Week 1-4 - Mechanics PDF
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Centennial College
Cristina Toma
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Summary
These are lecture notes on mechanics covering fluid properties, including density, specific weight, specific gravity, viscosity, and compressibility, along with problem-solving examples on various fluid properties.
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Identify the Units Systems Fluid Properties ◼ Objectives ◼ ◼ ◼ 1)Identify the units systems for common quantities used in Fluid Mechanics 2) Define Pressure 3) Fluid Properties ◼ ◼ ◼ ◼ ◼ Compressibility Density Specific weight Specific gravity Viscosity ◼ ◼ 12/22/2021 Dynamic (absolute) viscosity Ki...
Identify the Units Systems Fluid Properties ◼ Objectives ◼ ◼ ◼ 1)Identify the units systems for common quantities used in Fluid Mechanics 2) Define Pressure 3) Fluid Properties ◼ ◼ ◼ ◼ ◼ Compressibility Density Specific weight Specific gravity Viscosity ◼ ◼ 12/22/2021 Dynamic (absolute) viscosity Kinematic viscosity Cristina Toma 1 Identify the Units Systems Fluid Properties Objective 1 ◼ ◼ 1) Identify the units for the basic quantities of time, force and mass in SI System (metric) and in U.S. Customary System SI System (metric) ◼ ◼ ◼ ◼ Length=meter (m) Time= second (s) Mass=kilogram (Kg) or N/[m/s2 ] Force=newton (N) or Kg*m/s2 ◼ Derived from 12/22/2021 F = ma Cristina Toma 2 Identify the Units Systems Fluid Properties Objective 1 ◼ SI System (metric) ◼ SI unit Prefixes 12/22/2021 Cristina Toma 3 Identify the Units Systems Fluid Properties Objective 1 ◼ U.S. Customary System ◼ ◼ ◼ ◼ Length=foot (ft) Time= second (s) Force=pound (lb) Mass=slug or lb-s2/ft ◼ Derived from F = ma F lb lb s 2 m= = = = slug a ft ft / s 2 12/22/2021 Cristina Toma 4 Identify the Units Systems Fluid Properties Objective 1 ◼ S. I. and U. S. Units Systems 12/22/2021 Cristina Toma 1bar=100kPa 5 Identify the Units Systems Fluid Properties Objective 1 ◼ Useful Conversion Factors 12/22/2021 Cristina Toma 6 Identify the Units Systems Fluid Properties Objective 2 ◼ Define Pressure: ◼ The amount of force exerted on a unit area of a substance p ◼ F = A Pascal’s Laws: ◼ ◼ Pressure acts uniformly in all directions on a small volume of a fluid In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary 12/22/2021 Cristina Toma 7 Identify the Units Systems Fluid Properties Objective 2 ◼ Problem 1 A: 12/22/2021 Cristina Toma 8 Identify the Units Systems Fluid Properties Objective 2 Problem 2 A hydraulic cylinder must be able to exert a force of 38.8 kN. The piston diameter is 40 mm. Compute the required pressure in the oil. ◼ A: 12/22/2021 Cristina Toma 9 Identify the Units Systems Fluid Properties Objective 2 12/22/2021 Cristina Toma 10 Identify the Units Systems Fluid Properties Objective 2 12/22/2021 Cristina Toma 11 Identify the Units Systems Fluid Properties Objective 3 ◼ Fluid Properties ◼ Fluids can be either liquids or gases ◼ ◼ ◼ Gases are readily compressible Liquids are only slightly compressible Compressibility: refers to the change in Volume (V) of a substance that is subjected to a change in pressure on it ◼ E Measured with Bulk Modulus of Elasticity or Bulk Modulus (E) − p = (V ) / V 12/22/2021 Cristina Toma 12 Identify the Units Systems Fluid Properties Objective 3 ◼ Fluid Properties: Compressibility ◼ Problem 1 ◼ A: 12/22/2021 Cristina Toma E − p = (V ) / V 13 Identify the Units Systems Fluid Properties Objective 3 ◼ Fluid Properties: Compressibility E − p = (V ) / V Problem 2 Compute the pressure change required to cause a decrease in