Carmela Marie Lingad – Oliquino MSCE – Geotechnical, SO2 Fluid Statics Unit Pressure TOPIC OUTLINE Pascal’s Law Atmospheric, Absolute And Gage Pressure Instruments Pressure Variations Pressure Head TOPIC OUTLINE Piezometer Manometer Fluid Statics Unit Pressure TOPIC OUTLINE Pascal’s Law Atmospheric, Absolute And Gage Pressure Instruments Fluid Statics • Study of pressures throughout a fluid at rest and the pressure forces on finite surfaces. • Pressure at any point P on a plane surface. • The limiting value of the ratio of normal force to surface area as the area approaches zero size. • Newtons per square meter (Pascals, Pa) or in bars (1 bar = 105 N/m2 or 105 Pa) Fluid Statics Unit Pressure TOPIC OUTLINE Pascal’s Law Atmospheric, Absolute And Gage Pressure Instruments Unit Pressure Or Pressure, P • Pressure is the force per unit area exerted by a liquid or gas on a body or surface, with the force acting at right angles to the surface uniformity in all directions. • The intensity of pressure (known also as unit pressure or simply called pressure) is the amount of force exerted by a fluid distributed uniformly over a unit area Equation for Unit Pressure Or Pressure, P π π·= π¨ where: P – pressure in Pascal (Pa) F – force in Newton (N) A – area in square meter (m2) Equation for Unit Pressure Or Pressure, P • If the unit pressure is not uniform over the unit area, it can be expressed as the sum of differential pressure. π π π π=ΰΆ± π π π¨ Unit Pressure Or Pressure, P • Since fluid at rest cannot resist shearing stress, pressure is always at right angle to the area where it is acting. It is also worthy to note that the total hydrostatic force F = pA, which can be found by cross multiplication. • Units: pounds per square inch (psi) kg/cm2 or atmosphere (atm) N/m2 (Pascal, Pa) English system International usage Metric system Fluid Statics Unit Pressure TOPIC OUTLINE Pascal’s Law Atmospheric, Absolute And Gage Pressure Instruments Pascal’s Law • Developed by French mathematician, Blaise Pascal • Pascal's law states that the pressure at a point in a fluid at rest is the same in all directions. • This means it is independent of the orientation of the surface around the point. Pascal’s Law • Consider a small triangular prism of unit length surrounding the point in a fluid at rest. Pascal’s Law • Since the body is in static equilibrium, we can write π1 : π΄π΅ × π − π3 π΅πΆ × π cos π = 0 (ππ. π) π2 : π΄π΅ × π − π3 π΅πΆ × π sin π − π = 0 (ππ. ππ) • From Equation (i) π1 = π3 since cos π = π΄π΅π΅πΆ, and Equation (ii) gives π2 = π3 since sin π = π΄πΆπ΅πΆ,and π = 0 as the prism shrinks to a point. π1 → π2 → π3 Pascal’s Law • Consider an elementary cylindrical volume of fluid (of length, L and cross-sectional area, dA) within the static fluid mass (Figures 2.2), p being the pressure at an elevation of y and dp being the pressure variation corresponding to an elevation variation of dy. • For equilibrium of the elementary volume, πππ΄ − ππππ΄πΏ sin π − π + ππ ππ΄ = 0 Or ππ = − ππππ¦ since ππ¦ sin π = πΏ Pascal’s Law • The laws of fluid mechanics are observable in many everyday situations. For example, the pressure exerted by water at the bottom of a pond will be the same as the pressure exerted by water at the bottom of a much narrower pipe provided depth remains constant. Fluid Statics Unit Pressure TOPIC OUTLINE Pascal’s Law Atmospheric, Absolute And Gage Pressure Instruments Atmospheric Pressure, Patm • It is the weight of all gases above the surface in which it comes in contact. • Under normal conditions at sea level 1 atm = 1.03323 kg/cm2 = 14.696 psi = 760 mm Hg = 29.921 in Hg = 101,325.024 Pa = 760 Torr = 1.01325 bar Gage Pressure (Relative Pressure) • Gage Pressure are pressures above or below the atmosphere and can be measured by pressure gauges or manometers. • Negative gauge pressure indicates a vacuum which cannot go below –101.325 kPa. • Positive gauge pressure indicates that the pressure is above atmospheric. π·ππππ = πΈπ Absolute Pressure • Absolute pressure (pabs) is equal to gauge pressure plus atmospheric pressure. There is no such thing as negative absolute pressure. In the absence of all matter (complete vacuum), the absolute pressure is zero. π·πππ = π·ππππ + π·πππ • Note: The term pressure, unless otherwise specified, will be used to denote gage or relative pressure. Fluid Statics Unit Pressure TOPIC OUTLINE Pascal’s Law Atmospheric, Absolute And Gage Pressure Instruments Gage Pressure (Relative Pressure) – Bourdon Gauge • For a higher pressure differences, a Bourdon gauge, named after the French inventor Eugene Bourdon, is used. • This consists of a hollow metal tube with an oval cross section, bent in the shape of a hook. One end of the tube is closed, the other open and connected to the measurement region. Gage Pressure (Relative Pressure) – U-Tube Manometer • For small pressure differences, a U-Tube manometer is used. • It consists of a U-shaped tube with one end connected to the container and the other open to the atmosphere. • Filled with a liquid, such as water, oil, or mercury, the difference in the liquid surface levels in the two manometer legs indicates the pressure differences from local atmospheric conditions. Absolute Pressure – Barograph • a recording barometer