Chapter 4. Analysis of Flows in Pipes Learning Outcomes: Classification of pipe flows based on Reynolds number Evaluate energy losses due to friction, valves, fittings, bends, diameter expansion and contraction. Design the size of distribution pipes considering head loss.

4.1 Characterization of flow based on Reynolds number Osborne Reynolds (a British engineer) conducted a flow experiment, i.e. by injecting dye into pipe flow to classify the types of flow. Turbulent Near laminar

Three types of flow: Laminar flow Re < 2000 Transitional flow 2000 < Re < 4000 Turbulent flow Re > 4000 Reynolds number Re is the ratio of the inertia force on an element of fluid to the viscous force. Re is dimensionless. where V = average velocity, D = diameter of pipe,  = fluid density,  = dynamic viscosity, and  = kinematic viscosity Reynolds number is one of the important dimensionless number used in the dimensional analysis in fluid mechanics and hydraulics.

Example 4.1 Determine the range of average velocity of flow for which the flow would be in the transitional region if an oil of S.G. = 0.89 and dynamic viscosity = 0.1 Ns/m2 is flowing in a 2-in pipe. For transitional flow, When Re = 2000, When Re = 4000, Therefore, for the flow to be in transitional state, the average velocity V should be between 4.424 m/s and 8.847 m/s.

4.2 Head Loss due to Friction in Pipes of Constant Cross Section Head loss or energy loss in pipe flow maybe caused by friction, valves, fittings, expansion and contraction in pipe diameter, and others. Frictional head loss is known as the major head loss in pipe flow as the loss occurs along the pipe while other head losses such as due valves, fittings, expansion and contraction are known as minor losses. Friction head loss can be estimated using the Darcy-Weisbach equation: where hf = head loss due to friction, f = friction factor (dimensionless), L = length of flow (or pipe), D = diameter of pipe, V = average velocity of flow, and g = gravity acceleration. For laminar flow, the head loss due to friction can also be estimated using the Hagen-Poiseuille equation: where  = dynamic viscosity, L = length of flow (or pipe), D = diameter of pipe, V = average velocity of flow, and  = specific weight of liquid.

For laminar flow, the friction factor f can be calculated based on Reynolds number. Darcy-Weisbach equation: Hagen-Poiseuille equation: Equating both equations: Friction factor for laminar flow

Example 4.2 In a refinery oil (S.G. = 0.85,  = 1.8  105 m2/s) flows through a 30 m long, 100 mm diameter pipe at 0.50 L/s. Is the flow laminar or turbulent? Find the head loss per meter of pipe length. Reynolds number < 2000, therefore laminar flow For laminar flow, friction factor Head loss per meter pipe length

4.3 Friction factor and Moody Chart Friction factor can be estimated by the following equations: Darcy-Weisbach & Hagen-Poiseuille: for Re < 2000 Blasius (for smooth pipes): for 3000  Re  100000 Colebrook (for all pipes): for 4000  Re  108 For convenience, these friction factor equations are used to prepare a Moody diagram. Moody diagram provides the relationship between friction factor f, Reynolds number Re and pipe relative roughness e/D, where e is the absolute roughness. Pipe surface roughness measurement

Example 4.3 A 20-in diameter galvanized iron pipe 10560 ft long carries 4 cfs of water at 60F. Find the friction head loss using the Moody chart. L = 10560 ft, Q = 4 ft3/s,  = 1.217  105 ft2/s Reynolds number > 4000, turbulent flow Galvanized iron pipe, e = 0.0005 ft = 0.006 in Relative roughness From Moody diagram, Using Darcy-Weisbach equation, friction head loss

f = 0.0172 0.0003 2.51  105

Example 4.4 Determine the friction factor f if water at 70C is flowing at 9.14 m/s in stainless steel pipe having an inside diameter of 25 mm. Given velocity of flow = 9.14 m/s and diameter of pipe D = 0.025 m For water at 70C, kinematic viscosity  = 9.75  107 m2/s Reynolds number > 4000, turbulent flow Stainless steel pipe, e = 0.002 mm Relative roughness From Moody diagram,

f = 0.0159 0.00008 2.34  105

Example 4.5 Water at 20C flows in a 500-mm diameter welded steel pipe. If the friction loss gradient is 0.006, determine the flow rate. For welded steel pipe, surface roughness e = 0.045 mm Relative roughness For water at 20C, kinematic viscosity  = 1.003  106 m2/s Friction loss gradient Friction head loss

ftrial2 = 0.0131 ftrial1 = 0.0118 0.00009 1.113  106

Assume friction factor f by assuming turbulent flow condition, Based on , ftrial1 = 0.0118 Velocity Reynolds number Check the friction factor f with the corresponding e/D and Re, Friction factor ftrial2 = 0.0131 Velocity Reynolds number Check the friction factor f with the corresponding e/D and Re, Friction factor f = 0.0131 = ftrial2  converged!

