# Applied Fluid Mechanics

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1 Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and Bernoulli s Equation 7. General Energy Equation 8. Reynolds Number, Laminar Flow, Turbulent Flow and Energy Losses Due to Friction Applied Fluid Mechanics 9. Velocity Profiles for Circular Sections and Flow in Noncircular Sections 10.Minor Losses 11.Series Pipeline Systems 12.Parallel Pipeline Systems 13.Pump Selection and Application 14.Open-Channel Flow 15.Flow Measurement 16.Forces Due to Fluids in Motion 1

2 Applied Fluid Mechanics 17.Drag and Lift 18.Fans, Blowers, Compressors and the Flow of Gases 19.Flow of Air in Ducts Chapter Objectives Differentiate between a gas and a liquid. Define pressure. Identify the units for the basic quantities of time, length, force, and mass in the SI system (metric unit system). Identify the units for the basic quantities of time, length, force, and mass in the U.S. Customary System. Properly set up equations to ensure consistency of units. Define the relationship between force and mass. Define density, specific weight, and specific gravity. Identify the relationships between specific weight, specific gravity, and density, and solve problems using these relationships. Define surface tension. 2

3 Chapter Outline 1. Basic Introductory Concepts 2. The International System of Units (SI) 3. The US Customary System 4. Weight and Mass 5. Temperature 6. Consistent Units In an Equation 7. The Definition of Pressure 8. Compressibility 9. Density, Specific Weight and Specific Gravity 10. Surface Tension 1.1 Basic Introductory Concepts Figure 1.1 shows the typical piping system for fluid power. 3

4 1.1 Basic Introductory Concepts Pressure Pressure is defined as the amount of force exerted on a unit area of a substance or on a surface. This can be stated by the equation p = A F ( 1 1) 1.1 Basic Introductory Concepts Liquid and Gases Fluids can be either liquids or gases. The following general descriptions of liquids and gases that we will use in this book: 1. Gases are readily compressible. 2. Liquids are only slightly compressible. 4

5 1.1 Basic Introductory Concepts Weight and Mass Mass is the property of a body of fluid that is a measure of its inertia or resistance to a change in motion. It is also a measure of the quantity of fluid. Weight is the amount that a body of fluid weighs, that is, the force with which the fluid is attracted toward Earth by gravitation. We use the symbol m for mass and w for weight in this book. 1.1 Basic Introductory Concepts Fluid Properties Other fluid properties are specific weight, density, specific gravity, surface tension and viscosity. It is also important in determining the character of the flow of fluids and the amount of energy that is lost from a fluid flowing in a system. 5

6 1.2 The International System of Units (SI) The SI units for the basic quantities are An equivalent unit for force indicated above is derived from the relationship between force and mass, where a is the acceleration expressed (m/s 2 ) 1.2 The International System of Units (SI) Therefore, the derived unit for force is Similarly 6

7 1.2.1 SI Unit Prefixes Table 1.1 shows the SI unit prefixes SI Unit Prefixes Results of calculations should normally be adjusted so that the number is between 0.1 and times some multiple of Some examples follow. 7

8 1.3 The US Customary System U.S. Customary System defines the basic quantities as follows: It may help to note the relationship between force and mass, 1.4 Weight and Mass Weight is a force and mass is the quantity of a substance. We relate these two terms by applying Newton s law of gravitation stated as force equals mass times acceleration, or. Weight and mass relationship becomes w = mg ( 1 2) 8

9 1.4.1 Weight and Mass in the SI Unit System For example, consider a rock with a mass of 5.60 kg suspended by a wire. To determine what force is exerted on the wire, we use Newton s law of gravitation Weight and Mass in the SI Unit System We can also compute the mass of an object if we know its weight. For example, assume that we have measured the weight of a valve to be 8.25 N. What is its mass? 9

10 1.4.2 Weight and Mass in the US Unit System For an example of the weight mass relationship in the U.S. Customary System, assume that we have measured the weight of a container of oil to be 84.6 lb. What is its mass? We write Mass expressed as Ibm In the analysis of fluid systems, some professionals use the unit lbm (pounds-mass) for the unit of mass instead of the unit of slugs. When one tries to relate force and mass units using Newton s law, one obtains This is not the same as the lbf. In summary, because of the cumbersome nature of the relationship between lbm and lbf, we avoid the use of lbm in this book. Mass will be expressed in the unit of slugs when 2005 Pearson problems Education South Asia Pte are Ltd in the U.S. Customary System of units. 10

