Fluid Statics Fluid Mechanics Overview Fluid Mechanics Gas Liquids Statics Dynamics , Flows Water, Oils, Alcohols, etc. Stability Air, He, Ar, N2, etc. Buoyancy Pressure Compressible/ Incompressible Laminar/ Turbulent Surface Tension Steady/Unsteady Compressibility Density Viscosity Vapor Pressure Viscous/Inviscid Fluid Dynamics Introduction
Characteristics of Fluids • Gas or liquid state • “Large” molecular spacing relative to a solid • “Weak” intermolecular cohesive forces • Can not resist a shear stress in a stationary state • Will take the shape of its container • Generally considered a continuum • Viscosity distinguishes different types of fluids
Measures of Fluid Mass and Weight: Density The density of a fluid is defined as mass per unit volume. m = mass, and v = volume. • Different fluids can vary greatly in density • Liquids densities do not vary much with pressure and temperature • Gas densities can vary quite a bit with pressure and temperature • Density of water at 4° C : 1000 kg/m3 • Density of Air at 4° C : 1.20 kg/m3 Alternatively, Specific Volume:
Measures of Fluid Mass and Weight: Specific Weight The specific weight of fluid is its weight per unit volume. g = local acceleration of gravity, 9.807 m/s2 • Specific weight characterizes the weight of the fluid system • Specific weight of water at 4° C : 9.80 kN/m3 • Specific weight of air at 4° C : 11.9 N/m3
Measures of Fluid Mass and Weight: Specific Gravity The specific gravity of fluid is the ratio of the density of the fluid to the density of water @ 4° C. • Gases have low specific gravities • A liquid such as Mercury has a high specific gravity, 13.2 • The ratio is unitless. • Density of water at 4° C : 1000 kg/m3
Viscosity: Kinematic Viscosity • Kinematic viscosity is another way of representing viscosity • Used in the flow equations • The units are of L2/T or m2/s and ft2/s
Surface Tension At the interface between a liquid and a gas or two immiscible liquids, forces develop forming an analogous “skin” or “membrane” stretched over the fluid mass which can support weight. This “skin” is due to an imbalance of cohesive forces. The interior of the fluid is in balance as molecules of the like fluid are attracting each other while on the interface there is a net inward pulling force. Surface tension is the intensity of the molecular attraction per unit length along any line in the surface. Surface tension is a property of the liquid type, the temperature, and the other fluid at the interface. This membrane can be “broken” with a surfactant which reduces the surface tension.
Applied to Circumference Applied to Area Surface Tension: Liquid Drop The pressure inside a drop of fluid can be calculated using a free-body diagram: Real Fluid Drops Mathematical Model R is the radius of the droplet, s is the surface tension, Dp is the pressure difference between the inside and outside pressure. The force developed around the edge due to surface tension along the line: This force is balanced by the pressure difference Dp:
Surface Tension: Liquid Drop Now, equating the Surface Tension Force to the Pressure Force, we can estimate Dp = pi – pe: This indicates that the internal pressure in the droplet is greater that the external pressure since the right hand side is entirely positive.
Adhesion Cohesion Adhesion Cohesion Surface Tension: Capillary Action Capillary action in small tubes which involve a liquid-gas-solid interface is caused by surface tension. The fluid is either drawn up the tube or pushed down. “Wetted” “Non-Wetted” Cohesion > Adhesion Adhesion > Cohesion h is the height, R is the radius of the tube, q is the angle of contact. The weight of the fluid is balanced with the vertical force caused by surface tension.
Surface Tension: Capillary Action Free Body Diagram for Capillary Action for a Wetted Surface: Equating the two and solving for h: For clean glass in contact with water, q 0°, and thus as R decreases, h increases, giving a higher rise. For a clean glass in contact with Mercury, q 130°, and thus h is negative or there is a push down of the fluid.
