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Introduction to Fluid Mechanics 7th Edition
Fox, Pritchard, & McDonald
Answers to Selected Problems, Chapter 1
1.5 M = 5913 kg 1.7 L = 27.25 in. D = 13.75 in.
1.9 2
2
05.2tg
Wy =
1.11 d = 0.109 mm 1.15 y = 0.922 mm 1.17 a) N·m/s, lbf·ft/s b) N/m2, lbf/ft2 c) N/m2, lbf/ft2, d) 1/s, 1/s e) N·m, lbf·ft f) N·s, lbf·s g) N/m2, lbf/ft2 h) m2/s2·K, ft2/s2·R i) 1/K, 1/R j) N·m·s, lbf·ft·s 1.19 a) 6.89 kPa b) 0.264 gal 47.9 N·s/m2 1.21 a) 0.0472 m3/s b) 0.0189 m3 c) 29.1 m/s d) 2.19 x 104 m2 1.23 101 gpm 1.25 SG = 13.6 v = 7.37 x 10−5 m3/kg γE = 847 lbf/ft3, γM = 144 lbf/ft3 1.27 2.25 kgf/cm2
1.29 c = 0.04 K1/2·s/m 0
0max 36.2
TpAm t ⋅=& (ft2, psi, R)
1.31 CD is dimensionless 1.33 c: N·s/m, lbf·s/ft k: N/m, lbf/ft f: N, lbf 1.35 H(m) = 0.457 − 3450·(Q(m3/s))2 1.37 ρ = 0.0765 ± 2.66 x 10−4 lbm/ft3 (± 0.348%) 1.39 ρ = 1130 ± 21.4 kg/m3 SG = 1.13 ± 0.0214 1.41 ρ = 930 ± 27.2 kg/m3 1.43 t = 1, 5, 5 s Flow rate uncertainty = ± 5.0, 1.0, 1.0% 1.45 μ = 1.01 x 10−3 N·s/m2 ± 0.609% 1.47 δx = ± 0.158 mm 1.49 H = 57.7 ± 0.548 ft θmin = 31.4o
2.75 a = 0 b = 2U c = −U 2.77 V = 229 mph 2.79 M = 2.5 xtrans = 0.327 m 2.81 SG = 0.9 γ = 8830 N/m3 Laminar flow 2.83 V = 667 km/hr
Introduction to Fluid Mechanics 7th Edition
Fox, Pritchard, & McDonald
Answers to Selected Problems, Chapter 3
3.1 M = 62.4 kg t = 22.3 mm 3.3 z = 9303 ft Δz = 5337 ft 3.5 F = 45.6 N 3.9 Δp = 972 Pa ρ = 991 kg/m3 3.11 D = 0.477 in 3.13 Δρ/ρ0 = 4.34% Δp/p0 = 2.15% 3.15 p = − 0.217 psig 3.17 p = 6.39 kPa (gage) h = 39.3 mm 3.19 p = 128 kPa (gage) 3.21 Δp = 59.5 Pa 3.23 H = 30 mm 3.25 SG = 0.900 3.27 Δp = 1.64 psi
3.29 ( )[ ]2oil 1 D
d
pL+Δ
=ρ
L = 27.2 mm
3.31 h = 1.11 in 3.33 θ = 11.1o S = 5/SG 3.35 patm = 14.4 psi Shorter column at higher temperature 3.37 Δh = 38.1 mm Δh = 67.8 mm 3.39 Δh = 0.389 cm 3.43 Δz = 587 ft Δz = 3062 ft 3.45 p = 57.5 kPa p = 60.2 kPa 3.49 pA = 1.96 kPa pB = 8.64 kPa pC = 21.9 kPa pair = − 11.3 kPa pair = 1.99 kPa 3.51 FA = 79,600 lbf 3.53 W = 68 kN 3.55 FR = 0.407 lbf 3.57 FR = 8.63 MN R = (8.34 MN, 14.4 MN) 3.59 F = 600 lbf 3.63 D = 8.66 ft 3.65 d = 2.66 m 3.67 SG = 0.542 3.69 F = − 137 kN 3.71 FV = 7.62 kN x´FV = 3.76 kN·m FA = − 5.71 kN 3.73 FV = − ρgwR2π/4 x´ = 4R/3π 3.75 FV = 1.05 x 106 N x´ = 1.61 m 3.77 FV = 1.83 x 107 N α = 19.9o 3.79 FV = 416 kN FH = 370 kN α = 48.3o F = 557 kN 3.81 FV = 2.47 kN x´ = 0.645 m FH = 7.35 kN y´ = 0.217 m
