Articles on Hydraulics

8/6/2019 Articles on Hydraulics

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Darcy-Weisbach Formula

Flow of fluid through a pipe

The flow of liquid through a pipe is resisted by viscous shear stresses within the liquidand the turbulence that occurs along the internal walls of the pipe, created by theroughness of the pipe material. This resistance is usually known as pipe friction and ismeasured is feet or metres head of the fluid, thus the term head loss is also used toexpress the resistance to flow.

Many factors affect the head loss in pipes, the viscosity of the fluid being handled, thesize of the pipes, the roughness of the internal surface of the pipes, the changes inelevations within the system and the length of travel of the fluid.

The resistance through various valves and fittings will also contribute to the overall head

loss. A method to model the resistances for valves and fittings is described elsewhere.In a well designed system the resistance through valves and fittings will be of minorsignificance to the overall head loss, many designers choose to ignore the head loss forvalves and fittings at least in the initial stages of a design.

Much research has been carried out over many years and various formulae to calculatehead loss have been developed based on experimental data.Among these is the Chzy formula which dealt with water flow in open channels. Usingthe concept of wetted perimeter and the internal diameter of a pipe the Chzy formulacould be adapted to estimate the head loss in a pipe, although the constant C had tobe determined experimentally.

The Darcy-Weisbach equation

Weisbach first proposed the equation we now know as the Darcy-Weisbach formula orDarcy-Weisbach equation:

hf = f (L/D) x (v2/2g)

where:hf = head loss (m)

f = friction factorL = length of pipe work (m)d = inner diameter of pipe work (m)v = velocity of fluid (m/s)g = acceleration due to gravity (m/s)

or:

hf = head loss (ft)f = friction factorL = length of pipe work (ft)d = inner diameter of pipe work (ft)v = velocity of fluid (ft/s)g = acceleration due to gravity (ft/s)


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3/23

In 1944 LF Moody plotted the data from the Colebrook equation and this chart which isnow known as The Moody Chartor sometimes the Friction Factor Chart, enables auser to plot the Reynolds number and the Relative Roughness of the pipe and toestablish a reasonably accurate value of the friction factor for turbulent flow conditions.

The Moody Chart encouraged the use of the Darcy-Weisbach friction factor and thisquickly became the method of choice for hydraulic engineers. Many forms of head losscalculator were developed to assist with the calculations, amongst these a round slide

rule offered calculations for flow in pipes on one side and flow in open channels on thereverse side.

The development of the personnel computer from the 1980s onwards reduced the timeneeded to perform the friction factor and head loss calculations, which in turn haswidened the use of the Darcy-Weisbach formula to the point that all other formula arenow largely unused.


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Fanning Friction Factor

The frictional head loss in pipes with full flow may be calculated by using the followingformula and an appropriate Fanning friction factor.

hf = ff (L/Rh) x (v2/2g)

where:hf = head loss (m)

ff = Fanning friction factorL = length of pipe work (m)Rh = hydraulic radius of pipe work (m)v = velocity of fluid (m/s)g = acceleration due to gravity (m/s)

or:hf = head loss (ft)

ff = Fanning friction factorL = length of pipe work (ft)Rh = hydraulic radius of pipe work (ft)v = velocity of fluid (ft/s)g = acceleration due to gravity (ft/s)

The Fanning friction factor is not the same as the Darcy Friction factor (which is 4 timesgreater than the Fanning Friction factor)

The above formula is very similar to the Darcy-Weisbach formula but the HydraulicRadius of the pipe work must used, not the pipe diameter.

The hydraulic radius calculation involves dividing the cross sectional area of flow by thewetted perimeter.

For a round pipe with full flow the hydraulic radius is equal to of the pipe diameter.i.e. Cross sectional area of flow / Wetted perimeter = ( x d2 / 4) / ( x d) = d/4

Published tables of Fanning friction factors are usually only applicable to the turbulentflow of water at 60 F (15.5 C).

The development of The Moody Chart which enables engineers to plot the DarcyFriction factor and the use of the personnel computer to calculate the Darcy Friction

factor has led to a large reduction in the use of Fanning friction factors.


