Fluid Mechanics and Hydraulic Machinery (EEE) - Unit 1: Fluid Statics - PDF

FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) UNIT-I FLUID STATICS 1.1 INTRODUCTION: Fluid mechanics: The branch of science which deals with the behav...

FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) UNIT-I FLUID STATICS 1.1 INTRODUCTION: Fluid mechanics: The branch of science which deals with the behaviour of the fluids (liquids or gases) at rest as well as in motion is called Fluid mechanics. Thus this branch of science deals with the o Static o Kinematics o Dynamic aspects of fluids. Fluid statics: The study of fluids at rest is called fluid statics. Fluid kinematics: The study of fluids in motion, where pressure forces are not considered, is called fluid kinematics Fluid dynamics: If the pressure forces are also considered for the fluids in motion, that branch of science is called fluid dynamics. 1.2. PROPERTIES OF FLUIDS 1.2.1Density or Mass Density: Density or mass density of a fluid is defined as the ratio of the mass of a fluid to its volume. Thus mass per unit volume of a fluid is called density. It is denoted the symbol ρ (rho). The unit of mass density in SI unit is kg per cubic metre, i.e., kg/rn3. The density of liquids may be considered as constant while that of gases changes with the variation of pressure and temperature. mass of fluid Mathematically, mass density is written as ρ = volume of fluid The value of density of water is 1 gm/cm3 or 1000 kg/rn3. 1.2.2 Specific Weight or Weight Density: Specific weight or weight density of a fluid is the ratio between the weight of a fluid to its volume. Thus weight per unit volume of a fluid is called weight density and it is denoted by the symbol w. Thus Mathematically, Weight density is written as weight of fluid (Mass of fluid) × Acceleration due to gravity w= = Volume of fluid Volume of flud Mass of fluid × g = Volume of fluid = ρ w = ρ× g The value of specific weight or weight density (w) for water is 9.81 x 1000 Newton/rn3 in SI units. 1.2.3 Specific Volume: Specific volume of a fluid is defined as the volume of a fluid occupied by a unit mass or volume per unit mass of a fluid is called specific volume. Mathematically, it is expressed as Volume of fluid 1 1 Specific volume = = = Mass of fluid Mass of fluid ρ Volume Thus specific volume is the reciprocal of mass density. It is expressed as m3/kg. It is commonly applied to gases. 1  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) 1.2.4 Specific Gravity: Specific gravity is defined as the ratio of the weight density (or density) of a fluid to the weight density (or density) of a standard fluid. For liquids, the standard fluid is taken water and for gases, the standard fluid is taken air. Specific gravity is also called relative density. It is dimensionless quantity and is denoted by the symbol S. Mathematically, Weight density (density) of liquid S(for liquids) = Weight density (density) of water Weight density (density) of lgas S(for liquids) = Weight density (density) of air Thus Weight density of a liquid = S x Weight density of water = S x 1000 x 9.81 N/m3 The density of a liquid = S x Density of water = S x 1000 kg/m3 If the specific gravity of a fluid is known, then the density of the fluid will be equal to specific gravity of fluid multiplied by the density of water. For example the specific gravity of mercury is 13.6, hence density of mercury = 13.6 x 1000 = 13600 kg/rn3. 1.2.5 Viscosity Viscosity is defined as the property of a fluid which offers resistance to the movement of one layer of fluid over another adjacent layer of the fluid. When two layers of a fluid, a distance ‘dy’ apart, move one over the other at different velocities, say U and u + du as shown in Fig. 1.1, the viscosity together with relative velocity causes a shear stress acting between the fluid layers. The top layer causes a shear stress on the adjacent lower layer while the lower layer causes a shear stress on the adjacent top layer. This shear stress is proportional to the rate of change of velocity with respect to y. It is denoted by symbol τ called Tau. du Mathematically, τ ∝ dy du τ=μ dy τ From the above equation μ = du ( ) dy Fig. 1.1 Velocity variation near a solid boundary Thus viscosity is also defined as the shear stress required to produce unit rate of shear strain. where μ (called mu) is the constant of proportionality and is known as the co-efficient of dynamic viscosity or only viscosity represents the rate of shear strain or rate of shear deformation or velocity gradient. 