the volume of mercury by 1.00 percent. Express the result in both psi and MPa. ◼ 12/22/2021 Cristina Toma 14 Identify the Units Systems Fluid Properties Objective 3 ◼ Fluid Properties: Compressibility 12/22/2021 Cristina Toma E − p = (V ) / V 15 Identify the Units Systems Fluid Properties Objective 3 ◼ Fluid Properties: Compressibility 12/22/2021 Cristina Toma E − p = (V ) / V 16 Identify the Units Systems Fluid Properties Objective 3 ◼ Density=is the amount of mass per unit volume of a substance m = V ◼ Where: m= mass V= volume Relationship between Weight (w) and Mass (m) w = mg Where: g= acceleration due to gravity g= 9.81 m/s2 or 32.2 ft/s2 12/22/2021 Cristina Toma 17 Identify the Units Systems Fluid Properties Objective 3 ◼ Specific Gravity = sg ◼ = ratio of the density of a substance to the density of the water at 4° C (at this temperature the water has the greatest density) sg = ◼ s @ 4C w The properties of water at 4° are constant, having the following values: kg w @ 4C = 1000 3 m 12/22/2021 Cristina Toma w @ 4C = 1.94 slugs ft 3 18 Identify the Units Systems Fluid Properties Objective 3 ◼ Specific weight=is the amount of weight per unit volume of a substance w = V Where: w= weight V= volume 12/22/2021 Cristina Toma 19 Identify the Units Systems Fluid Properties Objective 3 ◼ Problems 1) Calculate the weight of a reservoir of oil if it has a mass of 825 Kg. A: w=8.09 kN 12/22/2021 Cristina Toma 20 Identify the Units Systems Fluid Properties Objective 3 ◼ Problems 2) If the reservoir from previous problem has a volume of 0.917 m3, compute the density and the specific gravity of the oil. kg A: = 900 m3 12/22/2021 sg = 0.900 Cristina Toma 21 Identify the Units Systems Fluid Properties Objective 3 ◼ Problems 3) The specific gravity of benzene is 0.876. Calculate its density in SI units. A: = 876 12/22/2021 kg m3 Cristina Toma 22 Identify the Units Systems Fluid Properties Objective 3 ◼ Viscosity ◼ Dynamic (absolute) Viscosity= (eta) ◼ ) (tau) is developed in As a fluid moves, a shear stress ( it: v = y Where: = dynamic viscosity v =velocity gradient y (shear rate) 12/22/2021 Cristina Toma 23 Identify the Units Systems Fluid Properties Objective 3 ◼ Dynamic (absolute) Viscosity ◼ A fundamental condition that exist when a real fluid is in contact with a boundary surface is that the fluid has the same velocity as the boundary (see last figure ). ◼ ◼ If the distance between the 2 surfaces is small, then the rate of change of velocity with position y is linear. Example: stirring a fluid with a rod=>velocity gradient will be created ◼ A greater force is required to stir a cold oil having a high viscosity ( eta ) than is required to stir water, which has a low viscosity 12/22/2021 v = y Cristina Toma 24 Identify the Units Systems Fluid Properties Objective 3 ◼ Units for Dynamic (absolute) Viscosity v = y = v y N 1 N s = = = Pa s unit m2 m / s m2 m unit _ obsolete unit _ obsolete 12/22/2021 unit = Pa s = poise = 0.1 Pa s = centipoise( cP ) = poise / 100 = 0.001 Pa s = 1mPa s Cristina Toma 25 Identify the Units Systems Fluid Properties Objective 3 ◼ Values for Dynamic (absolute) Viscosity ◼ ◼ Appendixes A, B Appendix D: Variation of viscosity with temperature 12/22/2021 Cristina Toma 26 Identify the Units Systems Fluid Properties Objective 3 ◼ Kinematic Viscosity (the Greek letter nu) is defined as: ◼ Where: = (eta)= dynamic viscosity = density SI units for kinematic viscosity N m3 m3 = ( Pa s ) = 2 s = kg kg m 𝑚2 𝜐 =◼ 𝑠 ◼ 12/22/2021 m kg s2 m2 m3 s kg 𝑓𝑡 2 ( ) in US Customary System 𝑠 Cristina Toma 27 Identify the Units Systems Fluid Properties