CHAPTER 1 PROPERTIES OF FLUIDS 1.1
Introduction
Fluids mechanics deals with with study of fluids, fluids, liquid and gases. gases. The study can be behavior of liquid liquid fluids fluids at rest rest (stati (static) c) and in motion motion (dynam (dynamic) ic)..
The study of fluid mechanic mechanicss is
important because our life depend on them. The air we breathe, flight of birds in air, air, the motion of fish in water, circulation of blood in veins of human body, flow of oil and gas in pipelines, transportat transportation ion of water in pipe, all follow the principles principles of fluid mechanics. mechanics. Engineers Engineers have applied these principles in the design of dams, construction of ships, airplanes, turbo-machinery etc. Fluids Fluids in motion are potential potential sources sources of energy and can be converted into into useful wor to drive a water turbine turbine or windmill. windmill. The principles principles of fluid mechanics mechanics are also applied to fluid power system in which pressured fluid is used to transmit power. !ydraulic drives and controls have become more and important due to automation and mechani"ation. Today, a very large large part or modern machinery is controlled completely or pa rtly by fluid power.
Fluid can be defined as substance that has ability to flow flow.. #ases e$pand whereas liquids do not. %iquid have no shape of it&s it&s own but rather rather tae the shape of the container in which it is placed. That means if liquid or volume less than volume of the container is poured into container the fluids will occupy a volume of the container and will have a free surface. surface. #ases e$pand and occupy full volume of the the container. #ases are compressible which means their volume changes with pressure pressure where as liquids liquids are incompress incompressible. ible. 'ompressib 'ompressible le flows are again divided divided into subsonic and supersonic depending on gas velocity less or greater than sound velocity. Their application is in et propulsion system, aircraft and rocets. 1.2 1.2
Inte Intern rnat atio iona nall Sys Syste te o! Unit Unitss "SI "SI##
n the te$t we shall use * units. The dimension in any system can be considered as either primary or secondary dimensions. n the * units there are + primary dimensions.
a) ri rimar mary ni nitts
Diension ass %ength Time Temperature Electric 'urrent
International Sy$ol % T / 0
Unit /ilogram (g) eter (m) *econd (s) /elvin (/) 0mpere (0)
b) *econdary nits *econdary units is a combination of primary units such as 1ewton (1 or gm2s3), 4oule (4 or 1m), 5att 5att (5 or 1m2s) etc.
1.% 1.%
S&ec S& eci! i!ic ic 'ei( 'ei()t )t and and ass ass den density sity
Two important parameters that tend to indicate heaviness of the substance are specific weight and mass density. density. The specific weight in the weight of substance substance per unit volume and is commonly designated by #ree letter 6gamma& ( γ ). n equation form,
γ =
Weight W = Volume V
ass density is the mass per unit volume of the substance. t commonly designated by a #ree letter 6rho& ( ρ ). n the equation form,
ρ=
mass m = Volume V
7r n thermodynamics normally referred as specific volume, 89
ν=
Volume V 1 = = mass m ρ
There e$ist an important relation between specific weight and mass density. density.
5eight of the substance w : mg
5eight of the substance2unit volume,
γ =
mg = ρg V
For ideal gases, the density of gas is depended on the pressure and temperature of the gas. The density can be obtained by the gas equation;
PV =mRT
or
P ρ RT =
R=
Universal gas
where
Coefficient
molar gas
=
ℜ
M
Thus the specific weight is the product of mass density and acceleration due to gravity. n the * units, will be e$pressed in 12m< and = in g2m<. The values of specific weight and mass density of water at different temperature are given in Table >.> and Table >.3 gives Table >.> hysical roperties of 5ater
Temperatur
Specific
Mass
Dynamic
Kinematic
Surface
e
Weight
Density
Viscosity
tension
T
γ
ρ
μ
Viscosity ν 2
−6
N/m 0.0756
−7
0.0712
−7
0.0662
−7
0.0608
(°C) 0
(kN/m³) 9.81
(kg/m³) 1000
30
9.77
996
60
9.65
984
90
9.47
956
N-s/m² 1.75 x 10 8.00 x 10 4.60 x 10 3.11 x 10
m /s −3 −4 −4 −4
1.75 x 10 1.02 x 10 4.67 x 10 3.22 x 10
σ
the mass density for common fluids. From this table, other fluids can be compared with water in terms of density and specific of weight.
