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mechanical engineering

Questions and Answers of Mechanical Engineering

By making a few algebraic substitutions, show that Eq. (9.74), or the relation in Prob. 9.96, may be written in the density form Why is this formula awkward if one is trying to solve for the mass
Compressible laminar flow, f ≈ 64/Re, may occur in capillary tubes. Consider air, at stagnation conditions of 100°C and 200 kPa, entering a tube 3 cm long and 0.1 mm in diameter. If the
A compressor forces air through a smooth pipe 20 m long and 4 cm in diameter, as in Fig. P9.99. The air leaves at 101 kPa and 200°C. The compressor data for pressure rise versus mass flow are shown
Modify Prob. 9.99 as follows: Find the length of 4-cm-diameter pipe for which the pump pressure rise will be exactly 200 kPa.
How do the compressible-pipe-flow formulas behave for small pressure drops? Let air at 20°C enter a tube of diameter 1 cm and length 3 m. If f = 0.028 with p1 = 102 kPa and p2 = 100 kPa, estimate
Air at 550 kPa and 100°C enters a smooth 1-m-long pipe and then passes through a second smooth pipe to a 30-kPa reservoir, as in Fig. P9.102. Using the Moody chart to compute f, estimate the mass
Natural gas, with k ≈ 1.3 and a molecular weight of 16, is to be pumped through 100 km of 81-cm-diameter pipeline. The downstream pressure is 150 kPa. If the gas enters at 60°C, the mass flow
A tank of oxygen (Table A.4) at 20°C is to supply an astronaut through an umbilical tube 12 m long and 1.5 cm in diameter. The exit pressure in the tube is 40 kPa. If the desired mass flow is 90
Air enters a 5-cm-diameter pipe at p1 = 200 kPa and T1 = 350 kPa. The downstream receiver pressure is 74 kPa and the friction factor is 0.02. If the exit is choked, what is (a) The length of the
Air at 300 K flows through a duct 50 m long with f = 0.019 . What is the minimum duct diameter which can carry the flow without choking if the entrance velocity is (a) 50 m/s, (b) 150 m/s, and
A fuel-air mixture, assumed equivalent to air, enters a duct combustion chamber at V1 = 104 m/s and T1 = 300 K. What amount of heat addition in kJ/kg will cause the exit flow to be choked? What will
What happens to the inlet flow of Prob. 9.107 if the combustion yields 1500 kJ/kg heat addition and po1 and To1 remain the same? How much is the mass flow reduced?
A jet engine at 7000-m altitude takes in 45 kg/s of air and adds 550 kJ/kg in the combustion chamber. The chamber cross section is 0.5 m2, and the air enters the chamber at 80 kPa and 5°C. After
Compressible pipe flow with heat addition, Sec. 9.8, assumes constant momentum (p + ρV2) and constant mass flow but variable stagnation enthalpy. Such a flow is often called Rayleigh flow, and a
Add to your Rayleigh line of Prob. 9.110 a Fanno line (see Prob. 9.94) for stagnation enthalpy equal to the value associated with state 1 in Prob. 9.110. The two curves will intersect at state 1,
Air enters a duct subsonically at section 1 at 1.2 kg/s. When 650 kW of heat is added, the flow chokes at the exit at p2 = 95 kPa and T2 = 700 K. Assuming frictionless heat addition, estimate (a)
Air enters a constant-area duct at p1 = 90 kPa, V1 = 520 m/s, and T1 = 558°C. It is then cooled with negligible friction until it exists at p2 = 160 kPa. Estimate (a) V2; (b) T2; and (c) The
We have simplified things here by separating friction (Sec. 9.7) from heat addition (Sec. 9.8). Actually, they often occur together, and their effects must be evaluated simultaneously. Show that, for
Air flows subsonically in a duct with negligible friction. When heat is added in the amount of 948 kJ/kg, the pressure drops from p1 = 200 kPa to p2 = 106 kPa. Using one-dimensional theory, estimate
An observer at sea level does not hear an aircraft flying at 12000 ft standard altitude until it is 5 statute miles past her. Estimate the aircraft speed in ft/sec.
A particle moving at uniform velocity in sea-level standard air creates the two disturbance spheres shown in Fig. P9.118. Compute the particle velocity and Mach number
An observer at sea level does not hear an aircraft flying at 6000-m standard altitude until 15 seconds after it has passed overhead. Estimate the aircraft speed in m/s.
The particle in Fig P9.119 is moving supersonically in sea-level standard air. From the two disturbance spheres shown, compute the particle (a) Mach number; (b) Velocity; and (c) Mach angle.