3.83 ( )a
dHLM3
4 23
−=
ρ M = 583 kg
3.85 ( ) ( ) ( )⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ++−=
42sin
2sin2 maxmax
maxθθθρ RRdLRM M = 631 kg
3.87 γ = 8829 N/m3 h = 0.292 m 3.89 VNot submerged/ VSubmerged = 10.5% 3.91 SG = Wa/(Wa − Ww) 3.93 FB = 1.89 x 10−11 lbf V = 1.15 x 10−3 ft/s (0.825 in/min) 3.95 L/VHe = 0.0659 lbf/ft3 L/VH2 = 0.0712 lbf/ft3 L/Vair = 0.0172 lbf/ft3/0.0249 lbf/ft3 3.97 D = 116 m M = 703 kg 3.99 θ = 9.1o (with A = 25 cm2 not A = 20 cm2) 3.101 x = 0.257 m f = 6.1 N 3.103 D = 2.57 ft 3.105 f = 0.288 cycle/s (ω = 1.81 rad/s) 3.107 F = 34.2 lbf 3.113 a = g(h/L) 3.115 ω = 185 rad/s (1764 rpm) 3.117 Δp = ρω2R2/2 ω = 7.16 rad/s 3.119 dy/dx = − 0.25 p = 105 − 1.96x (p: kPa, x: m) 3.121 α = 30o dy/dx = 0.346 3.123 T = 47.6 lbf p = 55.3 lbf/ft2 (gage)
Introduction to Fluid Mechanics 7th Edition
Fox, Pritchard, & McDonald
Answers to Selected Problems, Chapter 4
4.1 x = 0.934 m 4.3 x = 747 m t = 23.9 mm 4.5 V0 = 87.5 km/hr 4.7 τ = 1.50 hr
4.31 U = 1.5 m/s 4.33 Q = 1.05 x 10−5 m3/s (10.45 mL/s) Vave = 0.139 m/s umax = 0.213 m/s 4.35 vmin = 5.0 m/s 4.37 ∂Voil/∂t = − 2.43 x 10−2 ft3/s (0.18 gal/s) 4.39 dh/dt = − 8.61 mm/s 4.41 dh/dt = − 0.289 mm/s 4.43 Q = 1.5 x 104 gal/s A = 4.92 x 107 ft2 4.45 t = 22.2 s 4.47 dy/dt = − 9.01 mm/s 4.49 Qcd = 4.50 x 10−3 m3/s Qad = 6.0 x 10−4 m3/s Qbc = 1.65 x 10−3 m3/s
4.51 ( )Ag
yt2
tan52 25
02 θπ
= 0
0
56
QVt =
4.53 mf = (− 2406, 2113) lbf 4.55 mf2/mf1 = 1.33 4.57 mf = (− 320, 332) N 4.61 T = 3.12 N 4.63 F = 35.7 lbf 4.65 slbm2.311 =m& slbm0.322 =m& (because of weight plus momentum loss) 4.67 V = 51 m/s V = 18.0 m/s V = 67.1 m/s 4.69 F = 1.81 kN (tension)
4.71 ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−+−=
222 1sin1
4 DdDVRx θπρ Rx = − 314 N
4.73 F = 11.6 kN 4.75 F = (− 714, 498) N 4.77 F = 1.70 lbf 4.79 F = 22.7 kN 4.81 d/D = 0.707 No-dimensional pressure = 0.5 4.83 t = 1.19 mm F = 3.63 kN 4.85 Rx = − 4.68 kN Ry = 1.66 kN 4.87 Rx = − 1040 N Ry = − 667 kN 4.89 F = 2456 N 4.91 Q = 0.141 m3/s Rx = − 1.65 kN Ry = − 1.34 kN 4.93 F = 37.9 N
0 −= h = 4.28 m 4.111 V = 175 ft/s F = 2.97 lbf 4.113 p1 = 68.4 kPa (gage0 F = 209 N
