharge coefficient C is introduced.
Now letting β = d 2 / d 1 that is β 2 = A 2 / A 1. The equation for flow through an orifice becomes
Note: This equation is very similar to the equation provided in BS EN ISO 5167:2 except that an expansion coefficient (ε )is introduced to cater for the measurement of compressible fluids. The equation provided in the standards is .
Values of the discharge coefficient C are provided in BS EN ISO 5167:2 for the different meter tapping arrangements, for different values of β against Reynold number ranges.
Small table showing C values for different Reynold numbers and β values . Detailed tables are provided in BS EN ISO 5167:2
β Re
5 x 103 1 x 105 1 x 108<
0.25 0,6102 0,6025 0,6013
0.5 0,6284 0,6082 0,6036
0.75 0,6732 0,6171 0,6025
Flow Conditioning
The accuracy of the flow measuring devices is very much affected by uniformity of the approaching fluid flow. Therefore ideally there should be a straight length of piping before the flow measuring device. It is generally accepted that for accurate flow readings there should be 50 pipe diameters of straight piping before the metering device following any pipe bend, valve, tee, reducer etc. The relevant standard provides a range of recommended minimum straight lengths depending on the nearest upstream fittings varying from 5 to 44 lengths. This length can be reduced if flow straighteners or flow conditioning devices systems are used upstream of the flow measuring device. A flow straightener is designed to remove swirl from the flowing fluid. A flow conditioner is a device which removes swirl and also redistributes the velocity profile to produce near ideal metering conditions.
Losses resulting from flow metering devices
The orifice plate and, to a lesser extent the nozzle has significant kinetic energy losses downstream of the metering device as the locally generated kinetic energy is dissipated. The figure below illustrates the extent of these losses.
Fluid Machines
Introduction
The fluid machines covered on this page are primarily hydraulic machines.
The notes below only relate to the general principles involved - more specific notes and information on steam and pneumatic machines, are provided on the linked pages below.
A fluid machine is a device for converting energy held in a fluid ( dynamic or potential ) into mechanical energy or it can convert mechanical energy into fluid based energy.
Notes on Pump Types and Operation Notes on Air Compressors and MotorsNotes on Steam TurbinesNotes on pneumatics and hydraulics
The main types of fluid machines are listed as
Turbines and pumps..A turbine directly converts fluid energy into rotating shaft energy.
If the fluid motion is converted, initially to reciprocating mechanical motion the machine is an engine e.g and internal combustion engine or a steam engine ).
A machine for converting mechanical energy into fluid flow is called a pump...
Compressors or Fan ..If the machine converts mechanical energy to increase the potential energy of a compressible fluid by increasing its pressure the machine is called a compressor. If the machine is primarily provided to increase the kinetic energy of a compressible fluid e.g. air, the machine is a fan. With a fan or blower the pressure head developed is usually relatively small and fluid calculations can often be done assuming the fluid is incompressible.
Positive Displacement Machines ..A pump can be a positive displacement machine or a rotodynamic machine Ref. Pumps ..Positive displacement machines are designed such that there is virtually zero fluid slippage in the energy transfer process. The general principle of these type of pumps is that fluid is drawn into a chamber at a low pressure. The inlet to the chamber is closed and the outlet is opened, and the fluid is then forced out of the chamber by reducing its volume.
The type of pump can be used to generate very high pressures in a compact mechanical envelope. The main disadvantage is that the operation is an intermittent one resulting in a high level of pressure fluctuation throughout the operating cycle.
Rotodynamic Machines ..All rotodynamic machines have a rotating component through which the fluid passes. In a turbine this is called the rotor which has a number of vanes or blades. The fluid passes through the blades and drives the rotor round transferring tangential momentum to the rotor. In a pump the tangential motion of the rotor as it rotates results in an increase in the tangential momentum of the fluid . This increase in kinetic energy is converted to pressure by decelerating the fluid in the discharge route from the pump. In a turbine the fluid passes over /through the impeller and loses energy (momentum and pressure) the energy being transferred to the rotor.
Rotodynamic machines are smooth and continuous in action with a consequent pulsation free flow from pumps and smooth rotation from turbines. In the event of pump discharge flow being suddenly stopped there are no high shock loads. Positive displacement machines can easily be damaged if a discharge valve is suddenly closed. Rotodynamic pumps are ideal for high flow low discharge head duties and provide compact reliable solutions.
