Fundamentals of Structural Analysis

Chapter 2 PROPERTIES OF FLUIDS 2-1_Introduction Property: Any characteristic of a system. Some familiar properties are pressure P, temperature T, volume V, and mass m. Properties are considered to be either intensive or extensive. Intensive properties: Those that are independent of the mass of a system, such as temperature, pressure, and density. Extensive properties: Those whose values depend on the size—or extent—of the system. Specific properties: Extensive properties per unit Criterion to differentiate intensive mass. and extensive properties. 2-2_Density and Specific Gravity Density m  V  kg m 3  Specific Volume v V m  1  Density is mass per unit volume; specific volume is volume per unit mass. Specific gravity: The ratio of the density of a substance to the density of some standard substance at a specified temperature (usually  water at 4°C). SG  H 2 O Specific weight: The weight of a unit volume of a 3 substance.   g N m s   TABLE 2–1 The specific gravity of some substances at 20°C and 1 atm unless stated otherwise Substance SG Water 1.0 Blood (at 37°C) 1.06 Seawater 1.025 Gasoline 0.68 Ethyl alcohol 0.790 Mercury 13.6 Balsa wood 0.17 Dense oak wood 0.93 Gold 19.3 Bones 1.7–2.0 Ice (at 0°C) 0.916 Air 0.001204 Density of Ideal Gases Equation of state: Any equation that relates the pressure, temperature, and density (or specific volume) of a substance. Ideal-gas equation of state: The simplest and best-known equation of state for substances in the gas phase. Pv  RT or P   RT R  Ru /M Ru  8.314 kj/kmol  K PV  mRT or PV  NRuT P1V1 /T1  P2V2 /T2 For a fixed mass The universal gas constant Example - Density Example – 1 Determine the density of air at atmospheric conditions (Pressure = 1 atm, Temperature = 20 °C). Temperature Scales Kelvin scale: The thermodynamic temperature scale in the SI system. Rankine scale: The thermodynamic temperature scale in English system. T (K)  T (C)  273.15  T (R)/1.8 T (R)  T (F)  459.67  1.8 T (K) • An ideal gas is a hypothetical substance that obeys the relation Pv = RT. • At low pressures and high temperatures, the density of a gas decreases and the gas behaves like an ideal gas. • In the range of practical interest, many familiar gases such as air, nitrogen, oxygen, hydrogen, helium, argon, neon, and carbon dioxide can be treated as ideal gases. • Dense gases such as water vapor in steam power plants and refrigerant vapor in refrigerators, however, should not be treated as ideal gases since they usually exist at a state near saturation. 2-3_Vapor Pressure and Cavitation Saturation temperature Tsat: The temperature at which a pure substance changes phase at a given pressure. Saturation pressure Psat: The pressure at which a pure substance changes phase at a given temperature. Vapor pressure (Pv): The pressure exerted by its vapor in phase equilibrium with its liquid at a given temperature. It is identical to the saturation pressure Psat of the liquid (Pv = Psat). Partial pressure: The pressure of a gas or vapor in a mixture with other gases. For example, atmospheric air is a mixture of dry air and water vapor, and atmospheric pressure is the sum of the partial pressure of dry air and the partial pressure of water vapor. TABLE 2–2 Saturation (or vapor) pressure of water at various temperatures Temperature T, °C The vapor pressure (saturation pressure) of a pure substance (e.g., water) is the pressure exerted by its vapor molecules when the system is in phase equilibrium with its liquid molecules at a given temperature. Saturation Pressure Psat, kPa −10 0.260 −5 0.403 0 0.611 5 0.872 10 1.23 15 1.71 20 2.34 25 3.17 30 4.25 40 7.38 50 12.35 100 101.3 (1 atm) 150 475.8 200 1554 250 3973 300 8581 There is a possibility of the liquid pressure in liquid-flow systems dropping below the vapor pressure at some locations, and the resulting unplanned vaporization. The vapor bubbles (called cavitation bubbles since they form “cavities” in the liquid) collapse as they are swept away from the low-pressure regions, generating highly destructive, extremely highpressure waves. This phenomenon, which is a common cause for drop in performance and even the erosion of impeller blades, is called cavitation, and it is an important consideration in the design of hydraulic turbines and pumps. Photograph by David Stinebring, ARL/ Pennsylvania State University. Used by permission. Cavitation damage on a 16-mm by 23-mm aluminum sample tested at 60 m/s for 2.5 h. The sample was located at the cavity collapse region downstream of a cavity generator specifically designed to produce high damage potential. 2-4_Energy and Specific Heats Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential, electric, magnetic, chemical, and nuclear, and their sum constitutes the total energy, E of a system. Macroscopic forms of energy: Those a system possesses as a whole with respect to some outside reference frame, such as kinetic and potential energies. Microscopic forms of energy: Those related to the molecular structure of a system and the degree of the molecular activity. 