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1.3 Analysis of Fluid Behavior
Fluid Statics : When the fluid is at rest.
Fluid Dynamics : When the fluid is moving.
Governing equations :
mics) themodynaof law(First energy ofon Conservati
law) second (Newtons momentum ofon Conservati
mass ofon Conservati
1.4 Measures of Fluid Mass and Weight
Density, : Mass of a fluid per unit volume
[slug/ft3, kg/m3]
For water at 5oC,
water = 1.940 slugs/ft3 = 1000 kg/m3
For air at standard pressure and at 20oC,
air = 2.377 10-3 slugs/ft3 = 1.225 kg/m3
V
mlim
VV
Specific volume, : Volume per unit mass.
1
Specific weight, : Weight per unit volume
For water at 5oC,
water = 62.4 lb/ft3 = 9.8 kN/m3
For air at standard pressure and at 20oC,
air = 7.64 10-2 lb/ft3 = 12.01 N/m3
g [lb/ft3, N/m3]
Specific gravity, SG : The ratio of the density of the given
fluid to the density of water at some specified temperature,
usually at 4oC(39.2oF).
Specific gravity of gases is usually based on dry air as the
reference fluid.
33
C4 at OHm/kg 1000 or ,ft/slugs 94.1
SGo
2
Pressure : the normal compressive force per unit area acting
on a real or imaginary surface in the fluid.
Microscopically, pressure represents molecular momentum
and intermolecular forces within the fluid.
A
Flimp n
AA
pressure zero torelative defined)(or measured Pressure : pressure Absolute
pressure atmosphere local torelative measured Pressure : pressure Gage
Absolute pressure
= Gage pressure + Atmosphere pressure in vicinity of gage
Vacuum pressure : pressure below local atmosphere pressure
Vacuum pressure = Atmosphere pressure – Absolute pressure
= - Gage pressure
The subscripts “g” and “a” indicates whether the pressure is
gage or absolute.
(ex. 10 psig = 10 pounds per square inch, gage ;
10 psia = 10 pounds per square inch, absolute)
Standard value of
atmospheric pressure is
101.333 kPa (14.496 psia,
29.92 in. Hg, abs)
[Pascal ; Pa=N/m2]
Temperature : defined as a measure of (not equal to) the
energy contained in the molecular motion of the fluid
T (Rankine) = T (Fahrenheit) + 459.67
T (Kelvin) = T (Celsius) + 273.15
T (Rankine) = 1.80 T (Kelvin)
Internal energy (U) : Energy contained in random molecular motions and intermolecular forces, U = U(T)
Specific internal energy ( ) : Internal energy per unit massu
Specific heat at constant volume,
Specific heat at cont. pressure,
For an incompressible fluid, all processes are constant specific volume and
So cp = cv ( for incompressible fluid)
v
vT
uc
p
pT
hc
pvup
uh ; enthalpy specific
0
T
vp
T
pv
pp
1.5 Ideal (Perfect) Gas Law(=Equation of State for an ideal gas)
R is the specific gas constant and is equal to the universal gas
constant (R0) devided by the molecular weight (MW) of the
gas :
v
RT RT p
R.lbm/lb.ftMW
1545K.kg/m.N
MW
8314
MW
RR O0
Liquids exhibit slight variation of density with temperature
and pressure.
No simple, exact equations are available for properties of
liquids. For most practical purposes, liquids are treated as
incompressible fluids.
1.6 Viscosity
Newton’s law of viscosity
where the constant of proportionality, , is called the absolute
viscosity, dynamic viscosity, and simply viscosity of the fluid.
Dimension = [lb.s/ft2], [N.s/m2]
dy
du
viscosity
[Pa·s]
viscosity
[cP]
liquid nitrogen @ 77K 1.58 × 10−4 0.158
acetone* 3.06 × 10−4 0.306
methanol* 5.44 × 10−4 0.544
benzene* 6.04 × 10−4 0.604
water 8.94 × 10−4 0.894
ethanol* 1.074 × 10−3 1.074
mercury* 1.526 × 10−3 1.526
nitrobenzene* 1.863 × 10−3 1.863
propanol* 1.945 × 10−3 1.945
Ethylene glycol 1.61 × 10−2 16.1
sulfuric acid* 2.42 × 10−2 24.2
olive oil .081 81
glycerol* .934 934
castor oil* .985 985
corn syrup* 1.3806 1380.6
HFO-380 2.022 2022
pitch 2.3 × 108 2.3 × 1011
viscosity
[cP]
honey 2,000–10,000
molasses 5,000–10,000
molten glass 10,000–1,000,000
chocolate syrup 10,000–25,000
molten
chocolate*45,000–130,000 [19]
ketchup* 50,000–100,000
peanut butter ~250,000
shortening* ~250,000
Viscosity of Liquids at 25oC
Sutherland equation : (for gases)
where C and S are empirical constants and T is absolute
temperature.