The particle in Fig P9.120 is moving in sea-level standard air. From the two disturbance spheres shown, estimate(a) The position of the particle at this instant; and(b) The temperature in °C at
A thermistor probe, in the shape of a needle parallel to the flow, reads a static temperature of –25°C when inserted in the stream. A conical disturbance of half-angle 17° is formed. Estimate
Supersonic air takes a 5° compression turn, as in Fig. P9.122. Compute the downstream pressure and Mach number and wave angle, and compare with small disturbance theory.
Modify Prob. 9.122 as follows: Let the 5° turn be in the form of five separate compression turns of 1° each. Compute the final Mach number and pressure, and compare the pressure with an isentropic
When a sea-level air flow approaches a ramp of angle 20°, an oblique shock wave forms as in Figure P9.124 Calculate(a) Ma1;(b) P2;(c) T2; and (d) V2.
Show that, as the upstream Mach number approaches infinity, the Mach number downstream of an attached oblique-shock wave will have the value
Consider airflow at Ma1 = 2.2. Calculate, to two decimal places, (a) The deflection angle for which the downstream flow is sonic; and (b) The maximum deflection angle.
Do the Mach waves upstream of an oblique-shock wave intersect with the shock? Assuming supersonic downstream flow, do the downstream Mach waves intersect the shock? Show that for small deflections
Air flows past a two-dimensional wedge-nosed body as in Fig. P9.128. Determine the wedge half-angle δ for which the horizontal component of the total pressure force on the nose is 35 kN/m of
Air flows at supersonic speed toward a compression ramp, as in Fig. P9.129. A scratch on the wall at a creates a wave of 30° angle, while the oblique shock has a 50° angle. What is(a) The
Modify Prob. P9.129 as follows: If the wave angle φ is 42°, determine (a) The shock wave angle (it is not 50°) and (b) The deflection angle θ.
The following formula has been suggested as an alternate to Eq. (9.86) to relate upstream Mach number to the oblique shock wave angle β and turning angle θ: Can you prove or disprove this
Air flows at Ma = 3 and p = 10 psia toward a wedge of 16° angle at zero incidence, as in Fig. P9.132(a) If the pointed edge is forward, what is the pressure at point A? If the blunt edge is
Air flows supersonically toward the double-wedge system in the figure. The (x, y) coordinates of the tips are given. Both wedges have 15° deflection angles. The shock wave of the forward wedge
When an oblique shock strikes a solid wall, it reflects as a shock of sufficient strength to cause the exit flow Ma3 to be parallel to the wall, as in Fig. P9.134. For airflow with Ma1 = 2.5 and p1 =
A bend in the bottom of a supersonic duct flow induces a shock wave which reflects from the upper wall, as in Fig. P9.135. Compute the Mach number and pressure in region 3
Figure P9.136 is a special application of Prob. 9.135. With careful design, one can orient the bend on the lower wall so that the reflected wave is exactly canceled by the return bend, as shown. This
A 6° half-angle wedge creates the reflected shock system in Fig. P9.137. If Ma3 = 2.5, find(a) Ma1; and(b) The angle α.
The supersonic nozzle of Fig P9.138 is over expanded (case G of Fig. 9.12) with Ae/At = 3.0 and a stagnation pressure of 350 kPa. If the jet edge makes a 4° angle with the nozzle centerline, what
Airflow at Ma = 2.2 takes a compression turn of 12° and then another turn of angle θ in Fig. P9.139. What is the maximum value of θ for the second shock to be attached? Will the two
The solution to Prob. 9.122 is Ma2 = 2.750 and p2 = 145.5 kPa. Compare these results with an isentropic compression turn of 5°, using Prandtl-Meyer theory.
Supersonic airflow takes a 5° expansion turn, as in Fig. P9.141. Compute the downstream Mach number and pressure and compare with small-disturbance theory.
A supersonic airflow at Ma1 = 3.2 and p1 = 50 kPa undergoes a compression shock followed by an isentropic expansion turn. The flow deflection is 30° for each turn. Compute Ma2 and p2 if (a) The
Airflow at Ma1 = 3.2 passes through a 25° oblique-shock deflection. What isentropic expansion turn is required to bring the flow back to(a) Ma1 and(b) p1?
Consider a smooth isentropic compression turn of 20°, as in Fig. P9.144. The Mach waves thus generated will form a converging fan. Sketch this fan as accurately as possible, using five equally
Air at Ma1 = 2.0 and p1 = 100 kPa undergoes an isentropic expansion to a downstream pressure of 50 kPa. What is the desired turn angle in degrees?
