PetE 211 Introduction to Fluid Mechanics COURSE CONTENT Hizmete Ozel Scope of PetE 211 1. FUNDAMENTALS 2. FLUID STATICS TODAY 3. KINEMATICS OF FLUID MOTION 4. REYNOLDS TRANSPORT THEOREM /CONSERVATION LAWS / INCOMPRESSIBLE BERNOULLI EQUATION 5. FLOW OF A COMPRESSIBLE IDEAL FLUID LABS 6. FLOW OF A REAL FLUID 7. FLUID FLOW IN PIPES 8. TRANSPORTATION AND METERING OF FLUIDS Hizmete Ozel PetE 211 Introduction to Fluid Mechanics FUNDAMENTALS Hizmete Ozel CHAPTER 1: FUNDEMENTALS 1.1 Definition of Fluid Matter exists in three states in nature: MATTER Solid state Liquid state Gaseous state FLUID Hizmete Ozel The spacing & free motion of molecules in each state, are completely different from each other: the spacing & free motion of molecules gas > the spacing & Free motion of molecules > liquid the spacing & free motion of molecules solid Thus it follows that intermolecular cohesive forces are larger for a solid, smaller in a liquid, & exteremely small in a gas. These fundamental facts cause for the compactness & rigidity of the solids, the ability of liquid molecules to move freely within the liquid mass, capacity of gases to fill the containers in which they are placed, while a liquid has a definite volume & well-defined surface. Hizmete Ozel Some Characteristics of Gases, Liquids & Solids & the Microscopic Explanation for the Behavior GAS LIQUID SOLID takes the shape & volume of its container particles can move past one another takes the shape of the container which it occupies particles can move/slide past one another retains a fixed volume & shape rigid - particles locked into place compressible lots of free space between particles not easily compressible little free space between particles not easily compressible little free space between particles flows easily particles can move past one another flows easily particles can move/slide past one another do not flow rigid - particles cannot move/slide past one another Ref. → http://www.chem.purdue.edu/gchelp/liquids/character2.html Hizmete Ozel solid liquid gas Then what makes the liquid & gas to be combined in the same group as fluids? Hizmete Ozel It is their reaction to the applied shear stresses. A FLUID is a substance which deforms continuously (forever) under the application of a shearing stress, no matter how small the shearing stress is → even a very small shear stress results in motion in the fluid. A solid, on the other hand, will deform by an amount proportional to the stress applied, after which static equlibrium will result (i.e. assume a new stationary shape). Hizmete Ozel plate solid fluid = F/A t>0 at t = 0 plate F t=0 t=t’t=t’’ t=t’’’ 1 2 3 4 F t Hizmete Ozel t = = Rate of deformation lim t t →0 t Therefore t Since the deformation in fluids is continuous, we talk about the rate of deformation, rather than the deformation itself. Hizmete Ozel The fluids at rest cannot contain shearing stresses, This means that the forces in static fluids must be transmitted to solid boundaries or arbitrary sections normal to these boundaries, or sections at every point. In other words, if there exists shear stresses, then the fluid must be in motion. Arbitrary volume Hizmete Ozel 1.2. Scope of Fluid Mechanics Engineering Applications Utilize Encounter • Energy production thermal power plants hdyro-electric power • Energy Storage • Transportation of fluids, of material, as waterways • etc. • Pipelines • Groundwater flow • Hydrocarbons in porous structures • Wind forces on structures • Batteries, FuelCells • HVAC Systems • etc. Hizmete Ozel Weather & Climate Tornadoes Thunderstorm Global Climate Hurricanes Hizmete Ozel Vehicles Aircraft Surface ships High-speed rail Submarines Hizmete Ozel Environment Air pollution River hydraulics Hizmete Ozel Physiology & Medicine Blood pump Digital human lung Hizmete Ozel Sports & Recreation Water sports Auto racing Cycling Offshore racing Surfing Hizmete Ozel Water Supply http://img1.loadtr.com/b-404583-sevimli_kuzular.jpg Husbandary Agriculture http://www.tombaki.com/blog/wp-content/plugins/wp-o-matic/ cache/d969d_ziraatvehayvan_tarim_takvimi.jpg http://www.googleevdeneve.com/resim/termik_by.jpg Industry http://www.tnemec.com/resources/ project/392/mirimartank_1.jpg Domestic Hizmete Ozel Dams Atatürk Dam Karakaya Dam Hizmete Ozel Flood Control Hizmete Ozel Hydropower Plants http://science.howstuffworks.com/ Hizmete Ozel Energy Storage www.batteries18650.com/2011/11/probably-best-18650-li-ion-battery-in.html Hizmete Ozel Hydrogen Fuelcells https://www.power-technology.com/comment/standing-at-the-precipice-of-thehydrogen-economy/ Hizmete Ozel Especially as future petroleum engineers, you need to know fluid mechanics because: 1. Pertoleum and Natural gas are fluids -> You will work with fluids. 