Andrade’s equation : (for liquids)
where D and B are constants. T is absolute temperature.
Kinematic viscosity : [ft2/s, m2/s]
In CGS (centimeter-gram-second) unit, the dynamic viscosity
has the unit of dyne.s/cm2 (=poise, abbreviated as P).
The kinematic viscosity has the unit of cm2/s (=stoke, St)
** 1 dyne = (1g) x (1cm/s2)
ST
CT 2/3
T/BDe
Newtonian and Non-Newtonian Fluid
Newtonian fluid :
Fluids that obey the Newton’s law of viscosity.
(The shearing stress is linearly related to the rate of shearing
strain)
Most common fluids, both liquids and gases, are Newtonian.
Non-Newtonian fluid :
Fluids for which the shearing stress is not linearly related to
the rate of shearing strain
Shear-thinning fluid
The coefficient of resistance decreases
with increasing strain rate.
Ex. Ketchup (It all comes out of the
bottle at once)
Colloidal suspensions
Polymer solutions,
Latex paint (It does not drip from the
brush because the shear rate is small
and the shear stress is large. However,
it flows smoothly onto the wall because
the thin layer of paint between the wall
and brush causes a large shear rate
(large du/dy) and a small shear stress.)
Shear-thickening fluid
Fluids having the characteristics that
the shear stress increases with
increasing the shear strain. The
harder the fluid is sheared, the more
viscous it becomes.
Ex. Water-corn starch mixture
Water-sand mixture (quicksand):
The difficulty in removing an
object from quicksand increases
dramatically as the speed of removal
increases.
Bingham plastic
This is neither a fluid nor a solid.
This material can withstand a finite
shear stress without motion ( hence,
not a fluid), but once the yield
stress is exceeded it flows like a
fluid (i.e., not a solid).
Ex. Toothpaste, Mayonnaise
No-slip condition : Whenever a fluid is in contact with asolid surface, the velocity of the fluid at the surface is equal tothe velocity of the surface; that is, the fluid “sticks” to thesurface and does not “slip” relative to it.
This condition is true regardless of the type of the fluid, typeof surface, or surface roughness, so long as the continuumhypothesis is valid.
Inviscid fluid : the fluid with zero viscosity, i.e., = 0.Consequently, = 0.
The assumption of an inviscid fluid is often useful foranalyzing flow remote from the solid boundaries.
Ideal fluid : = 0 and = constant(incompressible)
1.7 Compressibility of Fluids
Bulk modulus, Ev : measure of the compressibility of fluid
[psi, Pa]
Large values of the bulk modulus indicate that the fluid is
relatively incompressible-that is, it takes a large pressure
change to create a small change in volume.
Common liquids have large value of Ev, For example, at
atmospheric pressure and a temperature of 60oF it would
require a pressure of 3120 psi to compress a unit volume of
water 1%. i.e., Ev=3.12x105 psi (=2.15x109 Pa) for water.
For most practical engineering problems, we consider the
liquids are incompressible.
d
dp
VdV
dpE
mV
v
For isothermal process, =constant,
For isentropic process, constant
where (for air k=1.4) and R=cp-cv
RTp
pRT/d
RTd
/d
dpE
RTddpv
k
p
kpp
kk)const(/d
dk)const(
/d
dpE k
k
k1k
dk)const(dpv 1k
vp cck
For air under standard atmospheric conditions with p=14.7 psi
and k=1.4, the isentropic bulk modulus is 20.6 psi.
Comparing this value with that of water (Ev,water=312,000
psi), the air is approximately 15,000 times as compressible as
water.
Speed of Sound
Speed of sound : defined as
Since the disturbance is small, there is negligible heat transfer and the process is assumed to be isentropic.
Thus, for ideal gases the speed of sound is proportional to the square root of the absolute temperature.
d
dpc
kRTkp
E
d
dpc
processisentropicundergoinggasfor
v
gas ideal
For example, for air at 60oF with k=1.4 and R=1716
ft.lb/slug.oR, c=1117 ft/s(340 m/s).
For water at 20oC, Ev=2.19 gN/m2 and =998.2 kg/m3 so that
c=1481 m/s or 4860 ft/s.
The speed of sound in water is much higher than in air.
If a fluid is truly incompressible (Ev=), the speed of sound
would be infinite.