Objective 3 ◼ ◼ ◼ Problems 1) Appendix D gives dynamic viscosity for a variety of fluids as a function of temperature. Using Appendix D, give the value of the viscosity for the following fluids: ◼ 1) (2.18M) Water at 40°C ◼ 2) (2.19M) Water at 5°C 9 ◼ 3) (2.23M) Glycerine at at 20°C T F = TC + 32 5 ◼ 4) (2.25E) Water at 150°F ◼ 5) (2.29E) Glycerine at 110°F A: 1) ◼ Ns 6.5 10−4 2 m lbs − 3 5) 4.1 10 2 12/22/2021 ft Ns 2) 1.5 10−3 Ns 3) 1.9 2 m m2 Cristina Toma −6 4) 9.8 10 lbs ft 2 28 Identify the Units Systems Fluid Properties Objective 3 12/22/2021 Cristina Toma 29 Identify the Units Systems Fluid Properties Objective 3 12/22/2021 Cristina Toma 30 Identify the Units Systems Fluid Properties Objective 3 12/22/2021 Cristina Toma 31 Pressure: Types and measurement ◼ Objectives ◼ ◼ ◼ ◼ 1) Absolute and Gage Pressure 2) Relationship between Pressure and Elevation 3) Pressure Measurement ◼ Manometers ◼ Pressure gages and transducers 4) Problems 1/31/2022 Cristina Toma 1 Pressure: Types and measurement Objective 1 ◼ Pressure: Types and Measurement ◼ Absolute and Gage Pressure ◼ ◼ ◼ When making calculations involving pressure in a fluid, you must make the measurements relative to some reference pressure. Normally the reference pressure is that of the atmosphere, and the resulting measured pressure is called gage pressure. Pressure measured relative to a perfect vacuum is called absolute pressure. 1/31/2022 Cristina Toma 2 Pressure: Types and measurement Objective 1 ◼ Pressure: Types and Measurement ◼ Absolute and Gage Pressure pabs = pgage + patm where pabs = Absolute pressure= pressure measured relative to a perfect vacuum pgage = Gage pressure patm = Atmospheric pressure, patm=~101 kPa, [95-105kPa] 1/31/2022 Cristina Toma 3 Pressure: Types and measurement Objective 1 ◼ ◼ Pressure: Absolute and Gage Pressure pabs = Problems: ◼ pgage + patm 1) Express a pressure of 155 kPa (gage) as an absolute pressure. The local atmospheric pressure is 98 kPa(abs). Notice that the units in this calculation are kilopascals (kPa) for each term and are consistent. The indication of gage or absolute is for convenience and clarity. 1/31/2022 Cristina Toma 4 Pressure: Types and measurement Objective 1 ◼ ◼ Pressure: Absolute and Gage Pressure pabs = pgage + patm Problems: ◼ 2) Express a pressure of 225 kPa(abs) as a gage pressure. The local atmospheric pressure is 101 kPa(abs). A: Pgage= 124 kPa 1/31/2022 Cristina Toma 5 Pressure: Types and measurement Objective 1 ◼ ◼ Pressure: Absolute and Gage Pressure Problems: pabs = ◼ pgage + patm 3) Express an absolute pressure of 75.2 kPa as a gage pressure. The local atmospheric pressure is 103.4 kPa. A: Pgage= - 28.2 kPa Notice that the result is negative. This can also be read “28.2 kPa below atmospheric pressure” or “28.2 kPa vacuum.” 1/31/2022 Cristina Toma 6 Pressure: Types and measurement Objective 2 ◼ Relationship Elevation ◼ between Pressure and The change in pressure in a homogeneous liquid at rest due to a change in elevation can be calculated from: p = h p = change _ in _ pressure = specific _ weight _ of _ liquid h = change _ in _ elevation 1/31/2022 Cristina Toma w = V Where: w=weight V=volume 7 Pressure: Types and measurement Objective 2 ◼ Relationship between Pressure and Elevation ◼ ◼ The term elevation means the vertical distance from some reference level to a point of interest and is called z. A change in elevation between two points is called h. Elevation will always be measured positively in the upward direction. ◼ ◼ In other words, a higher point has a larger elevation than a lower point. Fig 3.2 shows the