11 1.5 Temperature Temperature is most often indicated in (degrees Celsius) or (degrees Fahrenheit). The following values at sea level on Earth is as follow: Given the temperature in F the temperature in C is Given the temperature in C the temperature in F is Absolute Temperature The absolute temperature is defined so the zero point corresponds to the condition where all molecular motion stops. This is called absolute zero. In the SI unit system, the standard unit of temperature is the kelvin, for which the standard symbol is K and the reference (zero) point is absolute zero. We can then make the conversion from the Celsius to the kelvin scale by using 11

12 1.6 Consistent Units in an Equation A simple straightforward procedure called unit cancellation will ensure proper units in any kind of calculation, not only in fluid mechanics, but also in virtually all your technical work. Table 1.2 shows the SI units for common quantities used in fluid mechanics. Table 1.3 shows the U.S. customary units for common quantities used in fluid mechanics. 1.6 Consistent Units in an Equation 12

13 1.6 Consistent Units in an Equation 1.6 Consistent Units in an Equation The six steps of the procedure are listed below. UNIT-CANCELLATION PROCEDURE 1) Solve the equation algebraically for the desired term. 2) Decide on the proper units for the result. 3) Substitute known values, including units. 4) Cancel units that appear in both the numerator and the denominator of any term. 5) Use conversion factors to eliminate unwanted units and obtain the proper units as decided in Step 2. 6) Perform the calculation. 13

14 Example 1.1 Imagine you are traveling in a car at a constant speed of 80 kilometers per hour (km/h). How many seconds (s) would it take to travel 1.5 km? For the solution, use the equation where s is the distance traveled, is the speed, and t is the time. Using the unit-cancellation procedure outlined above, what is the first thing to do? Example 1.1 The first step is to solve for the desired term. Because you were asked to find time, you should have written Step 2 is to decide on the proper units for the result, in this case time. From the problem statement the proper unit is seconds. If no specification had been given for units, you could choose any acceptable time unit such as hours. Proceed to Step 3. The result should look something like this: 14

15 Example 1.1 For the purpose of cancellation it is not convenient to have the units in the form of a compound fraction as we have above. To clear this to a simple fraction, write it in the form After some practice, equations may be written in this form directly. Now perform Step 4 of the procedure. Example 1.1 The result should now look like this: The equation in the preceding panel showed the result for time in hours after kilometer units were cancelled. Although hours is an acceptable time unit, our desired unit is seconds as determined in Step 2. Thus the conversion factor 3600 s/1 h is required. 15

16 Example 1.1 The units determine this. Our objective in using the conversion factor was to eliminate the hour unit and obtain the second unit. Because the unwanted hour unit was in the numerator of the original equation, the hour unit in the conversion factor must be in the denominator in order to cancel. Now that we have the time unit of seconds we can proceed with Step 6. The correct answer is t = 67.5 s. 1.7 The Definition of Pressure Pressure is defined as the amount of force exerted on a unit area of a substance. This can be stated by the equation p = A F ( 1 3) Two important principles about pressure were described by Blaise Pascal, a seventeenth-century scientist: 1. Pressure acts uniformly in all directions on a small volume of a fluid. 2. In a fluid confined by solid boundaries, pressure acts perpendicular to the boundary. 16

17 1.7 The Definition of Pressure Fig 1.2 shows the pressure acting uniformly in all directions on a small volume of fluid. 1.7 The Definition of Pressure Fig 1.3 shows the direction of fluid pressure on boundaries. 17

18 1.7 The Definition of Pressure The bar is another unit used by some people working in fluid mechanics and thermodynamics. The bar is defined as 10 5 Pa or 10 5 N/m 2. Another way of expressing the bar is 1 bar = 100 x 10 3 N/m 2, which is equivalent to 100 kpa. Example 1.2 Figure 1.4 shows a container of liquid with a movable piston supporting a load. Compute the magnitude of the pressure in the liquid under the piston if the total weight of the piston and the load is 500 N and the area of the piston is 2500 mm 2. 18

19 Example 1.2 It is reasonable to assume that the entire surface of the fluid under the piston is sharing in the task of supporting the load. The second of Pascal s laws states that the fluid pressure acts perpendicular to the piston. Then, using Eq. (1 3), we have Example 1.2 The standard unit of pressure in the SI system is the N/m 2 called the pascal (Pa) in honor of Blaise Pascal. The conversion can be made by using the factor 10 3 mm = 1 m. We have Note that the pressure in N/mm 2 is numerically equal to pressure in MPa. It is not unusual to encounter pressure in the range of several megapascals (MPa) or several hundred kilopascals (kpa). 19

20 Example 1.3 A load of 200 pounds (lb) is exerted on a piston confining oil in a circular cylinder with an inside diameter of 2.50 inches (in). Compute the pressure in the oil at the piston. Example 1.3 To use Eq. (1 3), we must compute the area of the piston 20

