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mechanical engineering

Thermodynamics An Interactive Approach 1st edition Subrata Bhattacharjee - Solutions

Use potential flow to approximate the flow of air being sucked into a vacuum cleaner through a 2-D slit attachment, as in the figure. Model the flow as a line sink of strength (−m), with its axis in the zdirection at height a above the floor.(a) Sketch the streamlines and locate any
Consider three-dimensional, incompressible, irrotational flow. Use two methods to prove that the viscous term in the Navier-Stokes equation is zero: (a) Using vector notation; and (b) Expanding out the scalar terms using irrotationality.
Reconsider the lift-drag data for the NACA 4412 airfoil from Prob. 8.83.(a) Again draw the polar lift-drag plot and compare qualitatively with Fig. 7.26.(b) Find the maximum value of the lift-to-drag ratio.(c) Demonstrate a straight-line construction on the polar plot which will immediately yield
Find a formula for the stream function for flow of a doublet of strength λ at a distance a from a wall, as in Fig. C8.7(a) Sketch the streamlines.(b) Are there any stagnation points?(c) Find the maximum velocity along the wall and its position.
An ideal gas flows adiabatically through a duct. At section 1, p1 = 140 kPa, T1 = 260°C, and V1 = 75 m/s. Farther downstream, p2 = 30 kPa and T2 = 207°C. Calculate V2 in m/s and s2 − s1 in J/(kg ⋅ K) if the gas is(a) Air, k = 1.4, and(b) Argon, k = 1.67. Fig P9.1
Solve Prob. 9.1 if the gas is steam. Use two approaches: (a) An ideal gas from Table A.4; and (b) Real steam from the steam tables [15].
If 8 kg of oxygen in a closed tank at 200°C and 300 kPa is heated until the pressure rises to 400 kPa, calculate (a) The new temperature; (b) The total heat transfer; and (c) The change in entropy.
Compressibility becomes important when the Mach number > 0.3. How fast can a two-dimensional cylinder travel in sea-level standard air before compressibility becomes important somewhere in its vicinity?
Steam enters a nozzle at 377°C, 1.6 MPa, and a steady speed of 200 m/s and accelerates isentropically until it exits at saturation conditions. Estimate the exit velocity and temperature.
Helium at 300°C and 200 kPa, in a closed container, is cooled to a pressure of 100 kPa. Estimate (a) The new temperature, in °C; and (b) The change in entropy, in J/(kg⋅K).
Carbon dioxide (k = 1.28) enters a constant-area duct at 400°F, 100 lbf/in2 absolute, and 500 ft/s. Farther downstream the properties are V2 = 1000 ft/s and T2 = 900°F. Compute (a) P2, (b) the heat added between sections, (c) The entropy change between sections, and (d) The mass flow per unit
Atmospheric air at 20°C enters and fills an insulated tank which is initially evacuated. Using a control-volume analysis from Eq. (3.63), compute the tank air temperature when it is full.
Liquid hydrogen and oxygen are burned in a combustion chamber and fed through a rocket nozzle which exhausts at exit pressure equal to ambient pressure of 54 kPa. The nozzle exit diameter is 45 cm, and the jet exit density is 0.15 kg/m3. If the exhaust gas has a molecular weight of 18, estimate
A certain aircraft flies at the same Mach number regardless of its altitude. Compared to its speed at 12000-m Standard Altitude, it flies 127 km/h faster at sea level. Determine its Mach number.
At 300°C and 1 atm, estimate the speed of sound of (a) Nitrogen; (b) Hydrogen; (c) Helium; (d) Steam; and (e) Uranium hexafluoride 238UF6 (k ≠ 1.06).
