# Ch2_Fluid_properties

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```					FLUID PROPERTIES
Chapter 2

CE319F: Elementary Mechanics of Fluids   1
Fluid Properties
• Define “characteristics” of a specific fluid
•Properties expressed by basic “dimensions”
– length, mass (or force), time, temperature
• Dimensions quantified by basic “units”

We will consider systems of units, important fluid properties
(not all), and the dimensions associated with those properties.

2
Systeme International (SI)
•   Length = meters (m)
•   Mass = kilograms (kg)
•   Time = second (s)
•   Force = Newton (N)
– Force required to accelerate 1 kg @ 1 m/s2
– Acceleration due to gravity (g) = 9.81 m/s2
– Weight of 1 kg at earth’s surface = W = mg = 1 kg (9.81 m/s2) =
9.81 kg-m/s2 = 9.81 N
•   Temperature = Kelvin (oK)
– 273.15 oK = freezing point of water
– oK = 273.15 + oC

3
Système International (SI)
• Work and energy = Joule (J)
J = N*m = kg-m/s2 * m = kg-m2/s2

• Power = watt (W) = J/s

• SI prefixes:
G = giga = 109                     c = centi = 10-2
M = mega = 106                     m = milli = 10-3
k = kilo = 103                     m = micro = 10-6
4
English (American) System
•   Length = foot (ft) = 0.3048 m
•   Mass = slug or lbm (1 slug = 32.2 lbm = 14.59 kg)
•   Time = second (s)
•   Force = pound-force (lbf)
– Force required to accelerate 1 slug @ 1 ft/s2
•   Temperature = (oF or oR)
– oRankine = oR = 460 + oF
• Work or energy = ft-lbf                                  Banana Slug
• Power = ft-lbf/s                                    Mascot of UC Santa Cruz
– 1 horsepower = 1 hp = 550 ft-lbf/s = 746 W

5
Density
• Mass per unit volume (e.g., @ 20 oC, 1 atm)
– Water         rwater = 1,000 kg/m3 (62.4 lbm/ft3)
– Mercury       rHg = 13,500 kg/m3
– Air           rair = 1.205 kg/m3

• Densities of gases = strong f (T,p) = compressible
• Densities of liquids are nearly constant
(incompressible) for constant temperature
• Specific volume = 1/density = volume/mass
6
Example: Textbook Problem 2.8
•   Estimate the mass of 1 mi3 of air in slugs and kgs.
Assume rair = 0.00237 slugs/ft3, the value at sea level for standard conditions

7
Example
•   A 5-L bottle of carbon tetrachloride is accidentally spilled onto a laboratory
floor. What is the mass of carbon tetrachloride that was spilled in lbm?

8
Specific Weight

• Weight per unit volume (e.g., @ 20 oC, 1 atm)

gwater   = (998 kg/m3)(9.807 m2/s)
= 9,790 N/m3
[= 62.4 lbf/ft3]
gair     = (1.205 kg/m3)(9.807 m2/s)
= 11.8 N/m3
[= 0.0752 lbf/ft3]

9
Specific Gravity
• Ratio of fluid density to density of water @
4oC

Water                    SGwater = 1
Mercury                  SGHg = 13.55
Note: SG is dimensionless and independent of system of units
10
Example
•   The specific gravity of a fresh gasoline is 0.80. If the gasoline fills an
8 m3 tank on a transport truck, what is the weight of the gasoline in the
tank?

11
Ideal Gas Law (equation of state)
P = absolute (actual) pressure (Pa = N/m2)
V = volume (m3)
n = # moles
Ru = universal gas constant = 8.31 J/oK-mol
T = temperature (oK)

R = gas-specific constant
R(air) = 287 J/kg-oK (show)

12
Example
• Calculate the volume occupied by 1 mol of any ideal gas at a
pressure of 1 atm (101,000 Pa) and temperature of 20 oC.

13
Example
• The molecular weight of air is approximately 29 g/mol. Use this
information to calculate the density of air near the earth’s
surface (pressure = 1 atm = 101,000 Pa) at 20 oC.

