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Chapter 2

# ENB221 Fluid Mechanics Chapter 2 Notes.docx

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Course
ENB205
Professor
Negareh Ghasemi
Semester
Spring

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Chapter 2 Notes Fluid Mechanics, 1 Edition Tom Barwell Properties of Fluids Intensive properties are those that are independent of the mass of a system i.e. temperature, pressure and density. Extensive properties are those whose values depend on the size of the system. Generally uppercase letters are used to denote extensive properties and lowercase letters are used for intensive properties. Extensive properties per unit mass are called specific properties. 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. A continuum is a continuous, homogeneous matter with no holes. The continuum idealization allows us to treat properties as point functions and to assume 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. Density and specific gravity Density is defined as mass per unit volume: The reciprocal of density is the specific volume which is defined as volume per unit mass. The density of a substance depends on temperature and pressure. The density of most gases is proportional to pressure and inversely proportional to temperature. Liquids and solids are essentially incompressible substances and the variation of their density with pressure is usually negligible. 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. That is The specific gravity of a substance is a dimensionless quantity. However in SI units the numerical value of the specific gravity of a substance is exactly equal to its density. Substances with specific gravities less than 1 are lighter than water, and thus will float on water. The weight of a unit volume of a substance is called specific weight and is expressed as: (N/m ) where G is gravitational acceleration. Recall that densities of liquids are essentially constant and thus they can often be approximated as being incompressible substances during most processes. Density of Ideal Gases Any equation that relates the pressure, temperature and density (or specific volume) of a substance is called an equation of state. The simplest is the ideal-gas equation of state: Where P is absolute pressure, v is specific volume, T is thermodynamic temperature, is density and R is gas constant. The gas constant R is different for each gas and is determined from Where R iU the universal gas constant which equals 8.314kJ/kmol K and M is the molar (molecular) mass. P a g e | 1 Chapter 2 Notes Fluid Mechanics, 1 Edition Tom Barwell In SI the thermodynamic temperature scale is the Kelvin Scale (K) and the English system uses the Rankine scale (R) Any gas that obeys the ideal-gas equation of state (or ideal-gas relation) is called an ideal gas. For an ideal gas of Volume V, mass m and number of moles N =m/M the ideal gas equation of state can also be written as: For a fixed mass m, the properties of an ideal gas at two different states are related to each other by: An ideal gas is a hypothetical substance that obeys the ideal gas relation; however it has been experimentally observed that it closely approximates the P-v-T behaviour of real gases at low densities. At low pressures and high temperatures the density of a gas decreases and the gas behaves much like an ideal gas. Some familiar gases such as air, nitrogen, oxygen, hydrogen, helium, argon, neon and krypton and even heavier gases such as carbon dioxide can be treated as ideal gases with neglible error. Dense gases such as water vapour and refrigerant vapour however should not be treated as ideal gases since they usually exist at a state near saturation. Vapor Pressure and Cavitation Temperature and pressure are dependent properties for pure substances during phase change processes, and there is one-to-one correspondence between temperatures and pressures. At a given pressure the temperature at which a pure substance changes phase is called the saturation temperature T . Similiarly the pressure at which a pure sat substance changes phase is called the saturation pressure P . Thsatapor pressure P of a puve substance is defined as the pressure exerted by its vapour in phase equilibrium with its liquid at a given temperature. Vapor pressure is a property of the pure substance and turns out to be identical the to the saturation pressure of the liquid (P =P ). V sat Partial pressure is defined as the pressure of a gas or vapour in a mixture with other gases. The partial pressure of a vapour must be less than or equal to the vapour pressure if there is no liquid present. However, when both vapour and liquid are present and the system is in phase equilibrium the partial pressure of the vapour must equal the vapour pressure and the system is said to be saturated. For phase-change processes between the liquid and vapour phases of a pure substance, the saturation pressure and the vapour pressure are equivalent