5/23

Hazen-Williams Formula

Empirical formulae are sometimes used to calculate the approximate head loss in a pipewhen water is flowing and the flow is turbulent. Prior to the availability of personalcomputers the Hazen-Williams formula was very popular with engineers because of the

relatively simple calculations required.

Unfortunately the results depend upon the value of the friction factor C hw which must beused with the formula and this can vary from around 80 up to 130 and higher,depending on the pipe type, pipe size and the water velocity.

The imperial form of the Hazen-Williams formula is:

hf = 0.002083 L (100/C)1.85 x (gpm1.85/d4.8655)

where:

hf = head loss in feet of waterL = length of pipe in feetC = friction coefficientgpm = gallons per minute (USA gallons not imperial gallons)d = inside diameter of the pipe in inches

The empirical nature of the friction factor C hw makes the Hazen-Williams formulaunsuitable for accurate prediction of head loss.

The results are only valid for fluids which have a kinematic viscosity of 1.13 centistokes,

where the fluid velocity is less than 10 feet per sec and the pipe size is greater than 2diameter. Water at 60 F (15.5 C) has a kinematic viscosity of 1.13 centistokes.

Common Friction Factor Values of C hw used for design purposes are:

Asbestos Cement 140Brass tube 130Cast-Iron tube 100Concrete tube110Copper tube130Corrugated steel tube 60Galvanized tubing 120Glass tube130Lead piping130Plastic pipe140PVC pipe 150General smooth pipes 140Steel pipe 120Steel riveted pipes 100Tar coated cast iron tube 100Tin tubing130Wood Stave 110

These factors include some allowance to provide for the effects of changes to theinternal pipe surface due to the build up of deposits or pitting of the pipe wall during long

periods of use.


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Non-Circular Pipe Friction

The frictional head loss in circular pipes is usually calculated by using theDarcy-Weisbach formula with a Darcy Friction factor. For circular pipes the inner pipediameter is used is used to calculate the Reynolds number and to calculate the relativeroughness of the pipe, which are both used to calculate the Darcy Friction factor.

To calculate the frictional head loss non-circular pipes the method must be adapted touse the Hydraulic Diameter instead of the internal dimensions of the pipe.

Hydraulic Diameter = 4 x cross sectional area of flow / wetted perimeter

For a round pipe the Dh = 4 x ( x d2 / 4) / ( x d) = d

For a rectangular duct the Dh = 4 x (w x h) / 2 x (w + h) where w = width, h = height

For an elliptical duct the Dh = 4 x ( x a x b) / x [(2 x (a2 + b2)) ((a - b)2/2)]

where a = major diameter / 2, b = minor diameter /2 ,Note: the formula uses an approximation for the circumference of an elliptical duct.

For an annulus formed by placing a smaller diameter pipe inside a larger diameter pipethe cross sectional area of flow will be the cross sectional area of the larger pipecalculated using the inner pipe diameter minus the cross sectional area of the smallerpipe calculated using the outer pipe diameter. The wetted perimeter will be the innercircumference of the larger pipe plus the outer circumference of the smaller pipe.Dh = 4 x ( x (d1

2 d22) / 4) / ( x d1 + d2)

where d1 = inner diameter of larger pipe, d2 = outer diameter of smaller pipe

Example calculation of pipe friction factors:

1. Round pipe:

A round steel pipe 0.4 m internal diameter x 10.0 m long carries a water flow rate of349.1 litres/sec (20.946 m3/min). The temperature of the water is 10o C (50o F).

Dh = Internal diameter of pipe = 0.4 mPipe cross sectional area = x 0.4002/4 = 0.1256 m2Flow velocity = 20.94/0.1256/60 = 2.778 m/sRelative roughness = 0.000046/0.4 = 0.000115Re = v x Dh/ (kinematic viscosity in m

2/s) = 2.778 x 0.4 / 0.000001307 = 850191Friction factor = 0.014 (plotted from Moody chart)

hf = f (L / Dh) x (v2 / 2g) = 0.014 x (10 / 0.4) x (2.7782 / (2 x 9.81)) = 0.138 m head

where:hf = frictional head loss (m)f = friction factorL = length of pipe work (m)

Dh = Hydraulic diameter (m)v = velocity of fluid (m/s)g = acceleration due to gravity (m/s )


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2. Rectangular duct:

A rectangular steel duct 0.6 m wide x 0.3 m high x 10.0 m long carries a water flow rateof 500 litres/sec (30 m3/min). The temperature of the water is 10o C (50o F).