Symbols A = Area (m2)a = Speed of sound (m/s)CV = Coefficient of Velocity for nozzle. ρ = density (kg/m3)ρ1 = density at inlet conditions(kg/m3)g = acceleration due to gravity (m/s2 )ε = Expansion factor h = fluid head (m)L = Pipe length (m )m = mass (kg)m = mass flow rate (kg/s)..
p s= surface pressure (N /m2 )Q = Volume flow rate (m3 /s)Qm = Mass flow rate = Qρ (kg /s)Re = Reynolds Number = u.ρD/μ r = Turbine wheel radius (m)u1 = initial fluid velocity (m/s) u2 = final fluid velocity (m/s) uW1 =initial fluid whirl velocity (m/s) uW2 =final fluid whirl velocity (m/s) VB = Bucket velocity (m/s) VR1 = Inlet Fluid Velocity relative to bucket(m/s) w = Turbine shaft work / unit mass flow rate (Nm /(kg/s)θ =slope (radians)μ = viscosity (Pa.s) ν = kinematic viscosity (m2�s-1) υ = Specific volume (m3 / kg)γ= Ratio of Specific Heats ω = Angular Velocity of wheel ( radians /s)
Reciprocating Pumps
The general principles of operation of the reciprocating pump is illustrated by the figure below. The motion of the piston is such that it is driving the fluid out of the cylinder through the one way valve into the upper tank. The fluid, in this case, is incompressible and the flow rate is almost totally dependent on the velocity of the piston. As the driving arm continues to rotate the piston commences to move outwards. Reverse flow in the discharge line results in the discharge valve closing and fluid being drawn up from the lower reservoir causes the suction valve to open. Fluid is therefore drawn into the cylinder from the lower reservoir.
The design of the inlet and outlet valves as shown are simple flap valves. Other valve types include ball valves and poppet valves. The valves may be gravity/flow operated or spring operated or mechanically linked to the position of the piston.
A plot showing the absolute pressure against piston displacment volume is shown above. The piston has an area A and the pressure within the cylinder is p.
The work done by the piston in the discharge stroke and on the piston in the suction stroke is F dx. = p A dx. = p dV. The area enclosed within the pressure- volume curve therefore represents the work done by the cylinder in one cycle.
A-B-C-D represents the operating cycle with no losses. A'-F-B'-C'-G-D' is the curve allowing for the inertia of the fluid i.e. the fluid has to be accelerated and the start of the stroke and slowed at the end of the stroke. A'-F'-B'-C'-G'-D' also includes for the friction in the piping and the inlet and outlet valves- this will be greatest at the middle of the stroke when the velocity of the piston is highest ( due to the motion of the driving arm). Note: the differences between the ideal and real plots are exaggerated. The deviations from the no-loss operating cycles are clearly more pronounced as the pump speed increases
The design of the valves can have a significant effect on the efficiency of operation of the piston pump. Some systems have valves which are large compared to the volume of the cylinders. Spring loaded lift valves working at relatively high frequencies can bounce and the effect on the operating curve can be significant as shown in the figure above.
The motion of the piston (plunger) and consequently the output flow is sinusoidal. In practice it is desirable to have a smooth continuous flow. There are a number of methods of increasing the delivered flow including. Using double acting pumps and using mult-cylinder pumps. These arrangements are shown below.
The inflows and the outflows can be further smoothed and the pulsations can be largely eliminated by installing surge tanks as shown below
In practice the discharge resulting from each cycle is slightly different from the volume displaced by the piston (plunger). This is because a certain leakage takes place and the valves do not open and close instantly. This is identified by using a coefficient of discharge. The coefficient of discharg Cd is the Discharge per cycle / The Swept volume. The percentage slip is also used to identify this source of inefficiency.
Unlike the reciprocating positive displacement pumps, which work primarily under hydrostatic conditions, turbines transfer energy to the fluid under hydrodynamic conditions by converting the kinetic energy of the blades to kinetic energy of the fluid or vice versa. In radial flow machines the fluid flows mainly in the plane of the rotation. The fluid enters the machine at one radius and exits the machine at a different radius. This type of machine includes the Francis turbine and the centrifugal pump. In axial flow machines the fluid moves generally parallel to the axis of the pumps. If the flow is partly radial and partly axial the machine is said to be a mixed flow machine. A rotodynamic pump can often be used as a motor. This option is often used for hydraulic pump storage systems
For any turbine the energy in the fluid is initially pressure energy. For water turbines, this pressure energy is developed as a head of fluid in a high level reservoir. For steam turbines the pressure is developed by the addition of heat in boilers..
The impulse turbine has one or more fixed nozzles. The nozzles turn the pressure energy into kinetic energy as high velocity jets of fluid. These jets then impinge on the moving blades of the runner where the fluid loses almost all of its kinetic energy and the momentum is transferred to the blade. A typical impulse turbine is the pelton wheel
The reaction turbine fluid transfers its energy by tangential slippage across the blades literally pushing the blades sideways out of its path. The energy transfer is a gradual process, the fluid loses it kinetic energy progressively. The fluid literally fills all of the passages through turbine blades. A typical reaction turbine is the steam or gas turbine.
Impulse Turbines - Pelton Wheel.... Reference Jets on moving plates
In a typical pelton wheel, as shown, below the fluid kinetic energy is transferred to the rotating wheel according to the momentum equation. The fluid velocity, momentum, and the resulting force on the buckets are all vector quantities and the system is best analysed using vectors.
