Lecture 2 Properties of Fluids

Properties of a System: 

Properties of a System Any characteristic of a system is called a property. Extensive properties are those whose value depends on the size of the system. Examples: Total mass, total volume, and total momentum. Intensive properties are independent of the mass of the system. Examples: temperature, pressure, and density. Extensive properties per unit mass are called specific properties. Examples : specific volume v = V/m and specific total energy e=E/m.

Properties of a fluid: 

Properties of a fluid S.No Properties Examples 1 Kinematic Properties Linear velocity, angular velocity, vorticity, acceleration, strain rate 2 Transport Properties Viscosity, thermal conductivity, mass diffusivity 3 Thermodynamic Properties Pressure , density, temperature, enthalpy, entropy, bulk modulus, coefficient of thermal expansion 4 Other miscellaneous properties Surface tension, vapour pressure, eddy-diffusion coefficients

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In a given flow situation, the determination of the properties of the fluid either by experiment or theory as a function of position and time is considered to be the solution to the problem In almost all cases, the emphasis is on the space-time (x,y,z,t) distribution of the fluid properties

Continuum: 

Continuum Atoms are widely spaced in the gas phase. However, we can disregard the atomic nature of a substance and 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.

Primary Dimensions and Units in SI System: 

Primary Dimensions and Units in SI System

Secondary Dimensions in Fluid Mechanics: 

Secondary Dimensions in Fluid Mechanics

Density and Specific Gravity: 

Density and Specific Gravity Density is defined as the mass per unit volume  = m/V. Density has units of kg/m 3 Specific volume is defined as v = 1/  = 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=  /  H 2 0 . SG is a dimensionless quantity. The specific weight or weight density is defined as the weight per unit volume, i.e.,  s =  g where g is the gravitational acceleration. g s has units of N/m 3 .

Density of Ideal Gases: 

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 =  R T Ideal-gas equation holds for most gases. At low pressures and high temperatures, the density of a gas decreases, and the gas behaves as an ideal gas under these conditions 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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Partial Pressure It is the pressure of the gas or vapour P p in a mixture with the other gases Condition 1: P p of vapour < P v when no liquid present Condition 2: P p of vapour = P v when liquid is saturated P v = vapour or saturation pressure

Vapor Pressure and Cavitation: 

Vapor Pressure and Cavitation Vapor Pressure P v is defined as the pressure exerted by its vapor in phase equilibrium with its liquid at a given temperature If P drops below P v , liquid is locally vaporized, creating cavities of vapor. Vapor cavities collapse when local P rises above P v . Collapse of cavities is a violent process which can damage machinery. Cavitation is noisy, and can cause structural vibrations. Cavitation Number

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Cavitation must avoided in flow systems since it reduces performance, generates annoying vibrations and noise and causes damage to equipment . The large number of bubbles collapsing near the solid surface over a long period of time may cause erosion, surface pitting, fatigue failure and the destruction of the components or machinery. The presence of cavitation in a flow system can be sensed by its characteristic tumbling sound

Energy and Specific Heats: 

Energy and Specific Heats 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=V 2 /2 Potential energy pe=gz In the absence of electrical, magnetic, chemical, and nuclear energy, the total energy is e flowing =h+V 2 /2+gz.

How does fluid volume change with P and T ?: 

How does fluid volume change with P and T ? Fluids expand as T ↑ or P ↓ Fluids contract as T ↓ or P ↑ The amount of volume change is different for different fluids 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

Coefficient of Compressibility: 

Coefficient of Compressibility The fluids act like elastic solids with respect to pressure. It is also called as bulk modulus of compressibility or bulk modulus of elasticity A larger value of k indicates that a large change in pressure is required to cause very small change in volume and thus a fluid with a large k is essentially incompressible. coefficient of compressibility of an ideal gas k ideal gas = P (Pa)

Water hammer: 

Water hammer Small density changes in liquid causes water hammering in piping system. When a liquid flowing in a pipe is restricted by closing a valve, it is locally compressed. Due to that acoustic waves are produced which strikes the pipe surface, bends and valves causing the pipe to vibrate and produce the familiar sound