4.115 gzVV 220 −=
20
0
21Vgz
AA−
=
4.117 22
0 ⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛−=
Lx
whQpp ρ
4.119 ( )tVhrVV
00
0
2 −=
4.123 Rx = − 2400 N Ry = 1386 N
4.125 jVkQ
kQ
kQV ρρρ
+⎟⎠⎞
⎜⎝⎛+−=
2
22 Vj = 80 m/s
4.127 F = 3840 lbf 4.129 F = 4.24 kN t = 4.17 s 4.131 α = 30o F = 10.3 kN 4.133 a = 13.5 m/s2 4.135 t = 0.680 s
4.137 ⎟⎟⎠
⎞⎜⎜⎝
⎛−
=VAtM
MVU
ρ0
0ln V = 0.61 m/s
4.139 amax at t = 0 θ = 90o U → V
4.141 ( )232
atVaMA−
=ρ
A = 111 mm2
4.143 h = 17.9 mm
4.145 ( )M
kUAUVa222 −−
=ρ a = 5.99 m/s2 U/Ut = 0.667
4.147 a = 14.2 m/s2 t = 15.2 m/s
4.149 ( )M
AUVa2+
−=ρ
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
0
1UVVA
Mtρ
4.151 V = 5 m/s xmax = 1.93 m t = 2.51 s 4.153 a = 2.28 m/s2
4.155 ⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
00 1ln
MtmVUU e&
U = 227 m/s
4.157 Vmax = 834 m/s amax = 96.7 m/s2 4.159 mf = 82.7 kg 4.161 a = 83.3 m/s2 U = 719 m/s 4.163 V = 3860 ft/s Y = 33,500 ft 4.165 V = 641 m/s 4.167 θ = 19o
4.169 t
MAU
UU
0
0
0
21 ρ+
=
4.171 Vmax = 138 m/s ymax = 1085 m 4.175 h = 10.7 m 4.181 M = − 192 N·m 4.183 T = 0.193 N·m 2rad/s2610=ω&
9.1 xp = 18.6 cm xm = 10.3 mm 9.3 xp = 10.4 cm xp = 7.47 mm 9.5 ReD = 1 (reasonable) ReD = 2.5 x 105 (not reasonable) Use water and D = 10 cm 9.7 L increases with elevation
9.9 A = U δπ2
=B C = 0
9.11 375.0*=
δδ 139.0=
δθ
9.13 396.0*=
δδ 152.0=
δθ
9.15 Linear: 167.0=δθ Sinusoidal: 137.0=
δθ Parabolic: 133.0=
δθ
9.17 Power: 125.0*=
δδ , 0972.0=
δθ Parabolic: 333.0*
=δδ , 133.0=
δθ
9.19 skg4.50=abm& FD = 50.4 N 9.21 dexit = 3.13 mm ΔU = 3.91% 9.23 U2 = 13.8 m/s Δp = 20.6 Pa 9.25 Δp = – 1.16 lbf/ft2 9.27 U2 = 24.6 m/s p2 = – 44.5 mm H2O 9.29 *
2δ = 2.54 mm Δp = – 107 Pa FD = 2.00 N
9.35 y = 0.121 in dy/dx = 0.00326 x
wURe
3321.02ρτ =
LD
bLUFRe
6642.02ρ
=
θL = 0.0454 in 9.39 θL = 0.0454 in FD = 0.850 N 9.41 FD = 26.3 N FD = 45.5 N 9.43 FD = 8.40 x 10–4 N (or FD = 1.68 x 10–3 N for two sides) (Higher than Problem 9.42) 9.45 FD = 3.45 x 10–3 N (or FD = 6.90 x 10–3 N for two sides) (Higher than Problem 9.44) 9.51 FD = 0.557 N 9.53 U = 1.81, 2.42, 3.63, and 7.25 m/s
9.55 5
1
2
Re0297.0
x
wUρτ =
51
2
Re0360.0
L
DbLUF ρ
= FD = 2.34 N
9.57 FD = 4.57 x 10–3 N (or FD = 9.14 x 10–3 N for two sides) 9.59 FD = 55.8 N (or FD = 112 N for two sides)