1.2.6 Units of Viscosity: The units of viscosity is obtained by putting the dimensions of the quantities in above equation Shear stress Force/Area μ= = Change of velocity (Length/Ti me) × (1/Length) Change of distance 2  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Force/(Length) 2 Force × Time = = 1 (Length) 2 Time In MKS system, force is represented by kgf and length by metre (m), in CGS system, force is represented by dyne and length by cm and in SI system force is represented by Newton (N) and length by metre (m). kgf - sec MKS unit of viscosity = m2 dyne - sec CGS unit of viscosity = cm2 2 In the above expression N/rn is also known as Pascal which is represented by Pa. Hence N/rn2 = Pa = Pascal SI unit of viscosity = Ns/m2 = Pa s. Newton - sec Ns SI unit of viscosity = 2 = 2 m m The unit of viscosity in CGS is also called Poise which is equal to (dyne-sec)/cm The numerical conversion of the unit of viscosity from MKS unit to CGS unit is given below: 3  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) 1.2.7 Kinematic Viscosity: It is defined as the ratio between the dynamic viscosity and density of fluid. It is denoted by the Greek symbol (v) called ‘nu’. Thus, mathematically, 1.2.8 Newton’s Law of Viscosity: It states that the shear stress (t) on a fluid element layer is directly proportional to the rate of shear strain. The constant of proportionality is called the du co-efficient of viscosity. Mathematically, it is expressed as τ = μ dy Fluids which obey the above relation are known as Newtonian fluids and the fluids which do not obey the above relation are called Non-newtonian fluids. 1.2.9 Variation of Viscosity with Temperature. Temperature affects the viscosity. The viscosity of liquids decreases with the increase of temperature while the viscosity of gases increases with the increase of temperature. This is due to reason that the viscous forces in a fluid are due to cohesive forces and molecular momentum transfer. In liquids the cohesive forces predominates the molecular momentum transfer, due to closely packed molecules and 4  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) with the increase in temperature, the cohesive forces decreases with the result of decreasing viscosity. But in case of gases the cohesive force are small and molecular momentum transfer predominates. With the increase in temperature, molecular momentum transfer increases and hence viscosity increases. The relation between viscosity and temperature for liquids and gases are: 1.3 Types of Fluids: The fluids may be classified into the following five types: 1. Ideal fluid, 2. Real fluid. 3. Newtonian fluid, 4. Non-Newtonian fluid, and 5. Ideal plastic fluid. 1. Ideal Fluid. A fluid, which is incompressible and is having no viscosity, is known as an ideal fluid. Ideal fluid is only an imaginary fluid as all the fluids, which exist, have some viscosity. 2. Real Fluid. A fluid, which possesses viscosity, is known as real fluid. All the fluids, in actual practice, are real fluids. 3. Newtonian Fluid. A real fluid, in which the shear stress is directly, proportional to the rate of shear strain (or velocity gradient), is known as a Newtonian fluid. 4. Non-Newtonian Fluid. A real fluid, in which the shear stress is not proportional to the rate of shear strain (or velocity gradient), known as a Non-Newtonian fluid. 5. Ideal Plastic Fluid. A fluid, in which shear stress is more than the yield value and shear stress is proportional to the rate of shear strain (or velocity gradient), is known as ideal plastic fluid. Fig. 1.2 Types of fluids Fig.1.3 1.4 Compressibility and Bulk Modulus 5  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Compressibility is the reciprocal of the bulk modulus of elasticity, K which is defined as the ratio of compressive stress to volumetric strain. Consider a cylinder fitted with a piston as shown in Fig.1.3 Let V = Volume of a gas enclosed in the cylinder p = Pressure of gas when volume is V Let the pressure is increased to p + dp, the volume of gas decreases from V to V — dV. Then increase in pressure = dp kgf/m2 Decrease in volume = dV 1.5 Surface Tension and Capillarity Surface tension is defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two immiscible liquids such that the contact surface behaves like a membrance under tension. The magnitude of this force per unit length of the free surface will have the same value as the surface energy per unit area. It is denoted by Greek letter a (called sigma). In MKS units, it is expressed as kgf/m while in SI units as N/rn. The phenomenon of surface tension is explained by Fig. 1.4. Consider three molecules A, B, C of a liquid in a mass of liquid. The molecule A is attracted in all directions equally by the surrounding molecules of the liquid. Thus the resultant force acting on the molecule A is zero. But the molecule B, which is situated near the free surface, is acted upon by upward and downward forces which are unbalanced. Thus a net resultant force on molecule B is acting in the downward direction. The molecule C, situated on the free surface of liquid. does experience a resultant downward force. All the molecules on the free surface experience a downward force. Thus the freesurface of the liquid acts like a very thin film under tension of the surface of the liquid act as though it is an elastic membrance under tension. Fig. 1.4 Surface Tension. Fig. 1.5 Forces on droplet. 