Assume that water follows Eq. (1.19) with n ≈ 7 and B ≈ 3000. Compute the bulk modulus (in kPa) and the speed of sound (in m/s) at (a) 1 atm; and (b) 1100 atm (the deepest part of the ocean). (c) Compute the speed of sound at 20°C and 9000 atm and compare with the measured value of
Assume that the airfoil of Prob. 8.84 is flying at the same angle of attack at 6000 m standard altitude. Estimate the forward velocity, in mi/h, at which supersonic flow (and possible shock waves) will appear on the airfoil surface.
Assume steady adiabatic flow of a perfect gas. Show that the energy Eq. (9.21), when plotted as a versus V, forms an ellipse. Sketch this ellipse; label the intercepts and the regions of subsonic, sonic, and supersonic flow; and determine the ratio of the major and minor axes.
A weak pressure wave (sound wave), with a pressure change Δp ≈ 40 Pa, propagates through still air at 20°C and 1 atm. Estimate (a) The density change; (b) The temperature change; and (c) The velocity change across the wave.
A weak pressure wave (sound wave) Δp propagates through still air. Discuss the type of reflected pulse which occurs, and the boundary conditions which must be satisfied, when the wave strikes normal to, and is reflected from,(a) A solid wall; and(b) A free liquid surface.
A submarine at a depth of 800 m sends a sonar signal and receives the reflected wave back from a similar submerged object in 15 s. Using Prob. 9.12 as a guide, estimate the distance to the other object.
Race cars at the Indianapolis Speedway average speeds of 185 mi/h. After determining the altitude of Indianapolis, find the Mach number of these cars and estimate whether compressibility might affect their aerodynamics.
Concorde aircraft flies at Ma ≈ 2.3 at 11-km standard altitude. Estimate the temperature in °C at the front stagnation point. At what Mach number would it have a front stagnation point temperature of 450°C?
A gas flows at V = 200 m/s, p = 125 kPa, and T = 200°C. For (a) Air and (b) Helium, compute the maximum pressure and the maximum velocity attainable by expansion or compression.
CO2 expands isentropically through a duct from p1 = 125 kPa and T1 = 100°C to p2 = 80 kPa and V2 = 325 m/s. Compute (a) T2; (b) Ma2; (c) To; (d) po; (e) V1; and (f) Ma1.
Given the Pitot stagnation temperature and pressure and the static-pressure measurements in Fig P9.22, estimate the air velocity V, assuming(a) Incompressible flow and(b) Compressible flow.
A large rocket engine delivers hydrogen at 1500°C and 3 MPa, k = 1.41, R = 4124 J/kg⋅K, to a nozzle which exits with gas pressure equal to the ambient pressure of 54 kPa. Assuming isentropic flow, if the rocket thrust is 2 MN, estimate (a) The exit velocity; and (b) The mass flow of
For low-speed (nearly incompressible) gas flow, the stagnation pressure can be computed from Bernoulli€™s equation(a) For higher subsonic speeds, show that the isentropic relation (9.28a) can be expanded in a power series as follows:(b) Suppose that a Pitot-static tube in air measures the
If it is known that the air velocity in the duct is 750 ft/s, use that mercury manometer measurement in Fig. P9.25 to estimate the static pressure in the duct, in psia.
Show that for isentropic flow of a perfect gas if a Pitot-static probe measures p0, p, and T0, the gas velocity can be calculated from what would be a source of error if a shock wave were formed in front of the probe?
In many problems the sonic (*) properties are more useful reference values than the stagnation properties. For isentropic flow of a perfect gas, derive relations for p/p*, T/T*, and ρ/ρ* as functions of the Mach number. Let us help by giving the density-ratio formula:
A large vacuum tank, held at 60 kPa absolute, sucks sea-level standard air through a converging nozzle of throat diameter 3 cm. Estimate (a) The mass flow rate; and (b) The Mach number at the throat.
Steam from a large tank, where T = 400°C and p = 1 MPa, expands isentropically through a small nozzle until, at a section of 2-cm diameter, the pressure is 500 kPa. Using the Steam Tables, estimate (a) the temperature; (b) the velocity; and (c) the mass flow at this section. Is the flow subsonic?
