Reading Materials: Chapter 6

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Reading Materials Chapter 6
Fluid Flow
LECTURES 16-18
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What is a Fluid

• Material that continually deforms under a shear
  stress
• Divided into two groups
• Liquid
• Gases

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GASES

• Characterized as loosely-associated molecules
  which are normally not close together and which
  travel through space for long distances before
  colliding with each other. The velocity of their
  travel depends on the temperature of the gas.
• Characteristic of gases

• Readily compressible
• Expand quickly and fill a container

To convert from volume to mass/mole use
PVnRT
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LIQUIDS

• Characterized by molecules which are very close
  together and which are in collision with each
  other very frequently as they move around each
  other. The velocity of that motion and the rate
  of that collision depend on the temperature of
  the liquid.
• Characteristic of liquids

• Slightly compressible
• Takes shape of container sides and bottom

To convert from volume to mass/mole use
Density mass/volume
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Fluid-Like Systems

• Solids present in fluids
• Slurries
• fine particles suspended in liquid
• Solids in a fluidized bed
• particles moving with the fluid in a tall reactor.

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Variables associated with fluids

• Others
• Thermal conductivity
• Electrical conductivity
• Boiling point
• Freezing point
• Heat capacity
• Enthalpy

• Density
• Flow rate
• Pressure
• Viscosity
• Surface tension

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Flow Rate

• Rate at which a material is transported through a
  process line.
• Mass flow rate
• Molar flow rate
• Volume flow rate
• where

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Pressure

• The pressure of the fluid is defined as the total
  force (exerted on the boundary by the fluid
  molecules) divided by the surface area of the
  boundary it is acting upon.

F A P
SI N m2 Nm-2 (Pa)
cgs dyne cm2 dyne?cm-2
American lbf in2 lbfin-2 (psi)
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Puzzle
A womans high heels sink into the soft ground,
but the larger shoes of the much bigger man do
not.
Pressure force/area
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Pressure

• We express pressure in two ways absolute and
  gauge pressures
• Gauge Pressure fluid pressure that is measured
  relative to the atmospheric pressure.
• Absolute Pressure the total magnitude of force
  exerted per unit area
• In process calculations
• Pabs Pgauge Patmosphere
• Pabs 0 in complete vacuum
• Letter a or g is added to designate absolute
  or gauge, thus psia or psig

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Standard Atmospheric Pressure

• At sea level, 0oC and 45o latitude
• Patm 1 atm
• 101,325 Pa
• 14.7 psi
• 1.01325 bars
• 760 mmHg
• 10.333 m H2O 33. 9 ft H2O

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Standard Atmospheric Pressure
How much does the atmosphere heigh? Answer the
same as 76 cm of mercury. How?
Torricelli filled a tube with mercury and
inverted it into an open container of mercury.
Air pressure acting on the mercury in the dish
can support a column of mercury 76 cm in height.
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Example 7.1

• A man pumps his automobile tire until the tire
  gauge reads 34.0 psi. If the atmosphere in his
  community is 14.2 psia, what is the absolute
  pressure of the air in the tire?
• Solution
• Pg 34.0 psig
• Patm 14.2 psia
• Pabs 34.0 14.2 48.2 psia

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Hydrostatic Pressure

• It is the pressure (P) of the fluid at the base
  of the column. That is the force (F) exerted on
  the base divided by the base area (A). F thus
  equals the force on the top surface plus the
  weight of the fluid in the column.
• P P0 ?gh
• h height of a hypothetical column of the
• fluid
• A pressure may also be expressed
• as a head of a particular fluid (Ph)
• Ph P0 h

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Example 7.2

• For the tank depicted in Fig. 7.2, if the NaOH
  solution is 8 ft high, what is the pressure at
  the bottom of the tank? Assume that the density
  of the NaOH solution is the same as that of
  water. Perform the calculation in metric units.

Solution
P1 0 Pa (gauge), so P2 23,896 Pa (gauge) or
P2 23,896 101,325 125,221 Pa (absolute)
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Quick Quiz 1

• What is the pressure 30.0 m below the surface of
  a lake? Assume the atmospheric pressure (the
  pressure at the surface) is 10.4 m H2O? Express
  your answer in atm.
• Solution

Ph P0 h 10.4 30.0 40.4 m H2O
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Figure 3.4-3 (p. 57) of Felder and
RousseauBourdon gauge. It is used to measured
fluid pressure from nearly perfect vacuums to
about 7000 atm.
Fluid Pressure Measurement
A hollow tube closed at one end and bent into a C
configuration. The open end of the tube is
exposed to the fluid whose pressure is to be
measured. As the pressure increases, the tube
tends to straighten, causing a pointer attached
to the tube to rotate.
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Figure 3.4-4 (p. 58)Manometers. It is used for
more accurate measurements of pressure (below
about 3 atm)
Fluid Pressure Measurement
Gauge pressure
Absolute pressure
Pressure difference
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Figure 3.4-5 (p. 58)Differential Manometer
variables.
Fluid Pressure Measurement
If ? is gas then ? may be neglected
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Illustration 1

• A manometer reading gives 100 mmHg, calculate the
  absolute pressure.
• Solution
• Measured gauge pressure
• Pabs 13,328 101,325 114,653 Pa
• 115 kPa

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How does pressure relate to flow?

• In the absence of other forces, fluids tend to
  flow from regions of high pressure to regions
  where the pressure is lower. Therefore, pressure
  differences provide a driving force for fluid
  flow.
• Example when a tire is punctured, air flows out
  of the high pressure tire to the atmosphere,
  which is a low pressure.