21 Example 1.3 Although the standard unit for pressure in the U.S. Customary System is pounds per square foot (lb/ft 2 ), it is not often used because it is inconvenient. Length measurements are more conveniently made in inches, and pounds per square inch (lb/in 2 ) abbreviated psi, is used most often for pressure in this system. The pressure in the oil is 40.7 psi. This is a fairly low pressure; it is not unusual to encounter pressures of several hundred or several thousand psi. 1.8 Compressibility Compressibility refers to the change in volume (V) of a substance that is subjected to a change in pressure on it. The usual quantity used to measure this phenomenon is the bulk modulus of elasticity or, simply, bulk modulus, E: E = Δp / V ( ΔV ) ( 1 4) 21

22 1.8 Compressibility Table 1.4 shows the Values for bulk modulus for selected liquids at atmospheric pressure and 68 F (20 C). Example 1.4 Compute the change in pressure that must be applied to water to change its volume by 1.0 percent. The 1.0-percent volume change indicates that ΔV/V = Then, the required change in pressure is 22

23 1.9 Density, Specific Weight and Specific Gravity Density is the amount of mass per unit volume of a substance. Therefore, using the greek letter ρ (rho) for density, we write ( 1 5) E = m / V where V is the volume of the substance having a mass m. The units for density are kilograms per cubic meter in the SI system and slugs per cubic foot in the U.S. Customary System. 1.9 Density, Specific Weight and Specific Gravity Specific weight is the amount of weight per unit volume of a substance. Using the Greek letter γ (gamma) for specific weight, we write ( 1 6) γ = w/ V where V is the volume of a substance having the weight w. The units for specific weight are newtons per cubic meter in the SI system and pounds per cubic foot in the U.S. Customary System. 23

24 1.9 Density, Specific Weight and Specific Gravity Specific gravity can be defined in either of two ways: a. Specific gravity is the ratio of the density of a substance to the density of water at 4 C. b. Specific gravity is the ratio of the specific weight of a substance to the specific weight of water at 4 C. These definitions for specific gravity (sg) can be shown mathematically as where the subscript s refers to the substance whose specific gravity is being determined and the subscript w refers to water. γ s ρs sg = = C C w w ( 1 7) Relation Between Density and Specific Weight Quite often the specific weight of a substance must be found when its density is known and vice versa. The conversion from one to the other can be made using the following equation: γ = ρg ( 1 9) The definition of specific weight is By multiplying g on numerator and denominator, we reduced to 24

25 Example 1.5 Calculate the weight of a reservoir of oil if it has a mass of 825 kg. We have Example 1.6 If the reservoir from Example Problem 1.5 has a volume of 0.917m 3, compute the density, the specific weight, and the specific gravity of the oil. 25

26 Example 1.7 Glycerine at 20 C has a specific gravity of Compute its density and specific weight. Example 1.8 A pint of water weighs N. Find its mass. Because w = mg, the mass is Remember that the units of kg and Ns 2 /m are the same. 26

27 Example 1.9 One gallon of mercury has a mass of 51.2 kg. Find its weight. Write This is correct, but the units may seem confusing because weight is normally expressed in N. The units of mass may be rewritten as Ns 2 /m, and we have Specific Gravity in Degrees Baume or Degree API The reference temperature for specific gravity measurements on the Baumé or American Petroleum Institute (API) scale is 60 F rather than 4 C as defined before. To emphasize this difference, the API or Baumé specific gravity is often reported as 27

28 1.9.2 Specific Gravity in Degrees Baume or Degree API For liquids heavier than water, 145 sg = 145 deg Baume ( 1 10) 145 deg Baume = 145 sg For liquids lighter than water, 140 sg = deg Baume 140 deg Baume = 130 sg ( 1 11) ( 1 12) ( 1 13) Specific Gravity in Degrees Baume or Degree API The API has developed a scale that is slightly different from the Baumé scale for liquids lighter than water. The formulas are sg = deg API deg API = sg ( 1 14) ( 1 15) 28

29 1.9.2 Specific Gravity in Degrees Baume or Degree API Fig 1.5 shows the Hydrometer with built-in thermometer (thermohydrometer) Surface Tension Surface tension acts somewhat like a film at the interface between the liquid water surface and the air above it. Surface tension is also the reason that water droplets assume a nearly spherical shape. The movement of liquids within small spaces depends on this capillary action. Wicking is the term often used to describe the rise of a fluid from a liquid surface into a woven material. 29

30 1.10 Surface Tension Table 1.5 shows the surface tension of water Surface Tension Table 1.6 shows the surface tension of some common liquids. 30

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