illustration of reference level for elevation 1/31/2022 Cristina Toma 8 Pressure: Types and measurement Objective 2 Relationship between Pressure and Elevation Problems 1) Calculate the change in water pressure from the surface to a depth of 5 m. Use Appendix A: Properties of water. ◼ A: p = 49.05kPa 1/31/2022 Cristina Toma 9 Pressure: Types and measurement Objective 2 Relationship between Pressure and Elevation Problems 2) Calculate the change in water pressure from the surface to a depth of 15 ft. Use Appendix A: Properties of water. ◼ A: lb p = 6.5 2 in 1/31/2022 Cristina Toma 10 Pressure: Types and measurement Objective 2 ◼ Relationship between Pressure and Elevation ◼ Conclusions ◼ ◼ ◼ ◼ p = h The equation is valid only for a homogeneous liquid at rest Points on the same horizontal level have the same pressure A decrease in elevation causes an increase in pressure (this is what happens when you go deeper in a swimming pool) An increase in elevation causes a decrease in pressure 1/31/2022 Cristina Toma 11 Pressure: Types and measurement Objective 2 p = h Problems: 2) (3.37E) When you dive to a depth of 12.5 ft in seawater, what is the pressure? ◼ A: p=5.56 psig 1/31/2022 Cristina Toma 12 Pressure: Types and measurement Objective 2 ◼ p = h Problems: 5) (3.44E) For the tank shown in next figure, determine the reading of the bottom pressure gage in psi if the top of the tank is vented to the atmosphere and the depth of the oil h is 28.5 ft. A: p=11.73 psig 1/31/2022 Cristina Toma 13 Pressure: Types and measurement Objective 2 ◼ p = h Problems: 1/31/2022 Cristina Toma 14 Pressure: Types and measurement Objective 2 ◼ p = h Problems: 3) (3.41E) Figure 3.20 shows a diagram of the hydraulic system for a vehicle lift. An air compressor maintains pressure above the oil in the reservoir. What must the air pressure be if the pressure at point A must be at least 180 psig. A: p=177.9 psig 1/31/2022 Cristina Toma 15 Pressure: Types and measurement Objective 2 ◼ p = h Problems: 1/31/2022 Cristina Toma 16 Pressure: Types and measurement Objective 2 ◼ p = h Problems: 6) (3.45E) For the tank shown in next figure, determine the reading of the bottom pressure gage in psi if the top of the tank is sealed, the top gage reads 50.0 psig, and the depth of the oil h is 28.5 ft A: p=61.73 psig 1/31/2022 Cristina Toma 17 Pressure: Types and measurement Objective 2 ◼ p = h Problems:3.56/58 A: p=237.3 kPa 1/31/2022 Cristina Toma 18 Pressure: Types and measurement Objective 2 ◼ p = h Problems: 1/31/2022 Cristina Toma 19 Pressure: Types and measurement Objective 2 ◼ p = h Problems: 1/31/2022 Cristina Toma 20 Pressure: Types and measurement Objective 3 ◼ ◼ Manometers (U-tube) uses the relationship between a change in pressure and a change in elevation in a static fluid. ◼ Fig 3.10 shows the U-tube manometer. 1/31/2022 Cristina Toma 21 Pressure: Types and measurement Objective 3 p = h Manometers (U-tube) Because the fluids in the manometer are at rest, the equation Δp=γh can be used to write expressions for the changes in pressure that occur throughout the manometer. These expressions can then be combined and solved algebraically for the desired pressure. 1/31/2022 Cristina Toma 22 Pressure: Types and measurement Objective 3 ◼ Manometers (U-tube) Using Fig. 3.10, calculate the pressure at point A. ◼ The only point for which the pressure is known is the surface of the mercury in the right leg of the manometer, point 1. ◼ p1 = patm = 0 _ Pa ( gage) p2 = p1 + m 0.25m 1/31/2022 p2 = p3 Cristina Toma 23 Pressure: Types and measurement Objective 3 ◼ Manometers(U-tube) Using Fig. 3.10, calculate the pressure at point A. ◼ p = p3 − p4 = w h = w 0.40m p3 = p4 + w 0.40m sg = s w@ 4C = w = 9.81 1/31/2022 s w@ 4C kN m3 W s g V s = = g W w@ 4C V w @ 4C m = (sg ) m 9.81 kN kN kN = 13. 