1 Hydraulic Engineering Lecture Notes for CIVE 401 Part I Hydrostatic Review and Closed Conduits By Pierre Y. Julien with the assistance of Joonhak Lee 2020 Fall Semester 8-17-20 2 Table of Contents 1. Review of Hydrostatics (Week 1) 3 1.1. Dimensions and Units 3 1.2. Properties of Water 6 1.3. Specific Gravity 9 1.4. Atmospheric Pressure 10 1.5. Hydrostatic Pressure 11 1.6. Vapor Pressure 12 1.7. Piezometric Head 14 1.8. Hydrostatic Force on a Plate 15 1.9. Hydrostatic Force on a Dam 18 2. Flow in Pipes (Week 2) 24 2.1. Continuity/Conservation of Mass 24 2.2. Friction Losses in Pipes 25 2.3. Three Reservoirs 30 2.4. Pipe Networks 32 2.5. Minor Losses in Pipes 35 2.6. Negative Pressure/Suction 40 3. Hydrodynamic Forces (Week 3) 41 3.1. Momentum Force on a Plate 41 3.2. Hydrodynamic Force on a Pipe Bend 43 4. Flow Meters (Week 4) 46 4.1. Manometer 46 4.2. Venturi Meter 47 4.3. Flow Nozzle 48 4.4. Various Flow Meters 49 3 1. Review of Hydrostatics (Week 1) We review hydrostatics and start to solve problems! 1.1. Dimensions and units Physical water properties are usually expressed in terms of the following fundamental dimensions: mass (M), length (L), time (T), and temperature (T°) is also considered. In the SI system of units, the units for mass, length, time, and temperature are the kilogram (kg), the meter (m), the second (s), and degrees Celsius (°C). A Newton (N) is the force to accelerate 1 kg at 1 m/s2, or 1N = 1 kg m/s2. The gravitational acceleration at the Earth’s surface is g = 9.81 m/s2. The weight of one kilogram is F = mass g = 1 kg 9.81 m/s2 = 9.81 N. The pressure is a perpendicular force per unit area given in pascals from 1 Pa = 1 N/m2. The unit of work (or energy) is the joule (J), which equals the product of 1 N 1 m. Table 1.1 gives the unit of power is a watt (W), which is 1 J/s. In English units, the time unit is a second, the units for mass, length, time, and temperature are the slug (slug), the foot (ft), the second (s), and degrees Fahrenheit (°F). The force to accelerate a mass of one slug at 1 ft/s2 is a pound force (lb). A pound always refers to a force, not a mass. The temperature in degrees Celsius T°C is converted to degrees Fahrenheit T°F as T°F = 32.2 °F + 1.8 T°C. A useful conversion factor for weight is 1 lb = 4.448 N, and for discharge 1 m3/s = 35.32 ft3/s, with abbreviation 1 cms = 1m3/s and 1 cfs = 1ft3/s. Other conversions are found in Table 1.2. Prefixes indicate multiples or fractions of units by powers of 10: (micro)== 10−63 , k(kilo) 10 , m(milli)== 10−36 , M(mega) 10 , c(centi)== 10−29 , G(giga) 10 . For example, one megawatt (MW) equals one million watts (1,000,000 W or 106 W). 4 Table 1.1. Geometric, kinematic, dynamic, and dimensionless variables Variable Symbol Fundamental SI Units dimensions Geometric (L) Length L, x, h L m Area A L2 m2 Volume ∀ L3 m3 Kinematic (L, T) Velocity V LT–1 m/s Acceleration a, g LT–2 m/s2 Kinematic viscosity L2T–1 m2/s Unit discharge q L2T–1 m2/s Discharge Q L3T–1 m3/s Dynamic (M, L, T) Mass m M 1 kg Force F = ma, mg MLT–2 1 kg m/s2 =1N Pressure p = F/A ML–1T–2 1 N/m2 = 1 Pa Work or energy E = Fd ML2T–2 1 Nm= J Mass density ρ ML–3 kg/m3 Specific weight γ = ρ g ML–2T–2 N/m3 Dynamic viscosity μ = ρ v ML–1T–1 1 kg/m s = 1 Pa s Dimensionless Slope S0, Sf – – Specific gravity G = γs /γ – – Reynolds number Re = V h / – – Froude number Fr = u / (g h)0.5 – – 5 Table 1.2. Useful unit conversions Unit kg, m, s N, Pa, W 1 acre 4,047 m2 1 atmosphere (atm) 101,3 kg/m s2 101.3 kPa 1 cubic foot per second (ft3/s) 0.0283 m3/s 1 degree Fahrenheit (°F) = 32+1.8 T°C 