9.61 5
1Re
353.0
xx=
δ 5
1Re
0612.0
x
fc = FD = 2.41 N
9.63 Lδ = 31.3 mm =Lwτ 0.798 Pa FD = 0.700 N
9.65 w2 = 80.3 mm 9.67 Δp = 6.16 Pa L = 0.233 m 9.69 Petroleum used ≈ 0.089% (about 15% of pipeline use)
9.71 δρ 2lam 525.0 Um =& δρ 2
turb 777.0 Um =& 9.73 a = 0 b = 0 c = 3 d = – 2 H = 3.89 9.75 U2 = 2.50 m/s Δp = 0.00370 in H2O
9.77 ( ) ⎥⎦
⎤⎢⎣
⎡+
+= xHc
UU f
21
1 θ (constant wτ ) ( )
4
41
11 200583.01
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
Hx
UUU
θδν ( ≠wτ constant)
9.79 FD = 5.58 N (11.2 N for both sides) One system: FD = 4.23 N (8.46 N for both sides) 9.81 FD = 1500 lbf P = 2000 hp 9.85 xlam/L = 0.0352% FD = 5.49 x 105 N P = 7.63 MW 9.87 FD = 3.02 x 104 N Savings = FD = 7.94 x 104 kg/yr 9.91 FD = 3610 lbf P = 77.6 hp 9.93 di = 96.5 mm 9.95 t = 9.29 s, x = 477 m (t = 7.93 s, x = 407 m for three parachutes) “g” = – 3.66 9.97 B is 20.8% better than A (H > D) 9.99 =DC 0.299 9.101 V = 24.7/35.8 km/hr New tires: V = 26.8/32.6/39.1 km/hr Plus fairing: V = 29.8/35.7/42.1 km/hr 9.105 M = 0.0451 kg
9.107 ⎥⎦
⎤⎢⎣
⎡=
θρθ2cos
sin2ACmgV
D
t = 1.30 mm
9.109 t = 2.95 s x = 624 ft 9.111 V = 47.3 mph (1970’s car) V = 59.0 (current car) 9.113 FE = 6.72 mpg ΔQ = 1720 gal/yr (8.78%) 9.115 CD = 1.17
9.117
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
12
12
2
1
2
2
1
AA
AAA
mgVρ
9.119 ( )221 UVACF DD −= ρ ( ) RUVACT D
2
21
−= ρ ( ) UUVACP D2
21
−= ρ R
V3opt =ω
9.121 M = 11.0 N·m 9.123 P = 3.00 kW 9.125 V = 23.3 m/s Re = 48,200 FD = 0.111 N 9.127 x = 13.9 m 9.129 CD = 61.9 =sρ 3720 kg/m3 V = 0.731 m/s 9.131 M = 0.0471 kg 9.133 CL = 1.01 CD = 0.0654
9.135 DHUCF DD2
21
97 ρ= 22
21
167 DHUCM D ρ=
97
uniform
=D
D
FF
87
uniform
=M
M
9.137 D = 7.99 mm y = 121 mm 9.139 t = 4.69 s x = 70.9 m 9.141 xmax = 48.7 m (both methods) 9.143 CD = 0.606 V = 37.4 mph
9.145 ( )bDbuDu
Rwb ACAC
FVV+
−=ρ
2 Vb = 4.56 m/s (16.4 km/hr)
9.147 x ≈ 203 m
9.151 ΔP = 16.3 kW (94%) 9.157 Vmin = 5.62 m/s (10.9 kt) Pmin = 31.0 kW Vmax = 19.9 m/s (38.7 kt) 9.159 Vmin = 144 m/s R = 431 m 9.161 M = 37.9 kg P = 1.53 kW (or 3.02 kW if treated as two wings) 9.163 T = 17,300 lbf 9.165 FD = 524 lbf P = 209 hp 9.167 θ = 3.42o L = 168 km 9.169 For a race car, effective; for a passenger car, not effective 9.175 FL = 0.00291 lbf 9.177 FL = 50.9 kN FD = 18.7 kN F = 54.2 kN P = 5.94 kW 9.179 ω = 14,000 – 17,000 rpm x = 3.90 ft
