ASE 320 - HW 4 Solutions

ASE 320 - HW 4 Solutions - r- _ \N-A I . Z—Z‘ .\ [:3...

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C m\f\ «\(fixg _ HZ. 614 Solutions Manual 0 Fluid Mechanics, Fifth Edition Solution: For sea—level air, take ,0 = 0.00237 slug/ft3. Convert 0.7 lbf per mi/h rolling friction to 0.477 lbf per ft/s of speed. Then the power relationship for the cycle is ' Power = (Fd, + quw = [CD/igvz + Cm,le V, ft-lbf s or: 115*550 3 = [(5.5 ft2)—0'00237251“g/ft V2 + [0.477 %] V]V S Solve this cubic equation, by iteration or EES, to find Vmax z 192 ft/s z 131 mi/h. Ans. The helium-filled balloon in Fig. P7.75 is tethered at 20°C and 1 atm with a string of negligible weight and drag. The diameter is 50 cm, and the balloon material weighs 0.2 N, not including the helium. The helium pressure is 120 kPa. Estimate the tilt angle 6 if the airstream velocity U is (a) 5 m/s or (b) 20 m/s. Solution: For air at 20°C and 1 atm, take p = 1.2 kg/m3 and ,u = 1.8E—5 kg/m-s. For helium, R = 2077 J/kg~°K. The helium density = (120000)/[2077(293)] z 0.197 kg/m3. The balloon net buoyancy is independent of the flow velocity: 7f 3 7T 3 BM = (pm-r — pHe)ggD = (1.2— 0.197)(9.81)g(0.5) z 0.644 N The net upward force is thus Fz = (Bnet — W) = 0.644 — 0.2 = 0.444 N. The balloon drag does depend upon velocity. At 5 m/s, we expect laminar flow: F, 7/6 W25 2. Reb=%=mm Blimpm S . — Drag = CD §U2 $1? = 0.4%?) (5)2 $0.5? z 1384 N Then Q, = tan“ { Drag = tan—1 = 72° Ans. (a) 1 F2 1 0.444 Chapter 7 o Flow Past Immersed Bodies 615 (b) At 20 m/s, Re = 667000 (turbulent), Table 7.3: CD z 0.2: 1.2 7: 9.43 D =0.2[—] 202— 0.52:9.43N, a =t "[—]=87° A . b mg 2 ( ) 4( ) b an 0.444 "S 0 These angles are too steep——the balloon needs more buoyancy and/or less drag. P7.76 The recent movie The World ’3 Fastest Indi tells the story of Burt Munro, a New Zealander who, in 1937, set a motorcycle recor of 201 mi/h on the Bonneville Salt Flats. Using the data of Prob. P7.74, (a) estimate he horsepower needed to drive this fast. (1)) What horsepower would have gotten Burt u to 250 mi/h? Solution: Prob. P7.74 suggests CDA = 5.5 ft2 and Fromng = 0.7 lbf per mi/h of speed. Convert 201 mi/hr to 295 ft/s. Bonneville is at 43 0 ft altitude, so take p = 1.0784 kg/m3 = 0.00209 slug/ft3 from Table A.6. Now compu the total resistance force: F=Fdrag+Fr 011mg = (CDA)£2)—V2 + '7Vmi/h 3 = (5.5fi2)(W)(295 /s)2 + 0.7(201) = 500 +141 = 64llbf Power = FV = (64llbf)(295ft/s) = 189,000 fi—lbf/s +550 = 343hp Ans.(a) A lot of power! Presumably Burt di some streamlining to reduce drag. ([2) Repeat this for V: 250 mi/h = 67 ft/s to get F = 914 lbf, Power = 610 hp. Ans. (b) P7.77 To measure the rag of an upright person, without violating human-subject protocols, a life-sized m nequin is attached to the end of a 6-m rod and rotated at Q = 80 rev/min, as in Fig. P .77. The power required to maintain the rotation is 60 kW. By including rod-drag p er, which is significant, estimate the drag—area CDA of the mannequin, in m2. Chapter 7 0 Flow Past Immersed Bodies 607 _ . V A sphere of density ps and diameter D is dropped from rest in a fluid of density p and Viscosity y. Assuming a constant DmgrBUOY-ancyl drag coefficient Cdo, derive a differential D equation for the fall velocity V(t) and show that the solution is 1/2 4 D — I V = —g—(S—1) tanh Ct PM 3ng (5—1) “2 C = ———°2 La 45 D where S = ps/p is the specific gravity of the sphere material. Solution: Newton’s law for downward motion gives ‘ V deOWanadown’ or: W_B_CD—p—V2A=Ed—, Where A=£D2 ‘ 2 g dt 4 and W—B=p(S—l)g%D3. Rearrangeto i—Y=IB—avz, 1] ngDA = 1—— and (2: fl g[ S W Separate the variables and integrate from rest, V = 0 at t = 0: i dt = i dV/(fl — aVz), or: V = F tanh (tfi) = Vfinaltanh(Ct) Ans. a 1/2 1/2 where Vfinal = w and C = , s = 10—5 > 1 3CD 48 D ,0 P7.68 baseball weighs 145 g and is 7.35 cm in diameter. It is dropped from rest from a 35—m—h1gh tower at approximately sea level. Assuming a laminar—flow drag coefficient, estimate (a) its terminal velocity and (b) whether it will reach 99 percent of its terminal veloc1ty before it hits the ground. Chapter 7 o Flow Past Immersed Bodies 609 Solution: For sea—level air, take ,0 = 1225 kg/m3 and y = 1.78E—5 kg/m-s. Assume a laminar drag coefficient CD z 0.47 from Table 7.3. The terminal velocity is VFW]: ———2W 2= mflomxggm 2~34.1E Ans. (a) CDp(7r/4)D O.47(l.225)(fl/4)(0.0735) s Now establish the “specific gravity” of the ball, relative to air: pball =E=——w——=697.4 E “S”: pba” =69_7'4____569 U (7r/6)(0.0735)3 m3 ’ p,ir 1.225 Then the constant C from Prob. 7.66 gives the time history of velocity and displacement: C I [3ch6 —1)]”2 = [3(9.81)(0.47)(569 — 1) 1/2 2 2 a 0.287 s", v = vf tanh(Ct), 45 D 4(569) (0.0735) 34.1 0.287 Check ReD (max) = 1.225(34.1)(0.0735)/(1.78E—5) z 172000 (OK, CD z 0.47) or: V = 34.1 tanh (0.287t), Z =J th = ln[cosh (02870] We can now find the time and velocity when the balls hits Z = 35 m: Z = 35 = 34'1 ln[cosh(0.287t)], solve for t z 2.81 s, whence 0.287 V(at Z = 35 m) = 34.1 tanh[0.287(2.81)] z 22.8 E Ans. (b) S This is only 67% of terminal velocity. If we try the formulas again for V = 99% of terminal velocity (about 33.8 m/s), we find that tz 9.22 s and Z w 230 m. P7.69 Two baseballs from Prob. 7.68 are connected to a rod 7 mm in diameter and 56 cm long, as in Fig. P7.69. What power, in W, is required to keep the system spinning at 400 r/min? Include the drag of the rod, and assume sea—level standard air. ...
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ASE 320 - HW 4 Solutions - r- _ \N-A I . Z—Z‘ .\ [:3...

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