6  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) 1.5.1 Surface Tension on Liquid Droplet. Consider a small spherical droplet of a liquid of radius ‘r’. On the entire surface of the droplet, the tensile force due to surface tension will be acting. Let σ = Surface tension of the liquid p = Pressure intensity inside the droplet (in excess of the outside pressure intensity) d = Dia. of droplet. Let the droplet is cut into two halves. The forces acting on one half (say left half) will be (i) tensile force due to surface tension acting around the circumference of the cut portion as shown in Fig. 1.5 (b) and this is equal to = σ x Circumference =σx πd π 2 π (ii) pressure force on the area d and =p x d 2 as shown in Fig. 1.5 (C). These two forces 4 4 will be equal and opposite under equilibrium conditions, i.e., The above equation shows that with the decrease of diameter of the droplet, pressure intensity inside the droplet increases. 1.5.2 Surface Tension on a Hollow Bubble. A hollow bubble like a soap bubble in air has two surfaces in contact with air, one inside and other outside. Thus two surfaces are subjected to surface tension. In such case, we have 1.5.3 Surface Tension on a Liquid jet. Consider a liquid jet of diameter ‘d’ and length ‘L’ as shown in Fig. 1.12. Let p = Pressure intensity inside the liquid jet above the outside pressure σ = Surface tension of the liquid. Consider the equilibrium of the semi jet, we have Force due to pressure = p x area of semi jet =pxLxd Force due to surface tension = σ x 2L. Equating the forces, we have P x L x d = σ x 2L Fig. 1.6 Forces on liquid jet capillary depression. It is expressed in terms of cm or mm of liquid. Its value depends upon the specific weight of the liquid, diameter of the tube and surface tension of the liquid. 7  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Expression for Capillary Rise. Consider a glass tube of small diameter ‘d’ opened at both ends and is inserted in a liquid, say water. The liquid will rise in the tube above the level of the liquid. Let h = height of the liquid in the tube. Under a state of equilibrium, the weight of liquid of height h is balanced by the force at the surface of the liquid in the tube. But the force at the surface of the liquid in the tube is due to surface tension. Let a = Surface tension of liquid θ= Angle of contact between liquid and glass tube. The weight of liquid of height h in the tube = (Area of tube x h) x p x g The value of θ between water and clean glass tube is approximately equal to zero and hence cos θ is equal to unity. Then rise of water is given by Fig. 1.7 Capillary rise. Fig. 1.8 Capillary depression Expression for Capillary Fall. If the glass tube is dipped in mercury, the level of mercury in the tube will be lower than the general level of the outside liquid as shown in Fig. 1.8. Let h = Height of depression in tube. Then in equilibrium, two forces are acting on the mercury inside the tube. First one is due to surface tension acting in the downward direction and is equal to a σ x π d x Cos θ Second force is due to hydrostatic force acting upward and is equal to intensity of pressure at a depth ‘h’x Area 8  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) 1.6 Vapour Pressure And Cavitation. A change from the liquid state to the gaseous state is known as vaporization. The vaporization (which depends upon the prevailing pressure and temperature condition) occurs because of continuous escaping of the molecules through the free liquid surface. Consider a liquid (say water) which is confined in a closed vessel. Let the temperature of liquid is 20°C and pressure is atmospheric. This liquid will vaporise at 100°C. When vaporization takes place, the molecules escapes from the free surface of the liquid. These vapour molecules get accumulated in the space between the free liquid surface and top of the vessel. These accumulated vapours exert a pressure on the liquid surface. This pressure is known as vapour pressure of the liquid. Or this is the pressure at which the liquid is converted into vapours. Again consider the same liquid at 20°C at atmospheric pressure in the closed