Properties of Fluids and Mechanics

Properties of Fluids and Mechanics

• Fluids are liquids and gases
• Mechanics is the study of fluids
• Statics studies fluids at rest
• Kinematics studies fluids in motion, focusing on velocities
• It does not focus on the causes of motion
• Dynamics studies fluids in motion, considering velocities, accelerations, forces, energies, impulse, and momentum
• Fluid mechanics uses the same principles as solid mechanics
• Fluid mechanics is more difficult, because there are no separate elements to distinguish

Development of Fluid Mechanics

• Classical hydrodynamics is largely theoretical and deals with imaginary ideal fluids
• The results have limited practical application
• Hydraulics relies on experiments and empirical formulas to solve practical problems
• Hydraulics was initially limited to water-related applications
• The need for broader applications in aeronautics, chemical engineering, and petroleum industries led to combining classical hydrodynamics and hydraulics

Distinction Between Solids and Fluids

• Solids have closer molecules and stronger attractive forces than fluids
• Solids maintain a fixed shape
• Static fluids have a variable shape
• Solids always have tangential stresses
• Tangential stresses in fluids depend on the magnitude of velocity

Distinction Between Gases and Liquids

• Gas molecules are farther apart than liquid molecules
• Gases are highly compressible
• Liquids are almost incompressible
• Gases expand indefinitely when external pressure is removed
• Cohesive forces in liquids prevent indefinite expansion when external pressure is removed, except for vapor pressure
• Gas volume depends on temperature and pressure
• Liquid volume is dependent only on temperature under normal conditions
• It is independent of pressure

Vapor vs Gas

• Vapor is a gas near its liquid phase
• Steam is a gaseous substance that is in this vapor form
• Gases can be defined as highly superheated vapour whose state is far removed from liquid phase
• Example: Air
• Vapors can be liquefied by pressure alone if its temperature is below its critical temperature
• Gas is a state of matter, whereas vapor is a substance in a gaseous state under conditions where it is ordinarily a liquid or solid
• At room temperature, a gas remains a gas in its natural state
• At room temperature, a vapor's natural state would be liquid or solid
• All vapors can be gases, but not all gases can be vapors

Critical Temperature and Pressure

• The critical temperature of water is 374 degrees Celsius
• The critical pressure of water is 22,060 kPa

Density

• Density (ρ) is a fluid's mass per unit volume
• Density = mass/volume
• Mass is the amount of matter contained, measured in kg or lb
• 1 kg equals 2.204 lb
• Density is absolute, as it depends on mass, which is independent of location
• Density is used to find the mass of a fluid
• Mass of fluid = density x volume occupied by that fluid
• Volume = length x width x height

Specific Weight

• Specific weight is the weight of fluid per unit volume
• It is the force exerted by gravity on a unit volume of fluid
• Specific weight = weight/volume
• Weight is the measurement of gravitational pull
• Weight = m*g
• Specific Weight = W/V
• Specific weight is not absolute because it depends on gravity (g)
• Gravity varies with location, latitude, and elevation above mean sea level
• Specific weight is used to find the weight of a fluid
• Weight of fluid = specific weight x volume
• Volume = length x width x height

Values of Gravity

• Gravity on the moon is 1.625 m/s²
• Gravity at the equator is 9.780 m/s²
• Gravity at the poles is 9.832 m/s²

Relationship Between Density and Specific Weight

• Specific Weight = weight/volume
• Specific weight = (mass x acceleration due to gravity) / Volume
• Specific weight = (mass/volume) x acceleration due to gravity
• = ρg or ρ = /g

Specific Volume

• Specific volume (v) is the volume occupied by a unit mass of fluid
• commonly applied to gases
• specific volume = volume/mass
• v = V/m
• Relationship between density (ρ) and specific volume (v) is reciprocal
• v = 1/ρ