Oxygen flows in a duct of diameter 5 cm. At one section, To = 300°C, p = 120 kPa, and the mass flow is 0.4 kg/s. Estimate, at this section, (a) V; (b) Ma; and (c) ρo.
Air flows adiabatically through a duct. At one section, V1 = 400 ft/s, T1 = 200°F, and p1 = 35 psia, while farther downstream V2 = 1100 ft/s and p2 = 18 psia. Compute (a) Ma2; (b) Umax; and (c) po2/po1.
The large compressed-air tank in Fig P9.32 exhausts from a nozzle at an exit velocity of 235 m/s. The mercury manometer reads h = 30 cm. Assuming isentropic flow, compute the pressure (a) In the tank and (b) In the atmosphere. (c) What is the exit Mach number?
Air flows isentropically from a reservoir, where p = 300 kPa and T = 500 K, to section 1 in a duct, where A1 = 0.2 m2 and V1 = 550 m/s. Compute (a) Ma1; (b) T1; (c) P1; (d) M; and (e) A*. Is the flow choked?
Steam in a tank at 450°F and 100 psia exhausts through a converging nozzle of throat area 0.1-in2 to a 1-atm environment; Compute the initial mass flow rate (a) For an ideal gas; and (b) From the Steam Tables.
Helium, at to = 400 K, enters a nozzle isentropically. At section 1, where A1 = 0.1 m2, a Pitot-static arrangement (see Fig. P9.25) measures stagnation pressure of 150 kPa and static pressure of 123 kPa. Estimate (a) Ma1; (b) Mass flow; (c) T1; and (d) A*.
An air tank of volume 1.5 m3 is at 800 kPa and 20°C when it begins exhausting through a converging nozzle to sea-level conditions. The throat area is 0.75 cm2. Estimate (a) The initial mass flow; (b) The time to blow down to 500 kPa; and (c) The time when the nozzle ceases being choked.
Make an exact control volume analysis of the blowdown process in Fig. P9.37, assuming an insulated tank with negligible kinetic and potential energy, Assume critical flow at the exit and show that both po and to decrease during blowdown. Set up first-order differential equations for po (t) and To
Prob. 9.37 makes an ideal senior project or combined laboratory and computer problem, as described in Ref. 30, sec. 8.6. In Bober and Kenyon’s lab experiment, the tank had a volume of 0.0352 ft3 and was initially filled with air at 50 lb/in2 gage and 72°F. Atmospheric pressure was 14.5 lb/in2
Consider isentropic flow in a channel of varying area, between sections 1 and 2. Given Ma1 = 2.0, we desire that V2/V1 equal 1.2. Estimate (a) Ma2 and (b) A2/A1. (c) Sketch what this channel looks like, for example, does it converge or diverge? Is there a throat?
Air, with stagnation conditions of 800 kPa and 100°C, expands isentropically to a section of a duct where A1 = 20 cm2 and p1 = 47 kPa. Compute (a) Ma1; (b) The throat area; and (c) M. At section 2, between the throat and section 1, the area is 9 cm2. (d) Estimate the Mach number at section
Air, with a stagnation pressure of 100 kPa, flows through the nozzle in Fig. P9.41, which is 2 m long and has an area variation approximated by A≈20−20x+10x2 with A in cm2 and x in m, It is desired to plot the complete family of isentropic pressures p(x) in this nozzle, for the range of
A bicycle tire is filled with air at 169.12 kPa (abs) and 30°C. The valve breaks, and air exhausts into the atmosphere of 100 kPa (abs) and 20°C. The valve exit is 2-mmdiameter and is the smallest area in the system. Assuming one-dimensional isentropic flow, (a) Find the initial Mach number,
Air flows isentropically through a duct with To = 300°C. At two sections with identical areas of 25 cm2, the pressures are p1 = 120 kPa and p2 = 60 kPa. Determine (a) The mass flow; (b) The throat area, and (c) Ma2.