F Pressure = A Pressure Pressure is the force on an object that is spread over a surface area. The equation for pressure is the force divided by the area where the force is applied. Although this measurement is straightforward when a solid is pushing on a solid, the case of a solid pushing on a liquid or gas requires that the fluid be confined in a container. The force can also be created by the weight of an object. Unit of pressure is Pa
Force Equilibrium of a Fluid Element • Fluid static is a term that is referred to the state of a fluid where its velocity is zero and this condition is also called hydrostatic. • So, in fluid static, which is the state of fluid in which the shear stress is zero throughout thefluid volume. • In a stationary fluid, the most important variable is pressure. • For any fluid, the pressure is the same regardless its direction. As long as there is no shear stress, the pressure is independent of direction. This statement is known as Pascal’s law
Fluid surfaces Figure 2.1 Pressure acting uniformly in all directions Figure 2.2: Direction of fluid pressures on boundaries Force Equilibrium of a Fluid Element
P-vapor P P atm atm Standard Atmosphere 1 atm = 101325 Pa = 760 mmHg = 760 torr = 1 bar h = 76 cm Mercury Mercury barometer
P P P P P = + abs gages atm gages atm
A pressure is quoted in its gauge value, it usually refers to a standard atmospheric pressure p0. A standard atmosphere is an idealised representation of mean conditions in the earth’s atmosphere. • Pressure can be read in two different ways; the first is to quote the value in form of absolute pressure, and the second to quote relative to the local atmospheric pressure as reference. • The relationship between the absolute pressure and the gauge pressure is illustrated in Figure 2.6.
The pressure quoted by the latter approach (relative to the local atmospheric pressure) is called gauge pressure, which indicates the ‘sensible’ pressure since this is the amount of pressure experienced by our senses or sensed by many pressure transducers. • If the gauge pressure is negative, it usually represent suction or partially vacuum. The condition of absolute vacuum is reached when only the pressure reduces to absolute zero.
Pressure Measurement Based on the principle of hydrostatic pressure distribution, we can develop an apparatus that can measure pressure through a column of fluid (Fig. 2.7)
Pressure Measurement We can calculate the pressure at the bottom surface which has to withstand the weight of four fluid columns as well as the atmospheric pressure, or any additional pressure, at the free surface. Thus, to find p5, Total fluid columns =(p2 – p1) + (p3 – p2) + (p4 – p3) + (p5 – p4) p5 – p1 =og (h2 – h1) + wg (h3 – h2) +gg (h4 – h3) + mg (h5 – h4) The p1 can be the atmospheric pressure p0 if the free surface at z1 is exposed to atmosphere. Hence, for this case, if we want the value in gauge pressure (taking p1=p0=0), the formula for p5 becomes p5 =og (h2 – h1) + wg (h3 – h2) + gg (h4 – h3) + mg (h5 – h4) The apparatus which can measure the atmospheric pressure is called barometer (Fig 2.8).
For mercury (or Hg — the chemical symbol for mercury), the height formed is 760 mm and for water 10.3 m. patm =760 mm Hg (abs) = 10.3 m water (abs) By comparing point A and point B, the atmospheric pressure in the SI unit, Pascal, pB =pA+ gh pacm =pv+ gh = 0.1586 + 13550 (9.807)(0.760) 101 kPa
This concept can be extended to general pressure measurement using an apparatus known as manometer. Several common manometers are given in Fig. 2.9. The simplest type of manometer is the piezometer tube, which is also known as ‘open’ manometer as shown in Fig. 2.9(a). For this apparatus, the pressure in bulb A can be calculated as: pA =p1+ p0 = 1gh1 + p0 • Here, p0 is the atmospheric pressure. If a known local atmospheric pressure value is used for p0, the reading for pA is in absolute pressure. If only the gauge pressure is required, then p0 can be taken as zero.
Although this apparatus (Piezometer) is simple, it has limitations, i.e. It cannot measure suction pressure which is lower than the atmospheric pressure, The pressure measured is limited by available column height, It can only deal with liquids, not gases. The restriction possessed by the piezometer tube can be overcome by the U-tube manometer, as shown in Fig. 2.9(b). The U-tube manometer is also an open manometer and the pressure pA can be calculated as followed: p2 =p3 pA +1gh1= 2gh2 + p0 pA =2gh2- 1gh1 + p0
If fluid 1 is gas, further simplification can be made since it can be assumed that 12, thus the term 1gh1 is relatively very small compared to 2gh2 and can be omitted with negligible error. Hence, the gas pressure is: pA p2 =2gh2- p0 There is also a ‘closed’ type of manometer as shown in Fig. 2.9(c), which can measure pressure difference between two points, A and B. This apparatus is known as the differential U-tube manometer. For this case, the formula for pressure difference can be derived as followed: p2 =p3 pA +1gh1= pB + 3gh3 + 2gh2 pA - pB =3gh3+ 2gh2 - 1gh1
P P P = + abs gages atm Piezometer tube Open h
U-tube manometer Pressure is defined as a force per unit area - and the most accurate way to measure low air pressure is to balance a column of liquid of known weight against it and measure the height of the liquid column so balanced. The units of measure commonly used are inches of mercury (in. Hg), using mercury as the fluid and inches of water (in. w.c.), using water or oil as the fluid
Example • An underground gasoline tank is accidentally opened during raining causing the water to seep in and occupying the bottom part of the tank as shown in Fig. E2.1. If the specific gravity for gasoline 0.68, calculate the gauge pressure at the interface of the gasoline and water and at the bottom of the tank. Express the pressure in Pascal and as a pressure head in metres of water. Use water = 998 kg/m3 and g = 9.81 m/s2.