Isothermal Compressibility: 

Isothermal Compressibility The inverse of the coefficient of compressibility is called the isothermal compressibility The isothermal compressibility of a fluid represents the fractional change in volume or density corresponding to a unit change in pressure

Coefficient of Volume Expansion: 

Coefficient of Volume Expansion The density of fluid depends more on temperature rather than pressure The variation of density with temperature is responsible for numerous natural phenomenon such as - rise of plumes in chimney - use of ventilators in room - operation of hot - air balloons - heat transfer by natural convection To quantify these effects we need to define a property called coefficient of volume expansion A large value of  for a fluid means a large change in density with temperature

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From the above equations, the natural convection are initiated by the buoyancy force, which is proportion to the density difference, which is proportional to temperature difference at constant pressure. Contd., Volume expansion coefficient of an ideal gas

Combined effect : 

Combined effect The combined effects of pressure and temperature changes on the volume change of a fluid is given by

Viscosity: 

Viscosity 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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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  = 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:  =  du/dy  is the dynamic viscosity and has units of kg/m·s, Pa·s, or poise.

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Dynamic Viscosity  =  (du/dy) =  Dynamic viscosity is also called as absolute viscosity or coefficient of viscosity. Unit of dynamics viscosity kg/ms or Ns/m 2 or poise Poise = 0.1 Ns/m 2 Kinematic viscosity  = dynamic viscosity / density Unit of Kinematic Viscosity is m 2 /s 1 Stoke = 1 cm 2 /s = 10 -4 m 2 /s

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Types of fluids Fluids which do not follow the linear law of viscosity are called nonnewtonian and also called rheological fluids .

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Types of Non-Newtonian fluids : Dilatant, or shear-thickening fluid increases resistance with increasing applied stress. Ex: Solutions with suspended starch and sand Pseudoplastic, or shear-thinning fluid decreases resistance with increasing stress. Ex: paints , polymer solutions If the thinning effect is very strong, as with the dashed-line curve, the fluid is termed plastic. The limiting case of a plastic substance is one which requires a finite yield stress before it begins to flow. Bingham plastic Flow behaviour after yield may also be nonlinear. An example of a yielding fluid is toothpaste, which will not flow out of the tube until a finite stress is applied by squeezing

Variation of Viscosity with Temperature: 

Variation of Viscosity with Temperature

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A further complication of nonnewtonian behavior is the transient effect shown in Fig below. Some fluids require a gradually increasing shear stress to maintain a constant strain rate and are called rheopectic. The opposite case of a fluid which thins out with time and requires decreasing stress is termed thixotropic.

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In Liquids, viscosity is caused by the cohesive forces between the molecules. Viscosity of liquids decrease with increase in temperature. This is because in a liquid the molecules possess more energy at high temperature, so liquids can oppose cohesive intermolecular forces more strongly. As a result, the energized liquid molecules can move more freely In gases, Viscosity is caused by the molecular collisions between molecules. The intermolecular forces are negligible, so the gas molecules at high temperature move randomly at high velocities. As a result molecular collision per unit volume per unit time increases. The viscosity of a fluid is directly related to the pumping power needed to transport a fluid in pipe or to move a body through a fluid.

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Gas viscosity increases with temperature. Two common approximations are the power law and the Sutherland law: Liquid viscosity decreases with temperature and is roughly exponential, ; but a better fit is the empirical result that ln is quadratic in 1/T, where T is absolute temperature

Viscometer: 

Viscometer Torque T = FR Wetted surface area of inner cylinder = 2 RL L – length of the cylinder - Number of revolutions per min R – Radius of the inner cylinder l - distance between two cylinder where fluid whose viscosity is to be measured is filled

Flow between Plates: 

Flow between Plates

Surface Tension: 

Surface Tension .