In Prob. 3.34 we knew nothing about compressible flow at the time so merely assumed exit conditions p2 and T2 and computed V2 as an application of the continuity equation. Suppose that the throat diameter is 3 in. For the given stagnation conditions in the rocket chamber in Fig P3.34 and assuming k
At a point upstream of the throat of a converging-diverging nozzle, the properties are V1 = 200 m/s, T1 = 300 K, and p1 = 125 kPa. If the exit flow is supersonic, compute, from isentropic theory, (a) M; and (b) A1. The throat area is 35 cm2.
If the writer did not falter, the results of Prob. 9.43 are(a) 0.671 kg/s,(b) 23.3 cm2, and(c) 1.32. Do not tell your friends who are still working on Prob. 9.43. Consider a control volume which encloses the nozzle between these two 25-cm2 sections. If the pressure outside the duct is 1 atm,
In wind-tunnel testing near Mach 1, a small area decrease caused by model blockage can be important. Let the test section area be 1 sq.m. And unblocked conditions are Ma = 1.1 and T = 20°C. What model area will first cause the test section to choke? If the model cross-section is 0.004 sq.m what %
A force F = 1100 N pushes a piston of diameter 12 cm through an insulated cylinder containing air at 20°C, as in Fig. P9.48. The exit diameter is 3 mm, and pa = 1 atm. Estimate(a) Ve,(b) Vp, and(c) Me
Consider the venturi nozzle of Fig. 6.40c, with D = 5 cm and d = 3 cm. Air stagnation temperature is 300 K, and the upstream velocity V1 = 72 m/s. If the throat pressure is 124 kPa, estimate, with isentropic flow theory, (a) p1; (b) Ma2; and (c) the mass flow.
Argon expands isentropically at 1 kg/s in a converging nozzle with D1 = 10 cm, p1 = 150 kPa, and T1 = 100°C. The flow discharges to a pressure of 101 kPa. (a) What is the nozzle exit diameter? (b) How much further can the ambient pressure be reduced before it affects the inlet mass flow?
Air, at stagnation conditions of 500 K and 200 kPa, flow through a nozzle, at section 1, where A = 12 cm2, the density is 0.32 kg/m3. Assuming isentropic flow, (a) Find the mass flow. (b) Is the flow choked? Is so, estimate A*. Also estimate (c) P 1; and (d) Ma1.
A converging-diverging nozzle exits smoothly to sea-level standard atmosphere. It is supplied by a 40-m3 tank initially at 800 kPa and 100°C. Assuming isentropic flow, estimate (a) The throat area; and (b) The tank pressure after 10 sec of operation. NOTE: The exit area is 10 cm2 (this was
Air flows steadily from a reservoir at 20°C through a nozzle of exit area 20 cm2 and strikes a vertical plate as in Fig. P9.53 the flow is subsonic throughout. A force of 135 N is required to hold the plate stationary. Compute (a) Ve, (b) Mae, and (c) p0 if pa = 101 kPa.
For flow of air through a normal shock, the upstream conditions are V1 = 600 m/s, To1 = 500 K, and po1 = 700 kPa. Compute the downstream conditions Ma2, V2, T2, p2, and po2.
Air, supplied by a reservoir at 450 kPa, flows through a converging-diverging nozzle whose throat area is 12 cm2. A normal shock stands where A1 = 20 cm2. (a) Compute the pressure just downstream of this shock. Still farther downstream, where A3 = 30 cm2, estimate (b) P3; (c) A3*; and (d)
Air from a reservoir at 20°C and 500 kPa flows through a duct and forms a normal shock downstream of a throat of area 10 cm2. By an odd coincidence it is found that the stagnation pressure downstream of this shock exactly equals the throat pressure. What is the area where the shock wave stands?
Air flows from a tank through a nozzle into the standard atmosphere, as in Fig. P9.57. A normal shock stands in the exit of the nozzle, as shown. Estimate(a) The tank pressure; and(b) The mass flow.