For gasoline: g = 0.68(998) = 678.64kg/m3 At the free surface, take the atmospheric pressure to be zero, or p0 = 0 (gauge pressure). p1 = p0 + pgghg = 0 + (678.64)(9.81)(5.5) = 36616.02 N/m2 = 36.6 kPa The pressure head in metres of water is: h1 = p1 – p0 = 36616.02 - 0 pwg (998)(9.81) = 3.74 m of water At the bottom of the tank, the pressure: p2 = p1 + pgghg = 36616.02 + (998)(9.81)(1) = 46406.4 N/m2 = 46.6 kPa And, the pressure head in meters of water is: h2 = p1 – p0 = 46406.4 - 0 pwg (998)(9.81) = 4.74 m of water
Strain and Strain (Shear) Rate • Strain • a dimensionless quantity representing the relative deformation of a material Normal Strain Shear Strain
Solids: Elastic and Shear Moduli • When a solid material is exposed to a stress, it experiences an amount of deformation or strain proportional to the magnitude of the stress • Stress () Strain ( or ) • Stress = Modulus Strain • Normal stress: elastic modulus (E) • Shear stress: shear modulus (G)
Fluid Viscosity • Newtonian fluids • viscosity is constant (Newtonian viscosity, ) • Non-Newtonian fluids • shear-dependent viscosity (apparent viscosity, or a)
Viscosity: Introduction The viscosity is measure of the “fluidity” of the fluid which is not captured simply by density or specific weight. A fluid can not resist a shear and under shear begins to flow. The shearing stress and shearing strain can be related with a relationship of the following form for common fluids such as water, air, oil, and gasoline: • is the absolute viscosity or dynamics viscosity of the fluid, u is the velocity of the fluid and y is the vertical coordinate as shown in the schematic below: “No Slip Condition”
y = 0 y = y-max = Coefficient Viscosity ( Pa s) Viscosity Viscosity is a property of fluids that indicates resistance to flow. When a force is applied to a volume of material then a displacement (deformation) occurs. If two plates (area, A), separated by fluid distance apart, are moved (at velocity V by a force, F) relative to each other, Newton's law states that the shear stress (the force divided by area parallel to the force, F/A) is proportional to the shear strain rate . The proportionality constant is known as the (dynamic)viscosity
Shear stress The unit of viscosity in the SI system of units is pascal-second (Pa s) In cgs unit , the unit of viscosity is expressed as poise Shear rate 1 poise = 0.1 Pa s 1 cP = 1 m Pa s
Example : Shear stress in soybean oil • The distance between the two parallel plates is 0.00914 m and the lower plate is being pulled at a relative velocity of 0.366 m/s greater than the top plate. The fluid used is soybean oil with viscosity of 0.004 Pa.s at 303 K • Calculate the shear stress and the shear rate • If water having a viscosity of 880x10-6 Pa.s is used instead of soybean oil, what relative velocity in m/s needed using the same distance between plates so that the same shear stress is obtained? Also, what is the new shear rate?
Dynamic viscosity Kinematics Viscosity = Density 2 The unit of kinematics viscosity are m /s Kinematics Viscosity
Inertial Forces Kinematics = Viscous Forces 3 = Density of liquid kg / m = Density of liquid Pa s V= Flow Velocity m/s D= Diameter of the pipe m o m= mass flow rate kg /sec Reynolds Number
Experiment for find Re Laminar flow Turbulent flow