Other Friction Losses Valves and Fittings CBE 150A Transport Spring Semester Goals Calculate frictional losses in a system containing valves, fittings, and sudden expansions and contractions Express frictional losses in terms of velocity head Assess relative contributions of different sources to total viscous dissipation CBE 150A Transport Spring Semester Sudden Expansion

Frictional losses occur as result of turbulence generated immediately downstream of the expansion CBE 150A Transport Spring Semester Sudden Expansion Assume 2 Va h fe K e 2 Ke is the expansion loss coefficient which we will attempt to describe in terms of flow properties. CBE 150A Transport Spring Semester Sudden Expansion Mass Balance aVa S a bVb Sb Sa Vb Va Sb

CBE 150A Transport Spring Semester Sudden Expansion Momentum Balance Assume turbulent: 12 m bVb aVa pa Sb pb Sb Fw Fg 0 Replaced Sa with Sb because pa is at the point of expansion. CBE 150A Transport Spring Semester 0 Momentum Balance Sb pa pb m Vb Va m p Vb Va

Sb Vb Vb Va 2 Vb VaVb CBE 150A Transport Spring Semester Mechanical Energy Balance Assume turbulent: 12 p p 1 2 2 b a W bVb aVa gz hf 2

0 0 2 2 2 2 Va Vb pa pb hf 2 Va Vb p 2 CBE 150A Transport

Spring Semester Combining 2 2 Va Vb 1 2 hf Vb VaVb 2 2 Va 2VaVb Vb 2 V a

CBE 150A Transport Vb 2 2 2 Spring Semester Final Result Sa Vb Va Sb Recall Mass Balance Result: 2 2 S a Va h f 1 Sb 2 Notes: Velocity head is based on smaller cross section

What if flow becomes laminar in large pipe? CBE 150A Transport Spring Semester For Tank Filling Sa 1 1 Sb for S b S a Sa Va 2 hf 2 CBE 150A Transport K e 1.0 Sb Va

Spring Semester Sudden Contractions At sudden contractions, flow streamlines converge causing the downstream developed flow to have an area smaller than the downstream pipe diameter. This flow constriction is called the vena contracta. Viscous dissipation occurs in the vortices developed in this area. CBE 150A Transport Spring Semester Sudden Contraction Development of an expression for sudden contraction proceeds in much the same way as that for sudden expansion with the definition of a contraction coefficient. 2 Vb h fc K c 2 For laminar flow experimentally, Kc < 0.1 and hfc is usually neglected Turbulent (empirical): Sb

K c 0.4 1 Sa Note: Calculations again based on small cross section. CBE 150A Transport Spring Semester Tank Emptying Sb 1 1 Sa for S a S b Sa Vb2 h f 0.4 2 CBE 150A Transport K c 0.4 Sb Vb

Spring Semester Velocity Heads 2 pa pb L V g z a zb 4 f K c K e K f D 2 The above expression shows that friction loss in a complicated flow system can be expressed as a number of velocity heads. It is a measure of momentum loss resulting from flow through the system. For instance in making a 90 turn all xmomentum is turned into y-momentum. KTee 1 CBE 150A Transport K Globe 6 Spring Semester Alternate Method The previous equation can be manipulated to change the Kf values into equivalent lengths of pipe (see attached

table) of diameter D. When this method is used the equivalent lengths are add to the length of the actual pipe sections and the equation becomes. 2 Ltotal V h f 4 f D 2 Note: The values in the table are L/D and must be multiplied by D to get equivalent lengths. CBE 150A Transport Spring Semester CBE 150A Transport Spring Semester Example Water is pumped at 250 gpm from tank 1 to tank 2 as shown. Calculate the required power input to the pump assuming a pump efficiency of 70%. e d

Pe = 30 psig Tank 2 L2=10 ft 5 Sch. 40 Steel Zab = -10 ft Pa = 0 psig Zbc = +0.5 ft L2=90 ft 4 Sch. 40 Steel a Tank 1 b Zcd = +75 ft Zde = +15 ft c gate valve (open) CBE 150A Transport Spring Semester V5" 250

N RE gal 1 min ft 3 1 ft 4.0 2 min 60s 7.48gal 0.1390 ft s ft lb 5.047 ft 4.0 62.4 m3 s ft 12 1.56 105 lb 6.7197 10 4 m ft s

k 0.00015 k/D 0.00015 ft 0.00036 5.047 ft 12 f 0.0045 CBE 150A Transport Spring Semester gal 1 min ft 3 1 ft V4" 250 6 . 3

min 60s 7.48 gal 0.0884 ft 2 s N RE lbm ft 4.026 ft 6 . 3 62 . 4 3 s ft 12 1.96 105

lb 6.7197 10 4 m ft s k 0.00015 k/D 0.00015 ft 0.00045 4.026 ft 12 f 0.0042 CBE 150A Transport Spring Semester Pe Pa V 2 gz h f W p

2gc gc h f 5" V2 4f L fittings contraction 2gc D h f 5" 4.0 ft 4 0.0045 (10 ft ) ft lb f s 0 0.4 0.206 lbm 32.2 ft lbm

5.047 ft 2 lb s 2 12 f 2 V2 h f 4" 2gc 4f L D fittings exp ansion 2 6.3 ft

ft lb f 4 0.0042 (90 ft ) s h f 4" 2 0.75 0.17 1.0 4.423 lbm 32.2 ft lbm 4.026 ft 2 lb s 2 12 f CBE 150A Transport Spring Semester Pe gz

h f W p gc 30 lb f 144 in 2 ft lb f in 2 ft 2 10 0.5 75 15 ft 4.629 W p lbm lb m 62.4 3 ft 154.4 ft lb f lbm m 250 gal min 8.33 lbm lb 34.71 m

min 60s gal s ft lb f lbm 154.4 s lbm ft lb f 550 (0.70) s Hp 34.71 P CBE 150A Transport W p 13.9 Hp Spring Semester 10 Minute Problem The Alaskan pipeline is 48 in. ID, 800 miles long and carries crude

oil at a rate of 1.2 million bbl/day (1 bbl = 42 gallons). Assuming North Slope crude oil to be a Newtonian fluid with a viscosity of 25 cP and a specific gravity of 0.87, what total pumping horsepower is required to operate the pipeline ? The oil enters and leaves the pipeline at sea level and the line contains the equivalent of 150 90 degree elbows and 100 fully open gate valves. Assume inlet and discharge pressures are equal to 1 atm. CBE 150A Transport Spring Semester