Specific Gravity

• Specific gravity (s) of a liquid is the ratio of its density to that of pure water at a standard temperature
• Physicists use 4°C as the standard, while engineers often use 15°C
• Specific gravity of liquid = liquid density / water density
• For a gas, specific gravity (s) is the ratio of its density to that of either hydrogen or air at a specified temperature and pressure
• Specific gravity of gas = gas density / air or hydrogen density
  • Formulas are simplified if gravity remains constant
  • Measured at specified temperature and pressure

Summary Formulas

• ρg or ρ = /g
• v = 1/ρ

Standard Values (Earth Surface)

• Density of water: ρ = 1000 kg/m³ = 1 g/cm³ = 1 Mg/m³
• Specific weight of water: = 9810 N/m³ = 9.81 kN/m³
• Specific gravity of water: s = 1
• Specific gravity of mercury: s = 13.55 or 13.6

Types of Fluids: Compressible vs. Incompressible

• Compressible fluids have a density that changes
• Gases
• Incompressible fluids have a constant density
• Liquids
• In reality, no fluid is truly incompressible
• The term is used when density changes with pressure are negligible
• Liquids are generally considered incompressible, but sound waves and water hammer phenomena show compressibility
• Gases are compressible, but may behave as incompressible when pressure changes are negligible
• Ventilation systems

Compressibility of Liquids

• Compressibility is the change in volume due to pressure changes (dv/dp)
• It is inversely proportional to the bulk modulus of elasticity.
• Increasing the pressure of 2200 Mpa water by 7 MPa will only compress it by 0.3% of its original volume
• Water is justified as incompressible
• Volume modulus of mild steel is about 170,000MN/m2
• Volume modulus of water is 2,200MN/m2
• Water is 80 times more compressible than steel
• Mercury is 8 times more compressible than water
• Nitric acid is 6 times more compressible than water

Specific Weight of Liquids

• The value to use for water in the problems is 9.81 unless otherwise specified or implied by some specific temperature
• If gravity remains constant:
• Where g varies with location, primarily latitude and elevation above mean sea level

Equations of State for Gases

• No perfect gas exists, but real gases far from the liquid phase are considered as such
• Equation of state for a perfect gas: (1.4)
• Where absolute pressure, density, and specific volume are constants
• R is a gas constant that depends on the particular gas
• T is absolute temperature in degrees Rankine or Kelvin
• For air, R = 287 N.m/(kg.K) and equation (1.4) may be written as (1.5)
• Any gas's density at a specific temperature and pressure can be computed if R and g are known
• Avogadro's law: all gases at the same T and p under a specific g have the same number of molecules per unit volume
• where m denotes molecular weight
• These only apply to perfect gases
• Another fundamental equation for a perfect gas is (1.6)
• Where n may have any nonnegative value from zero to infinity, depending upon the process to which the gas is subjected
• For constant temperature process (isothermal), n = 1
• For air and diatomic gases under isentropic process (frictionless adiabatic process), n = k = 1.4
• By combining Eqs. (1.4) and (1.6), it is possible to obtain other useful relations, (1.7)

Compressibility of Gases

• Differentiating Eq. (1.6) npvn-1 dv+vn
• The Isothermal modulus of elasticity for a gas is 100 kPa at 100kPa
• The Isentropic Process is 1.4 x 100 kPA.

Ideal Fluids

• Ideal fluids have zero viscosity
• Internal forces at any section are always normal to the section, even during motion
• Real fluids develop tangential or shearing forces, creating fluid friction due to motion relative to a body

Ideal vs Real Fluids

• Ideal Fluids:*
• Incompressible (density is constant).
• Irrotational (flow is smooth, no turbulence).
• Non-viscous (inviscid), meaning the fluid has no internal friction
• Continuity and Bernoulli's equations rely on ideal fluids.
• Internal forces at any section are normal to the cross-section, even during motion.
• Real Fluids:*
• Compressible.
• Rotational (turbulence and flow is not smooth).
• Viscous (the fluid has viscosity as internal friction is present).
• Tangential or shearing forces always develop whenever there is motion relative to the body.

