monroe l. weber-shirk s chool of civil and environmental engineering fluid properties and units cee...
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Monroe L. Weber-Shirk
School of Civil and
Environmental Engineering
Fluid Properties and Units
Fluid Properties and Units
CEE 331
April 18, 2023
CEE 331
April 18, 2023
Dimensions and UnitsDimensions and Units
The dimensions have to be the same for each term in an equation
Dimensions of mechanics are length time mass force temperature
The dimensions have to be the same for each term in an equation
Dimensions of mechanics are length time mass force temperature
aF m aF m
L
T
MMLT-2
Dimensions and UnitsDimensions and Units
Quantity SymbolDimensionsVelocity V LT-1
Acceleration a LT-2
Area A L2
Volume L3
Discharge Q L3T-1
Pressure p ML-1T-2
Gravity g LT-2
Temperature T’ Mass concentration C ML-3
Quantity SymbolDimensionsVelocity V LT-1
Acceleration a LT-2
Area A L2
Volume L3
Discharge Q L3T-1
Pressure p ML-1T-2
Gravity g LT-2
Temperature T’ Mass concentration C ML-3
Show this!
Dimensions and UnitsDimensions and Units
Quantity Symbol DimensionsDensity ML-3
Specific Weight ML-2T-2
Dynamic viscosity ML-1T-1
Kinematic viscosity L2T-1
Surface tension MT-2
Bulk mod of elasticity E ML-1T-2
These are _______ properties!fluid
How many independent properties? _____4
mn
r=m
nr
=
gg r= gg r=
Definition of a FluidDefinition of a Fluid
“a fluid, such as water or air, deforms continuously when acted on by shearing stresses of any magnitude.” - Young, Munson, Okiishi
“a fluid, such as water or air, deforms continuously when acted on by shearing stresses of any magnitude.” - Young, Munson, Okiishi
Why isn’t steel a fluid?Why isn’t steel a fluid?
Fluid Deformation between Parallel Plates
Fluid Deformation between Parallel Plates
Side viewSide view
Force F causes the top plate to have velocity U.Force F causes the top plate to have velocity U.What other parameters control how much force What other parameters control how much force is required to get a desired velocity?is required to get a desired velocity?
Distance between plates (t)Distance between plates (t)
Area of plates (A)Area of plates (A)
F
t
U
Viscosity! ()Viscosity! ()
F =AUt
mAUt
m
If this parameter increases, what does F do?
Shear StressShear Stress
change in velocity with respect to distancechange in velocity with respect to distance
AFAF
2mN
2mN
tU t
U t
Ut
U
dydu dydu
tAU
F t
AUF
AUFt
AUFt
2msN
2msN
dimension of
s1
s1
Tangential force per unit area
Rate of angular deformation
rate of shear
Our general equation relating shear and viscosity
Fluid ViscosityFluid Viscosity
Examples of highly viscous fluids ______________________________
Fundamental mechanisms Gases - transfer of molecular momentum
Viscosity __________ as temperature increases. Viscosity __________ as pressure increases.
Liquids - cohesion and momentum transfer Viscosity decreases as temperature increases. Relatively independent of pressure (incompressible)
Examples of highly viscous fluids ______________________________
Fundamental mechanisms Gases - transfer of molecular momentum
Viscosity __________ as temperature increases. Viscosity __________ as pressure increases.
Liquids - cohesion and momentum transfer Viscosity decreases as temperature increases. Relatively independent of pressure (incompressible)
molasses, tar, 20w-50 oil, glycerin
increases
_______
increases
Example: Measure the viscosity of water
Example: Measure the viscosity of water
The inner cylinder is 10 cm in diameter and rotates at 10 rpm. The fluid layer is 2 mm thick and 20 cm high. The power required to turn the inner cylinder is 100x10-6 watts. What is the dynamic viscosity of the fluid?
The inner cylinder is 10 cm in diameter and rotates at 10 rpm. The fluid layer is 2 mm thick and 20 cm high. The power required to turn the inner cylinder is 100x10-6 watts. What is the dynamic viscosity of the fluid?
Outer Outer cylindercylinder
Thin layer of waterThin layer of water
Inner Inner cylindercylinder
dydu dydu
tAU
F t
AUF
Solution SchemeSolution Scheme
Restate the goal Identify the given parameters and represent the
parameters using symbols Outline your solution including the equations describing
the physical constraints and any simplifying assumptions
Solve for the unknown symbolically Substitute numerical values with units and do the
arithmetic Check your units! Check the reasonableness of your answer
Restate the goal Identify the given parameters and represent the
parameters using symbols Outline your solution including the equations describing
the physical constraints and any simplifying assumptions
Solve for the unknown symbolically Substitute numerical values with units and do the
arithmetic Check your units! Check the reasonableness of your answer
Viscosity Measurement: SolutionViscosity Measurement: Solution
hr
Pt322
hr
Pt322
-6-3 2
2 3
(100 10 W) (0.002 m)1.16x10 N s/m
2 (1.047/s) (0.05 m) (0.2 m)x
mp
= = ×-6
-3 22 3
(100 10 W) (0.002 m)1.16x10 N s/m
2 (1.047/s) (0.05 m) (0.2 m)x
mp
= = ×
tAU
F t
AUF U U A A
thr
F22
thr
F22
P P
thr
P322
thr
P322
Outer Outer cylindercylinder
Thin layer of waterThin layer of water
Inner Inner cylindercylinder
r = 5 cmt = 2 mmh = 20 cmP = 100 x 10-6 W10 rpm
r 2rh
Fr
Role of ViscosityRole of Viscosity
Statics Fluids at rest have no relative motion between
layers of fluid and thus du/dy = 0 Therefore the shear stress is _____ and is
independent of the fluid viscosity Dynamics
Fluid viscosity is very important when the fluid is moving
Statics Fluids at rest have no relative motion between
layers of fluid and thus du/dy = 0 Therefore the shear stress is _____ and is
independent of the fluid viscosity Dynamics
Fluid viscosity is very important when the fluid is moving
zerozero
Dynamic and Kinematic Viscosity
Dynamic and Kinematic Viscosity
Kinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density
Kinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density
3mkg
smkg
3mkg
smkg
mÞmÞ [ ]N =[ ]N =
[m2/s]
Connection to Reynolds number!