2-4_Energy and Specific Heats Internal energy, U: The sum of all the microscopic forms of energy. In daily life, we frequently refer to the sensible and latent forms of internal energy as heat. In engineering, however, those forms of energy are usually referred to as thermal energy. Kinetic energy, KE: The energy that a system possesses as a result of its motion relative to some reference frame. Potential energy, PE: The energy that a system possesses as a result of its elevation in a gravitational field. © Creatas/PunchStock RF Comstock Images/Jupiterimages RF At least six different forms of energy are encountered in bringing power from a nuclear plant to your home, nuclear, thermal, mechanical, kinetic, magnetic, and electrical. h  u  Pv  u  eflowing P Enthalpy  V2  P /  e  h  ke  pe  h   gz 2 du  cv dT and u  cv ,avg T (kJ/kg) dh  c p dT and h  cp,avg T h  u  P /  cavg T  P / h  u  cavg T for P = const. process h  p / Energy of a flowing fluid P/ is the flow energy, also called the flow work, which is the energy per unit mass needed to move the fluid and maintain flow. For T = const. process Specific heat at constant volume, cv: The energy required to raise the temperature of the unit mass of a substance by one degree as the volume is maintained constant. Specific heat at constant pressure, cp: The energy required to raise the temperature of the unit mass of a substance by one degree as the pressure is maintained constant. The internal energy u represents the microscopic energy of a non-flowing fluid per unit mass, whereas enthalpy h represents the microscopic energy of a flowing fluid per unit mass. 2-5_Compressibility and Speed of Sound Coefficient of Compressibility We know from experience that the volume (or density) of a fluid changes with a change in its temperature or pressure. Fluids usually expand as they are heated or depressurized and contract as they are cooled or pressurized. Fluids, like solids, compress when the applied pressure is increased from P1 to P2. 2-5_Compressibility and Speed of Sound But the amount of volume change is different for different fluids, and we need to define properties that relate volume changes to the changes in pressure and temperature. Two such properties are: the bulk modulus of elasticity   P   P    v        v  T    T the coefficient of volume expansion  P P    v /v  /  Pa  (T  constant) Coefficient of compressibility (bulk modulus of compressibility or bulk modulus of elasticity) for fluids The coefficient of compressibility represents the change in pressure corresponding to a fractional change in volume or density of the fluid while the temperature remains constant. What is the coefficient of compressibility of a truly incompressible substance (v = constant)? A large value of  indicates that a large change in pressure is needed to cause a small fractional change in volume, and thus a fluid with a large  is essentially incompressible. This is typical for liquids, and explains why liquids are usually considered to be incompressible. Water hammer: Characterized by a sound that resembles the sound produced when a pipe is “hammered.” This occurs when a liquid in a piping network encounters an abrupt flow restriction (such as a closing valve) and is locally compressed. The acoustic waves that are produced strike the pipe surfaces, bends, and valves as they propagate and reflect along the pipe, causing the pipe to vibrate and produce the familiar sound. Water hammering can be quite destructive, leading to leaks or even structural damage. The effect can be suppressed with a water hammer arrestor. Water Hammer: https://www.youtube.com/watch?v=6ydsAIHWVNM Photo provided courtesy of Oatey Company Photo by Arris S. Tijsseling, visitor of the University of Adelaide, Australia. Used by permission Water hammer arrestors: (a) A large surge tower built to protect the pipeline against water hammer damage. (b) Much smaller arrestors used for supplying water to a household washing machine. For an ideal gas, P   RT and (P /)T  RT  P /, and thus  ideal gas  P (Pa) The coefficient of compressibility of an ideal gas is equal to its absolute pressure, and the coefficient of compressibility of the gas increases with increasing pressure.  P Ideal gas:   P T  constant  The percent increase of density of an ideal gas during isothermal compression is equal to the percent increase in pressure. Isothermal compressibility: The inverse of the coefficient of compressibility. The isothermal compressibility of a fluid represents the fractional change in volume or density corresponding to a unit change in pressure. 