Fluid Mechanics Chapter 2 PDF

Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu CHAPTER 2 PROPERTIES OF FLUIDS INTRODUCTION  Any characteristic of a system is called a property. Properties are considered to be either intensive or extensive. Intensive properties...

Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu CHAPTER 2 PROPERTIES OF FLUIDS INTRODUCTION  Any characteristic of a system is called a property. Properties are considered to be either intensive or extensive. Intensive properties are those that are independent of the mass of the system, such as temperature, pressure, and density. Extensive properties are those whose values depend on the size—or extent—of the system. Extensive properties per unit mass are called specific properties.  Specifying a certain number of properties is sufficient to fix a state. The number of properties required to fix the state of a system is given by the state postulate: The state of a simple compressible system is completely specified by two independent, intensive properties. Continuum  It is convenient to disregard the atomic nature of the fluid and view it as continuous, homogeneous matter with no holes, that is, a continuum.  The continuum idealization allows us to treat properties as point functions and to assume that the properties vary continually in space with no jump discontinuities.  This idealization is valid as long as the size of the system we deal with is large relative to the space between the molecules.  The continuum model is applicable as long as the characteristic length of the system (such as its diameter) is much larger than the mean free path of the molecules. Quantitatively, a dimensionless number called the Knudsen number Kn = 𝜆/L is defined, where 𝜆 is the mean free path of the fluid molecules and L is some characteristic length scale of the fluid flow. If Kn is very small (typically less than about 0.01), the fluid medium can be approximated as a continuum medium.  Otherwise, the rarefied gas flow theory should be used, and the impact of individual molecules should be considered. In this course, we limit our consideration to substances that can be modeled as a continuum. 1 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu DENSITY AND SPECIFIC GRAVITY Specific gravity or Density Specific volume specific weight or relative density weight density 𝑃  Recall that for an ideal gas, 𝜌 = 𝑅𝑇 VAPOR PRESSURE AND CAVITATION  At a given pressure, the temperature at which a pure substance changes phase is called the saturation temperature 𝑇𝑠𝑎𝑡. Likewise, at a given temperature, the pressure at which a pure substance changes phase is called the saturation pressure 𝑃𝑠𝑎𝑡.  The vapor pressure 𝑃𝑣 of a pure substance is defined as the pressure exerted by its vapor in phase equilibrium with its liquid at a given temperature.  𝑃𝑣 is a property of the pure substance, and turns out to be identical to the saturation pressure 𝑃𝑠𝑎𝑡 of the liquid (𝑃𝑣 = 𝑃𝑠𝑎𝑡 ).  Partial pressure is defined as the pressure of a gas or vapor in a mixture with other gases. The partial pressure of a vapor must be less than or equal to the vapor pressure if there is no liquid present. However, when both vapor and liquid are present and the system is in phase equilibrium, the partial pressure of the vapor must equal the vapor pressure, and the system is said to be saturated.  The reason for our interest in vapor pressure is the 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 high-pressure 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. On the other hand, we note that some flow systems use The VA-111 Shkval torpedo cavitation to their advantage, e.g., high-speed “supercavitating” torpedoes. 2 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu ENERGY AND SPECIFIC HEATS  Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential, electrical, magnetic, chemical, and nuclear and their sum constitutes the total energy E (or e on a unit mass basis) of a system. The forms of energy related to the molecular structure of a system and the degree of the molecular activity are referred to as the microscopic energy. The sum of all microscopic forms of energy is called the internal energy of a system, and is denoted by U (or u on a unit mass basis).  The macroscopic energy of a system is related to motion and the influence of some external effects such as gravity, magnetism, electricity, and surface tension. The energy that a system possesses as a result of its motion is called kinetic energy. The energy that a system possesses as a result of its elevation in a gravitational field is called potential energy.  In daily life, we frequently refer to the sensible and latent forms of internal energy as heat, and we talk about the heat content of bodies. In engineering, however, those forms of energy are usually referred to as thermal energy to prevent any confusion with heat transfer. Enthalpy flow energy or flow work The total energy of a flowing fluid on a unit-mass basis for a simple compressible system The differential and finite changes for ideal gases For incompressible substances (Liquids) For incompressible substances if pressure = constant if temperature = constant 3 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu COMPRESSIBILITY AND SPEED OF SOUND Coefficient of Compressibility In an analogous manner to Young’s modulus of elasticity for solids, it is appropriate to define a coefficient of compressibility κ (also called the bulk modulus of compressibility or bulk modulus of elasticity) for fluids as approximately in terms of finite changes Also, For an ideal gas, 𝜕𝑃 𝑃 𝜅 = 𝜌( ) = 𝜌 𝜕𝜌 𝑇 𝜌 ∆𝑃 𝜅= ∆𝜌/𝜌 𝜕𝑃 𝜅 = 𝜌 ( ) = 𝜌𝑘𝜌 𝑘−1 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝜅𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 = 𝑘𝑃 (𝑘 = 𝑐𝑝 ⁄𝑐𝑣 ) 𝜕𝜌 The inverse of the coefficient of compressibility is called the isothermal compressibility α. Coefficient of Volume Expansion The density of a fluid, in general, depends more strongly on temperature than it does on pressure. To quantify these effects, we need a property that represents the variation of the density of a fluid with temperature at constant pressure. The property that provides that information is the coefficient of volume expansion (or volume expansivity) β. approximately in terms of finite changes For an ideal gas, 𝑃 = 𝜌𝑅𝑇 and (𝜕𝑣 ⁄𝜕𝑇)𝑃 = 𝑅/𝑃 𝛽 = 𝑅/(𝑃𝑣) 4 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu In the study of natural convection currents, from the Boussinesq approximation, 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. Speed of Sound and Mach Number An important parameter in the study of compressible flow is the speed of sound (or the sonic speed), defined as the speed at which an infinitesimally small pressure wave travels through a medium. To simplify the analysis, consider a control volume that encloses the wave front and moves with it. Mass Balance Neglecting the higher-order terms, this equation reduces to Energy Balance Neglecting the higher-order terms, this equation reduces to The propagation of a sonic wave is not only adiabatic but also very nearly isentropic. 