Viscosity

• Viscosity measures a fluid's resistance to shear or angular deformation
• Motor oil has high viscosity where gasoline has low viscosity
• Friction in fluids results from cohesion and momentum interchange between molecules
• As temperature increases, liquid viscosity decreases, while gas viscosity increases
• In liquids, cohesion diminishes with temperature
• Gases increase with temperature
• In gases, molecular interchange is the dominant factor
• A rapidly moving gas molecule entering a slower layer speeds it up, and vice versa -This interchange creates shear and friction

Classic Viscosity Case

• With 2 parallel plates where the space between is filled by fluid which are close enough, the fluid's velocity must be U where touching the upper plate and zero at the lower
• The behaviour is that if fluid is a series of thin layers, each that slips a little
• Expressing this Shearing force between fluid:
• Known as Newton's Law
  • This is also the coeficient of viscosity + absolute viscosity + Dynamic Viscosity

Viscosity Units and Types

• Absolute viscosity dimensions are force per unit area divided by velocity gradient
• A widely used unit for viscosity in the metric system is the poise (P) (named after Jean Léonard Marie Poiseuille)
• One poise equals 0.10 Ns/m
• The centipoise (cP) is frequently used
  • 0.01 P = 1 mN s/
• 20°C water, the viscosity is 1 cP
• Kinematic viscosity (nu) is absolute viscosity divided by density
• Called kinematic because force is not involved
• Only dimensions are length and time, as in kinematics
• SI units measure it in /s
• Metric system’s common units were /s, also called the stoke (St), after Sir George Stokes
• 0.01 St = /s, a centistoke (cSt)
• Absolute viscosity is virtually independent of pressure in engineering
  • High pressure increases slightly
• Kinematic viscosity of gases varies strongly with pressure due to density changes
  • If you need it, lookup the value and calculate

Dynamic vs Kinematic Viscosity

• Dynamic viscosity indicates the force needed for a certain flow rate
• Kinematic viscosity indicates how fast the fluid moves when a certain force is applied

Reynolds Number

• The Reynolds number is the ratio of inertial force and viscosity
• R = (VL)/ν
  • V = velocity of flow in feet per second (fps)
  • L = characteristic length in feet (ft)
  • ν (nu) = kinematic viscosity of water in ft²/sec
• If R < 500, the flow is laminar
• If R > 2000, the flow is turbulent
• If R is between 500 and 2000, the flow is transitional

Surface Tension and Capillarity

• Liquids have cohesion and adhesion (forms of molecular attraction)
• Cohesion enables a liquid to resist tensile stress
• Adhesion enables it to stick to another body
• At the interface between a liquid and a gas or between two immiscible liquids, imbalanced attraction between molecules develops a tension force exerted on a surface
• surface tension
• Measured by tension force per unit length
• When a second fluid is not specified at the interface, use air
• The surface tensions of liquids vary widely and decrease slightly with increasing temperature
• Water value is 0.0756-0.0589 N/m;
• Capillarity is the property of exerting forces on fluids by fine tubes or porous media
• This is both a result of cohesion and adhesion
• Less cohesion than adhesion: liquid wets a solid surface and rises at the point of contact
• If more cohesion: the liquid surface lowers at the point of contact
• Capillarity makes water rise in a glass tube
• Mercury depresses below true level, the curved liquid surface that develops in a tube is a meniscus
• The expression in Eq (1.11) can be used to estimate approximate capillary rise or depression in a tube
• Capillary effects are negligible for tube diameters larger than 12mm

Vapor Pressure of Liquids

• Liquids can evaporate, which projects molecules into the space above their surfaces
• If confined, the partial pressure increases until it equals to the vapor pressure
• Known as the saturation pressure.
• The rate of molecular evaporation increases with increasing temperature and decreasing pressure
• Boiling occurs when the pressure on the surface falls below the saturation pressure
• Boiling pressure is the saturation pressure
• Rapid vaporization and recondensation is called cavitation
• This is damaging
• Mercury is suitable for barometers