14
Example: Textbook Problem 2.4
• Given: Natural gas stored in a spherical tank
– Time 1: T1=10oC, p1=100 kPa
– Time 2: T2=10oC, p2=200 kPa
• Find: Ratio of mass at time 2 to that at time 1
• Note: Ideal gas law (p is absolute pressure)

15
Viscosity

16
Some Simple Flows
• Flow between a fixed and a moving plate
Fluid in contact with plate has same velocity as plate
(no slip condition)
u = x-direction component of velocity
y
Moving plate                      u=V
V

B                             Fluid

x
Fixed plate                       u=0
17
Some Simple Flows
• Flow through a long, straight pipe
Fluid in contact with pipe wall has same velocity as wall
(no slip condition)
u = x-direction component of velocity

r
R
x
V
Fluid

18
Fluid Deformation
• Flow between a fixed and a moving plate
• Force causes plate to move with velocity V
and the fluid deforms continuously.
y
Moving plate           u=V

t0   t1 t2

Fluid
x
Fixed plate            u=0
19
Fluid Deformation
For viscous fluid, shear stress is proportional
to deformation rate of the fluid (rate of strain)

y
dL     Moving plate                    u=V+dV
t   da           t+dt
dy
dx                           Fluid
x
Fixed plate
u=V
20
Viscosity
•   Proportionality constant = dynamic (absolute) viscosity

•   Newton’s Law of Viscosity
V+d
•   Viscosity                                                                v
V
•   Units

•   Water (@ 20oC):   m = 1x10-3 N-s/m2

•   Air (@ 20oC): m = 1.8x10-5 N-s/m2                         Kinematic viscosity:   m2/s

•   Kinematic viscosity                                          1 poise = 0.1 N-s/m2

1 centipoise = 10-2 poise = 10-3 N-s/m2
21
Shear in Different Fluids
•   Shear-stress relations for different fluids
•   Newtonian fluids: linear relationship
•   Slope of line = coefficient of
proportionality) = “viscosity”

Shear thinning fluids (ex): toothpaste, architectural coatings;
Shear thickening fluids = water w/ a lot of particles, e.g., sewage
sludge; Bingham fluid = like solid at small shear, then liquid at
greater shear, e.g., flexible plastics                                22
Effect of Temperature
Gases:
greater T = greater interaction
between molecules = greater
viscosity.

Liquids:
greater T = lower cohesive forces
between molecules = viscosity
down.

23
24
Typical Viscosity Equations

T = Kelvin
S = Sutherland’s constant
Gas:                        Air = 111 oK
+/- 2% for T = 170 – 1900 oK

C and b = empirical constants
Liquid:

25
Flow between 2 plates
Force is same on top
and bottom
Thus, slope of velocity
profile is constant and
velocity profile is a st. line
y
Moving plate                        u=V
V

Fluid       Force acting
B                                         ON the plate

x
Fixed plate                         u=0
26
Flow between 2 plates
Shear stress anywhere
between plates

y
Moving plate                  u=V
V
t
Shear
B                                        on fluid
t
x
Fixed plate                  u=0
27
Flow between 2 plates
• 2 different coordinate systems

r
B
x
V
y
x

28
Example: Textbook Problem 2.33
Suppose that glycerin is flowing (T = 20 oC) and that the pressure
gradient dp/dx = -1.6 kN/m3. What are the velocity and shear stress at a
distance of 12 mm from the wall if the space B between the walls is 5.0
cm? What are the shear stress and velocity at the wall? The velocity
distribution for viscous flow between stationary plates is

29
30
Example: Textbook Problem 2.34
A laminar flow occurs between two horizontal parallel plates under a
pressure gradient dp/ds (p decreases in the positive s direction). The upper
plate moves left (negative) at velocity ut. The expression for local velocity
is shown below. Is the magnitude of the shear stress greater at the moving
plate (y = H) of at the stationary plate (y = 0)?