since the vapour is pure. Note that the pressure would be the same whether it is measured in the vapour or liquid phase, provided that it is measured at a location close to the liquid-vapor interface to avoid hydrostatic effects). Vapor pressure increases with temperature. The reason for our interest in vapour pressure is the possibility of the liquid pressure in liquid-flow systems dropping below the vapour pressure at some locations, and the resulting unplanned vaporization. For example, water at 10°C will flash into vapour and form bubbles at locations (such as the tip of impellers or suction sides of pumps) where the pressure drops below 1.23kPa. The vapour bubbles (Cavitation bubbles since they create cavities in the liquid) collapse as they are swept away from the low-pressure regions, generating highly destructive, extremely high pressure waves. This phenomenon is called cavitation and it is an important consideration in the design of hydraulic turbines and pumps. Cavitation must be avoided or at least minimized in flow systems since it reduces performance, generates annoying vibrations and noise and causes damage to equipment. It can be characterised by a tumbling sound. P a g e | 2 Chapter 2 Notes Fluid Mechanics, 1 Edition Tom Barwell Energy and specific heats Total Energy E The summation of all other forms of energy within a given system Internal Energy U The sum of all microscopic forms of energy of a system Kinetic Energy Ke The energy that a system possesses as a result of its motion relative to some reference frame, when all parts of the system move with the same velocity Ke = V /2 2 Potential Energy Pe The energy that a system possesses as a result of its elevation in a gravitational field can be expressed as pe = gz (g = Gravity, z = elevation above an arbitrary axis) Thermal Energy The sensible and latent forms of internal energy (heat) simililarly known as thermal energy The international unit of energy is the Joule or kilojoule. In the analysis of systems that involve fluid flow we frequently encounter the combination of properties u and Pv. For convenience, this combination is called Enthalpy, h. where ( ). Which is the energy per unit mass needed to move the fluid and maintain flow. In the energy analysis of flowing fluids, it is convenient to treat the flow energy as part of the energy of the fluid and to represent the microscropic energy of a fluid stream by enthalpy. Enthalpy is a quantity per unit mass, and thus it is a specific property. In the absence of magnetic, electric and surface tension a system is called a simple compressible system. The total energy of a simple compressible system consists of three parts: Internal, kinetic and potential energies. Then the total energy of a flowing fluid on a unit-mass basis becomes. (kJ/kg) where h = is the enthalpy. V is velocity, z is elevation. By using enthalpy instead of the internal energy to represent the energy of a flowing fluid, one does not need to be concerned about the flow work. The energy associated with pushing the fluid is automatically taken care of by enthalpy. The differential and finite changes in the internal energy and enthalpy of an ideal gas can be expressed in terms of the specific heats as and Where C avd C arp the constant-volume and constant pressure specific heats of the ideal gas. Using specific heat values at the average temperature, the finite changes in internal energy and enthalpy can be expressed approximately as and For liquids and are identical therefore and the change in the internal energy of liquids can be expressed as . Enthalpy change can be written as Therefore for constant pressure processes and for constant-temperature processes of liquids. P a g e | 3 Chapter 2 Notes Fluid Mechanics, 1 Edition Tom Barwell Coefficient of compressibility To determine the amount of volume change you need to define properties that relate volume changes to the changes in pressure and temperature. Two such properties are the bulk modulus of elasticity κ and the coefficient of volume expansion β. It is known that fluids act like solids with respect to pressure. Therefore the coefficient of compressibility κ (also called the bulk modulus of compressibility or bulk modulus of elasticity) for fluids is: ( ) ( ) ( ) In terms of finite changes it can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) ( )are dimensionless, must have the dimension of pressure. Also, 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. Then it follows that the coefficient of compressibility of a truly incompressible substance (v = constant) is infinity. A large coefficient of compressibility ( ) indicates that a large change in pressure is needed to cause a small fractional change in volume, and thus a fluid with a laris essentially incompressible. Note that volume and pressure are inversely proportional. Also differentiating pressure gives that Is, the fractional changes in the specific volume and the d
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