1 1  v  1           1/ Pa    v  P  T   P  T Coefficient of Volume Expansion The density of a fluid depends more strongly on temperature than it does on pressure. The variation of density with temperature is responsible for numerous natural phenomena such as winds, currents in oceans, rise of plumes in chimneys, the operation of hot-air balloons, heat transfer by natural convection, and even the rise of hot air and thus the phrase “heat rises”. To quantify these effects, we need a property that represents the variation of the density of a fluid with temperature at constant pressure. © G.S. Settles, Gas Dynamics Lab, Penn State University. Used with permission Natural convection over a woman’s hand. The coefficient of volume expansion (volume expansivity): The variation of the density of a fluid with temperature at constant pressure. 1  v  1          1/K   v  T  P   T  P v /v  /    at constant P T T A large value of  for a fluid means a large change in density with temperature, and the product  T represents the fraction of volume change of a fluid that corresponds to a temperature change of T at constant pressure. The volume expansion coefficient of an ideal gas (P = RT ) at a temperature T is equivalent to the inverse of the 1 temperature: idealgas  1/K  The coefficient of volume expansion is a measure of the change in volume of a substance with temperature at constant pressure. In the study of natural convection currents, the condition of the main fluid body that surrounds the finite hot or cold regions is indicated by the subscript “infinity” to serve as a reminder that this is the value at a distance where the presence of the hot or cold region is not felt. In such cases, the volume expansion coefficient can be expressed approximately as      /   T  T or      T  T  The combined effects of pressure and temperature changes on the volume change of a fluid can be determined by taking the specific volume to be a function of T and P.  v   v  d v    dT    dP  ( dT   dP )v  T  P  P  T The fractional change in volume (or density) due to changes in pressure and temperature can be expressed approximately as v     T    P v  The variation of the coefficient of volume expansion of water with temperature in the range of 20°C to 50°C. Speed of Sound and Mach Number Speed of sound (sonic speed): The speed at which an infinitesimally small pressure wave travels through a medium. Control volume moving with the small pressure wave along a duct.  P  c  k     T 2 For any fluid c  kRT For an ideal gas Propagation of a small pressure wave along a duct. Mach number Ma: The ratio of the actual speed of the fluid (or an object in still fluid) to the speed of sound in the same fluid at the same state. V Ma  c The Mach number depends on the speed of sound, which depends on the state of the fluid. The speed of sound in air increases with temperature. At typical outside temperatures, c is about 340 m/s. The speed of sound changes with temperature and varies with the fluid. © Purestock/SuperStock RF 2-6_Viscosity • Viscosity is due to the internal frictional force that develops between different layers of fluids as they are forced to move relative to each other. Top: © Photodisc/Getty Images RF Bottom: © Digital Vision/Getty Images RF Viscosity: A property that represents the internal resistance of a fluid to motion or the “fluidity”. Drag force: The force a flowing fluid exerts on a body in the flow direction. The magnitude of this force depends, in part, on viscosity. • The viscosity of a fluid is a measure of its “resistance to deformation.” Newtonian fluids: Fluids for which the rate of deformation is proportional to the shear stress.  d dt or  Shear stress:    du dy du dy  N/m  2 The behavior of a fluid in laminar flow Shear force: F     du  N dy between two parallel plates when the upper plate moves with a constant velocity.  coefficient of viscosity Dynamic y du V F u  y  V and   (absolute) viscosity dy A kg/ms or Ns/m2 or Pas da Vdt du d  du 1 poise = 0.1 Pas d   tan d     dt  dy dt dy The rate of deformation (velocity gradient) of a Newtonian fluid is proportional to shear stress, and the constant of proportionality is the viscosity. Variation of shear stress with the rate of deformation for Newtonian and non-Newtonian fluids (the slope of a curve at a point is the apparent viscosity of the fluid at that point). Kinematic viscosity: v   / m2/s or stoke 1 stoke = 1 cm2/s For liquids, both the dynamic and kinematic viscosities are practically independent of pressure, and any small variation with pressure is usually disregarded, except at extremely high pressures. For gases, this is also the case for dynamic viscosity (at low to moderate pressures), but not for kinematic viscosity since the density of a gas is proportional to its pressure. b / T c  aT 1/2   a 10 For liquids For gases   1  b /T Dynamic viscosity, in general, does not depend on pressure, but kinematic