vessel. If the pressure above the liquid surface is reduced by some means, the boiling temperature will also reduce. If the pressure is reduced to such an extent that it becomes equal to or less than the vapour pressure, the boiling of the liquid will start, though the temperature of the liquid is 20°C. Thus a liquid may boil even at ordinary temperature, if the pressure above the liquid surface is reduced so as to be equal or less than the vapour pressure of the liquid at that temperature. Now consider a flowing liquid in a system. If the pressure at any point in this flowing liquid becomes equal to or less than the vapour pressure, the vaporization of the liquid starts. The bubbles of these vapours are carried by the flowing liquid into the region of high pressure where they collapse, giving rise to high impact pressure. The pressure developed by the collapsing bubbles is so high that the material from the adjoining boundaries gets eroded and cavities are formed on them. This phenomenon is known as cavitation. Hence the cavitation is the phenomenon of formation of vapour bubbles of a flowing liquid in a region where the pressure of the liquid falls below the vapour pressure and sudden collapsing of these vapour bubbles in a region of higher pressure. When the vapour bubbles collapse, a very high pressure is created. The metallic surfaces, above which the liquid is flowing, is subjected to these high pressures, which cause pitting action on the surface. Thus cavities are formed on the metallic surface and hence the name is cavitation. 9  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) 1.7 FLUID PRESSURE AT A POINT Consider a small area dA in large mass of fluid. If the fluid is stationary, then the force exerted by the surrounding fluid on the area dA will always be perpendicular to the surface dA. Let dF is the force acting on the area dA in the normal direction. Then the ratio of is known as the intensity of pressure or simply pressure and this ratio is represented by p. Hence mathematically the pressure at a point in a fluid at rest is If the force (F) is uniformly distributed over the area (A), then pressure at any point is given by F Force ∴ Force or pressure force. F = p x A. The units of pressure are : (i) kgf/m2 and kgf/cm2 in MKS units, (ii) Newton/m2 or N/m2and N/mm2 in SI units. N/m2 is known as Pascal and is represented by Pa. Other commonly used units of pressure are: kPa = kilo pascal = 1000 N/rn2 bar = 100 kPa = 105 N/rn2. Fig. 1.9 Forces on a fluid element. 1.8 PASCAL’S LAW It states that the pressure or intensity of pressure at a point in a static fluid is equal in all directions. This is proved as: The fluid element is of very small dimensions i.e., dx, dy and ds. Consider an arbitrary fluid element of wedge shape in a fluid mass at rest as shown in Fig. 1.9. Let the width of the element perpendicular to the plane of paper is unity and px, py and pz. are the pressures or intensity of pressure acting on the face AB, AC and BC respectively, let ∠ ABC = 0. Then the forces acting on the element are: 1. Pressure forces normal to the surfaces. 2 Weight of element in the vertical direction. The (forces on the faces are Force on the face AB = px x Area of face AB = px x dv x 1 Similarly force on the face AC = py x dx x I Force on the face BC = py x ds x I 10  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) The above equation shows that the pressure at any point in x, v and c directions is equal. Since the choice of fluid element was completely arbitrary, which means the pressure at any point is the same in all directions. 1.9 PRESSURE VARIATION IN A FLUID AT REST The pressure at any point in a fluid at rest is obtained by the Hydro-static Law which states that the rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at that point. This is proved as: a small tluid element as shown in Fig 1.10 Let ΔA = Cross-sectional area of element p = Pressure on face Al? Z = Distance of fluid element from free surface. The forces acting on the fluid element are Fig. 1.10 Forces on a fluid element 11  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Above equation states that rate of increase of pressure in a vertical direction is equal to weight density of the fluid at that point. This is Hydrostatic Law. By integrating the above equation for liquids, we get where p is the pressure above atmospheric pressure and Z is the height of the point from free surfaces. From equation (2.5), we have Here Z is called pressure head. 1.10 ABSOLUTE, GAUGE, ATMOSPHERIC AND VACUUM PRESSURES The atmospheric air exerts a normal pressure upon all surfaces with which it is in contact, and it known as atmospheric pressure. At sea level under normal conditions the equivalent values of the atmospheric pressure are 10.1043 x I 0 N/m2 or 1.03 kg(f)/cm2.The atmospheric pressure head is 760 mm of mercury or 10.33 m of water. Thus: 1. Absolute pressure is defined as the pressure which is measured with reference to absolute vacuum pressure. 