Dh = 4 x (0.6 x 0.3) / 2 x (0.6 + 0.3) = 0.4 m

Duct cross sectional area = 0.6 x 0.3 = 0.18 m2

Flow velocity = 30.00/0.18/60 = 2.778 m/sRelative roughness = 0.000046/0.4 = 0.000115Re = v x Dh/ (kinematic viscosity in m



where:hf = frictional head loss (m)f = friction factorL = length of pipe work (m)Dh = Hydraulic diameter (m)v = velocity of fluid (m/s)g = acceleration due to gravity (m/s )

Pseudo check calculation: A steel pipe with an internal diameter of 0.400 m x 10 m longcarrying a water flow rate of 349.1 litres/sec (20.946 m3/min) will have the same flowvelocity as the rectangular duct. If the water temperature is 10o C (50o F) the calculatedfrictional pressure drop through the steel pipe is 0.138 m head.

3. Elliptical duct:

An elliptical duct made from aluminium has internal dimensions of 0.8 m at its widestpoint and 0.3 m at is highest point. The duct is 10.0 m long and carries a water flow rateof 400 litres/sec (24 m3/min). The temperature of the water is 10o C (50o F).

a = major diameter / 2 = 0.800 / 2 = 0.400b = minor diameter / 2 = 0.300 / 2 = 0.150Duct cross sectional area = x a x b = x 0.400 x 0.150 = 0.1885 m2Duct circumference = x [(2 x (a2 + b2)) ((a - b)2/2)]= x [(2 x (0.42 + 0.152)) ((0.4 0.15)2/2)] = x [0.365 0.03125] = 1.8149 mDh = 4 x 0.1885 / 1.8149 = 0.415 mFlow velocity = 24.00 / 0.1885 / 60 = 2.1220 m/sRelative roughness = 0.0000015 / 0.415= 0.000003615Re = v x Dh/ (kinematic viscosity in m



where:hf = frictional head loss (m)f = friction factorL = length of pipe work (m)Dh = Hydraulic diameter (m)

v = velocity of fluid (m/s)g = acceleration due to gravity (m/s )


8/23

Pseudo check calculation: An aluminium pipe with an internal diameter of 0.415 m x 10m long carrying a water flow rate of 287.1 litres/sec (17.226 m3/min) will have the sameflow velocity as the elliptical duct. If the water temperature is 10o C (50o F) the calculatedfrictional pressure drop is 0.069 m head.

4. Annulus:

An annulus section is formed by placing a stainless steel pipe with an outer diameter of

350 mm inside a stainless steel pipe with an inner diameter of 600. The annulussection is 10 m long and carries a water flow rate of 600 litres/sec (36.00 m3/min). Thewater temperature is 20o C (68o F).

Inner cross sectional area of the larger pipe = x 0.6002/ 4 = 0.2827 m2Outer cross sectional area of the smaller pipe = x 0.3502/ 4 = 0.0962 m2Cross sectional area of the annulus = 0.2827 - 0.0962 = 0.1865 m2

Inner circumference of the larger pipe = x 0.600= 1.8850 mOuter circumference of the smaller pipe = x 0.350= 1.0995 mWetted perimeter = 1.8850 + 1.0995 = 2.9845 m

Dh = 4 x 0.1865 / 2.9845 = 0.250 mFlow velocity = 36.00 / 0.1865 / 60 = 3.217 m/sRelative roughness = 0.000045 / 0.250 = 0.000180Re = v x Dh/ (kinematic viscosity in m



where:hf = frictional head loss (m)

f = friction factorL = length of pipe work (m)Dh = Hydraulic diameter (m)v = velocity of fluid (m/s)g = acceleration due to gravity (m/s )

Pseudo check calculation: A stainless steel pipe with an internal diameter of 0.250 m x10 m long carrying a water flow rate of 157.917 litres/sec (9.475 m3/min) will have thesame flow velocity as the annulus. If the water temperature is 20o C (68o F) thecalculated frictional pressure drop through the steel pipe is 0.307 m head.