2. You have to know great deal about both liquid and gas forms of petroleum and their motion. Hizmete Ozel History: Faces of Fluid Mechanics Archimedes Newton (C. 287-212 BC) (1642-1727) Navier Stokes (1785-1836) (1819-1903) Leibniz Bernoulli Euler (1646-1716) (1667-1748) (1707-1783) Reynolds Prandtl Taylor (1842-1912) (1875-1953) (1886-1975) Hizmete Ozel 1.3. Concept of Continuum From Cengel Molecules are widely spaced in the gas phase. However, we can disregard the molecular nature of a substance. Instead we can 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. Hizmete Ozel CONCEPT OF CONTINUUM Actual molecular structure A hypothetical medium m → d = lim d Hizmete Ozel Fluid Particle A fluid particle is defined as the mass contained in the smallest fluid volume for which the continuum assumption is not violated. m = lim → d d Hizmete Ozel 1.4. Dimensions & System of Dimensions Dimensions & Units DESCRIBING PHYSICAL ENTITIES QUALITATIVE DESCRIPTION QUANTITATIVE DESCRIPTION DIMENSIONS UNITS Hizmete Ozel 1.5. Units & System of Units Primary Units Quantity Dimensions in FLT Dimensions in MLT SI units Mass FT2L-1 M kg Length L L m Time T T s Temperature C Derived Units F = ma Area L2 L2 m2 Velocity LT-1 LT-1 m/s Acceleration LT-2 LT-2 m/s2 Force F MLT-2 N Pressure FL-2 ML-1T-2 Pa Energy FL ML2T-2 Joule Power FLT-1 ML2T-3 Watt Angle 1 1 radian Dimensions & Units Dimensional Homogenity: An equation is said to be dimensionally homogenous if it does not depend on the system of units used. A dimensionally homogenous equation has therefore the same dimensions for each additive term on both sides of the equations. In practice this means that the numerical constants appearing in the equation are dimensionless. Example-1: h=(1/2)gt2 where h is the distance, g is the gravitational acceleration, t is time and ½ is a constant. 1 h = gt 2 2 1 (L) = (LT−2 )(T 2 ) 2 Since the constant ½ is dimensionless 1 L the equation is 2 = L = 1 dimensionally homogenous. Hizmete Ozel Dimensions & Units h= 1 2 gt 2 g = 9.81 m / s 2 h = 4.905 t 2 ( L) = 4.905(T 2 ) 4.905 = L2 1 T Not dimensionally homogenous! Hizmete Ozel 1.6. Physical Properties of Fluids A property is a characteristic of a substance in a particular state. Properties Extensive: depends on the amount of substance present, examples: volume, energy, weight, momentum Intensive: independent on the amount of substance, examples: specific values, namely mass per unit volume (density), energy per unit mass, etc. The intensive properties are the values which apply to a particle of a fluid. These are: Density Specific weight Specific Gravity Specific Volume Viscosity Surface Tension Vapor Pressure Compressibility Hizmete Ozel Density, Density, , of a substance is a measure of concentration of matter, and expresed as mass per unit volume. = m/, actually m = lim → 0 Where m is the mass of a substance contained in volume . =(P,T), where P=pressure, and T=temperature Unit in SI=? []=ML-3 and the unit in SI system is kg/m3 Hizmete Ozel Specific Weight, Specific Weight, , is the force due to gravity on the mass contained in a unit volume of a substance, i.e. weight per unit volume. = W/=Mg/= g Therefore: = g []=FL-3, and the unit in SI system is N/m3 Hizmete Ozel Specific Gravity, SG: The specific gravity is a term used to compare the density of a substance with that of water if the fluid is liquid, & with that of air or hydrogen if the fluid is gas. Thus: liq liq gas SG = = , or SG = w w air ( H ) 2 Since the density depends on the temperature & pressure, for precise values of specific gravity, temperature & pressure values must be specified. Hizmete Ozel Densities & Specific Weights of Some Fluids (g=9.81 m/s2) Liquids Fluid Gases G as es Temperature ( C ) Density (kg/m3) Specific Weight (N/m3) Water 4.0 1000. 9810. Mercury 20.0 13600. 133416. Gasoline 15.6 680. 6671. Alcohol 20.0 789. 