Table >.3 hysical roperties of 'ommon fluids at *tandard 0tmospheric ressure (at 3?o')
Fluids
Air
Specific
Specific
Mass
Dynamic
Kinematic
Gravity
Weight
Density
Viscosity
s
γ
ρ
μ
Viscosity ν
-
(kN/m³)
(kg/m³)
N-s/m²
0.0012
11.8
1.20
1.81 x 10
m
2
/s
1.51 x 10
−5
−5
Ammoni
0.830
8.31
829
!"#$%rin%
1.263
12.34
1258
&%ros%n%
0.823
8.03
819
2.20 x 10
4
−3
950 x 10 1.92 x 10
2.65 x 10 7.55 x 10 2.34 x 10
−7 −4 −6
−3
'%r$r#
13.60
133.1
13570
1.56 x 10
1.14 x 10
−7
−3
'%*no"
0.79
7.73
788
+A, 10 i"
0.87
8.71
869
4
5.98 x 10 8.14 x 10
5.58 x 10 9.36 x 10
−7
−5
−2
+A, 30 i"
0.89
8.71
888
4.40 x 10
4.95 x 10
−4
−1
r%nin%
0.87
8.51
868
1.38 x 10
1.58 x 10
−6
−3
%r
1.00
9.79
998
1.02 x 10
1.02 x 10
−6
−3
+% %r
1.03
10.08
1028
1.07 x 10
1.04 x 10
−6
−3
1.*
S&eci!ic (ra+ity
t is the ratio of specific weight of the substance to the specific weight of water at +@'. 0 convenient method to measure specific gravity is by means of a hydrometer. t is dipped into the
liquid and a calibrated scale gives the specific gravity. t should be noted that specific gravity is a dimensionless number and its value for a particular substance is the same regardless of the system of units. t is abbreviated as (s).
Specific gravit
specific !eight =
of flui"
specific !eight
C
of !ater at
=
ρ f ρ!
o #
The specific gravity can also be e$pressed as ratio of mass density of the substance to mass density of water at +@'.
E,a&le 1.1
0 tan of glycerol has a mass of >3AAg and volume of A.B?m<. Cetermine9 (a) 5eight of glycerol (b) Censity (c) *pecific weight (d) *pecific gravity
Solution-
(a) From 1ewton&s %aw;
5: mg
Thus, 5: >3AA $ B.D> : >>.1
(b) From equation (>.3)
ρ=
m 1200 3 = =1265 $g / m V 0.95
(c) From equation (>.<)
γ = ρg =1263× 9.81=12.38 $% / m
3
(d) From equation (>.+)
s=
1.
ρ s ρ W
=
1265 1000
= 1.26
/iscosity
Fluids offer resistance to shearing force. Giscosity is the property of the fluid that determine amount of this resistance.
'onsider a fluid in between two parallel plates Figure >.> where the upper plate is moving with velocity G and lower plate is stationery. The distance between the plates is y. The layer in contact with the upper plate is moving with velocity G where as the layer is contact with lower plate which is fi$ed will have "ero velocity. The deformation of the fluid under the action of shear stress is assumed proportional to the rate of change of velocity, may be e$pressed in the equation form,
Figure >.> The displaced fluid due to shear stress acted and the shear stress is assumed proportional with the velocity gradient. t can be illustrated in the equation;
& =μ
∂ v ∂
(>.?)
5here, H: shear stress I: proportional coefficient ∂v ∂ : velocity gradient
The shear force, FC acting on the lower plate surface is given by;
' (= & × )
5here 0: surface area of the lower plate
The unit of dynamic viscosity, J is * unit is (1-sm -3 or as). /inematics viscosity which is usually, denoted by the #ree letter 6nu& (v) is determined by dividing dynamic viscosity (J) by mass density of the fluid ( ρ ). n the equation form;
μ ν= ρ
(>.)
5hen the fluid is at rest the velocity gradient dv/dy is "ero and therefore no shearing force e$ists. The viscosity varies with temperature therefore values of J for given fluid are usually tabulated at various temperatures. There are e$perimental methods to calculate viscosity.