mm
nn
ReVD VDrm n
= =ReVD VDrm n
= =
nu
2mN s×é ù
ê úë û2mN s×é ù
ê úë û2s
kg m×é ùê úë û2skg m×é ù
ê úë û
Density and Specific WeightDensity and Specific Weight
Density (mass/unit volume) density of water: density of air at
atmospheric pressure and 15 C:
Specific Weight of water (weight per unit volume) __________________
Density (mass/unit volume) density of water: density of air at
atmospheric pressure and 15 C:
Specific Weight of water (weight per unit volume) __________________
950960970980990
1000
0 50 100Temperature (C)
Den
sity
(kg
/m3 )
997
998
999
1000
0 10 20
Temperature (C)
Den
sity
(kg
/m3 )
1000 kg/m3
1.22 kg/m3
= g = 9806 N/m3
Specific mass
Perfect Gas LawPerfect Gas Law
PV = nRT R is the universal gas constant T is in Kelvin
PV = nRT R is the universal gas constant T is in Kelvin
Note deviation from the text!Note deviation from the text!
R
8 314.N m
mol K
RR
Mtextgas
Mgas is molecular mass
Mgas for air is 0.029 kg/mole
Why is this Mgas for air reasonable?
N2 28 g/mol, O2 32 g/mol
Bulk Modulus of ElasticityBulk Modulus of Elasticity
Relates the change in volume to a change in pressure changes in density at
high pressure pressure waves
_________ ______ __________
Relates the change in volume to a change in pressure changes in density at
high pressure pressure waves
_________ ______ __________ 2.00
2.05
2.10
2.15
2.20
2.25
2.30
2.35
0 20 40 60 80 100
Temperature (C)
Bul
k M
odul
us o
f el
asti
city
(G
Pa)
soundsoundwater hammerwater hammer
Edp
dv /
Edp
dV Vv /
Water
v
dV dpV E
=How much does water compress?
Compression and Expansion of Gases
Compression and Expansion of Gases
Isothermal (constant temperature)
Isentropic (no heat exchanged)
Isothermal (constant temperature)
Isentropic (no heat exchanged)
p
constant
Ck
pr
= kc
cp
v
where (specific heat ratio)
pVn
constantdpd r
= Edp
dv / vE p=
RT
E kpv 1kdpCk
dr
r-= 1k
k
dp pk
dr
r r-=
dp pk
d r r=
Speed of Sound (c)Speed of Sound (c)
cEv
cdpd
Edp
dv /
E dpd
v
E kpv
For gasses, if no heat exchanged (isentropic) then we have
It can be shown that (homework) ckRTM gas
Connection to Mach number!V
Mac
=
. Solve for dpd r
and
Ev is large for fluids that are difficult to compress
Vapor PressureVapor Pressure
0
1000
2000
3000
4000
5000
6000
7000
8000
0 10 20 30 40
Temperature (C)
Vap
or p
ress
ure
(Pa)
liquid
What is vapor pressure of water at 100°C?101 kPa
Cavitation! When absolute pressure drops below vapor pressure
pR2 = 2R
Surface TensionSurface Tension
Pressure increase in a spherical droplet
Pressure increase in a spherical droplet
Rp
2R
p2
pR2
2R
Surface moleculesSurface molecules
0.0500.0550.0600.0650.0700.0750.080
0 20 40 60 80 100
Temperature (C)
Sur
face
tens
ion
(N/m
)Fp=
F=
Example: Surface TensionExample: Surface Tension
Estimate the difference in pressure (in Pa) between the inside and outside of a bubble of air in 20ºC water. The air bubble is 0.3 mm in diameter.
Estimate the difference in pressure (in Pa) between the inside and outside of a bubble of air in 20ºC water. The air bubble is 0.3 mm in diameter.
Rp
2R
p2
R = 0.15 x 10-3 mR = 0.15 x 10-3 m
= 0.073 N/m = 0.073 N/m
mx
mNp
31015.0
/073.02
mx
mNp
31015.0
/073.02
970 Pap =970 Pap =
What is the difference between pressure in a water droplet and in an air bubble?