7740. Air 15.0 1.23 12.0 Oxygen 20.0 1.33 13.0 Hydrogen 20.0 0.0838 0.822 Methane 20.0 0.667 6.54 Hizmete Ozel Viscosity Viscosity is the property of a fluid by virtue of which it offers resistance (friction) to the flow. Although the two fluids shown look alike (both are clear liquids and have a specific gravity of 1), they behave very differently when set into motion. The very viscous silicone oil is approximately 10,000 times more viscous than the water. Hizmete Ozel In study of fluid flow, viscosity requires the greatest consideration among the other properties of fluid. In order to understand the effect of viscosity, let us consider the motion of a fluid along a stationary solid boundary: Free surface y Flow A y1 u1 Solid boundary Hizmete Ozel Observations show that, while the fluid clearly has a finite velocity, u, at a finite distance from the boundary, the velocity is zero at the solid boundary. Thus velocity increases with the increasing distance from the stationary boundary. That is: if we measure the velocities at different distances from the boundary, we might obtain a picture as follows: y u1+Δu y2 Δy y1 u1 at at y=0 u=0 and y=y1 u=u1 This is called a velocity profile Hizmete Ozel Example: As a fluid flows near a solid surface, it "sticks" to the surface, i.e., the fluid matches the velocity of the surface. This so-called "no-slip" condition is a very important one that must be satisfied in any accurate analysis of fluid flow phenomena. Hizmete Ozel Let us consider two such layers, the lower layer is moving with velocity u1, and the upper layer with (u1+Δu). u1+Δu 2 Δy 1 u1 2’ 1’ Two particles (1) and (2), starting on the same vertical line, will move to different distances during the time interval Δt: S2=(u1+Δu).Δt S1=u1.Δt Thus, the fluid is distorted or sheared, as the line connecting (1) and (2) acquires an increasing slope & length as time t increases. Hizmete Ozel In other words, the faster moving fluid tries to speed up the slower moving layer, while the slower moving layer tries to reduce the speed of faster one. Therefore, it is evident that a frictional or shearing force must exist between the fluid layers. This shearing force may be expressed as a shear stress (shearing force per unit contact area). The shear stress is directly proportional to the velocity difference & inversely proportional to the distance between two layers, that is: u y u at the limit: = y Hizmete Ozel The constant of proportionality is called the dynamic viscosity & it is a property of the fluid. It shows the effect of different fluids. Ex: honey shows more resistance to flow than water. Therefore honey must have a higher . u NEWTON’S LAW = y OF VISCOSITY Hizmete Ozel All real fluids posses viscosity & therefore exhibit certain frictional resistance when moving. The term, u/y, is called the velocity gradient & it shows the angular velocity of the line ab & hence equal to the rate of deformation. u+Δu b Δy a Δs Δ u a’ b’ S = y = u t u Therefore : = y t Remember that : u = t y Hizmete Ozel Newton’s Law of viscosity states that the shear stress at any point is proprotional to the velocity gradient at that point: y u 2 = y y = y2 2 y2 Δy Δu y1 1 u 1 = y y = y1 Δy Δu u Hizmete Ozel The magnidute of the shear stress on the solid boundary is called the wall-shear stress. That is: u w = y y=0 If we integrate the wall-shear stress over the area on which it is acting, we obtain the total frictional resistance (force) on the fluid due to solid boundary. On the other hand, the reaction force, that is the force which has the same magnitude but opposite in direction to the frictional resistance is called the drag force. That is: The force exerted by the flowing fluid on a solid boundary in the direction of flow is called the DRAG FORCE. Hizmete Ozel A typical variation of shear stress y U u(y) (y) Δy Δu u(y) 1 2 w Shear stress from 1 on 2 w Shear stress from 2 on 1 Wall shear stress Frictional drag force Fd = w dA du w = dy y =0 The magnitude of the shear stress on a solid boundary is called as the wall shear stress. Hizmete Ozel Viscosity & Drag Force Viscosity is a property that represents the internal resistance of a fluid to motion. A flowing fluid exerts a force on a body in the flow direction is called the drag force & the magnitude of this force