viscosity does. • The viscosity of a fluid is directly related to the pumping power needed to transport a fluid in a pipe or to move a body through a fluid. • Viscosity is caused by the cohesive forces between the molecules in liquids and by the molecular collisions in gases, and it varies greatly with temperature. • In liquid, the molecules possess more energy at higher temperatures, and they can oppose the large cohesive intermolecular forces more strongly. As a result, the energized liquid molecules can move more freely. • In gas, the intermolecular forces are negligible, and the gas molecules at high temperatures move randomly at higher velocities. This results in more molecular The viscosity of liquids decreases and the viscosity of collisions per unit volume per unit time and therefore in greater resistance to flow. gases increases with temperature. The variation of dynamic (absolute) viscosity of common fluids with temperature at 1 atm (1 Ns/m2 = 1 kg/ms = 0.020886 lbfs/ft2) Data from EES and F. M. White, Fluid Mechanics 7e. Copyright © 2011 The McGraw-Hill Companies, Inc. TABLE 2–3 Dynamic viscosity of some fluids at 1 atm and 20°C (unless otherwise stated) Fluid Glycerin: -20°C 0°C 20°C 40°C Engine oil: SAE 10W SAE 10W30 SAE 30 SAE 50 Mercury Ethyl alcohol Water: 0°C 20°C 100°C (liquid) 100°C (vapor) Blood, 37°C Gasoline Ammonia Air Hydrogen, 0°C Dynamic Viscosity 𝜇, kg/m⋅s 134.0 10.5 1.52 0.31 0.10 0.17 0.29 0.86 0.0015 0.0012 0.0018 0.0010 0.00028 0.000012 0.00040 0.00029 0.00015 0.000018 0.0000088 L length of the cylinder 𝑛 number of revolutions per unit time T=FR =  2 R 3 L  4 2 R 3 nL    n This equation can be used to calculate the viscosity of a fluid by measuring torque at a specified angular velocity. Therefore, two concentric cylinders can be used as a viscometer, a device that measures viscosity. 2-7_Surface Tension and Capillary Effect • Liquid droplets behave like small balloons filled with the liquid on a solid surface, 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 is due to the attractive forces between the molecules of the liquid. • The magnitude of this force per unit length is called surface tension (or coefficient of surface tension) and is usually expressed in the unit N/m. • This effect is also called surface energy [per unit area] and is expressed in the equivalent unit of Nm/m2. (a) © Don Paulson Photography/Purestock/SuperStock RF (b) NPS Photo by Rosalie LaRue (c) © G.S. Settles, Gas Dynamics Lab, Penn State University. Used with permission. Some consequences of surface tension: (a) drops of water beading up on a leaf, (b) a water strider sitting on top of the surface of water, and (c) a color schlieren image of the water strider revealing how the water surface dips down where its feet contact the water (it looks like two insects but the second one is just a shadow). Attractive forces acting on a liquid molecule at the surface and deep inside the liquid. F W  Force  Distance  F  x  2b  x   A Stretching a liquid film with a s s s  U-shaped wire, and the forces 2b Surface tension: The work done per unit increase in the surface area of the liquid. acting on the movable wire of length b. TABLE 2–4 The free-body diagram of half a droplet or air bubble and half a soap bubble. Surface tension of some fluids in air at 1 atm and 20°C (unless otherwise stated) Surface Tension 𝜎s, N/m* Fluid †Water: 0°C 0.076 20°C 0.073 1000C 0.059 3000C 0.014 Glycerin 0.063 SAE 30 oil 0.035 Mercury 0.440 Ethyl alcohol 0.023 Blood, 37°C 0.058 Gasoline 0.022 Ammonia 0.021 Soap solution 0.025 Kerosene 0.028 Droplet or air bubble: Soap bubble:    2 R   s   R 2 Pdroplet  Pdroplet  Pi  Po    2  2 R   s   R 2 Pbubble  Pbubble  Pi  Po  4 s R 2 s R Capillary Effect Capillary effect: The rise or fall of a liquid in a small-diameter tube inserted into the liquid. Capillaries: Such narrow tubes or confined flow channels. The capillary effect is partially responsible for the rise of water to the top of tall trees. Meniscus: The curved free surface of a liquid in a capillary tube. The strength of the capillary effect is quantified by the contact (or wetting) angle, defined as the angle that the tangent to the liquid surface makes with the solid surface at the point of contact. The contact angle for wetting and nonwetting fluids. Photo by Gabrielle Trembley, Pennsylvania State University. Used by permission. The meniscus of colored water in a 4-mm-inner-diameter glass tube. Note that the edge of the meniscus meets the wall of the capillary tube at a very small contact angle. The capillary rise of water and the capillary fall of mercury in a smalldiameter glass tube. Capillary rise : 2 s h cos   gR The forces acting on a liquid column that has risen in a tube due to the capillary effect.  R  constant  Capillary rise is inversely proportional to the radius of the tube and density of the liquid.