2. Gauge pressure is defined as the pressure which is measured with the help of a pressure measuring instrument, in which the atmospheric pressure is taken as datum. The atmospheric pressure on the scale is marked as zero. 3. Barometric pressure the atmospheric pressure varies with the altitude and it can be measured by means of a barometer. As such it is also called the barometric pressure 12  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Fig 1.11 Relationship between absolute, gage and vacuum pressures 4. Local atmospheric pressure when it is measured either above or below atmospheric pressure as a datum, it is called gage pressure. This is because practically all pressure gages read zero when open to the atmosphere and read only the difference between the pressure of the fluid to which they are connected and the atmospheric pressure. 5. Vacuum pressure If the pressure of a fluid is below atmospheric pressure it is designated as vacuum pressure (or suction pressure on negative gage pressure) ; and its gage value is the amount by which it is below that of the atmospheric pressure. A gage which measures vacuum pressure is known as vacuum gage. All values of absolute pressure are positive, since in the case of fluids the lowest absolute pressure which can possibly exist corresponds to absolute zero or complete vacuum. However, gage pressures are positive if they are above that of the atmosphere and negative if they are vacuum pressures. Fig. 1.11 illustrates the relation between absolute, gage and vacuum pressures. From the foregoing discussion it can be seen that the following relations hold: Absolute Pressure = Atmospheric Pressure + Gage Pressure Absolute Pressure = Atmospheric Pressure - Vacuum Pressure 1.11 MEASUREMENT OF PRESSURE The various devices adopted for measuring fluid pressure may be broadly classified under the following two heads: (1) Manometers (2) Mechanical Gages. Manometers. Manometers are those pressure measuring devices which are based on the principle of balancing the column of liquid (whose pressure is to be found) by the same or another column of liquid. The manometers may be classified as (a) Simple Manometers. (b) Differential Manometers. 13  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Simple Manometers are those which measure pressure at a point in a fluid contained in a pipe or a vessel. On the other hand Differential Manometers measure the difference of pressure between, any two points in a fluid contained in a pipe or a vessel. (1) Simple Manometers. In general a simple manometer consists of a glass tube having one of its ends connected to the gage point where the pressure is to be measured and the other remains open to atmosphere. Some of the common types of simple manometers are as noted below: (i) Piezometer (ii) U-tube Manometer. (iii) Single Column Manometer. (i) Piezometer A piezometer is the simplest form of manometer which can be used for measuring moderate pressures of liquids. It consists of a glass tube (Fig. 1.12) inserted in the wall of a pipe or a vessel, containing a liquid whose pressure is to be measured. The tube extends vertically upward to such a height that liquid can freely rise in it without overflowing. The pressure at any point in the liquid is indicated by the height of the liquid in the tube above that point, which can be read on the scale attached to it. Thus, if w is the specific weight of the liquid, then the pressure at point m in Fig. 1.12 (a) is pm = Whm In other words, hm is the pressure head at m. Piezometers measure gage pressure only, since the surface of the liquid in the tube is subjected to atmospheric pressure. From the foregoing principles of pressure in homogeneous liquid at rest, it is obvious that the location of the point of insertion of a piezometer makes no difference. Hence as shown in Fig. 1.12 (a) piezometers may be inserted either in the top, or the side, or the bottom of the container, but the liquid will rise to the same level in the three tubes. (a) (b) (c) Fig 1.12 Piezometer Negative gage pressures (or pressures less than atmospheric) can be measured by means of the piezometer shown in Fig. 1.12 (b). It is evident that if the pressure in the container is less than the atmospheric no column of liquid will rise in the ordinary piezometer. But if the top of the tube is bent downward and its lower end dipped into a vessel containing water (or some other suitable