7ne such
e$perimental method is Falling *phere Giscometer. n this method a sphere of nown diameter is dropped into a liquid. Ky determining the time required for the sphere to fall through a certain distance, its terminal velocity (v) can be calculated. The stoes equation can be written as
*
( )
g" σ v= −1 18 ν ρ 5here
d : diameter of sphere L : sphere density = : fluid density 8 : inematics density
0 number of viscometers are available in the maret. These viscometers are electronic devices with digital panel and measured viscosity most of the liquids such paint, lubrication oil, polymer compound, chemical compositions etc.
Fluids obeying 1ewton&s law of viscosity (equation >.) and for which J has a constant value are nown as 1ewton&s fluids. ost common fluids such as air, water and oil come under this category for which shear stress in proportional to velocity gradient. The fluids that do not obey 1ewton&s law of viscosity are nown as non-1ewtonian fluids such as human blood, lubrication oils, molten rubber and sewage sludge etc. 0 general relation between shear stress and velocity gradient for non-1ewton&s fluids may be written as; n
( )
"v & = ) + + "
(>.)
5here 0 and K are constants. Kased on the value of power inde$ 6n& non M 1ewtonian fluids are classified as;
seudoplastic (such as mil, cement, clay)
nN>
Kingham-plastic (such as sewage sludge, toothpaste)
n:>
Cilatent (such as lubrication oil, butter, printing in)
nO>
0 1ewtonian fluid is a special case of nonM1ewtonian fluid for which 0 : A and power inde$ n : >.
The dynamic viscosity of various fluids at various temperatures is shown in Figure >.3.
3 m 2 s . 1 , y t i s o c s i v e t u l o s b 0
Figure >.3 Cynamic viscosities versus Temperature E,a&le 1.2
The viscosity of a fluid is to be measured by a viscometer constructed of two ?cm long concentric cylinders as shown in Figure E>.3. The outer radius of the inner cylinder is >?cm, and the gap between the two cylinders is A.>3cm. The inner cylinder is rotated at 3AArpm, and the torque is measured to be A.D1m. Cetermine the viscosity of the fluid.
Figure E>.3 Solution-
The shear force,
' ( = & × ) = μ
∂ v ×2 , R∂
5here shear force can be calculated by
' (=
Tor.ue 0 . 8 = =5.33 % R 0.15
G: 3PQf : 3(<.>+3)(A.>?)(3AA)2A : <.>+ m2s
∂ v V −u 3.14 −0 = = =2616.67 m / s 0.0012 ∂ t Thus, the dynamic viscosity9
μ=
1.0
(
2 , 0.15
5.33
)( 0.75 )( 2616.67 )
=0.0029 %s / m2
Co&ressi$ility and ul 3odulus
'onsider a mass of fluid m whose initial pressure and volume is and G respectively. %et the fluid be compressed by application of force such that final pressure is Rd and volume reduced to GMdG. !ence, change in pressure is dp and change in volume is MdG. Golumetric strain in defined as change in volume divided by original volume and is MdG2G. The bul modulus denoted by and is defined as change in pressure to volumetric strain;
: 'hanges in pressure2Golume strain or
$ =−V
"P "V
(>.D)
%et mass of fluid is m9
m : =G
(>.B)
0fter differentiation of equation (>.B)9
ρ "V +V" ρ =0 V ρ "V "ρ
− =
(>.>A)
*ubstitute MG2dG into equation (>.D), thus;
"P $ = ρ "ρ
(>.>>)
From equation (>.>>) the value of is dependent on the relationship between pressure and density. For liquids, changes of density with pressure are small and Kul modulus is high. These liquids can be considered incompressible. !owever, for gases the compressibility is so
large that value of is not a constant but proportional to pressure. For gases relation between pressure and mass density in obtained from characteristic equation of a gas and particular relation between pressure and density is established depending o n type of compression process.
(i)
For an isothermal process where the temperature is maintained constant the characteristic equation is written as
"P P = =const . "ρ ρ
P = const . ρ
"P P = = const . "ρ ρ
(>.>3)
(>.><)
*ubstitute d2d= into equation (>.) gives
:
(ii)
(>.>+)
For an adiabatic process where no heat is allowed to enter or leave during compression the relation between pressure and density is given by
P ρ γ
=const .
(>.>?)
0fter differentiation will give;
"P P = γ "ρ ρ
(>.>)
where S: ratio of specific heats at constant pressure and at constant volume or S: '2'G
0gain substitute d2d= into equation (>.>>) will give;
: S
(>.>)
The ratio of adiabatic bul modulus is equal to the ratio of specific heat of fluid as constant pressure to that at constant volume.