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Laminar Flow and Turbulent Flow of Fluids

Resistance to flow in a pipe

When a fluid flows through a pipe the internal roughness (e) of the pipe wall can createlocal eddy currents within the fluid adding a resistance to flow of the fluid. Pipes withsmooth walls such as glass, copper, brass and polyethylene have only a small effect onthe frictional resistance. Pipes with less smooth walls such as concrete, cast iron andsteel will create larger eddy currents which will sometimes have a significant effect onthe frictional resistance.

The velocity profile in a pipe will show that the fluid at the centre of the stream will movemore quickly than the fluid towards the edge of the stream. Therefore friction will occurbetween layers within the fluid.

Fluids with a high viscosity will flow more slowly and will generally not support eddycurrents and therefore the internal roughness of the pipe will have no effect on thefrictional resistance. This condition is known as laminar flow.

Reynolds Number

The Reynolds number (Re) of a flowing fluid is obtained by dividing the kinematicviscosity (viscous force per unit length) into the inertia force of the fluid (velocity xdiameter)

Kinematic viscosity = dynamic viscosityfluid density

Reynolds number = Fluid velocity x Internal pipe diameter _____________________________ ________________

Kinematic viscosity

Laminar Flow

Where the Reynolds number is less than 2300 laminar flow will occur andthe resistance to flow will be independent of the pipe wall roughness.

The friction factor for laminar flow can be calculated from 64 / Re.


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Turbulent Flow

Turbulent flow occurs when the Reynolds number exceeds 4000.

Eddy currents are present within the flow and the ratio of the internal roughness of thepipe to the internal diameter of the pipe needs to be considered to be able to determinethe friction factor. In large diameter pipes the overall effect of the eddy currents is lesssignificant. In small diameter pipes the internal roughness can have a major influenceon the friction factor.

The relative roughness of the pipe and the Reynolds number can be used to plot thefriction factor on a friction factor chart.

The friction factor can be used with the Darcy-Weisbach formula to calculate thefrictional resistance in the pipe. (See separate article on the Darcy-Weisbach Formula).

Between the Laminar and Turbulent flow conditions (Re 2300 to Re 4000) the flowcondition is known as critical. The flow is neither wholly laminar nor wholly turbulent.It may be considered as a combination of the two flow conditions.

The friction factor for turbulent flow can be calculated from the Colebrook-Whiteequation:

fD

ef

Re

35.9log214.1/1

10for Re > 4000


11/23

Internal roughness (e) of common pipe materials.

Cast iron (Asphalt dipped) 0.1220 mm 0.004800Cast iron 0.4000 mm 0.001575Concrete 0.3000 mm 0.011811Copper 0.0015 mm 0.000059PVC 0.0050 mm 0.000197

Steel 0.0450 mm 0.001811

Steel (Galvanised) 0.1500 mm 0.005906


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Net Positive Suction Head

Net positive suction head is the term that is usually used to describe the absolutepressure of a fluid at the inlet to a pump minus the vapour pressure of the liquid.The resultant value is known as the Net Positive Suction Head available.The term is normally shortened to the acronym NPSHa, the adenotes available.

A similar term is used by pump manufactures to describe the energy losses that occurwithin many pumps as the fluid volume is allowed to expand within the pump body.This energy loss is expressed as a head of fluid and is described as NPSHr (NetPositive Suction Head requirement) the r suffix is used to denote the value is arequirement.

Different pumps will have different NPSH requirements dependant on the impellordesign, impellor diameter, inlet type, flow rate, pump speed and other factors.A pump performance curve will usually include a NPSH requirement graph expressed inmetres or feet head so that the NPSHr for the operating condition can be established.

Pressure at the pump inlet

The fluid pressure at a pump inlet will be determined by the pressure on the fluidsurface, the frictional losses in the suction pipework and any rises or falls within thesuction pipework system.

NPSHa calculation

The elements used to calculate NPSHa are all expressed in absolute head units.The NPSHa is calculated from:

Fluid surface pressure + positive head pipework friction loss fluid vapour pressureorFluid surface pressure - negative head pipework friction loss fluid vapour pressure


13/23

Gas bubbles within the fluid (cavitation)

The Vapour pressure of a fluid is the pressure at which the fluid will boil at ambienttemperature.If the pressure within a fluid falls below the vapour pressure of the fluid,gas bubbles will form within the fluid (local boiling of the fluid will occur).