54 9. 81 = 132. 8 m3 m3 m3 Cristina Toma 24 Pressure: Types and measurement Objective 3 ◼ Manometers (U-tube) Using Fig. 3.10, calculate the pressure at point A. ◼ p A = p4 pA = p1 + m 0.25m − w 0.40m pA = 0 + (132.8kN / m3 ) (0.25m) − (9.81kN / m3 ) (0.40m) p A = 29.3kPa 1/31/2022 Cristina Toma 25 Pressure: Types and measurement Objective 3 ◼ Manometers (Differential) Using Fig. 3.12, calculate the pressure between points A and B: ◼ 1/31/2022 Cristina Toma 26 Pressure: Types and measurement Objective 3 ◼ Manometers Well-Type Manometer ◼ When a pressure is applied to a welltype manometer, the fluid level in the well drops a small amount while the level in the right leg rises a larger amount in proportion to the ratio of the areas of the well and the tube. A scale is placed alongside the tube so that the deflection can be read directly. The scale is calibrated to account for the small drop in the well level. 1/31/2022 Cristina Toma 27 Pressure: Types and measurement Objective 3 ◼ Manometers Inclined well-type manometer. ◼ It has the same features as the well-type manometer but offers a greater sensitivity by placing the scale along the inclined tube. The scale length is increased as a function of the angle of inclination of the tube,θ. where L is the scale length and h is the manometer deflection 1/31/2022 Cristina Toma 28 Pressure: Types and measurement Objective 3 ◼ Pressure gages, Pressure Transducers For those situations where only a visual indication is needed at the site where the pressure is being measured, a pressure gage is used. In other cases there is a need to measure pressure at one point and display the value at another. The general term for such a device is pressure transducer, meaning that the sensed pressure causes an electrical signal to be generated that can be transmitted to a remote location such as a central control station where it is displayed digitally. 1/31/2022 Cristina Toma 29 Pressure: Types and measurement Objective 3 Bourdon tube pressure gage ◼ For those situations where only a visual indication is needed at the site where the pressure is being measured, a pressure gage is used. The pressure to be measured is applied to the inside of a flattened tube, which is normally shaped as a segment of a circle or a spiral. The increased pressure inside the tube causes it to be straightened somewhat. The movement of the end of the tube is transmitted through a linkage that causes a pointer to rotate. 1/31/2022 Cristina Toma 30 Pressure: Types and measurement Objective 3 Strain gage pressure transducer and indicator. ◼ The pressure to be measured is introduced through the pressure port and acts on a diaphragm to which foil strain gages are bonded. As the strain gages sense the deformation of the diaphragm, their resistance changes. The readout device is typically a digital voltmeter, calibrated in pressure units. 1/31/2022 Cristina Toma 31 Pressure: Types and measurement Objective 4 ◼ Problems 1) (3.62) Water is in the pipe shown in the next figure. Calculate the pressure at point A in kPa (gage) ◼ A: -10.9 kPa (gage) 1/31/2022 Cristina Toma 32 Pressure: Types and measurement Objective 4 1/31/2022 Cristina Toma 33 Pressure: Types and measurement Objective 4 1/31/2022 Cristina Toma 34 Pressure: Types and measurement Objective 4 1/31/2022 Cristina Toma 35 Pressure: Types and measurement Objective 4 1/31/2022 Cristina Toma 36