0.5556 °K 1 foot (ft) 0.3048 m 1 gallon (U.S. gal) (1 U.S. gal = 3.785 liters) 0.003785 m3 1 horsepower (hp) = 550 lb ft/s 745.7 kg m2/s3 745.7 W 1 inch (in.) (1 ft = 12 in.) 0.0254 m 1 inch of mercury (in. Hg) 3,386 kg/m s2 3,386 Pa 1 inch of water 248.8 kg/m s2 248.8 Pa 1 mile (statute), (1mile = 5,280 ft) 1609 m 1 million gallons per day (1 mgd = 1.55 ft3/s) 0.04382 m3/s 1 fluid ounce (U.S.) 2.957 10–5 m3 1 pascal (Pa) 1 kg/m s2 1 N/m2 1 pound-force (lb) (1 lb = 1 slug 1 ft/s2) 4.448 kg m/s2 4.448 N 1 pound per square foot (lb/ft2 or psf) 47.88 kg/m s2 47.88 Pa 1 pound per square inch (lb/in.2 or psi) 6,895 kg/m s2 6,895 Pa 1 quart (U.S.) (1 qt = 2 pints) 0.0009463 m3 1 slug 14.59 kg 1 ton (S.I. metric) (1,000 kg = 1 Mg) 1,000 kg 1 ton (U.S. short) = 2,000 lb 8,900 kg m/s2 8.9 kN 1 yard (yd) (1 yd = 3 ft) 0.9144 m 6 1.2. Properties of water Mass density of water ρ. The mass of water per unit volume defines the mass density ρ. The maximum mass density of water at 4 °C is 1,000 kg/m3 and decreases slightly with temperature, as shown in Table 1.3. In comparison, the mass density of sea water is approximately 1,025 kg/m3 and the mass density of air at sea level is 1.29 kg/m3 at 0 °C. The conversion factor for mass density is 1 slug/ft3 = 515.4 kg/m3. Specific weight of water γ. The weight per unit volume is the specific weight γ. At 10 °C, water has a specific weight, γ = 9,810 N/m3 or 62.4 lb/ft3 (1 lb/ft3 = 157.09 N/m3). Specific weight varies with temperature as given in Table 1.3. The specific weight γ is the product of mass density ρ and gravitational acceleration g = 32.2 ft/s2 = 9.81 m/s2: = g Dynamic viscosity μ. As a fluid is brought into low rates of deformation, the velocity of the fluid at any boundary equals the velocity of the boundary. The fundamental dimension of the dynamic viscosity μ is M/LT. In Table 1.3, the dynamic viscosity of water decreases with temperature. The dynamic viscosity of water at 20 °C is 0.001 kg/m s = 0.001 N s/m2 = 0.001 Pa s. The conversion factor for the dynamic viscosity is 1 lb s/ft2 = 47.88 N s/m2 = 47.88 Pa s. Kinematic viscosity of water . The kinematic viscosity is obtained when the dynamic viscosity of a fluid is divided by its mass density . The kinematic viscosity v of water in L2/T in Table 1.3 decreases with temperature. The viscosity of clear water at 20 °C is 0.01 cm2/s = 1 10-6 m2/s. The conversion factor is 1 ft2/s = 0.0929 m2/s. The kinematic viscosity of water v depends on temperature T° as = = =[1.14 − 0.031(TT − 15) + 0.00068( − 15)2 ] 10− 6 m 2 / s CC 7 Table 1.3. Physical properties of water in SI Units at atmospheric pressure Temperature Vapor Density Specific Dynamic Kinematic Surface Modulus of Pressure Weight Viscosity Viscosity Tension Elasticity 푝 γ μ σ 퐸 T 푣 ρ 푣 (◦C) (kN/m2) (kg/m3 ) (kN/m3 ) (N-s/m2) (m2/s) (N/m) (kN/m2) × 106 absolute × 10−3 × 10−6 0 0.61 999.8 9.805 1.781 1.785 0.0756 2.02 5 0.87 1000.0 9.807 1.518 1.519 0.0749 2.06 10 1.23 999.7 9.804 1.307 1.306 0.0742 2.10 15 1.70 999.1 9.798 1.139 1.139 0.0735 2.14 20 2.34 998.2 9.789 1.002 1.003 0.0728 2.18 25 3.17 997.0 9.777 0.890 0.893 0.0720 2.22 30 4.24 995.7 9.764 0.798 0.800 0.0712 2.25 40 7.38 992.2 9.730 0.653 0.658 0.0696 2.28 50 12.33 988.0 9.689 0.547 0.553 0.0679 2.29 60 19.92 983.2 9.642 0.466 0.474 0.0662 2.28 70 31.16 977.8 9.589 0.404 0.413 0.0644 2.25 80 47.34 971.8 9.530 0.354 0.364 0.0626 2.20 90 70.10 965.3 9.466 0.315 0.326 0.0608 2.14 100 101.33 958.4 9.399 0.282 0.294 0.0589 2.07 Commonly used values of specific weight and viscosity are: Weight mass ퟏ, ퟎퟎퟎ 퐤퐠 9.81 m ퟗ.