A converging-diverging nozzle with a 4:1 exit-area ratio and p0 = 500 kPa operates in an under expanded condition (case 1 of Fig. 9.12) as in Fig. P9.147. The receiver pressure is pa = 10 kPa, which
Air flows supersonically over a surface which changes direction twice, as in Fig. P9.146 Calculate (a) Ma2; and (b) p3
Air flows supersonically over a circular-arc surface as in Fig. P9.148. Estimate (a) The Mach number Ma2 and (b) The pressure p2 as the flow leaves the circular surface.
Repeat Example 9.21 for an angle of attack of 2°. Is the lift coefficient linear with α in this range of 0°
A flat plate airfoil with chord C = 1.2 m is to have a lift of 30 kN/m when flying at 5000-m standard altitude with U∞ = 641 m/s. Using Ackeret theory, estimate (a) The angle of attack; and
A supersonic airfoil has a parabolic symmetric shape for upper and lower surfaces such that the maximum thickness is t at 12 x = C. Compute the drag coefficient at zero incidence by Ackeret theory,
Air flows at Ma = 2.5 past a half wedge airfoil whose angles are 4°, as in Fig. P9.151. Compute the lift and drag coefficients at α equal to(a) 0°; and(b) 6°.
A supersonic transport has a mass of 65 Mg and cruises at 11-km standard altitude at a Mach number of 2.25. If the angle of attack is 2° and its wings can be approximated by flat plates, estimate
A symmetric supersonic airfoil has its upper and lower surfaces defined by a sine wave shape: where t is the maximum thickness, which occurs at x = C/2. Use Ackeret theory to derive an expression for
For the sine-wave airfoil shape of Prob. 9.154, with Ma∞ = 2.5, k = 1.4, t/C = 0.1, and α = 0°, plot (without computing the overall forces) the pressure distribution p(x)/p∞
Prove from Ackeret theory that for a given supersonic airfoil shape with sharp leading and trailing edges and a given thickness, the minimum-thickness drag occurs for a symmetric double-wedge shape.
A thin circular-arc airfoil is shown in Fig. P9.156. The leading edge is parallel to the free stream. Using linearized (small turning- angle) supersonic-flow theory, derive a formula for the lift and
The formula for shallow-water wave propagation speed, Eq. (10.9) or (10.10), is independent of the physical properties of the liquid, i.e., density, viscosity, or surface tension. Does this mean that
A shallow-water wave 12 cm high propagates into still water of depth 1.1 m. Compute (a) The wave speed; and (b) The induced velocity δ V.
Narragansett Bay is approximately 21 (statute) mi long and has an average depth of 42 ft. Tidal charts for the area indicate a time delay of 30 min between high tide at the mouth of the bay (Newport,
The water-channel flow in Fig P10.4 has a free surface in three places. Does it qualify as an open-channel flow? Explain. What does the dashed line represent?
Water flows rapidly in a channel 25 cm deep. Piercing the surface with a pencil point creates a wedge-like wave of included angle 38°, as shown. Estimate the velocity V of the water flow.
Pebbles dropped successively at the same point, into a water-channel flow of depth 42 cm, create two circular ripples, as in Fig. P10.6. From this information, estimate(a) The Froude number; and,(b)
Pebbles dropped successively at the same point, into a water-channel flow of depth 65 cm, create two circular ripples, as in Fig. P10.7. From this information, estimate(a) The Froude number; and(b)
An earthquake near the Kenai Peninsula, Alaska, creates a single “tidal” wave (called a ‘tsunami’) which propagates south across the Pacific Ocean. If the average ocean depth is 4 km and
Equation (10.10) is for a single disturbance wave. For periodic small-amplitude surface waves of wavelength λ and period T, in viscid theory [5 to 9] predicts a wave propagation speed where y is
If surface tension U is included in the analysis of Prob. 10.9, the resulting wave speed is [Refs. 5 to 9]:
A rectangular channel is 2 m wide and contains water 3 m deep. If the slope is 0.85° and the lining is corrugated metal, estimate the discharge for uniform flow.
(a) For laminar draining of a wide thin sheet of water on pavement sloped at angle θ, as in Fig. P4.36, show that the flow rate is given by where b is the sheet width and h its depth(b) By
The laminar-draining flow from Prob. 10.12 may undergo transition to turbulence if Re > 500. If the pavement slope is 0.0045, what is the maximum sheet thickness, in mm, for which laminar flow is
The Chézy formula (10.18) is independent of fluid density and viscosity. Does this mean that water, mercury, alcohol, and SAE 30 oil will all flow down a given open channel at the same rate? Explain.