liquid) [Fig. 1.12 (b)], the atmospheric pressure will cause a column of the liquid to rise to a height h in the tube, from which the magnitude of the pressure of the liquid in the container can be obtained. Neglecting the weight of the air caught in the portion of the tube, the pressure on the free surface in the container is the same as that at free surface in the tube. 14  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Piezometers are also used to measure pressure heads in pipes where the liquid is in motion. Such tubes should enter the pipe in a direction at right angles to the direction of flow and the connecting end should be flush with the inner surface of the pipe. All burrs and surface roughness near the hole must be removed, and it is better to round the edge of the hole slightly. Also, the hole should be small, preferably not larger than 3 mm. In order to prevent the capillary action from affecting the height of the column of liquid in a piezometer, the glass tube having an internal diameter less than 12 mm should not be used. Moreover for precise work at low heads the tubes haying an internal diameter of 25 mm may be used. The rise of liquid gives the pressure head at that point. If at a point A. the height of liquid say water is ii in piezometer tube, then pressure at A (ii) U-tube Manometer. Piezometers cannot be used when large pressures in the lighter liquids are to be measured, since this would require very long tubes, which cannot be handled conveniently. Furthermore gas pressures cannot be measured by means of piezometers because a gas forms no free atmospheric surface. These limitations imposed on the use of piezometers may be overcome by the use of U-tube manometers. A U-tube manometer consists of a glass tube bent in U-shape, one end of which is connected to the gage point and the other end remains open to the atmosphere (Fig. 1.13). The tube contains a liquid of specific gravity greater than that of the fluid of which the pressure is to be measured. Sometimes more than one liquid may also be used in the manometer. The liquids used in the manometers should be such that they do not get mixed with the fluids of which the pressures are to be measured. Some the liquids that are frequently used in the manometers are mercury, oil, salt solution, carbon disuiphide, carbon tetrachioride, bromoform and alcohol. Water may also be used as a manometric liquid when the pressures of gases or certain coloured liquids (which are immiscible with water) are to be measured. The choice of the manometric liquid, however, depends on the range of pressure to be measured. For low pressure range, liquids of lower specific gravities are used and for high pressure range, generally mercury employed. When one of the limbs of the U-tube manometer is connected to the gage point, the fluid from the container or pipe A will enter the connected limb of the manometer thereby causing the manometric liquid to rise in the open limb as shown in Fig. 1.13. An air relief valve V is usually provided at the top of the connecting tube which permits the expulsion of all air from the portion A'B and its place taken by the fluid in A. This is essential because the presence of even a small air bubble in the portion A'B would result in an inaccurate pressure measurement. 15  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) (a) For gauge pressure (b)For gauge pressure (c) For gauge pressure (d)For vacuum pressure Fig. 1.13 U-tube Manometer. As the pressure is the same for the horizontal surface. Hence the pressure above the horizontal datum line A-A in the left column and in the right column of U-tube manometer should be same. (b) For Vacuum Pressure. For measuring vacuum pressure, the level of the heavy liquid in the manometer will be as shown in Fig. 1.13 (d). Then (iii) Single Column Manometer The U-tube manometers described above usually require readings of fluid levels at two or more points, since a change in pressure causes a rise of liquid in one limb of the manometer and a drop in the other. This difficulty may however be overcome by using single column manometers. A single column manometer is a modified form of a U-tube manometer in which a shallow reservoir having a large cross-sectional area (about 100 times) as compared to the area of the tube is introduced into one limb of the manometer, as shown in Fig 1.14. For any variation in pressure, the change in the liquid level in the reservoir will be so small that it may be neglected, and the pressure is indicated approximately by the height of the liquid in the other limb. As such only one reading in the narrow limb of the manometer need be taken for all pressure measurements. The narrow limb of the manometer may be 16  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) vertical as in Fig 1.14 (a) or it may be inclined as in Fig. 1.14 (b). Thus there are two types of single column manometer as a. Vertical Single Column Manometer b. Inclined Single Column Manometer. The inclined type is useful for the measurement of small pressures. As indicated later since no reading is required to be taken for the level of liquid in the reservoir, it need not be made of transparent material. 