Exam (elaborations) TEST BANK FOR Gas Dynamics 3rd Edition By James John, Theo Keith (Instructor's Solution Manual)

Exam (elaborations) TEST BANK FOR Gas Dynamics 3rd Edition By James John, Theo Keith (Instructor's Solution Manual) II N S T R U C T O R ’’ S S O L U T II O N S MA N UA L G A S D Y N A M II C S James E. A. John, Ph.D. President Kettering University Flint, Michigan Theo G. Keith, Jr., Ph.D. Distinguished University Professor Department of Mechanical, Industrial, and Manufacturing Engineering The University of Toledo Toledo, Ohio T H I R D E D I T I O N This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Table of Contents Chapter 1 Basic Equations of Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Chapter 2 Wave Propagation in Compressible Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 Chapter 3 Isentropic Flow of a Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Chapter 4 Stationary Normal Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 Chapter 5 Moving Normal Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 Chapter 6 Oblique Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 Chapter 7 Prandtl–Meyer Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 Chapter 8 Applications Involving Shocks and Expansion Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 Chapter 9 Flow with Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169 Chapter 10 Flow with Heat Addition or Heat Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207 Chapter 11 Equations of Motion for Multidimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236 Chapter 12 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .250 Chapter 13 Linearized Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272 Chapter 14 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .290 Chapter 15 Measurements in Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .339 This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. 1 Chapter One BASIC EQUATIONS OF COMPRESSIBLE FLOW Problem 1. – Air is stored in a pressurized tank at a pressure of 120 kPa (gage) and a temperature of 27°C. The tank volume is 1 m3. Atmospheric pressure is 101 kPa and the local acceleration of gravity is 9.81 m/s2. (a) Determine the density and weight of the air in the tank, and (b) determine the density and weight of the air if the tank was located on the Moon where the acceleration of gravity is one sixth that on the Earth. g 9.81m/ s R 0.287 kJ / kg K 1m T C P P P kpa 2 3 abs gage atm = = ⋅ ∀ = = + = ° = + = + = a) m3 2.5668 kg (0.287)(300) 221 RT ρ = P = = W = mg = ρ∀g = (2.5668)(1)(9.81) = 25.1801N b) moon earth 3 m ρ = ρ = 2.5668 kg W 4.1967N 6 W 1 g g W earth earth earth moon moon = = = Problem 2. – (a) Show that p/ρ has units of velocity squared. (b) Show that p/ρ has the same units as h (kJ/kg). (c) Determine the units conversion factor that must be applied to kinetic energy, V2/2, (m2/s2) in order to add this term to specific enthalpy h (kJ/kg). Air 2 a) 2 2 2 2 3 2 2 3 V s m N s 1 kg m kg N m kg m m p N m , kg m p N ≈ = ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ ⋅ − ⋅ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ ≈ ρ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ ≈ ρ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ ≈ b) kg kJ 1000 1 1000 J 1kJ N m 1 J kg P N m = ⎟⎠ ⎞ ⎜⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ ⎛ ⋅ ⋅ ≈ ρ c) c 2 2 2 2 1000g factor 1 kg kJ 1000 J 1 kJ N m 1 J kg m 1 N s s m 2 V ∴ = ≈ ⎟⎠ ⎞ ⎜⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ ⎛ ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ ≈ Problem 3. – Air flows steadily through a circular jet ejector, refer to Figure 1.15. The primary jet flows through a 10 cm diameter tube with a velocity of 20 m/s. The secondary flow is through the annular region that surrounds the primary jet. The outer diameter of the annular duct is 30 cm and the velocity entering the annulus is 5 m/s. If the flows at both the inlet and exit are uniform, determine the exit velocity. Assume the air speeds are small enough so that the flow may be treated as an incompressible flow, i.e., one in which the density is constant. m& i = m& e m& i = m& p + m& s = ρApVp + ρAsVs m& e = ρAeVe ∴ApVp + AsVs = AeVe So e p p s s e A A V A V V + = Ae = As + Ap 2p p D 4 A π = 2p 2 s o D 4 D 4 A π − π = 2 e Do 4 A π = i e s p 3 ( ) 2 ( p s ) o 2p 2 s o s 2p 2 p o 2p e p p s s e V V D D V D D V D D V A A V A V V = + − + − = + = (20 5) 6.6667m/ s 30 5 102 2 = + − = Problem 4. – A slow leak develops in a storage bottle and oxygen slowly leaks out. The volume of the bottle is 0.1 m3 and the diameter of the hole is 0.1 mm. The initial pressure is 10 MPa and the temperature is 20°C. The oxygen escapes through the hole according to the relation e Ae T m& = 0.04248 p where p is the tank pressure and T is the tank temperature. The constant 0.04248 