Or, trial-and-error in tabulated form, (m/s) Re = Trial f Check f 0.0118 2.233 1.113  106 0.0131 Try again 0.0131 2.120 1.057  106 0.0131 Converged! Flow rate

Example 4.6 A galvanized iron pipe 18000 ft long must convey ethyl alcohol ( = 2.3  105 ft2/s) at a rate of 135 gpm. If the friction head loss must be 215 ft, determine the pipe size using Moody chart. Note: 1000 gpm = 2.23 cfs. For galvanized iron pipe, surface roughness e = 0.0005 ft Relative roughness Velocity Reynolds number Friction head loss

Trial-and-error in tabulated form, (ft) Trial f Check f 0.03 0.3561 0.001404 46808.7 0.0253 Try again 0.0253 0.3442 0.001453 48431.4 0.0253 Converged! Assume mid-range value of f Therefore, the size of the pipe

ftrial1 = 0.03 ftrial2 = 0.0253 0.00145 0.0014 4.84  104 4.68  104

Example 4.7 In a chemical processing plant, benzene at 50C (S.G. = 0.86) must be delivered to point B with a pressure of 550 kPa. A pump is located at point A 21 m below point B, and the two points are connected by 240 m of plastic pipe having an inside diameter of 50 mm. If the volume flow rate is 110 L/min, calculate the required pressure at the outlet of the pump.  = 860 kg/m3  = 4.2  104 Pa.s = zB  zA Energy equation between A and B

zB  zA = 21 m  = 860 kg/m3 For plastic pipe, roughness e = 0 Reynolds number From Moody chart, f = 0.0181 Energy equation between A and B:

4.4 Minor Losses in Pipe Flow a. Loss of head at entrance Loss due to the entrance of flow into pipe where V = mean velocity in the pipe, ke = loss coefficient

b. Loss of head at submerged discharge Loss of head when pipe discharges fluid into filled reservoir or tank

c. Loss of head due to contraction Loss of head when flow contracts Loss coefficients for sudden contraction kc 0.0 0.50 0.1 0.45 0.2 0.42 0.3 0.39 0.4 0.36 0.5 0.33 0.6 0.28 0.7 0.22 0.8 0.15 0.9 0.06 1.0 0.00

Loss of head when flow contracts For smooth curved transition, kc can be as small as 0.05. For conical transducer with angle 20 to 40, a minimum kc is about 0.10.

d. Loss of head due to expansion Loss of head when flow expands. Loss of head due to sudden expansion Loss of head due to gradual expansion

Loss of head due to gradual expansion kx

e. Loss of head due to pipe fittings Loss of head due to pipe fittings such as valve, bend, and elbow Values of loss factors for pipe fittings Fitting k Globe valve, wide open 10 Angle valve, wide open 5 Close-return bend 2.2 T, through side outlet 1.8 Short-radius elbow 0.9 Medium-radius elbow 0.75 Long-radius elbow 0.60 45 elbow 0.42 Gate valve, wide open 0.19 Gate valve, half open 2.06

f. Loss of head in bends and elbows Loss of head due to bends and elbows Loss coefficient due to 90 bend

Example 4.8 Water at 15C is being pumped from a stream to a reservoir whose surface is 64 m above the pump. The pipe from the pump to the reservoir is 8 in steel pipe 762 m long. If 0.113 m3/s is being pumped, compute the pressure at the outlet of the pump. Consider the friction loss in the discharge line, but neglect other losses. If the pressure at the pump inlet is 16.27 kPa, compute the power delivered by the pump to the water. Water at 15C,  =1000 kg/m3,  = 1.15  106 m2/s B pB = 0, VB = 0  zB  zA = 64 m D = 8 in = 0.2032 m L = 762 m Steel pipe, e = 0.045 mm  A Q = 0.113 m3/s

B A From Moody chart, f = 0.0158 Energy equation between A and B:

Power delivered by pump to water

Example 4.9 The tanks, pump and pipelines of figure below have the characteristics noted. The suction line entrance from the pressure tank is flush, and the discharge into the open tank is submerged. If the pump P puts 2.0 hp into the liquid, (a) determine the flow rate, and (b) find the pressure in the pipe on the suction side of the pump. ke = 0.5 kd = 1.0 V1  0 p3 = 0 V3  0 Energy equation between 1 and 3:

Energy equation between 1 and 3:

To find the pressure in the pipe on the suction side of the pump, write the energy equation between 1 and 2.

Example 4.10 Water is being conveyed from 2 reservoirs through a steel pipe of diameter 20 cm. The difference of water level in the two reservoirs is 25 m. Based on your own assumptions, compute the flow rate through the pipe. The kinematic viscosity for water at 20C is 1.007  106 m2/s. D = 20 cm Steel pipe, e = 0.045 mm A zA  zB = 25 m  pA = 0 Gate valve, half open, kv = 2.06 VA  0 B  pB = 0 VB  0 Energy equation between A and B,

Energy equation between A and B, Assuming the length of the pipe is 100 m long and it is fully turbulent pipe flow, From Moody chart, f = 0.014

Assignment No. 4 (due March 18, 2011) 1. The tanks, pump and pipelines of figure below have the characteristics noted. The suction line entrance from the pressure tank is flush, and the discharge into the open tank is submerged. If the pump P puts 2.0 kW into the liquid, (a) determine the flow rate, and (b) find the pressure in the pipe on the suction side of the pump. ke = 0.5 Water m cm kPa kd = 1.0

Available that V1  0; V3  0; p3 = 0 and Energy equation between 1 and 3: Therefore, and To find the pressure in the pipe on the suction side of the pump, write the energy equation between 1 and 2.

2. A wrought iron pipe 18000 ft long must convey water at a rate of 35 m3/s at 20C. If the friction head loss must be 225 m, determine the pipe size using Moody chart. For wrought iron pipe, surface roughness e = 0.046 mm Relative roughness Velocity At 20C,  = 1.003  106 m2/s Reynolds number Friction head loss

Trial-and-error in tabulated form, Trial f Check f (m) 0.03 122.38 3.759  107 3.63  105 0.014 Try again 0.014 105.08 4.378  107 4.23  105 0.0135 Try again 0.0135 104.32 4.410  107 4.26  105 0.0135 Converged! Assume mid-range value of f Therefore, the size of the pipe