31
32
Elasticity (Compressibility)
• If pressure acting on mass of fluid increases: fluid contracts
• If pressure acting on mass of fluid decreases: fluid expands
• Elasticity relates to amount of deformation for a given
change in pressure

Ev = bulk modulus of elasticity

How does second part of

Small dV/V = large modulus of elasticity                             33
Example: Textbook Problem 2.45
•   Given: Pressure of 2 MPa is
applied to a mass of water that
initially filled 1000-cm3
(1 liter) volume.
•
•   Find: Volume after the
pressure is applied.

•   Ev = 2.2x109 Pa (Table A.5)

34
Example
• Based on the definition of Ev and the equation of state, derive an
equation for the modulus of elasticity of an ideal gas.

35
Surface Tension

•   Below surface, forces act equal in all
directions
Interface            air
•   At surface, some forces are missing, pulls
molecules down and together, like
membrane exerting tension on the surface
water               Net force
inward
•   Pressure increase is balanced by surface
tension, s
No net force
•    surface tension = magnitude of
tension/length

•   s = 0.073 N/m (water @ 20oC)
36
Surface Tension
• Liquids have cohesion and adhesion, both involving molecular
interactions
– Cohesion: enables liquid to resist tensile stress

• Capillarity = property of exerting forces on fluids by fine tubes
or porous media
–   due to cohesion and adhesion
–   If adhesion > cohesion, liquid wets solid surfaces at rises
–   If adhesion < cohesion, liquid surface depresses at pt of contact
–   water rises in glass tube (angle = 0o)
–   mercury depresses in glass tube (angle = 130-140o)

• See attached information
37
Example: Capillary Rise
• Given: Water @ 20oC, d = 1.6 mm
• Find: Height of water

W

38
Example: Textbook Problem 2.51
q
Find: Maximum capillary
rise between two vertical   s   s

glass plates 1 mm apart.    h

t

39
Examples of Surface Tension

40
Example: Textbook Problem 2.48
Given: Spherical soap bubble, inside
radius r, film thickness t, and surface
tension s.
Find: Formula for pressure in the
bubble relative to that outside.
Pressure for a bubble with a 4-mm
Should be soap bubble

41
Vapor Pressure (Pvp)
•   Vapor pressure of a pure liquid = equilibrium partial pressure of the gas
molecules of that species above a flat surface of the pure liquid
– Concept on board
– Very strong function of temperature (Pvp up as T up)
– Very important parameter of liquids (highly variable – see attached page)
•   When vapor pressure exceeds total air pressure applied at surface, the liquid
will boil.
•   Pressure at which a liquid will boil for a given temperature
– At 10 oC, vapor pressure of water = 0.012 atm = 1200 Pa
– If reduce pressure to this value can get boiling of water (can lead to “cavitation”)
•   If Pvp > 1 atm compound = gas
•   If Pvp < 1 atm compound = liquid or solid

42
Example
•   The vapor pressure of naphthalene at 25 oC is 10.6 Pa. What is the
corresponding mass concentration of naphthalene in mg/m3? (Hint:
you can treat naphthalene vapor as an ideal gas).

43
Vapor Pressure (Pvp) - continued

Vapor pressure of water (and other liquids) is a strong function of temperature.

44
Vapor Pressure (Pvp) - continued

Pvp,H2O = Pexp(13.3185a – 1.9760a2 – 0.6445a3 – 0.1299a4)

P = 101,325 Pa                a = 1 – (373.15/T)                      T = oK
valid to +/- 0.1% accuracy for T in range of -50 to 140 oC

Equation for relative humidity of air = percentage to which air is “saturated” with water vapor.
What is affect of RH on drying of building materials, and why? Implications?
45
Example: Relative Humidity
The relative humidity of air in a room is 80% at 25 oC.
(a) What is the concentration of water vapor in air on a volume percent
basis?
(b) If the air contacts a cold surface, water may condense (see effects on
attached page). What temperature is required to cause water
condensation?

46
47
Saturation Vapor Pressure

48
49

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