1.14 Single column manometer (a).Vertical Single Column Manometer Fig. 1.15 shows the vertical single column manometer. Let X-X be the datum line in the reservoir and in the right limb of the manometer, when it is not connected to the pipe. When the manometer is connected to the pipe, due to high pressure at A, the heavy liquid in the reservoir will be pushed downward and will rise in the right limb. Let Δh = Fall of heavy liquid in reservoir h2 = Rise of heavy liquid in right limb 1.15 Vertical Single Column Manometer h1 = Height of centre of pipe above X-X p = Pressure at A, which is to be measured A = Cross-sectional area of the reservoir a = Cross-sectional area of the right limb S1 = Sp. gr. of liquid in pipe S2 = Sp. gr. of heavy liquid in reservoir and right limb ρ1 = Density of liquid in pipe ρ2 = Density of liquid in reservoir Fall of heavy liquid in reservoir will cause a rise of heavy liquid level in the right limb. Now consider the datum line Y-Y as shown in Fig. 1.15. Then pressure in the right limb above Y-Y. 17  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) As the area A is very large as compared to a/A, hence ratio becomes very small and can be neglected. (b). Inclined Single Column Manometer Fig. 1.16 shows the inclined single column manometer. This manometer is more sensitive. Due to inclination the distance moved by the heavy liquid in the right limb will be more. Fig. 1.16 Inclined single column manometer PA= L sin θ x ρ2g - h1ρ1g (2) Differential Manometers. For measuring the difference of pressure between any two points in a pipeline or in two pipes or containers, a differential manometer is employed. In general.a differential manometer consists of a bent glass tube, the two ends of which are connected to each of the two gage points between which the pressure difference is required to be measured. Some of the common types of differential manometers are as noted below: (i) Two-Peizometer Manometer. (ii) U- Tube Differential Manometer. (iii) Inverted U-Tube Manometer. (iv) Micro-manometer. 18  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) (i) Two-Peizometer Manometer. As the name suggests this manometer consists of two separate piezometers which are inserted at the two gauge points between which the difference of pressure is required to be measured. The difference in the levels of the 1iquidraised in the two tubes will denote the pressure difference between the two points. Evidently this method is useful only if the pressure at each of the two points is small. Moreover it cannot be used to measure the pressure difference in gases, for which the other types of differential manometers described below may be employed. (ii) U-Tube Differential Manometer. It consists of glass tube bent in U-shape, the two ends of which are connected to the two gage points between which the pressure difference is required to be measured; Fig. 1.17 shows such an arrangement for measuring the pressure difference between any two points A and B. The lower part of the manometer contains a manometric liquid which is heavier than the liquid for which the pressure difference is to be measured and is immiscible with it. When the two limbs of the manometer are connected to the gage points A and B, then corresponding to the difference in the pressure intensities A and B the levels of manometric liquid in the two limbs of the manometer will be displaced through a distance x as shown in Fig. 1.17. By measuring this difference in the levels of the manometric liquid, the pressure difference (PA- PB) may be computed as indicated below. (a)Two pipes at different levels (b) A and B are at the same level Fig. 1.17 U-tube duff erential manometers. Fig. 1.17 (a). Let the two points A and B are at different level and also contains liquids of different sp. gr. These points are connected to the U-tube differential manometer. Let the pressure at A and B are PA and Let h = Difference of mercury level in the U-tube. y = Distance of the centre of B, from the mercury level in the right limb. x = Distance of the centre of A, from the mercury level in the right limb. ρ1 = Density of liquid at A. ρ2 = Density of liquid at B. ρg = Density of heavy liquid or mercury. 