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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
                                                           equation come about?




                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
   – Adhesion: enables liquid to adhere to other bodies

• 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
radius?
                                          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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