For liquids is almost equal to one, but for gases the
difference is large for e$ample for air : >.+. 1.4
3ac) no. and Co&ressi$ility
ach. 1o. is defined as ratio of velocity of flow ( v) to local velocity of sound (a) and is a measure of compressibility effects.
M =
v a
(>.>D)
The velocity of propagation of sound waves in a fluid, flow is e$pressed as
a=
√
"P "ρ
a2 = or
"P "ρ
(>.>B)
*ubstituting the value of d2d in equation (>.>>) we get
$ = ρa
2
(>.3A)
*ubstituting value of a in equation (>.>3) we get
√
v 2 ρ M = $
(>.3>)
For liquids the bul modulus is large and velocities small and, hence, ach. 1o. is negligible or effect of compressibility is neglected. #as velocities are high bul modulus is low, and hence,
ach. no. is high and compressibility cannot be neglected.
#ases can only be treated as
incompressible if pressure changes are small and ach. no. is less than A.<.
1.5
Sur!ace tension
The molecules of the liquid are attracted by the molecules of the same liquid by a force nown as 6'ohesion&. This force eeps the molecules bonded together.
The force of attraction between molecules of two different liquids that do not mi$ each other or between liquids molecules and solid boundary containing the liquid or between molecules or liquid on side and molecules of air (or gas) on the other side is nown as 6adhesion&.
Figure >.<
Figure >.< is shown a molecule of liquid at the surface is acted on by imbalance cohesion and adhesive forces giving rise to surface tension. t is commonly denoted by #ree letter, 6sigma& () and is defined as force per unit length of the surface. n the equation, it can be written as
' σ = -
(>.33)
The units of L in * units will be 12m. n many engineering problems surface tension forces are very small compared with other forces acting on the fluid and may therefore be
neglected. !owever, surface tension can cause serious errors in capillary effects particularly in manometer.
For a droplet or a half bubble, the surface tension effect can be illustrated by analy"ing a free body diagram as shown in Figure >.+.
Figure >.+
The pressure force e$erted in the droplet is given by
' = P,R
2
The force due to surface tension is
' = 2 ,Rσ
The pressure force and tension must be balance each other;
2
P,R = 2 ,Rσ
P=
1.6
Ca&illarity
2 σ
R
(>.3<)
f a small diameter glass tube is inserted into water through a free surface the water will rise in the tube. This phenomenon is nown as capillarity and is caused by cohesive force of the liquid molecules and adhesion of liquid surface to solid glass surface.
The rise in level of the capillarity tube will depend on L and angle of contact, as shown in Figure >.?.
Figure >.? %ength of line of contact of the liquid with the tube : Pd
Gertical component of the surface tension force : (Pd).L.cos
, 4
5eight of column of liquid, 5:
" 2 hγ
Thus, for the equilibrium of surfaces tension and gravity forces requires as
, ,"σ cos /= " 2 hγ 4
h=
4 σ cos /
γ"
P= or
4 σ cos /
"
(>.3+)
'onsequently, when one does not wish a meniscus to rise appreciably in a tube, a large value of diameter is chosen. t is believed that trees, even very tall ones, send water to their highest branches by means of capillarity effects. !ence, capillary passages must be e$tremely fine. n water and certain other liquids that e$hibit capillarity the meniscus is 6concave&. These liquids wet the glass and angle of contact is less than BA@. n some other liquids such as mercury the meniscus is conve$. The liquids do not wet the solid surface and angle of contact is more than BA@. #lass tubes are commonly used in manometer and capillary action is a serious source of error in reading levels in such tubes. They should have as large a diameter as is conveniently possible to minimi"e errors due to capillarity. 1.17
/a&or Pressure
'avitation is given to the phenomenon that occurs at the solid boundaries of liquid streams when the pressure of the liquid is reduced to vapor pressure of the liquid at the prevailing temperature (Table >.<). 0ny attempt to reduce the pressure still further merely causes the liquid to vapori"e more quicly and clouds of vapor bubble form. The bubbles of vapor formed in the region of cavitation move downstream to a region of higher pressure where they collapse (see Figure >.?). t is repeated formation and collapse of vapor bubbles which can have damaging effects upon the walls of the solid surface. The actual time between formation and collapse may not be more than >2>AA of a second, but dynamic force caused by this phenomenon may be very severe. t is only a matter of having enough bubbles formed over a sufficient period of time for the destruction of the metal begins.
