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1、Elementary Mechanics of Fluids,CE 319 F Daene McKinney,Flow in Pipes,Reynolds Experiment,Reynolds Number Laminar flow: Fluid moves in smooth streamlines Turbulent flow: Violent mixing, fluid velocity at a point varies randomly with time Transition to turbulence in a 2 in. pipe is at V=2 ft/s, so mos
2、t pipe flows are turbulent,Laminar,Turbulent,Shear Stress in Pipes,Steady, uniform flow in a pipe: momentum flux is zero and pressure distribution across pipe is hydrostatic, equilibrium exists between pressure, gravity and shear forces,Since h is constant across the cross-section of the pipe (hydro
3、static), and dh/ds0, then the shear stress will be zero at the center (r = 0) and increase linearly to a maximum at the wall. Head loss is due to the shear stress.,Applicable to either laminar or turbulent flow Now we need a relationship for the shear stress in terms of the Re and pipe roughness,Dar
4、cy-Weisbach Equation,Darcy-Weisbach Eq.,Friction factor,Laminar Flow in Pipes,Laminar flow - Newtons law of viscosity is valid:,Velocity distribution in a pipe (laminar flow) is parabolic with maximum at center.,Discharge in Laminar Flow,Head Loss in Laminar Flow,Nikuradses Experiments,In general, f
5、riction factor Function of Re and roughness Laminar region Independent of roughness Turbulent region Smooth pipe curve All curves coincide Re=2300 Rough pipe zone All rough pipe curves flatten out and become independent of Re,Rough,Smooth,Laminar,Transition,Turbulent,Blausius OK for smooth pipe,Mood
6、y Diagram,Pipe Entrance,Developing flow Includes boundary layer and core, viscous effects grow inward from the wall Fully developed flow Shape of velocity profile is same at all points along pipe,Entrance Loss in a Pipe,In addition to frictional losses, there are minor losses due to Entrances or exi
7、ts Expansions or contractions Bends, elbows, tees, and other fittings Valves Losses generally determined by experiment and then corellated with pipe flow characteristics Loss coefficients are generally given as the ratio of head loss to velocity head K loss coefficent K 0.1 for well-rounded inlet (h
8、igh Re) K 1.0 abrupt pipe outlet K 0.5 abrupt pipe inlet,Abrupt inlet, K 0.5,Elbow Loss in a Pipe,A piping system may have many minor losses which are all correlated to V2/2g Sum them up to a total system loss for pipes of the same diameter Where,EGL & HGL for Losses in a Pipe,Entrances, bends, and
9、other flow transitions cause the EGL to drop an amount equal to the head loss produced by the transition. EGL is steeper at entrance than it is downstream of there where the slope is equal the frictional head loss in the pipe. The HGL also drops sharply downstream of an entrance,Ex(10.2),Given: Liqu
10、id in pipe has g= 8 kN/m3. Acceleration = 0. D = 1 cm, m = 3x10-3 N-m/s2. Find: Is fluid stationary, moving up, or moving down? What is the mean velocity? Solution: Energy eq. from z = 0 to z = 10 m,1,2,Ex (10.4),Given: Oil (S = 0.97, m = 10-2 lbf-s/ft2) in 2 in pipe, Q = 0.25 cfs. Find: Pressure dr
11、op per 100 ft of horizontal pipe. Solution:,Ex. (10.8),Given: Kerosene (S=0.94, m=0.048 N-s/m2). Horizontal 5-cm pipe. Q=2x10-3 m3/s. Find: Pressure drop per 10 m of pipe. Solution:,Ex. (10.34),Given: Glycerin 20oC flows commercial steel pipe. Find: Dh Solution:,Ex. (10.43),Given: Figure Find: Estim
12、ate the elevation required in the upper reservoir to produce a water discharge of 10 cfs in the system. What is the minimum pressure in the pipeline and what is the pressure there? Solution:,Ex. (10.68),Given: Commercial steel pipe to carry 300 cfs of water at 60oF with a head loss of 1 ft per 1000
13、ft of pipe. Assume pipe sizes are available in even sizes when the diameters are expressed in inches (i.e., 10 in, 12 in, etc.). Find: Diameter. Solution: Assume f = 0.015,Relative roughness:,Get better estimate of f,f=0.010,Use a 90 in pipe,Ex. (10.81),Given: The pressure at a water main is 300 kPa
14、 gage. What size pipe is needed to carry water from the main at a rate of 0.025 m3/s to a factory that is 140 m from the main? Assume galvanized-steel pipe is to be used and that the pressure required at the factory is 60 kPa gage at a point 10 m above the main connection. Find: Size of pipe. Soluti
15、on:,Assume f = 0.020,Relative roughness:,Friction factor:,Use 12 cm pipe,Ex. (10.83),Given: The 10-cm galvanized-steel pipe is 1000 m long and discharges water into the atmosphere. The pipeline has an open globe valve and 4 threaded elbows; h1=3 m and h2 = 15 m. Find: What is the discharge, and what
16、 is the pressure at A, the midpoint of the line? Solution:,D = 10-cm and assume f = 0.025,So f = 0.025,Near cavitation pressure, not good!,Ex. (10.95),Given: If the deluge through the system shown is 2 cfs, what horsepower is the pump supplying to the water? The 4 bends have a radius of 12 in and the 6-in pipe is smooth. Find: Horsepower Solution:,So f = 0.0135,