If a fluid which contains gas bubbles is allowed to move through a pump, it is likely thatthe pump will increase the pressure within the fluid so that the gas bubbles collapse.This will occur within the pump and reduce the flow of delivered fluid. The collapse ofthe gas bubbles may cause vibrations which could result in damage to the pipeworksystem or the pump. This effect is known as cavitation.

To avoid cavitation the pressure within the fluid must be higher than the fluid vapourpressure at all times.

Avoiding cavitation

In a system where the pipe work layout provides a positive head, the motive force tomove the fluid to the pump will be the fluid surface pressure plus the positive head.

Incorrect sizing of the supply pipe work and isolating valves may result in high frictionallosses which can still lead to situations where the NPSHa is still too low to preventcavitation.

Understanding NPSHa and NPSHr


14/23

In a system where the fluid needs to be lifted to the pump inlet , the negative headreduces the motive force to move the fluid to the pump.In these instances it is essential to size the supply pipe work and isolating valvesgenerously so that high frictional losses do not reduce the NPSHa below the NPSHr.

Comparison of NPSHa and NPSHr

All calculated values must be in the same units either m hd or ft hd.

If the NPSHa is greater than the NPSHr cavitation should not occur.

If the NPSHr is lower than the NPSHr then gas bubbles will form in the fluid andcaviation will occur.

Increasing the NPSH available

Many systems suffer from initial poor design considerations.

To increase the NPSHa consider the following:

a. Increase the suction pipe work size to give a fluid velocity of about 1 m/sec or 3ft/sec

b. Redesign the suction pipework to eliminate bends, valves and fittings where

possible.c. Raise the height of the fluid container.d. Pressurise the fluid container, but ensure that the pressure in the container is

maintained as the fluid level is lowered.

High Fluid temperature

If the temperature of the fluid to be pumped is higher than normal the relative density ofthe fluid will reduce. This may result in a reduction in pipe work friction losses, but thisreduction may be offset (or exceeded) by an increase in Fluid Vapour Pressure.


15/23

An example:

Water at 20C (68F) has a density of 998 kg/m3 (62.303 lbs/ft3) the vapourpressure of the fluid is 2.339 kPa.a (0.339 psi.a).Water at 60C (140F) has a density of 984 kg/m3 (61.429 lbs/ft3) the vapourpressure of the fluid is 19.946 kPa.a (2.893 psi.a).

A comparison of the system with the two fluid temperatures:

The frictional resistance of the pipe work will reduce by about 10% due to the reduceddensity and viscsoity of the fluid, but the vapour pressure of the fluid will increase byabout 1.8 m head (5.9 ft head).It will be normally necessary to check the NPSHa for a normal ambient fluidtemperature and the higher fluid temperature under these circumstances.


16/23

Pumping Fluids and Getting Fluid to the Pump

Pump suction

The energy to move a fluid to the pump inlet is not provided by the pump.The popular view that a pump sucks fluid from the supply source is false.

Consider the case of Flooded suction where the supply container is positioned wellabove the pump inlet. If the suction pipework has been designed properly the fluidhead available will be sufficient to overcome the frictional resistance of the pipeworkallowing fluid to flow freely into the pump inlet.If the pump inlet connection is removed the fluid will still flow out of the suctionpipework.

In most instances where the fluid has to be lifted the fluid pressure at the pump inletwill be lower than atmospheric pressure. The energy to move the fluid is provided by thepressure on the fluid surface. The frictional losses in the suction pipework and rises inthe suction pipework system will reduce the fluid pressure at the pump inlet.If the pump inlet connection is removed the fluid will not flow out of the suctionpipework.

The pump moves fluid from the inlet port to the outlet port


17/23

This action allows the external forces acting on the fluid intake system to push some ofthe fluid in the supply system into the pump inlet port.If the pump type is not self-priming it will be necessary to fill the system with fluid, toallow pumping to commence.

The supply pipework system must be designed to provide enough pressure at the pumpinlet to avoid caviation occurring within the pump.See separate discussion about Net Positive Suction Head.