The finished-concrete channel of Fig P10.15 is designed for a flow rate of 6 m3/s at a normal depth of 1 m. Determine(a) The design slope of the channel and(b) The percentage of reduction in flow if
In Prob. 10.15, for finished concrete, determine the percentage reduction in flow if the channel is divided in the center by the proposed barrier in Fig. P10.15 above. How does your estimate change
The trapezoidal channel of Fig P10.17 is made of brickwork and slopes at 1:500. Determine the flow rate if the normal depth is 80 cm.
Modify Prob. 10.17 as follows: Determine the normal depth for which the flow rate will be 8 m3/s
Modify Prob. 10.17 as follows: Let the surface be clean earth, which erodes if V exceeds 1.5 m/s. What is the maximum depth to avoid erosion?
A circular corrugated-metal storm drain is flowing half-full over a slope of 4 ft/mile. Estimate the normal discharge if the drain diameter is 8 ft.
An engineer makes careful measurements with a weir (see Sect. 10.7 later) which monitors a rectangular unfinished concrete channel laid on a slope of 1°. She finds, perhaps with surprise, that when
A trapezoidal aqueduct has b = 5 m and θ = 40° and carries a normal flow of 60 m3/s when y = 3.2 m. For clay tile surfaces, estimate the required elevation drop in m/km.
It is desired to excavate a clean-earth channel as a trapezoidal cross-section with θ = 60° (see Fig. 10.7). The expected flow rate is 500 ft3/s, and the slope is 8 ft per mile. The uniform
A riveted-steel channel slopes at 1:500 and has a Vee shape with an included angle of 80°. Find the normal depth if the flow rate is 900 m3/h.
The equilateral-triangle in Fig P10.25 has constant slope and Manning factor n. Find Qmax and Vmax. Then, by analogy with Fig. 10.6b, plot the ratios Q/Qmax and V/Vmax as a function of y/a for the
In the spirit of Fig. 10.6b, analyze a rectangular channel in uniform flow with constant area A = by, constant slope, but varying width b and depth y. Plot the resulting flow rate Q, normalized by
A circular unfinished-cement water channel has a slope of 1:600 and a diameter of 5 ft. Estimate the normal discharge in gal/min for which the average wall shear stress is 0.18 lbf/ft2 and compare
Show that, for any straight, prismatic channel in uniform flow, the average wall shear stress is given by τavg ≈ρgRhSo Use this result in Prob. 10.27 also.
Suppose that the trapezoidal channel of Fig. P10.17 contains sand and silt which we wish not to erode. According to an empirical correlation by A. Shields in 1936, the average wall shear stress
A clay tile V-shaped channel, with an included angle of 90°, is 1 km long and is laid out on a 1:400 slope, when running at a depth of 2 m, the upstream end is suddenly closed while the lower end
A storm drain has the cross section shown in Fig. P10.31 and is laid on a slope 1.5 m/km. If it is constructed of brickwork, find the normal discharge when the water level passes through the center
A 2-m-diameter clay tile sewer pipe runs half full on a slope of 0.25°. Compute the normal flow rate in gal/min.
Five of the sewer pipes from Prob. 10.32 empty into a single asphalt pipe, also laid out at 0.25°. If the large pipe is also to run half-full, what should be its diameter?
In flood stage a natural channel often consists of a deep main channel plus two floodplains, as in Fig. P10.35. The floodplains are often shallow and rough. If the channel has the same slope
A brick rectangular channel, with a slope of 0.002, is designed to carry 230 ft3/s of water in uniform flow. There is an argument over whether the channel width should be 4 ft or 8 ft. Which design
The Blackstone River in northern Rhode Island normally flows at about 25 m3/s and resembles Fig. P10.35 with a clean-earth center channel, b1 ≈ 20 m and y1 ≈ 3 m. The bed slope is about 2
A triangular channel (see Fig E10.6) is to be constructed of corrugated metal and will carry 8 m3/s on a slope of 0.005. The supply of sheet metal is limited, so the engineers want to minimize the
A rectangular channel has b = 3 m and y = 1 m. If n and So are the same, what is the diameter of a semicircular channel which will have the same discharge? Compare the two wetted perimeters.
A trapezoidal channel has n = 0.022 and S0 = 0.0003 and is made in the shape of a half-hexagon for maximum efficiency. What should the length of the side of the hexagon be if the channel is to carry

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