is based on the gas constant and the ratio of specific heats of oxygen. The units are: pressure N/m2, temperature K, area m2 and mass flow rate kg/s. Assuming that the temperature of the oxygen in the bottle does not change with time, determine the time it takes to reduce the pressure to one half of its initial value. ∀ = 0.1 m3 p1 = 10 MPa T1 = 293K = T2 p2 = 5 MPa kg K 259.8219 J 32 R 8,314.3 ⋅ = = From the continuity equation me dt dm = − & but RT m p ∀ = so p T 0.04248 A m dt dp dt RT dm e = − e = − ∀ = & Integrating we get, O2 m(t) d = 0.1mm e Ae T m& = 0.04248 p 4 t 0.04248 R TA p p ln e 1 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∀ = − 46,713.4076sec 12.9759 hrs 2 ln 1 (259.8219) 293 1000 mm 0.1 mm m 4 (0.04248) 0.1 p p ln (0.04248)A R T t 2 1 2 e = = ⎟⎠ ⎞ ⎜⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ ⎛ π = − ∀ = − Problem 5. – A normal shock wave occurs in a nozzle in which air is steadily flowing. Because the shock has a very small thickness, changes in flow variables across the shock may be assumed to occur without change of cross-sectional area. The velocity just upstream of the shock is 500 m/s, the static pressure is 50 kPa and the static temperature is 250 K. On the downstream side of the shock the pressure is 137 kPa and the temperature is 343.3 K. Determine the velocity of the air just downstream of the shock. V1 = 500 m/ s V2 = ? p1 = 50 kPa p2 = 137 kPa T1 = 250 K T2 = 343.3 K A1 = A2 From the continuity equation m& 1 = m& 2 So ρ1A1V1 = ρ2A2V2 (500) 250.5839m/ s 250 343.3 137 V 50 T T p V p p / RT V V p / RT 1 1 2 2 1 1 2 2 1 1 1 2 1 2 = ⎟⎠ ⎞ ⎜⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ = = = ⎛ ρ ρ = 2 1 5 Problem 6. – A gas flows steadily in a 2.0 cm diameter circular tube with a uniform velocity of 1.0 cm/s and a density ρo. At a cross section farther down the tube, the velocity distribution is given by V = Uo[1-( r/R)2], with r in centimeters. Find Uo, assuming the gas density to be ρo[1+( r/R)2]. V1 = 1 cm/ s ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎠ ⎞ ⎜⎝ = − ⎛ 2 2 o R V U 1 r ρ1 = ρo ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎠ ⎞ ⎜⎝ ρ = ρ + ⎛ 2 2 o R 1 r m& 1 = m& 2 o 2 2 o 1 R 1 o 1 1 R 1 o 1 m& = ∫ ρ V dA = ∫ ρ V 2πrdr = ρ V πR = πR ρ ( ) o o 2 2 o o 1 o 2 2 5 o o 2 2 2 o R 2 o R 2 2 2 o o R U 3 2 6 1 2 2 U R 1 R U 2 R d where r 2 rdr R U 1 r R m V dA 1 r ρ π = ⎟⎠ ⎞ ⎜⎝ = πρ ⎛ − = ρ π ξ − ξ ξ ξ = π ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ρ = ρ + ∫ & ∫ ∫ o o 2 o 2 R U 3 R 2 ρ π ∴π ρ = so cm/ s 2 V 3 o = Problem 7. – For the rocket shown in Figure 1.6, determine the thrust. Assume that exit plane pressure is equal to ambient pressure. ( ) ( ) ( ) e e 2 H o e e H o e atm e e e H o A m m V m m p p A m V 0 m m ρ + = ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ ρ + = − + = + + & & & & T & & & r 1 2 6 Problem 8. – Determine the force F required to push the flat plate of Figure Pl.8 against the round air jet with a velocity of 10 cm/s. The air jet velocity is 100 cm/s, with a jet diameter of 5.0 cm. Air density is 1.2 kg/m3. Figure P1.8 To obtain steady state add + Vp to all velocities F = m& V ( ) (0.5) (1 0.1) 4 2 . 1 AV m 2 + ⎟⎠ ⎞ ⎜⎝ ⎛ π & = ρ = = 0. kg / s F = (0.)(1.1) = 0. N Problem 9. – A jet engine (Figure P1.9) is traveling through the air with a forward velocity of 300 m/s. The exhaust gases leave the nozzle with an exit velocity of 800 m/s with respect to the nozzle. If the mass flow rate through the engine is 10 kg/s, determine the jet engine thrust. Exit plane static pressure is 80 kPa, inlet plane static pressure is 20 kPa, ambient pressure surrounding the engine is 20 kPa, and the exit plane area is 4.0 m2. F x V = -10 s cm Vj = 100 s cm Vj = 110 s cm x V = 0 F 7 Figure P1.9 T = (pe − patm )Ae + m& (Ve − Vi ) = (80 − 20)(4)+ (10)(800 − 300) = 240 + 5 = 245kN Problem 10. – A high-pressure oxygen cylinder, typically found in most welding shops, accidentally is knocked over and the valve on top of the cylinder breaks off. This creates a hole with a cross-sectional area of 6.5 x 10-4 m2. Prior to the accident, the internal pressure of the oxygen is 14 MPa and the temperature is 27°C. Based on critical flow calculations, the velocity of the oxygen exiting the cylinder is estimated to be 300 m/s, the exit pressure 7.4 MPa and the exit temperature 250 K. How much thrust does the oxygen being expelled from the cylinder generate? What percentage is due to the pressure difference? What percentage due to the exiting momentum? Atmospheric pressure is 101 kPa. Also note that 0.2248 lbf = 1 N. Figure P1.10 Ve = 300 m/ s 4 2 Ae 6.5 10 m = × − pe = 7.4 MPa patm = 101 kPa = 0.101 MPa Te = 250 k e e e e e e e A V RT p m& = ρ A V = kg k R 259.82 J ⋅ = ( )( ) (6.5 10 )(300) 22.2 kg / s 259.82 250 m 7.4 10 N 2 4 6 × = ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ × = − 300 m/s 800 m/s 8