Atmospheric pressure

Standard pressure at sea level provided by the weight of the atmosphere is 101325 Pa.This value can be expressed as 0.0 bar gauge or 1.01325 bar absolute.The equivalent imperial values are 0.0 psi g or 14.696 psi a.

Atmospheric pressure applied against a perfect vacuum can support a column of waterof 10.33 m high (33.89 feet high).Mercury has a much higher density than water so that atmospheric pressure will support

a column of mercury 760 mm high (29.92 in. high) against a perfect vacuum.

Getting fluid to the pump

Atmospheric pressure on the fluid surface is the usual energy source used to push thefluid into the pump. The friction within the fluid and the pipework will oppose the fluidflow and reduce the pressure at the pump inlet.

If the suction pipework is resistance is too high it may not be possible to deliver the fulldesign flow rate to the pump, in these circumstances the system will operate at somereduced flow rate and cavitation may occur.A good design guide is to size the pipework to give a pump suction velocity of between0.75 m/sec and 1.25 m/sec (2.5 ft/sec and 4.0 ft/sec)

If the fluid has a high viscosity the resistance to flow may be mainly due to slidingbetween adjacent layers of fluid, the flow will probably be laminar and the pipeworkfriction will be negligible. In these instances the best solution is to raise the position ofthe supply container, to increase the positive head available.If the supply container cannot be raised some positive pressure (above atmospheric)will need to be applied to the fluid surface.


18/23

A sealed supply container could be pressurised to increase the force available to movethe fluid. This pressurisation must be maintained as the container is emptied, otherwise

the force to move the fluid will reduce.

Pump power calculations

The work performed in pumping a fluid will depend on the volume flow rate, the densityof the fluid, the additional head to be added to the fluid pressure and the efficiency ofthe pump.

Hydraulic power:Kw = Flow rate (m3/s) x m head x g x fluid density (kg/m3) 1000Hp = Flow rate (US gpm) x ft head x fluid density (lb/ft3) x 231 1728 33000

Using the example of Flooded suction shown above: The pump inlet pressure is 2 mhead and the resistance to flow in the outlet system is 30 m head, so the pump wouldneed to add the energy to raise the fluid pressure by 28 m head (91.86 ft head).

If the flow rate was 1136 litre/min (300 US gpm) and the fluid was water at 20C (68F)the hydraulic power required would be:

1136 1000 60 x 28.0 x 9.806 x 998 1000 = 5.188 Kw

or 300 x 91.89 x 62.303 x 231

1728

33000 = 6.955 Hp

If the efficiency of the pump was 70% the hydraulic power would have to be divided by0.70 to give the actual power consumed of 7.41 Kw (9.936 Hp)

Where the fluid inlet pressure is a suction value, as the case shown above where thefluid has to be Lifted the power provided by the pump must make up the differencebetween the suction value and the resistance to flow in the outlet system.In this instance the pump must add 34 m head (111.55 ft head).

If the flow rate was 30 litre/sec (475.5 US gpm) and the fluid was water at 10C (50F)

the hydraulic power required would be:

30 1000 x 34.0 x 9.806 x 1000 1000 = 10.002 Kwor 475.5 x 111.55 x 62.428 x 231 1728 33000 = 13.414 Hp

If the efficiency of the pump was 63% the hydraulic power would have to be divided by0.63 to give the actual power consumed of 15.876 Kw (21.292 Hp)


19/23

Viscosity and Density (Metric SI Units)

In the SI system of units the kilogram (kg) is the standard unit of mass, a cubic meter isthe standard unit of volume and the second is the standard unit of time.

Density p

The density of a fluid is obtained by dividing the mass of the fluid by the volume of thefluid. Density is normally expressed as kg per cubic meter.p= kg/m3

Water at a temperature of 20C has a density of 998 kg/m3Sometimes the term Relative Density is used to describe the density of a fluid.Relative density is the fluid density divide by 1000 kg/m3

Water at a temperature of 20C has a Relative density of 0.998

Dynamic Viscosity

Viscosity describes a fluids resistance to flow.Dynamic viscosity (sometimes referred to as Absolute viscosity) is obtained by dividingthe Shear stress by the rate of shear strain.The units of dynamic viscosity are: Force / area x timeThe Pascal unit (Pa) is used to describe pressure or stress = force per areaThis unit can be combined with time (sec) to define dynamic viscosity.