5 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu Combining the above equations yields the desired expression for the speed of sound as or ] Isentropic process 𝑃 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝜌𝑘 A second important parameter in the analysis of compressible fluid flow is the Mach number Ma. It is 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:  If Ma is less than about 1/3, the flow may be approximated as incompressible.  The flow is called sonic when Ma = 1, subsonic when Ma < 1, supersonic when Ma > 1, hypersonic when Ma >> 1, and transonic when Ma ≅ 1. VISCOSITY When two solid bodies in contact move relative to each other, a friction force develops at the contact surface in the direction opposite to motion. The situation is similar when a fluid moves relative to a solid or when two fluids move relative to each other. It appears that there is a property that represents the internal resistance of a fluid to motion or the “fluidity,” and that property is the viscosity. 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. Consider a fluid layer between two very large parallel plates. The fluid in contact with the upper plate sticks to the plate surface and moves with it at the same speed, and the shear stress 𝜏 acting on this fluid layer is 6 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu velocity profile velocity gradient During a differential time interval dt angular displacement or deformation (or shear strain) rate of deformation = velocity gradient Further, it can be verified experimentally that for most fluids the rate of deformation (and thus the velocity gradient) is directly proportional to the shear stress 𝜏. Fluids for which the rate of deformation is linearly proportional to the shear stress are called Newtonian fluids. The coefficient of viscosity or the dynamic (or absolute) viscosity  Note that viscosity is independent of the rate of deformation for Newtonian fluids. shear force The force F required to move the upper plate for the parallel plates above 7 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu For non-Newtonian fluids, the relationship between shear stress and rate of deformation is not linear. The slope of the curve on the 𝜏 versus du/dy chart is referred to as the apparent viscosity of the fluid. In fluid mechanics and heat transfer, the ratio of dynamic viscosity to density appears frequently. µ 𝑣= kinematic viscosity 𝜌  In general, the viscosity of a fluid depends on both temperature and pressure, although the dependence on pressure is rather weak. 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. The viscosity of liquids decreases with temperature, whereas the viscosity of gases increases with temperature. This is because in a liquid the molecules possess more energy at higher temperatures, and they can oppose the large cohesive intermolecular forces more strongly. In a gas, on the other hand, the intermolecular forces are negligible, and the gas molecules at high temperatures move randomly at higher velocities. This results in more molecular collisions per unit volume per unit time and therefore in greater resistance to flow. 8 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu Viscometer 𝑅 ≫ ℓ → The gap between the cylinders can be modeled as two parallel flat plates. 𝜔𝑅 T = 𝐹𝑅 = µ(2𝜋𝑅𝐿) 𝑅 ℓ SURFACE TENSION AND CAPILLARY EFFECT At the interface between a liquid and a gas, or between two immiscible liquids, forces develop in the liquid surface that cause the surface to behave as if it were a “skin” or “membrane” stretched over the fluid mass. 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 𝜎s 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 or J/m2. In this case, 𝜎s represents the stretching work that needs to be done to increase the surface area of the liquid by a unit amount. 9 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu To understand the surface tension effect better, consider a liquid film (such as the film of a soap bubble) suspended on a U-shaped wire frame with a movable side. Then, the surface tension, If 𝐹 ≈ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 over the small distance, The work done per unit increase in the surface area of the liquid  The surface tension varies greatly from substance to substance, and with temperature for a given substance.  The effect of pressure on surface tension is usually negligible.  The surface tension of a liquid, in general, decreases with temperature and becomes zero at the critical point (and thus there is no distinct liquid–vapor interface at temperatures above the critical point).  We speak of surface tension for liquids only at liquid–liquid or liquid– gas interfaces. Therefore, it is imperative that the adjacent liquid or gas be specified when specifying surface tension. (What about surface tension in vacuum?) A curved interface indicates a pressure difference (or “pressure jump”) across the interface with pressure being higher on the concave side. Consider, for example, a droplet of liquid in air, an air (or other gas) bubble in water, or a soap bubble in air. The excess pressure ΔP above atmospheric pressure can be determined. Droplet or air bubble Soap bubble 10 Course Code / Title: AEE 242 / FLUID MECHANICS Instructor: Asst. Prof. M. Mollamahmutoglu Capillary Effect Another interesting consequence of surface tension is the capillary effect, which is the rise or fall of a liquid in a small-diameter tube inserted into the liquid. The curved free surface of a liquid in a capillary tube is called the meniscus. The strength of the capillary effect is quantified by the contact (or wetting) angle ϕ, defined as the angle that the tangent to the liquid surface makes with the solid surface at the point of contact. A liquid is said to wet the surface when ϕ < 90° and not to wet the surface when ϕ > 90°. The phenomenon of the capillary effect can be explained microscopically by considering cohesive forces (the forces between like molecules, such as water and water) and adhesive forces (the forces between unlike molecules, such as water and glass). The liquid molecules at the solid–liquid interface are subjected to both cohesive forces by other liquid molecules and adhesive forces by the molecules of the solid. The relative magnitudes of these forces determine whether a liquid wets a solid surface or not. The magnitude of the capillary rise in a circular tube can be determined from a force balance on the cylindrical liquid column: Capillary rise Note that R = constant 11