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Viscosity (m)

• Characterises its resistance to flow
• A measure of stickiness of a fluid
• A frictional force

Low viscosity fluid
High viscosity fluid
Less energy required for mixing
More energy required for mixing
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Viscosity (m)

• Measurement - Shear stress/shear rate

• Units
• SI kg/(m.s) N.s/m2 Pa.s
• cgs cp (centipoise)
• 1 cP 10-2 poise 10-3 Pa.s 1 mPa.s

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Viscosity of various fluids
Fluid Temperature (oC) Viscosity Pa.s
Water 15.6 1.1x10-3
Gasoline 15.6 0.3x10-3
SAE 30 oil 15.6 383x10-3
Air 15 1.8x10-5
Methane 20 1.1x10-5
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Viscosity of various fluids
Substance (25C) Viscosity (Pa.s) Water
1 x 10-3 Mercury 1.5 x 10-3 Air 1.8 x
10-5 Castor oil 0.99
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Kinematic viscosity
The property viscosity may also be combined with
the fluids density to give the property
kinematic viscosity
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Types of viscous fluids
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Types of viscous fluids

• Newtonian fluid
• e.g. water, air, other gases.
• 2-4 Non-Newtonian fluids
• Bingham-plastic.
• e.g. toothpaste, margarine, soap
• Pseudo-plastic (shear thinning)
• e.g. mayonnaise, polymer melts, paints.
• Dilatant (shear thickening)
• e.g. wet beach sand, starch in water

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Newtonian fluids
Question How would you determine the viscosity
from a plot of shear stress against shear rate?
With Newtonian fluids, shear stress increases
proportionately with the shear rate
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Other types of Non-Newtonian fluids
Dilatant
Viscosity
Shear rate
Viscosity changes with power input
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Other types of Non-Newtonian fluids
Thixotropic
Rheopetic
Viscosity
Viscosity
Time
Time
Viscosity changes with time at constant shear rate
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Non-Newtonian fluids can have more than one
property
Non-Newtonian fluids

• Example
• Damp gypsum
• Pseudoplastic, Thixotropic and Viscoelastic
• Cream
• Dilatant, Rheopectic and not Viscoelastic

Viscoelastic Fluid returns to original viscosity
after power input ceases
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Ideal / Inviscid Fluid
Hypothetical fluid which is incompressible and
has zero viscosity.
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Fluid Flow Principles

• Flow patterns vary with
• velocity
• geometry of surface, and
• fluid properties such as viscosity, density.

Classic experiment by Osborne Reynolds (1883)
observed two types of fluid flow

• Laminar flow low flow rates
• Turbulent flow higher flow rates

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Reynolds experiment
(i) Low flow rates fluid moves in parallel
layers.
(ii) High flow rates cross currents (eddies)
develop
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Reynolds Number
Reynolds key variables
Arrange into single dimensionless group
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Reynolds Number
For pipe flow Re lt 2,100 - laminar flow 2,100
lt Re lt 10,000 - transition region Re gt
10,000 - turbulent flow Pipe-flow systems with
the same Re are said to be dynamically similar.
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Illustration 2
Fluid flow through a pipe
Water at 25oC flows through a pipe of internal
diameter 0.1 m at a velocity 0.2 m/s.

1. Is the flow laminar flow or turbulent?
2. What is the effect of reducing the velocity by a
   factor of 10?

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Solution
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Velocity Profiles Laminar Flow

1. Velocity profile is parabolic with the maximum
   velocity occurring in the centre of the pipe (r
   0)

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Velocity Profiles Laminar Flow
L
R
r
(i) Velocities of a fluid in laminar flow
through a circular pipe
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Velocity Profiles Laminar Flow

1. Volumetric flow rate

1. Mean velocity

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Velocity Profiles Turbulent Flow

• Difficult to mathematically model due to its
  complex and rapidly changing flow patterns.
• Experimental measurements show for time-averaged
  velocity and mean velocity

(ii) Turbulent Flow
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Velocity Profiles Plug Flow

• Common assumption for highly turbulent flow is
  that velocity does not vary over cross-section

(iii) Plug Flow
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Mass Conservation in Fluid Flow
Consider steady-state, one-dimensional fluid
through pipe
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Mass Conservation in Fluid Flow
Is v2 in (a) less than v2 in (b)?
No, they are both the same.
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Fluid Friction

• Frictional force cause pressure drop during fluid
  flow through a constant diameter horizontal pipe.
• Consider flow situation in pipe below.

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Fluid Friction
Momentum balance
f friction factor L length of pipe D
diameter of pipe
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Fluid Friction
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Example 7.7

• What value of the friction per mass of fluid (ef)
  is necessary to cause a decrease in pressure
  equal to
• 10 psi (answer in Btu/lbm)?

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Example 7.7
What value of the friction per mass of fluid (ef)
is necessary to cause a decrease in pressure
equal to b) 68,900 Pa (answer in J/kg)?
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Laminar Flow

• f determined analytically from velocity profile.

• Experimental data confirmed for Re lt 2,100 (see
  Figure 2.10-3)

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Figure 2.10-3 Friction Factor Chart
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Turbulent Flow

• f cannot be determined analytically.
• Experimental curves have been devised.
• f also depends on surface roughness factor, e

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Turbulent Flow

• For smooth pipe 2,100 lt Re lt 105
• Blassius equation may be used

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Illustration 3
Frictional losses in a pipe

• Pipeline 5 km long 30 cm internal diameter
• Conveys water at 25oC at a rate of 180 kg/s.
• Roughness factor ?e/D 0.001

Estimate the pressure drop across the pipe due to
friction.
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Solution
For water
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Solution
From Figure 2.10-3, f 0.005 Hence,
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