= Pas

1.00 Pas = 10 Poise = 1000 Centipoise

Centipoise (cP) is commonly used to describe dynamic viscosity because water at atemperature of 20C has a viscosity of 1.002 Centipoise.This value must be converted back to 1.002 x 10-3Pas for use in calculations.

Kinematic Viscosity v

Sometimes viscosity is measured by timing the flow of a known volume of fluid from aviscosity measuring cup. The timings can be used along with a formula to estimate thekinematic viscosity value of the fluid in Centistokes (cSt).The motive force driving the fluid out of the cup is the head of fluid.This fluid head is also part of the equation that makes up the volume of the fluid.Rationalizing the equations the fluid head term is eliminated leaving the units ofKinematic viscosity as area / time

v= m2/s

1.0 m2/s = 10000 Stokes = 1000000 Centistokes


20/23

Water at a temperature of 20C has a viscosity of 1.004 x 10-6m2/sThis evaluates to1.004000 Centistokes.This value must be converted back to 1.004 x 10-6m2/s for use in calculations.

The kinematic viscosity can also be determined by dividing the dynamic viscosity by thefluid density.

Kinematic Viscosity and Dynamic Viscosity Relationship

Kinematic Viscosity = Dynamic Viscosity / Densityv = / pCentistokes = Centipoise / Density

To understand the metric units involved in this relationship it will be necessary to use anexample:

Dynamic viscosity = PasSubstitute for Pa = N/m2 and N = kg m/s2

Therefore = Pas = kg/(ms)

Density p= kg/m3

Kinematic Viscosity = v =/p= (kg/(ms) x 10-3) / (kg/m3) = m2/s x 10-6

Viscosity and Density (Imperial Units)

In the Imperial system of units the pound (lb) is the standard unit of weight, a cubic foot

is the standard unit of volume and the second is the standard unit of time.The standard unit of mass is the slug.This is the mass that will accelerate by 1 ft/s when a force of one pound (lbf) is appliedto the mass. The acceleration due to gravity (g) is 32.174 ft per second per second.To obtain the mass of a fluid the weight (lb) must be divided by 32.174.

Density p

Density is normally expressed as mass (slugs) per cubic foot.

The weight of a fluid can be expressed as pounds per cubic foot.

p= slugs/ft 3

Water at a temperature of 70F has a density of 1.936 slugs/ft3(62.286lbs/ft3)

Dynamic Viscosity

The units of dynamic viscosity are: Force / area x time

= lbs/ft2


21/23

Water at a temperature of 70F has a viscosity of 2.04 x 10-5lbs/ft21.0 lbs/ft2= 47880.26 Centipoise

Kinematic Viscosity v

The units of Kinematic viscosity are area / time

v= ft2/s

1.00 ft 2/s = 929.034116 Stokes = 92903.4116 Centistokes

Water at a temperature of 70F has a viscosity of 10.5900 x 10-6ft2/s(0.98384713 Centistokes)

Kinematic Viscosity and Dynamic Viscosity Relationship

Kinematic Viscosity = Dynamic Viscosity / Densityv = / p

The imperial units of kinematic viscosity are ft2/s

To understand the imperial units involved in this relationship it will be necessary to usean example:

Dynamic viscosity = lbs/ft2

Density p= slugs/ft3

Substitute for slug = lb/32.174 fts2Density p= (lb/32.174 fts2)/ft3= (lb/32.174s2)/ft4Note: slugs/ft3 can be expressed in terms of lbs2/ft 4

Kinematic Viscosity v= (lbs/ft2)/(slugs/ft3)Substitute lbs2/ft 4 for slugs/ft3

Kinematic Viscosity v= (lbs/ft

2

)/(lbs

2

/ft4

) = ft2

/s


22/23

Proprties of Common Liquids and Gases

Properties of common liquids

Fluid C F kg/m3

lbs/ft3

ViscosityCentipoise

Vapor presskPa (abs)

Vapor presspsi (abs)

Acetic acid 20 68 1049 65.469 1.127 1.584 0.2297Acetone 20 68 780 48.680 0.325 24.220 3.5128