which provides a continuous record of pressure over a period of time Absolute Pressure – Aneroid Barometer • In an aneroid barometer, a partially evacuated metal drum expands or contracts in response to changes in air pressure. A series of levers and springs translates the up and down movement of the drum top into the circular motion of the pointers along the aneroid barometer’s face. Absolute Pressure – Mercury Barometer • A mercury barometer is an accurate and relatively simple way to measure changes in atmospheric pressure. • At sea level, the weight of the atmosphere forces mercury 760mm (29.9 in) up a calibrated glass tube. • Higher elevations yield lower readings because the atmosphere is less dense there, and the thinner air exerts less pressure on the mercury. Pressure Variations Pressure Head TOPIC OUTLINE Piezometer Manometer Variations on Pressure • The difference in pressure between any two points in a homogenous fluid at rest is equal to the product of the unit weight of the fluid (πΎ) to the vertical distance h) between the points. Variations on Pressure π·π = π·π + πΈπ • Any change in pressure at point 1 would cause an equal change to the pressure in point 2. • A pressure applied at any point in a liquid at rest is transmitted equally and undiminished to every point in the liquid. Variations on Pressure • Assume the point 1 lies on the free liquid surface, then the gauge pressure at point 1 is zero and the equation becomes: π· = πΈπ • The pressure at any point “h” below a free liquid surface is equal to the product of the unit weight of the fluid and the height. Variations on Pressure • Assume the points 1 and 2 lie on the same elevation, such that h = 0; then the equation becomes: π·π = π·π • The pressure along the same horizontal plane in a homogenous fluid at rest is equal. Pressure Variations Pressure Head TOPIC OUTLINE Piezometer Manometer Pressure Head • The equation P = γh may be written into the form π· π= πΈ • where h or its equivalent P/γ is in hydraulics called the pressure head. Pressure head is the height of column of homogeneous fluid of unit weight γ that will produce an intensity of pressure P. To convert pressure head of liquid A to equivalent pressure head of liquid B ππ¨ ππ¨ πΈπ¨ ππ© = ππ¨ = ππ¨ = ππ¨ ππ© ππ© πΈπ© To convert pressure head of any liquid to equivalent pressure head of water ππππππ = πππππππ × πππππππ where: s = specific gravity πΎ = unit weight π = density Pressure Variations Pressure Head TOPIC OUTLINE Piezometer Manometer Piezometer • Piezometer is the simplest form of manometer which is tapped into the wall of pressure conduit for the purpose of measuring pressure. • Though effective in many purposes, piezometer is not practical to use in lighter liquids with large pressure and cannot be used to measure gas pressure. Piezometer • From the figure, three piezometers A, B, and C are attached to a pressure conduit at bottom, top, and side, respectively. • The column of liquid at A, B, and C will rise at the same level above M indicating a positive pressure at M. • Also, the piezometer D measures the negative pressure at N. Pressure Variations Pressure Head TOPIC OUTLINE Piezometer Manometer Manometer • Manometer is a simple and inexpensive device of measuring pressure and pressure difference. It is usually bent to form a U-tube and filled with liquid of known specific gravity. The surface of the liquid will move in proportion to changes of pressure. Open Manometer • Open manometer is a tube bent into a Ushape to contain one or more fluids of different specific gravities. It is used to measure pressure. Differential Manometer • Differential manometer cannot measure pressure but can measure pressure difference. Frequently in hydraulic problems, difference in pressure is more useful information than the pressure itself. Draw a sketch of the manometer system approximately to scale. Number in order all levels of contact of the different fluids, starting with level 0 at the free surface. Decide on the fluid of which head are to be expressed. Water is more desirable. In most cases, we suggest to use head in water even if there is no water in the system Starting from level 0 (zero gage pressure) at free surface, proceed from level to level adding or subtracting as the elevation increases or decreases giving due regard to the specific gravity of the fluid. Carmela Marie Lingad – Oliquino MSCE – Geotechnical, SO2 βͺ A tank contains oil of SG = 0.80, gasoline of SG = 0.90 and seawater of SG = 1.03. If the depth of the liquids are 0.50 m, 0.80 m and 1 m for the oil, gasoline and seawater respectively. a. Find the pressure at a depth of 1.20m. (Ans. 10.10 kPa) b. Find the pressure at a depth of 1.8m. (Ans. 16.04 kPa) c. Find the pressure at the bottom of the tank. (Ans. 21.09 kPa) βͺ If the pressure at a point in the ocean is 60 kPa, what is the pressure 27 m below this point? π2 = π1 + πΎβ π2 = 60 πππ + 9.81 ππΰ΅ 3 1.03 27 π π π· = πππ. ππ ππ·π βͺ If the pressure in the air space above an oil (SG = 0.75) surface in a closed tank is 115 kPa absolute, what is the gage pressure 2 m below the surface? πππ‘π = 101.325 πππ π = ππ π’πππππ + πΎβ ππ π’πππππ = 115 πππ − 101.325 πππ ππ π’πππππ = 13.675 πππ π = 13.675 πππ + 9.81 ππΰ΅ 3 π π· = ππ. ππ ππ·π 2π