Introduction to Fluid Mechanics 7th Edition
Fox, Pritchard, & McDonald
Answers to Selected Problems, Chapter 11
11.1 Q = 3.18 m3/s 11.3 y = 2.61 ft 11.5 y2 = 0.507 m Fr2 = 2.51 11.7 S0 = 0.00186 11.9 S0 = 0.00160 11.11 Q = 0.194 m3/s 11.13 y = 2.47 ft 11.15 y = 0.775 m 11.19 y = 4.83 ft V = 3.69 ft/s 11.23 y = 7.38 ft 11.25 yc = 0.365 ft, Ec = 0.547 ft yc = 0.759 ft, Ec = 1.14 ft yc = 1.067 ft, Ec = 1.60 ft yc = 1.46 ft, Ec = 2.19 ft 11.27 yc = 0.637 m 11.31 Δx = 197 ft 11.33 y = 0.645 ft y = 4.30 ft
11.35 51
2
22⎟⎟⎠
⎞⎜⎜⎝
⎛=
gzQyc
11.37 Q = 3.24 ft3/s 11.39 y2 = 0.610 ft (– 32.2%) 11.41 y2 = 1.31 ft 11.43 Q = 49.5 ft3/s 11.45 Q = 10.6 m3/s yc = 0.894 m Hl = 0.808 m 11.47 y2 = 5.94 ft 11.49 Q = 54.0 ft3/s Hl = 1.62 ft 11.51 y2 = 4.45 m Hl = 9.31 m 11.53 Q = 26.6 ft3/s 11.55 H = 0.514 m 11.57 Cw = 1.45
Introduction to Fluid Mechanics 7th Edition
Fox, Pritchard, & McDonald
Answers to Selected Problems, Chapter 12
12.1 T = const. p decreases ρ decreases (Irreversible adiabatic process) 12.3 Δs > 0 so it is feasible for a real (irreversible) adiabatic process 12.5 T2 = 20oF p2 = 100 kPa 12.7 Δs = – 346 kJ/kg·K (ΔS = – 1729 J/K) Δu = – 143 kJ/kg (ΔU = – 717 kJ) Δh = – 201 kJ/k (ΔH = – 1004 kJ) 12.9 h = 57.5% 12.11 W = 176 MJ Ws = 228 MJ Ts (max) = 858 K Qs = – 317 MJ 12.13 m& = 36.7 kg/s T2 = 572 K V2 = 4.75 m/s W& = 23 MW 12.15 Δt = 828s (≈ 14 min) 12.17 Δρ = 1.70 x 10–4kg/m3 ΔT = 0.017 K ΔV = 0.049 m/s 12.19 Δt = 198 μs Ev = 12.7 GN/m2 12.21 x = 19.2 km 12.23 c = 299 m/s V = 987 m/s V/Vbullet = 1.41 12.29 c = 340 m/s (sea level) 12.31 V = 1471 mph α = 31.8o 12.33 V = 642 m/s (2110 ft/s) 12.35 V = 493 m/s Δt = 0.398 s 12.37 V = 515 m/s t = 6.92 s 12.39 Δx ≈ 1043 – 1064 m 12.41 Density change < 1.21%, so incompressible 12.43 M = 0.142 (1%) M = 0.322 (5%) M = 0.464 (10%) 12.45 Δρ/ρ = 48.5% (Not incompressible) 12.47 pdyn = 54.3 kPa p0 = 152 kPa 12.49 p0 = 546 kPa h0 – h = 178 kJ/kg T0 = 466 K 12.51 p0 – p = 8.67 kPa V = 195 m/s V = 205 m/s Error using Bernoulli = 5.13% 12.55 T0 = const (isoenergetic) p0 decreases (irreversible adiabatic) 12.57 V = 890 m/s T0 = 677 K p0 = 212 kPa 12.59 T0 = const = 294 K (20.6o) (isoenergetic)