Aniline 20 68 1022 63.784 4.565 0.400 0.0580

Benzene 20 68 879 54.859 0.654 10.100 1.4649

Bromine 20 68 3100 193.473 0.997 23.330 3.3837

Carbon disulphide 20 68 1293 80.697 0.366 14.000 2.0305Carbontetrachloride 20 68 1632 101.854 0.979 12.000 1.7405

Chloroform 20 68 1490 92.992 0.567 21.198 3.0745

Corn oil 20 68 922 57.543 71.400 0.000 0.0000

Ether diethyl 20 68 714 44.561 0.242 55.075 7.9880

Ethyl alcohol 20 68 789 49.242 1.190 7.900 1.1458

Gasoline (typical) 20 68 719 44.873 0.292 55.100 7.9916Glycerol 20 68 1262 78.762 1495.000 0.000 0.0000

Linseed oil 20 68 925 57.730 43.500 0.000 0.0000

Mercury 20 68 13546 845.415 1.559 0.000 0.0000

Methyl alcohol 20 68 791 49.367 0.594 12.930 1.8753

Nitrobenzene 20 68 1175 73.333 2.052 0.020 0.0029

Olive oil 20 68 920 57.418 83.676 0.000 0.0000

Phenol 20 68 1073 66.967 12.740 0.029 0.0042

Sunflower oil 20 68 920 57.418 64.073 0.000 0.0000

Toluene 20 68 867 54.110 0.585 2.914 0.4226

Turpentine 20 68 870 54.297 1.490 0.250 0.0363

Water 0 32 1000 62.411 1.792 0.611 0.0886

Water 5 41 1000 62.411 1.518 0.873 0.1266Water 10 50 1000 62.411 1.306 1.228 0.1781

Water 15 59 999 62.348 1.138 1.706 0.2474

Water 20 68 998 62.286 1.002 2.339 0.3392

Water 25 77 997 62.223 0.890 3.170 0.4598

Water 30 86 996 62.161 0.797 4.247 0.6160

Water 35 95 994 62.036 0.719 5.629 0.8164

Water 40 104 992 61.911 0.653 7.384 1.0710

Water 45 113 990 61.787 0.596 9.594 1.3915

Water 50 122 988 61.662 0.547 12.351 1.7914

Water 55 131 986 61.537 0.504 15.761 2.2859

Water 60 140 984 61.412 0.466 19.946 2.8929

Water 65 149 981 61.225 0.433 25.041 3.6319Water 70 158 978 61.038 0.404 31.201 4.5253

Water 75 167 975 60.850 0.378 38.595 5.5977

Water 80 176 971 60.601 0.354 47.415 6.8770

Water 85 185 968 60.414 0.333 57.867 8.3929

Water 90 194 965 60.226 0.314 70.182 10.1790

Water 95 203 962 60.039 0.297 84.609 12.2715

Water 100 212 958 59.789 0.282 101.325 14.6960

Water sea 10 50 1030 64.283 1.346 1.300 0.1885

Water sea 20 68 1028 64.158 1.070 2.340 0.3394


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Properties of some common gases(at atmospheric pressure)

Gas C F kg/m3

lbs/ft3

ViscosityCentipoise

Air 0 32 1.293 0.081 0.017Air 5 41 1.27 0.079 0.018

Air 10 50 1.247 0.078 0.018

Air 15 59 1.226 0.077 0.018

Air 20 68 1.205 0.075 0.018

Air 25 77 1.185 0.074 0.018

Air 30 86 1.165 0.073 0.019

Air 35 95 1.146 0.072 0.019

Air 40 104 1.128 0.070 0.019

Air 45 113 1.11 0.069 0.019

Air 50 122 1.093 0.068 0.020

Ammonia 20 68 0.708 0.044 0.010

Carbon dioxide 20 68 1.83 0.114 0.015Carbon monoxide 20 68 1.164 0.073 0.018

Helium 20 68 0.166 0.010 0.020

Hydrogen 20 68 0.084 0.005 0.009

Methane 20 68 0.667 0.042 0.011

Nitrogen 20 68 1.165 0.073 0.018

Oxygen 20 68 1.33 0.083 0.020

Sulphur dioxide 20 68 2.659 0.166 0.013

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