'avitation may occur in pumps, turbines,
hydrofoils, propellers, and in venture-meters. n the case of turbines, cavitation is most liely to occur on the blade surfaces near the tail race where as for pumps it is most liely to occur at inlet to the impeller. 'avitation can also occur if a liquid contains dissolved air or gases, since solubility of gases in liquid decreases as the pressure is reduced. #as and air bubbles will be released as vapor bubble with the same damaging effects.
'are should be taen to avoid
cavitation as far as possible but if this proves impracticable, than the parts liely to be affected by cavitations should be constructed of especially resistant metals such as stainless steel.
Table >.< *aturation vapor pressure of water
Temperature (U') A >A 3A +A A DA >AA
Gapor ressure (a) >? >3
A>
Figure >.? 'avitation phenomena inside the no""le
Pro$les
>.
Cetermine the density of air, hydrogen, and carbon dio$ide at an absolute pressure of
3.
'alculate the specific weight and density of o$ygen at absolute pressure of ?AA a and a temperature of +AW'. (D.3$>A-+ g2m<, D.A+$>A-< 12m<)
<.
f the volume of liquid decreases by A.3X for an increase of pressure from D 12mV to >?B 12mV, calculate the bul modulus of elasticity of the liquid. (+,+>+.?a)
+.
0 soap bubble ?> mm in diameter has an internal pressure in e$cess of outside pressure of 3.A $ >A-3 a, calculate the tension in the soap film.
?.
(A.><><<12m) f the pressure inside a water droplet is A.3 a in e$cess of e$ternal pressure, and given surface tension of water in contact with air at 3AW' is equal to A.A< 12m, determine the diameter of the droplet. (>.+3mm)
.
0ir is introduced a no""le into a tan of water to from a stream of bubbles. f the bubbles are intended to have diameter of 3 mm, determine the pressure of air at the no""le e$ceed that of surrounding water (given tension of water : A.A< 12m). (>+.312m3)
.
The air in an automobile tyre is at 3.B+< a absolute at 3.W'. 0ssuming no change in the volume of air if the temperature rises to rises to 3.3W', determine the air pressure. (<.+DBa)
D.
0 gas occupying a volume of ?A liters. 'alculate the initial and final bul modulus of elasticity. (A.<+12mm3, A.D12mm3)
B.
#iven v: >Ay>2, where v is the velocity of water in m2s and y is the distance from the boundary in mm, plot the velocity profile and determine the ratio of shear stress at y: 3mm to the shear stress at y: ?mm. (3.>+)
>A.
0 liquid flows between two fi$ed parallel boundaries. The velocity distribution near to lower wall is given in the following table9 y (mm) >.A 3.A <.A +.A ?.A
v (m2s) >.AA >.BB 3.BD <.AA <.AA
Cetermine the ma$imum and minimum shear stresses (The dynamic viscosity of fluid is A.A?as).
(?A12m3, A) >>.
Two plates are arranged as in Figure Y>> in the liquid. The top plate is moving with the velocity of A.?m2s and the middle plate is moving with the velocity of 3m2s in opposite direction. The area of both plates area A.3?m3. lot the velocity profile on all surfaces and determine the force acting on the middle plate (Tae the viscosity of liquid is A.A>as).
Figure Y>> (<.?1) >3. ercury does not adhere to a glass surface, so when a glass tube immersed in a pool of mercury, the meniscus is depressed, as a shown in Figure Y>3. The surface of mercury is A.?>+ 12m and the angle contact is +AW'. 'alculate the depression distance in a > mm glass tube.
Figure Y>3 (>>.D>mm) ><.
The vapor pressure of water at >AAW' is >A> 12mV, because water boils under these conditions. The vapor pressure of water decreases appro$imately linearly with decreasing temperature at a rate of <.> 12mV2W'. 'alculate the boiling temperature of water at an altitude of
>+.
'alculate the vacuum necessary to cause cavitation in a water flow at temperature of DAo' at the mount of /inabalu where the elevation is 3?AAm above sea level. (3.3a)
-ooo777ooo-
