Module 3c: Flow in Pipes Hazen-Williams Equation
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1 Module c: Flow in Pipes Based on experimental work Used to calculated velocity in a pipe based on the relative roughness and slope of the energy line V = kcr Robert Pitt University of Alabama And hirley Clark Penn tate - Harrisburg Where V = velocity C = factor for relative roughness R = hydraulic radius = slope of the energy line (head loss divided by pipe length) k = conversion factor for unit system k = for units of m/sec k = 1.18 for units of ft/sec Type of Pipe C Pipe Material C Pipes extremely straight and smooth 140 New cast iron 10 Pipes very smooth 10 5-year old cast iron mooth wood, smooth masonry 20-year old cast iron New riveted steel, vitrified clay 110 Average concrete 10 Old cast iron, ordinary brick New welded steel Old riveted steel 95 Asbestos cement 140 Old iron in bad condition Plastic 150 In: Metcalf & Eddy, Inc. and George Tchobanoglous. Wastewater Engineering: Collection and Pumping of Wastewater. McGraw-Hill, Inc Table 2-2. In: Warren Viessman, Jr. and Mark Hammer. Water upply and Pollution Control, ixth Edition. Addison-Wesley Table
2 Description of Pipe Value of C Cast iron 10 New 10 5 years old 10 years old years old 90 0 years old Concrete Cement lined 140 Welded steel As for cast iron 5 years older Riveted steel As for cast iron 10 years older Plastic 150 Asbestos Cement 140 Channel Characteristics: Hydraulic Radius and Wetted Perimeter Wetted Perimeter surface of pipe or channel where fluid is touching (accounts for areas where friction effects are occurring). Not used to describe any area that is open to the atmosphere since friction contact with the atmosphere is negligible. In: Terence McGhee. Water upply and ewerage, ixth Edition. McGraw-Hill Channel Characteristics: Hydraulic Radius and Wetted Perimeter Hydraulic Radius, R: R = A P Where A = cross-sectional area of flow P = wetted perimeter Hydraulic radius will have units of length. For circular pipes flowing full, the hydraulic radius is: A = area = πd 2 /4 P = circumference = πd R = A/P = (πd 2 /4)/(πD) = D/4 For circular pipes flowing full, the Hazen-Williams formula can be restated: For a circular pipe flowing full: By Continuity: R = D 4 Q = VA 2
3 For circular pipes flowing full, the Hazen-Williams formula can be restated: Conversion Factor: ( 7.48gal / ft) = /sec( 60sec/ min) 1ft/sec 7.48gal /sec 7.48gal = 448gal / min 448gal / min(60 min/ hr)(24hr / day) = 646,272gal / day 646,272gal / day million gal/day Conversion Factor = MGD / cfs ubstituting into Hazen-Williams: V[ ft /sec] = 1.18CR V[ ft /sec] A[ ft Q[ ft π 2 D /sec] = 1.18 D C D Q = 1.05D C Q = CD 4 Q = 0. 42CD ] = 1.18ACR [units of ft /sec and ft] ubstituting into Hazen-Williams: Q = 0.42CD Q = 0.42CD Q = 0.279CD [units of ft Where Q = flow (MGD) D = diameter (ft) (0.646MGD / cfs) /sec] For I units, the Hazen-Williams equation for pipes flowing full: Where Q = 0.278CD Q = flow (m /sec) D = pipe diameter (m)
4 From: Water upply and Pollution Control, ixth Edition. Warren Viessman, Jr., and Mark J. Hammer. Addison Wesley, Determine the head loss in a 0-m pipeline with a diameter of 500 mm that is discharging 0.25 m /sec. Assume that the Hazen-Williams coefficient for the pipe equals 10. Given: L = 0 m D = 0.5 m Q = 0.25 m /sec C = 10 Using the Hazen-Williams equation for flow: Q = By definition: = L ubstituting: Q = 0.278CD CD L olving for h f : Q = 0.278CD L Q = L 0.278CD Q = L 0.278CD ubstituting : (0.25m / sec) = (0m) 0.278(10)(0.5m) h = 2.94m f 1/ 1/
5 A 14-inch diameter schedule 80 pipe has an inside diameter of 12.5 inches (17.5 mm). What is the friction factor f if the pipe is flowing full and the allowable head loss is.5 m in a length of 200 m? Use Hazen-Williams equation to calculate velocity. V = kcr Assume that the pipe is cast-iron C = Calculate the slope of the energy gradient:.5m = = L 200m = Calculate the hydraulic radius, R, for pipe flowing full: D 0.175m R = = = m 4 4 ubstituting into the Hazen-William equation: For I units, k = V = (0.849) CR V = (0.849)()( m) V = 2.2m /sec (0.0175) Calculate a Reynolds number based on this velocity: VD 2.2m / sec(0.175m) Re = = 6 2 ν 1.00x10 m /sec 5 Re = 7.x10 Refer to Moody Diagram Reading from the Moody diagram (since this is based on a previous problem, ε/d has already been calculated and is ): f =
6 To correct for C factors not equal to when using the Hazen- Williams nomogram: Given flow and diameter, find from the nomogram: 1.85 C = C Given flow and slope, find D from the nomogram: D C = D Given diameter and slope, find flow from nomogram: Q C = Q C 0.8 C Example (using the nomogram and the conversions): A schedule 80 pipe has an inside diameter of 12.5 inches (17.5 mm). What is the velocity if the pipe is flowing full and the allowable head loss is.5 m in a length of 200 m? Use Hazen-William equation to calculate velocity. Assume C =. Given: D = 12.5 inches H L =.5 m L = 200 m Calculate slope of energy line. H L.5m = = L 200m = X From the nomogram, V = 6.0 ft/sec (0.048 m/ft) V = 1.8 m/sec To use the nomogram, need slope as feet/0 feet. 0 ft = ft / ft 0 ft = 17.5 ft /0 ft X 6
7 Modifying based on flow conversion (divide both sides by area) and substituting: C QC = Q Q V = A QC C V VC = = Q = A A C VC = V VC = (1.8m / sec) V = 2.2m / sec C Determine the head loss in a 46-cm concrete pipe with an average velocity of 1.0 m/sec and a length of 0 m. Using Hazen-Williams equation: V = kcr ince in I units, k = For a pipe flowing full: D 0.46m R = = 4 4 R = 0.115m By definition: lope of energy line = Head Loss/Length of Pipe Or Head Loss = h L = (lope)(pipe Length) Let C = 10 (concrete pipe use average of 140) olve Hazen-Williams for lope: V = kcr V = kcr V = kcr 1/ ubstituting: V = kcr 1/ 1m / sec = 0.849*10*0.115 = Head Loss = lope(pipe Length) H L = (0 m) = m
8 Find the discharge from a full-flowing cast iron pipe 24 in. in diameter having a slope of Assume: 5-year old cast iron C = Using Hazen-Williams: Q = 0.279CD The flow will be given in MGD, but it is needed in ft /sec. From earlier, conversion factor for MGD to cfs is: 1 ft /sec = MGD Find the discharge from a full-flowing cast iron pipe 24 in. in diameter having a slope of Assume: 5-year old cast iron C = Using Hazen-Williams: Q = 0.279CD The flow will be given in MGD, but it is needed in ft /sec. From earlier, conversion factor for MGD to cfs is: 1 ft /sec = MGD Modify Hazen-Williams: Q = 0.279CD Q = 0.42CD for flow in ft /sec ubstituting: 1ft /sec MGD 0.646MGD 24in Q = 0.42() 12in / ft Q = 16.27cfs ( 0.004) 8
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