# Lecturer, Department t of Mechanical Engineering, SVMIT, Bharuch

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1 Fluid Mechanics By Ashish J. Modi Lecturer, Department t of Mechanical Engineering, i SVMIT, Bharuch

2 Review of fundamentals Properties of Fluids

3 Introduction Any characteristic of a system is called a property. Familiar: pressure P, temperature T, volume V, and mass m. Less familiar: viscosity, thermal conductivity, modulus of elasticity, thermal expansion coefficient, vapor pressure, surface tension. Intensive properties are independent of the mass of the system. Examples: temperature, pressure, and density. Extensive properties are those whose value depends on the size of the system. Examples: Total mass, total volume, and total momentum. Extensive properties per unit mass are called specific properties. Examples include specific volume v = V/m and specific total energy e=e/m Fluid Mechanics 3

4 Continuum DofO = molecule 3x10 m mass of O 2 = 5.3x10-26 kg Mean free path = 6.3x10-8 m at 1 atm pressure and 20 C Atoms are widely spaced in the gas phase. However, we can disregard the atomic nature of a substance. View it as a continuous, homogeneous matter with no holes, that is, a continuum. This allows us to treat properties p as smoothly varying quantities. Continuum is valid as long as size of the system is large in comparison to distance between molecules. In this text we limit our consideration to substances that can be modeled as a continuum Fluid Mechanics 4

5 Density and Specific Gravity Density is defined as the mass per unit volume ρ = m/v. Density has units of kg/m 3 Specific volume is defined as v = 1/ρ = V/m. For a gas, density depends d on temperature t 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=ρ/ρ H 20. SG is a dimensionless quantity. The specific weight is defined as the weight per unit volume, i.e., γ s = ρg where g is the gravitational acceleration. γ s has units of N/m Fluid Mechanics 5

6 Density and Specific Gravity Fluid Mechanics 6

7 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 = ρ R T where P is the absolute pressure, v is the specific volume, T is the thermodynamic (absolute) temperature, ρ is the density, and R is the gas constant Fluid Mechanics 7

8 Density of Ideal Gases The gas constant R is different for each gas and is determined from R = R u /M, where R u is the universal gas constant t whose value is R u = kj/kmol K = Btu/lbmol R, and M is the molar mass (also called molecular weight) ) of the gas. The values of R and M for several substances are given in Table A Fluid Mechanics 8

9 Density of Ideal Gases The thermodynamic temperature scale In the SI is the Kelvin scale, designated by K. In the English system, it is the Rankine scale, and the temperature unit on this scale is the rankine, R. Various temperature scales are related to each other by (2 5) (2 6) It is common practice to round the constants and to 273 and 460, respectively Fluid Mechanics 9

10 Density of Ideal Gases 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 PV = mrt or PV = NR u T. For a fixed mass m, writing the ideal-gas relation twice and simplifying, i the properties of an ideal gas at two different states are related to each other by P 1 V 1 /T 1 = P 2 V 2 /T Fluid Mechanics 10

11 Density of Ideal Gases An ideal gas is a hypothetical substance that obeys the relation Pv = RT. Ideal-gas equation holds for most gases. However, dense gases such as water vapor and refrigerant vapor should not be treated as ideal gases. Tables should be consulted for their properties, e.g., Tables A-3E through A-6E in textbook. (Refer. Cengel & Cimbala) Fluid Mechanics 11

12 Vapor Pressure and Cavitation Pressure temperature relation at (liquid solid) phase change At a given pressure, the temperature at which a pure substance changes phase is called the saturation ti temperature T sat. Likewise, at a given temperature, the pressure at which a pure substance changes phase is called the saturation pressure P sat. At an absolute pressure of 1 standard atmosphere (1 atm or kpa), for example, the saturation temperature of water is 100 C. Conversely, at a temperature of 100 C C, the saturation pressure of water is 1 atm Fluid Mechanics 12

13 Vapor Pressure and Cavitation Water boils at 134 C in a pressure cooker operating at 3 atm absolute pressure, but it boils at 93 C in an ordinary ypan at a 2000-m elevation, where the atmospheric pressure is 0.8 atm. The saturation (or vapor) pressures are given in Appendices 1 and 2 for various substances Fluid Mechanics 13

14 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 Partial pressure is defined d as the pressure of a gas or vapor in a mixture with other gases. If P drops below P v, liquid id is locally ll 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 Fluid Mechanics 14