Practical Examples: 

Practical Examples Drop of blood forms a hump on a horizontal glass. Water droplets from rain A drop of mercury forms a near perfect square Dew hang from leaves of trees A soap released into air Liquid fuel injected into the engine

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In all these observations, the 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 tension acts parallel to the surface and it is due to cohesive forces between the molecules of the fluid. Repulsive forces from interior molecules causes the liquid to minimize its surface area and attain a spherical shape The magnitude of this force per unit length is called surface tension s (N/m). This effect is also called surface energy.

Slide 36: 

Surface tension decreases with increase in temperature Surface tension increase with increase in tensile strength of the surface film per unit width Surface tension varies from substance to substance Surface tension becomes zero at the critical point and thus there is no distinct liquid – vapour interface at temperature above critical point. Effect of pressure on surface tension is negligible Presence of impurities reduce surface tension Surface tension can be reduced by adding chemicals called surfactants ex., detergents for removing impurities Important remarks about surface tension effects

Capillary Effect: 

Capillary Effect Capillary effect is the rise or fall of a liquid in a small-diameter tube. The curved free surface in the tube is called 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.

Wetting or contact angle: 

Wetting or contact angle The strength of capillary effect is quantified by contact angle It is defined as the angle that the tangent to the liquid surface makes with solid surface at the point of contact A liquid is said to wet the surface if  < 90 o and not to wet the surface when  > 90 o Capillary rise/drop h = 2 scos  / gR

Pressure: 

Pressure Pressure is defined as a compressive force exerted by a fluid per unit area. Unit of pressure - N/m 2 , which is called a Pascal (Pa). Since the unit Pa is too small for pressures encountered in practice, kilopascal (1 kPa = 10 3 Pa) and MegaPascal (1 MPa = 10 6 Pa) are commonly used. Other units include bar, atm, kgf/cm 2 , lbf/in 2 =psi .

Absolute, gage, and vacuum pressures: 

Absolute, gage, and vacuum pressures

Slide 41: 

Actual pressure at a give point is called the absolute pressure. Most pressure-measuring devices are calibrated to read zero in the atmosphere, and therefore indicate gage pressure, P gage =P abs - P atm . Pressure below atmospheric pressure are called vacuum pressure, P vac =P atm - P abs .

Pressure at a Point: 

Pressure at a Point Pressure at any point in a fluid is the same in all directions. (Pascal’s Law) P x =P y =P z Pressure has a magnitude, but not a specific direction, and thus it is a scalar quantity.

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28 August 2011 44 In picture, pistons are at same height: Ratio A2/A1 is called ideal mechanical advantage

Basic Equation for Pressure Field: 

Basic Equation for Pressure Field 28 August 2011 45

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28 August 2011 46 Taylor Series

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For Incompressible Fluid 28 August 2011 48

For Compressible Fluid: 

For Compressible Fluid 28 August 2011 49

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Variation of Pressure with Depth: 

Variation of Pressure with Depth In the presence of a gravitational field, pressure increases with depth because more fluid rests on deeper layers. To obtain a relation for the variation of pressure with depth, consider rectangular element Force balance in z -direction gives

Slide 52: 

Evaluating Pressure changes through a column of multiple fluids

Variation of Pressure with Depth: 

Variation of Pressure with Depth Pressure in a fluid at rest is independent of the shape of the container. Pressure is the same at all points on a horizontal plane in a given fluid.

Slide 54: 

Points a , b , c , and d are at equal depths in water and therefore have identical pressures. Point D has a different pressure from A, B, and C because it is not connected to them by a water path

Slide 55: 

Manometry Pressure measuring devices based on liquid columns in vertical or inclined tubes are called manometers Piezometer Tube U-Tube Manometer

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Differential U-Tube Manometer

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Inclined-Tube Manometer Inclined-tube manometers can be used to measure small pressure differences accurately.

The Barometer: 

The Barometer Atmospheric pressure is often referred to as the barometric pressure. P C can be taken to be zero since there is only Hg vapor above point C, and it is very low relative to Patm. Change in atmospheric pressure due to elevation has many effects: Cooking, nose bleeds, engine performance, aircraft performance.

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