Chapter (1) (1) Fluids and their Properties . Page (1) Fluid Mechanics Fluids and their Properties
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Chapter 2: Properties of Fluids
2-1 Fluid as a Continuum 2-2 Density and Specific Gravity 2-3 Vapor Pressure and Cavitation 2-4 Energy and Specific Heats 2-5 Coefficient of Compressibility 2-6 Viscosity 2-7 Surface Tension 2-8 Description and Classification of Fluid Motions
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-1 Fluid as a Continuum (1)
Any characteristic of a system is called a property. Familiar: pressure P, temperature T, volume V, and mass m. Less familiar: viscosity, thermal conductivity, modulus of elasticity, thermal expansion coefficient, vapor pressure, surface tension.
Intensive properties are independent of the mass of the system. Examples: temperature, pressure, and density. Extensive properties are those whose value depends on the size of the system. Examples: Total mass, total volume, and total momentum. Extensive properties per unit mass are called specific properties. Examples include specific volume v = V/m and specific total energy e=E/m.
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Atoms are widely spaced in the gas phase. However, we can disregard the atomic nature of a substance. View it as a continuous, homogeneous matter with no holes, that is, a continuum. This allows us to treat properties as smoothly varying quantities. Continuum is valid as long as size of the system is large in comparison to distance between molecules.
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2-1 Fluid as a Continuum (2)
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Fluid as a Continuum:
Density at a point in a continuum 2-3
2-1 Fluid as a Continuum (3)
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Velocity Field: Consider also
Steady and Unsteady Flows 1D, 2D, and 3D Flows Timelines, Pathlines, and Streaklines
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2-1 Fluid as a Continuum (4)
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Density and Specific Gravity: Density is defined as the mass per unit volume ρ= m/V. Density has units of kg/m3
Specific volume is defined as v = 1/ρ = V/m. For a gas, density depends on temperature and pressure. Specific gravity, or relative density is defined as the ratio of the density of a substance to the density of some standard substance at a specified temperature (usually water at 4°C), i.e., SG=ρ/ρH20. SG is a dimensionless quantity. The specific weight is defined as the weight per unit volume, i.e., γ = ρg where g is the gravitational acceleration. γ has units of N/m3.
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2-2 Density and Specific Gravity (1)
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Density of Ideal Gases: Equation of State: equation for the relationship between pressure, temperature, and density. The simplest and best-known equation of state is the ideal-gas equation. P v = R T or P = ρ R T Ideal-gas equation holds for most gases. However, dense gases such as water vapor and refrigerant vapor should not be treated as ideal gases. Tables should be consulted for their properties.
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2-2 Density and Specific Gravity (2)
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Vapor Pressure Pv is defined as the pressure exerted by its vapor in phase equilibrium with its liquid at a given temperature If P drops below Pv, liquid is locally vaporized, creating cavities of vapor. Vapor cavities collapse when local P rises above Pv. Collapse of cavities is a violent process which can damage machinery. Cavitation is noisy, and can cause structural vibrations.
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Vapor Pressure and Cavitation:
2-3 Vapor Pressure and Cavitation
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Total energy E is comprised of numerous forms: thermal, mechanical, kinetic, potential, electrical, magnetic, chemical, and nuclear. Units of energy are joule (J) or British thermal unit (BTU). Microscopic energy
Internal energy u is for a non-flowing fluid and is due to molecular activity. Enthalpy h=u+Pv is for a flowing fluid and includes flow energy (Pv).
Macroscopic energy Kinetic energy ke=V2/2 Potential energy pe=gz
In the absence of electrical, magnetic, chemical, and nuclear energy, the total energy is eflowing=h+V2/2+gz. 2-8
Energy and Specific Heats:
2-4 Energy and Specific Heats
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
How does fluid volume change with P and T? Fluids expand as T ↑ or P ↓ Fluids contract as T ↓ or P ↑ Need fluid properties that relate volume changes to changes in P and T.
Coefficient of compressibility
Coefficient of volume expansion
Combined effects of P and T can be written as
T T
P Pvv
κ ρρ
∂ ∂ = − = ∂ ∂
1 1
P P
vv T T
ρβρ
∂ ∂ = = − ∂ ∂
P T
v vdv dT dPT P∂ ∂ = + ∂ ∂ 2-9
Coefficient of Compressibility:
2-5 Coefficient of Compresibility
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-6 Viscosity (1)
Viscosity is a property that represents the internal resistance of a fluid to motion. The force a flowing fluid exerts on a body in the flow direction is called the drag force, and the magnitude of this force depends, in part, on viscosity.