15 Energy and Specific Heats Total energy E (or e on a unit mass basis) 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) Fluid Mechanics 15

16 Energy and Specific Heats Microscopic energy Internal energy U (or u on a unit mass basis) is for a nonflowing fluid and is due to molecular l activity. it Enthalpy h=u+pv is for a flowing fluid and includes flow energy (Pv). where Pv is the flow energy, also called the flow work, which is the energy per unit mass needed to move the fluid and maintain flow. Note that enthalpy is a quantity per unit mass, and thus it is a specific property. p Fluid Mechanics 16

17 Energy and Specific Heats Macroscopic energy Kinetic energy ke=v 2 /2 Potential energy pe=gz 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 internal, kinetic, and potential energies. On a unit-mass basis, it is expressed as e =u + ke + pe. The fluid entering or leaving a control volume possesses an additional form of energy the flow energy P/ρ. Then the total energy of a flowing fluid on a unit-mass basis becomes e flowing = P/ρ + e =h + ke + pe = h +V 2 /2+gz Fluid Mechanics 17

18 Energy and Specific Heats By using the 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. In fact, this is the main reason for defining the property enthalpy. The changes of internal energy and enthalpy of an ideal gas are expressed as du=c v dt and dh=c p dt where c v and c p are the constant-volume and constantpressure specific heats of the ideal gas. For incompressible substances, c v and c p are identical Fluid Mechanics 18

19 Coefficient of Compressibility How does fluid volume change with P and T? Fluids expand as T or P Fluids contract as T or P Fluid Mechanics 19

20 Coefficient of Compressibility Need fluid properties that relate volume changes to changes in P and T. Coefficient of compressibility P P κ = v = ρ v ρ T T (or bulk modulus of compressibility or bulk modulus of elasticity) k must have the dimension of pressure (Pa or psi). What is the coefficient of compressibility of a truly incompressible substance?(v=constant). is infinity Fluid Mechanics 20

21 Coefficient of Compressibility A large κ implies incompressible. This is typical for liquids considered to be incompressible. For example, the pressure of water at normal atmospheric conditions must be raised to 210 atm to compress it 1 percent, corresponding to a coefficient of compressibility value of κ = 21,000 atm Fluid Mechanics 21

22 Coefficient of Compressibility Small density changes in liquids can still cause interesting phenomena in piping systems such as the water hammer characterized by a sound dthat tresembles the sound produced d 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 produced strike the pipe surfaces, bends, and valves as they ypropagate p and reflect along the pipe, causing the pipe to vibrate and produce the familiar sound Fluid Mechanics 22

23 Coefficient of Compressibility Differentiating ρ = 1/v gives dρ = - dv/v 2 ; therefore, dρ/ρ = - dv/v For an ideal gas, P = ρrt and ( P/ ρ) T =RT =P/ρ, and thus κ ideal gas = P (Pa) The inverse of the coefficient of compressibility is called the isothermal compressibility α and is expressed as Fluid Mechanics 23

24 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 Fluid Mechanics 24

25 Viscosity To obtain a relation for viscosity, consider a fluid layer between two very large parallel l plates separated by a distance l Definition of shear stress is τ = 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 Fluid Mechanics 25

26 Viscosity dβ tan dβ = da/ l = Vdt/l = (du/dy)dt Rearranging du/dy= dβ/dt τ dβ/dt or τ du/dy Fluids for which the rate of deformation is proportional to the shear stress are called Newtonian fluids, such as water, air, gasoline, and oils. Blood and liquid plastics are examples of non-newtonian fluids. In one-dimensional flow, shear stress for Newtonian fluid: τ = μdu/dy μ is the dynamic viscosity and has units of kg/m s, Pa s, or poise. kinematic viscosity ν = μ/ρ. Two units of kinematic viscosity are m 2 /s and stoke. 1 stoke = 1 cm 2 /s = m 2 /s Fluid Mechanics 26

27 Viscosity Non-Newtonian vs. Newtonian Fluid Fluid Mechanics 27

28 Viscosity Gas vs. Liquid Fluid Mechanics 28

29 Viscometry How is viscosity measured? A rotating viscometer. Two concentric cylinders with a fluid in the small gap l. Inner cylinder is rotating, outer one is fixed. Use definition of shear force: du F = τ A= μ A dy If l/r << 1, then cylinders can be modeled as flat plates. Torque T = FR, and tangential velocity V=ωR Wetted surface area A=2πRL. Measure T and ω to compute μ Fluid Mechanics 29

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