The velocity of the fluid leaving the jet is given by
CV = coefficient of velocity (between 0,97 and 0,99)H is the net head at the nozzle = (HT - Hf)= (the total head due gravity - The friction head which is generally negligible.)
The vector diagram for the fluid impinging on the bucket is shown below. Each bucket rotates slightly while it is in the path of the jet but this does not have a significant effect on the analysis.
The outlet flow flow from the bucket is split but the split is considered symmetrical and therefore one path only needs to be considered.
The velocity of the fluid approaching the bucket = u1.The velocity of the bucket = VB.The velocity of the approaching fluid to relative the bucket velocity is VR1 = u1 - VB
The velocity of the outlet fluid relative the bucket velocity is VR2 = k.VR1 (k being a friction loss factor)The velocity of the fluid in the direction of bucket movement -> is the whirl velocity.The inlet whirl velocity = u1
The outlet whirl velocity uw2 is simply VB - VR2(cos (π - θ)The radius of the wheel is r.The change in whirl component of the fluid velocity = ΔuW = u1 - [VB-VR2 (cos (π - θ)] = VR1 - k.VR1cos (π - θ)Therefore..
The force( F )driving the bucket round is equal to the rate of change of momentum of the fluid F = ρ QΔuW.The torque driving the wheel = F r The power from the wheel is F r ω = Q ρ ΔuW VB
The rate of kinetic energy arrival in the fluid = [ Q ρ u 12 ] /2
The efficiency of the Pelton Wheel is therefore expressed as follows
It can be easily proved that the maximum efficiency of the Pelton wheel is when VB = u1 / 2... (Assuming the friction factor is constant ). The above analysis is an idealised one. In practice there will be friction losses reflect in the value of k. There will be losses as the buckets move into and out of the line of the jet, and there will be windage losses as the wheel rotates. The graph below shows the effect of these losses on the efficiency curve
The principle methods of controlling the speed of the Pelton wheels is by using special valves associated with the nozzles or by diverting the jet away from the buckets or by simply starting and stopping the flow.
Pelton wheels are more suitable for operating under large heads...
Further notes on impulse turbines as related to steam as a fluid are found at Steam Turbines -Impulse Blading
Reaction Turbines - Francis Turbine - Centrifugal Pump ....
The reaction turbine is completely immersed in the fluid and the energy is converted from fluid to mechanical motion , and vice versa, gradually as the fluid passes over the blades. In a reaction turbine the fluid (water) supplied to the rotor (runner) possesses part pressure energy and part and kinetic energy . Both types of energy ar converted into work in the runner, resulting in a drop of the pressure and absolute velocity of the fluid. A typical reaction turbine is the Francis Turbine which is generally arranged with a vertical shaft.
This type of turbine is a mixed flow type - the fluid entering the runner in a radial direction and exiting the runner axially downwards.
The only movement of the runner blades is in the circumferential direction and therefore only circumferential fluid force components result in work transfer.The fluid velocity at the feed into the rotor tangential to the rotor is uW1.The fluid velocity leaving the rotor in the tangential direction is uW2.
Considering a particle of fluid δm at the rotor inlet . This has a tangential momentum uW1 δm and an angular momentum uW1 r1 δm. The mass particle has a angular momentum on leaving the rotor of uW2 r2 δm.If the total mass flow rate is m and considering a small part of this flow δm.The rate of momentum entering the rotor is uW1 r1 δm and the rate of momentum leaving the rotor is uW2 r2 δm. This assumes r1 , r2, uW1 and uW2 are constant over the area over which δm flows.
The total rate of fluid momentum loss across the turbine is equal to the torque acting on the fluid which is
By application of Newtons third law ( for revery action there is a equal and opposit reaction) the fluid exerts the same torque , in reverse on the rotors.
The above equation is known as Euler's equation and identifies a fundamental relationship for all forms of rotodynamic machinery including turbines, pumps, fans and compressors. It applies to rotors and stators.
If the products uW1 r1 and uW2 r2 are each constant at the inlet and outlets (vortex free) then the equation simplifies to
If the value of Tr is positive and the rotor is rotating in the same direction as the fluid then the fluid does work on the rotor and the machine is operating as a turbine. If the fluid resists rotation of the rotor or impeller and Tr from the above equation is negative then the machine is operating as a pump.
The rate at which shaft work is done on the rotor (power) is Tω .
Now if the turbine shaft work rate is divided by the mass flow rate the work done per unit mass ( w ) results
Referring to the figure below the net Head (H) across the Francis machine is the total head between the supply reservoir and the tail water , minus the pipe friction losses (hf ) and the kinetic head of the fluid at outlet of the draft tube. Therefore
The energy available per unit mass = Hg Therefore the efficiency of the Francis Turbine is