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Drag force--- friction force
Shear stress--- friction force per unit area
Shear stress is proportional to velocity gradient
)(N/m 2
dydvµτ =
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
The viscosity of a fluid is a measure of its “stickiness” or “resistance to deformation” Viscosity is caused by cohesive forces between molecules in liquids and by molecular collisions in gases, and it varies greatly with temperature
2-6 Viscosity (2)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
μ--- dynamic (or absolute) viscosity
unit is kg/m•s or N•s/m2 (or Pa•s) or
poise= 0.1 Pa•s or centipoise=0.01 poise
Newtonian fluids--- the fluids that obey the linear relationship above in the entire range of velocities, such as water, air, gasoline and oils, but, blood and liquid plastics are examples of non-Newtonian fluids
2-6 Viscosity (3)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
ν=µ/ρ--- kinematic viscosity
unit is m2/s or stoke(=1 cm2/s=0.0001m2/s)
In general, μ=f(P,T)
Kinematic viscosity is much less sensitive to temperature compared to dynamic viscosity
For liquids, ν and µ are usually independent of P, except at extremely high pressures.
2-6 Viscosity (4)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
For gases,
2-6 Viscosity (5)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-6 Viscosity (6)
To obtain a relation for viscosity, consider a fluid layer between two very large parallel plates separated by a distance ℓ Definition of shear stress is τ = F/A. Using the no-slip condition, u(0) = 0 and u(ℓ) = V, the velocity profile and gradient are u(y)= Vy/ℓ and du/dy=V/ℓ Shear stress for Newtonian fluid: τ = μ du/dy
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
y
x
y
x
Ayx dAdF
AF
y
==→ δδτ
δ 0lim
dtd
tt
αδδα
δ==
→0limDeformation rate
y
x
Ayx AF
y δδτ
δ 0lim
→=
tul δδδ =
δαδδ yl =
yu
t δδ
δδα
=dydu
dtd
=α
For small angles,
=>
2-6 Viscosity (7)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Newtonian Fluids Most of the common fluids (water, air, oil, etc.) “Linear” fluids
2-6 Viscosity (8)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Viscometry: How is viscosity measured? A rotating viscometer.
Two concentric cylinders with a fluid in the small gap ℓ. Inner cylinder is rotating, outer one is fixed.
Use definition of shear force: If ℓ/R << 1, then cylinders can be modeled as flat plates. Torque T = FR, and tangential velocity V=wR Wetted surface area A=2πRL. Measure T and w to compute μ
duF A Ady
τ µ= =
2-6 Viscosity (9)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Non-Newtonian Fluids Special fluids (e.g., most biological fluids, toothpaste,
some paints, etc.) “Non-linear” fluids
2-6 Viscosity (10)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
EXAMPLE2.1 Viscosity and Shear Stress in Newtonian Fluid
An infinite plate is moved over a second plate on a layer of liquid as shown. For small gap width, d, we assume a linear velocity distribution in the liquid. The liquid viscosity is 0.65 centipoise and its specific gravity is 0.88. Determine:
(a) The kinematic viscosity of the liquid, in m2/s. (b) The shear stress on the upper plate, in Pa. (c) The shear stress on the lower plate, in Pa. (d) The direction of each shear stress calculated in parts (b) and (c).
2-6 Viscosity (11)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-6 Viscosity (12)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-7 Surface Tension (1)
Liquid droplets behave like small spherical balloons filled with liquid, and the surface of the liquid acts like a stretched elastic membrane under tension. The pulling force that causes this is
due to the attractive forces between molecules called surface tension σs.
Attractive force on surface molecule is not symmetric. Repulsive forces from interior molecules causes the liquid to minimize its surface area and attain a spherical shape.
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Surface tension effects on water droplets
2-7 Surface Tension (2)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
Capillary effect is the rise or fall of a liquid in a small-diameter tube. The curved free surface in the tube is call the meniscus. Water meniscus curves up because water is a wetting fluid. Mercury meniscus curves down because mercury is a nonwetting fluid. Force balance can describe magnitude of capillary rise.
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2-7 Surface Tension (3)
Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-7 Surface Tension (4)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
EXAMPLE2.2 Analysis of Capillary Effect in a Tube Create a graph showing the capillary rise or fall of a column of
water or mercury, respectively, as a function of tube diameter D. Find the minimum diameter of each column required so that the height magnitude will be less than 1 mm.
2-7 Surface Tension (5)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-8 Description and Classification of Fluid Motions
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-8 Description and Classification of Fluid Motions (2)
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Fluid Mechanics Y.C. Shih February 2013 Chapter 2 Properties of Fluids
2-8 Description and Classification of Fluid Motions (3)
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