Speed of sound as a function of depth at north Hawaii
Sound during the Day Sound in the Evening
Sound during the Day Sound in the Evening
Warm
Cold Warm
Cold
Mach Number
Mach number, Ma : defined as Ma=
Subsonic flow regime : Ma < 1.0
Sonic flow : Ma = 1.0
Supersonic flow regime : Ma > 1.0
Transonic flow regime : 0.7~0.8 < Ma <1.2~1.5
(depends on the configuration of flying object)
c
V
1.8 Vapor Pressure
Evaporation takes place because some liquid molecules at the surface
have sufficient momentum to overcome the intermolecular cohesive
forces and escape into the atmosphere.
When the saturation is reached, the pressure that exerts on the liquid
surface is termed the Vapor Pressure.
Since the development of a vapor pressure is closely
associated with molecular activity, the value of vapor pressure
for a particular liquid depends on temperature (because the
molecular activity (internal energy) depends on temperature).
Generally, as the temperature increases, the vapor pressure
of a fluid also increases.
Boiling is initiated when the absolute pressure in the fluidreached the vapor pressure.
Water at standard atmospheric pressure will boil when thetemperature reaches 212oF (100oC)-that is, the vaporpressure of water at 212oF is 14.7 psi abs.
However, if at a higher elevation, say 10,000 ft above sealevel, where the atmospheric pressure is 10.1 psi abs, theboiling will start at about 193oF. At this temperature the vaporpressure is 10.1 psi abs.
Thus, boiling occurs at a given pressure acting on the fluid byraising the temperature, or at a given fluid temperature bylowering the pressure.
Cavitation phenomena in the pump, valve, marinepropeller, etc.
(mmHg)
760 mmHg
14.7 psia
Cavitation
Erosion by Cavitation Bubble
Supercavitating Torpedo
VA-111 Shkval Torpedo
Length: 8.2 m (27 feet)
Diameter: 533 mm
Weight: 2700 kg (5940 pounds)
Warhead weight: 210 kg
Speed Launch Speed: 50 kt (93 km/h)
Maximum Speed: 200+ kt (370 km/h)
Range: Around 7000 m to 13000 m (New version)
• Research is on going by PNU CFD lab.
German “Barracuda”
Western countries are not far behind though, with
Germany currently developing the "Barracuda",
which is guided and has been offically stated as
being capable of 360Km/h, but has been rumoured
to travel at up to 800km/h.
It looks like the Russians have been them to the
punch again though, with the Shkval-II already
deployed and rumoured of being cable of at least
720km/h whilst also being guided.
Iran
Underwater Express
DARPA (Defense Advanced Research Projects Agency) /ATO (Advanced Technology Office)
Period : April 2006 ~ August 2009 Technology development and demonstration program
(Model scale=1/4~1/2) Demonstrate stable and controllable high-speed
underwater transport through supercavitation for futurelittoral missions
Speed ~100 knots Size : 8 ft diameter, 60 tones for super-fast submerged
transport (SST)- comparable in size to current special purpose craft suchas the MK V Special Operations Craft and the AdvancedSeal Delivery Vehicle- Mark V: 82 feet long aluminum monohull surface craft,40 knots
• Research is on going by PNU CFD lab.
RAMICS (RAPID AIRBORNE MINE CLEARANCE SYSTEM)
AHSUM (Adaptable High-Speed Munitions)
1.9 Surface Tension
The surface tension is due to the unbalanced cohesive forces
acting on the liquid molecules at the fluid surface.
Molecules in the interior of the fluid mass are surrounded by
the molecules that are attracted to each other equally.
However, molecules along the surface are subjected to a net
force toward the interior.
This unbalanced force along the surface creates the
membrane. The tensile force along the surface is called the
surface tension.
Unit = [lf/ft], [N/m]
22 RpR
Rppp ei
2
Capillary : In fig (a), the attractive(adhesive) force between
the wall of the tube and liquid molecules is strong enough to
overcome the mutual attractive (cohesive) force of the
molecules. The liquid is said to “wet” the solid surface.
cos22 RhR
hh
cos2
Note that the height in a capillary tube is inversely
proportional to the tube radius, and thus the rise of a liquid
becomes increasingly pronounced as the tube radius is
decreased.
If adhesion of molecules to the solid surface is weak
compared to the cohesion between molecules, the liquid will
not wet the surface and the level in a tube placed in a
nonwetting liquid will be depressed as shown in fig.1.8(c).
Mercury is nonwetting liquid when it is contact with the glass,
130o.
Application:
- Detergent
- Washing in hot water