depends, in part, on viscosity. Hizmete Ozel Note that if a fluid is at rest, or in motion so that no layer moves relative to an adjacent layer, there will be NO u SHEAR STRESS, regardless the viscosity. Because: =0 y On the other hand, if the viscosity of the fluid is zero, regardless of the motion, no shear stress can develop. Such a fluid is called INVISCID FLUID. All the fluids in nature have a viscosity & are called REAL FLUIDS. Inviscid fluid is only an idealization. It simplifies the solution of many real fluid flow problem. Not all the real fluids obey the Newton’s Law of viscosity. If we plot shear stress , versus the velocity gradient ( or the rate of angular deformation) for various fluids, the plot looks like: Hizmete Ozel Newtonian & Non Newtonian Fluids e.g: sewage sludge, toothpaste, and jellies. Bingham Plastic n<1 n=1 Newtonian fluids n1 Non-Newtonian fluids n>1 Shear thickening (Dilatant) n<1 Shear thinning (Pseuda plastic) e.g: clay, milk, and cement n>1 ap e.g: quicksand, corn starch ap = 1 = 0 ideal fluid du = dy du dy n du dy Hizmete Ozel Examples on Non Newtonian Fluids Example: A mixture of water and corn starch, when placed on a flat surface, flows as a thick, viscous fluid. However, when the mixture is rapidly disturbed, it appears to fracture and behave more like a solid. The mixture is a non-Newtonian shear thickening fluid which becomes more viscous as the shearing rate is suddenly increased through the rapid action of the spoon. Hizmete Ozel Examples on Non Newtonian Fluids Bingham plastic: resist a small shear stress but flow easily under large shear stresses, e.g. sewage sludge, toothpaste, and jellies. Pseudo plastic: most non-Newtonian fluids fall under this group. Viscosity decreases with increasing velocity gradient, e.g. colloidal substances like clay, milk, and cement. Dilatants: viscosity increases with increasing velocity gradient, e.g. quicksand, cornstarch. Hizmete Ozel Dynamic & Kinematic Viscosity The proportionality constant is known as dynamic viscosity of the fluid. FL−2 −2 −1 −1 = = = FL T = ML T −1 du LT dy L Viscosity can be made independent of fluid density; kinematic viscosity is defined as the ratio: = ML−1T −1 2 −1 = = = L T −3 ML Hizmete Ozel Viscosities of air & water Fluid Temperature (C) (Ns/m2) (m2/s) Water 20 1.00E-03 1.01E-06 Air 20 1.80E-05 1.51E-05 Hizmete Ozel Effect of Temperature on Viscosity Gases: As temperature increases the viscosity increases. Because, the viscosity arises due to momentum exchange between the randomly moving molecules in gases. As temperature increases, the molecular activity will increase in gases, hence more momentum exchange will take place, & as a result viscosity will increase. Liquids: As temperature increases the viscosity decreases. Because, viscosity arises due to intermolecular cohesive forces. In liquids, as temperature increases, these forces will decrease, hence the viscosity will decrease. Hizmete Ozel Dynamic Viscosity For gases: μ as T For liquids: μ as T Hizmete Ozel Compressibility, K, andBulk Modulus of Elasticity, Ev Compresibility is a measure of change of volume & density when a substance is subject to normal pressure. Compressibility = % change in volume (or density) for a given pressure change: d 1 d / K=− =+ dP dP The (-) sign indicates a decrease in volume, with an increase in pressure. Hizmete Ozel The inverse of compressibility is known as the bulk modulus of elasticity: dP dP Ev = − = d / d / A 0 Bulk Modulus of Elasticity F Since fluids do not posses a rigid form, the modulus of elasticity must be defined on the basis of volume, & it is called BULK MODULUS. Hizmete Ozel Consider a cylinder full of a fluid which has a volume of 0. Application of a force F to the piston will the pressure (P=F/A) in the fluid & cause the volume to The slope of P vs /0 will give the bulk modulus of elasticity at that point. A 0 P F dP d/0 Ev 1 1 /0 Hizmete Ozel The steeping of the curve with increasing pressure shows that as fluids are compressed, they become increasingly difficult to compress further, which is logical consequence of reducing the space between the molecules. P The bulk modulus of elasticity of a fluid is not constant, but increases with pressure. dP E v → → fluid is d/0 incompressible Ev 1 1 /0 Hizmete Ozel (Ev )air = 1.42x105 Pa (Ev )water = 2.15x109 Pa (Ev )steel = 2.06 x1011 Pa Therefore: Water is 100 times more compressible than steel. Air is 20 000 times more compressible than water. For all practical purposes, water & all liquids may be assumed as incompressible unless very large pressure ranges are involved. Hizmete Ozel Surface Tension , Example: A heavier-than-water, double-edged steel razor blade can float on water. Without surface tension, the blade would sink because its weight is greater than its buoyant force. However, surface tension forces are not large enough to support a slightly heavier single edged blade. Hizmete Ozel Surface Tension , Surface tension is a property that results from the attractive forces between molecules. Intermolecular Attraction Forces http://www.tutorvista.com/physics/fluid-surface-tension Cohesive Forces (C) Liquid to liquid Gas to gas A>C Gas Liquid Solid Adhesive Forces (A) Liquid to solid Gas to liquid The intensity of the molecular attraction per unit length along any line on an interface is called the surface tension. = FL−1 Hizmete Ozel Although such forces are negligible in practice, they become important in capillary rise of liquids in narrow tubes, formation of liquid drops, etc. Consider a free liquid surface in contact with the atmosphere: A>C Gas molecules attract liquid molecules at free surface Liquid molecules attract liquid molecules at free surface Solid Little force attracts molecules away from the liquid because few molecules are present in the gas above the surface. Within the bulk liquid, the intermolecular attraction & repulsion forces are balanced in all directions. http://www.tutorvista.com/physics/fluid-surface-te Hizmete Ozel If Adhesive forces > Cohesive forces → The liquid wets the solid surface. Water rises up A>C Liquid (H2O) Solid If Cohesive forces > Adhesive forces → Then the solid surface is not wetted. Mercury depressed C>A Liquid (Hg) Solid Hizmete Ozel Different shapes of drops on a solid. The surface tensions, σ values, associated with each interaction interface are shown by red arrows. The contact angle θ Between the liquid drop and the solid surface is also shown. https://www.tau.ac.il/~phchlab/exp-tension-theory.html A hydrophilic surface is as a surface where 0°<θ<90 A hydrophobic surface is a surface where θ≥90° Hizmete Ozel Capillary Effects A>C h Capillary effect is the rise or fall of a liquid in a smalldiameter tube. The curved free surface in the tube is call the meniscus. Force balance can describe magnitude of capillary rise. Hizmete Ozel Wetting and Non-wetting Fluids C>A A>C h h Capillary rise (wetting fluid) Capillary drop (non-wetting fluid) The level h, that the water or mercury can rise or drop can be computed as follows: → Hizmete Ozel The force pulling the free surface up in the capillary tube is 2pRcos. This force will be balanced by the weight of water that rises in the capillary tube, which is pR2h. Therefore: 2pRcos= pR2h from here solving for h → 2 cos h= R Hizmete Ozel Vapor pressure, Pv Vapor P Water The pressure at which liquid boils is called the vapor pressure, Pv . Heat Boiling occurs when Pgas above Pvapor of liquid Pv=Pv(T) as T Pv where T is temperature Hizmete Ozel For boiling to occur, the equilibrium must be set either by 1. Raising the temperature to cause the vapor pressure to equal or exceed the total pressure applied at the free surface, or 2. By lowering the total pressure at the free surface until it is equal to or less than the vapor pressure. The more volatile the liquid, the higher its vapor pressure. http://media.photobucket.com/image/boiling/ whitetiger_2009/etc-frog-boiling.jpg Hizmete Ozel Vapor pressure of water Temperature C Pv (kPa) 0 0.61 10 1.23 25 3.17 60 19.92 100 101.33=Patm Hizmete Ozel Engineering importance of vapor pressure If P drops below Pv, liquid is locally vaporized, creating cavities of vapor. Vapor cavities collapse when local P rises above Pv. 3 1 2 P3 Vapor pockets P1>Pv P2Pv P3>Pv Hizmete Ozel Cavitation Damage Serious damage can result due to the impact of collapsing bubbles. Cavitation can affect the performance of hydraulic machinery such as pumps, turbines & propellers. http://getenecon.com/Portals/0/images/800px-Cavitation_Propeller_Damage.jpg Hizmete Ozel
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