19  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) Fig. 1.17 (b). A and B are at the same level and contains the same liquid of density ρ1. Then (iii) Inverted U-tube Manometer. It consists of a glass tube bent in U-shape and held inverted as shown in Fig. 1.18. Thus it is as if two piezometers described above are connected with each other at top. When the two ends of the manometer are connected to the points between which the pressure difference is required to be measured, the liquid under pressure will enter the two limbs of the manometer, thereby causing the air within the manometer to get compressed. The presence of the compressed air results in restricting the heights of the columns of liquids raised in the two limbs of the manometer. An air cock as shown in Fig. 1.18, is usually provided at the top of the inverted U-tube which facilitates the raising of the liquid columns to suitable level in both the limbs by driving out a portion of the compressed air. It also permits the expulsion of air bubbles which might have been entrapped somewhere in the pipeline. Fig. 1.18 shows an inverted U-tube differential manometer connected to the two points A and B. Let the pressure at A is more than the pressure at B Let h1 = Height of liquid in left limb below the datum line X-X h2 = Height of liquid in right limb h = Difference of light liquid ρ1 = Density of liquid at A ρ2 = Density of liquid at B ρs = Density of light liquid PA = Pressure at A PB = Pressure at B. Fig. 1.18 Inverted U-tube Manometer (iv) Micromanometers. For the measurement of very small pressure differences or for the measurement of pressure differences with very high precision, special forms of manometers called micromanometers are used. A wide variety of micrornanometers have been developed, which either magni1,t the readings or permit the readings to be observed with greater accuracy. One simple type of micromanometer consists of a glass U-tube, provided with two transparent 20  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) basins of wider sections at the top of the two limbs, as shown in Fig. 1.19. The manometer contains two manometric liquids of different specific gravities and immiscible with each other and with the fluid for which the pressure difference is to be measured. Before the manometer is connected to the pressure points A and B, both the limbs are subjected to the same pressure. As such the heavier manometric liquid of sp.gr. S1 will occupy the level DD’ and the lighter manometric liquid of sp.gr. S2 will occupy the level CC’. When the manometer is connected to the pressure points A and B where the pressure intensities are A and PB respectively, such that pA > pB then the level of the lighter manometric liquid will fall in the left basin and rise in the right basin by the same amount ‘Δy’. Fig. 1.19 Micromanometer Similarly the level of the heavier manometric liquid will fall in the left limb to point E and rise in the right limb to point F. If A and a are the cross-sectional areas of the basin and the tube respectively, then since the volume of the liquid displaced in each basin is equal to the volume of the liquid displaced in each limb of the tube the following expression may be readily obtained Further if w is specific weight of water, then, starting from point A the following gage equation in terms of water column may be obtained, Substituting the value of Δy from equation and simplifying the above equation it becomes The quantities within brackets on right side of above equation are constant for a particular manometer. Thus by measuring x and substituting in this equation the pressure difference between any two points can be known If the cross-sectional area of the basin is large as compared with the cross-sectional area of the tube, then the ratio a/A is very small and the above equation reduces to By selecting the two manometric liquids such that their specific gravities are very nearly equal then a measurable value of x may be achieved even for a very small pressure difference between the two points. 21  FLUID MECHANICS AND DEPARTMENT OF MECHANICAL ENGINEERING HYDRAULIC MACHINERY (EEE) In a number of other types of micromanometers the pressure difference to be measured is balanced by the slight raising or lowering (on a micrometer screw) of one arm of the manometer whereby a meniscus is brought back to its original position. The micromanometers of this type are those invented by Chattock, Small and Krell, which are sensitive to pressure differences down to less than 0.0025 mm of water. However the disadvantage with such manometers is that an appreciable time is required to take a reading and they are therefore suitable only for completely steady pressures. 22