Argon (Table A.4) approaches a normal shock with V1 = 700 m/s, p1 = 125 kPa, and T1 = 350 K. Estimate (a) V2, and (b) p2. (c) What pressure p2 would result if the same velocity change V1 to V2 were accomplished isentropically?
Air, at stagnation conditions of 450 K and 250 kPa, flows through a nozzle. At section 1, where the area = 15 cm2, there is a normal shock wave. If the mass flow is 0.4 kg/s, estimate (a) The Mach number; and (b) The stagnation pressure just downstream of the shock.
When a Pitot tube such as Fig (6.30) is placed in a supersonic flow, a normal shock will stand in front of the probe. Suppose the probe reads p0 = 190 kPa and p = 150 kPa. If the stagnation temperature is 400 K, estimate the (supersonic) Mach number and velocity upstream of the shock.
Repeat Prob. 9.56 except this time let the odd coincidence be that the static pressure downstream of the shock exactly equals the throat pressure. What is the area where the shock wave stands?
An atomic explosion propagates into still air at 14.7 psia and 520°R. The pressure just inside the shock is 5000 psia. Assuming k = 1.4, what are the speed C of the shock and the velocity V just inside the shock?
Sea-level standard air is sucked into a vacuum tank through a nozzle, as in Fig. P9.63. A normal shock stands where the nozzle area is 2 cm2, as shown. Estimate(a) The pressure in the tank; and(b) The mass flow. Solution: The flow at the exit section (€œ3€) is subsonic (after a
Air in a large tank at 100°C and 150 kPa exhausts to the atmosphere through a converging nozzle with a 5-cm2 throat area, compute the exit mass flow if the atmospheric pressure is (a) 100 kPa; (b) 60 kPa; and (c) 30 kPa.
Air flows through a converging diverging nozzle between two large reservoirs, as in Fig. P9.65. A mercury manometer reads h = 15 cm. Estimate the downstream reservoir pressure. Is there a shock wave in the flow? If so, does it stand in the exit plane or farther upstream?
In Prob. 9.65 what would be the mercury manometer reading if the nozzle were operating exactly at supersonic €œdesign€ conditions?
In Prob. 9.65 estimate the complete range of manometer readings h for which the flow through the nozzle is isentropic, except possibly in the exit plane.
Air in a tank at 120 kPa and 300 K exhausts to the atmosphere through a 5-cm2-throat converging nozzle at a rate of 0.12 kg/s, what is the atmospheric pressure What is the maximum mass flow possible at low atmospheric pressure?
With reference to Prob. 3.68, show that the thrust of a rocket engine exhausting into a vacuum is given by where Ae = exit area Mae = exit Mach number p0 = stagnation pressure in combustion chamber Note that stagnation temperature does not enter into the thrust.
Air, at stagnation temperature 100°C, expands isentropically through a nozzle of 6-cm2 throat area and 18-cm2 exit area. The mass flow is at its maximum value of 0.5 kg/s. Estimate the exit pressure for (a) Subsonic; and (b) Supersonic exit flow.
For the nozzle of problem 9.70, allowing for non-isentropic flow, what is the range of exit tank pressures pb for which (a) The diverging nozzle flow is fully supersonic; (b) The exit flow is subsonic; (c) The mass flow is independent of pb; (d) The exit plane pressure pe is independent of
A large tank at 500 K and 165 kPa feeds air to a converging nozzle. The back pressure outside the nozzle exit is sea-level standard. What is the appropriate exit diameter if the desired mass flow is 72 kg/h?
Air flows isentropically in a converging-diverging nozzle with a throat area of 3 cm2. At section 1, the pressure is 101 kPa, T1 = 300 K, and V1 = 868 m/s. (a) Is the nozzle choked? Determine (b) A1; and (c) The mass flow. Suppose, without changing stagnation conditions of A1, the flexible
The perfect-gas assumption leads smoothly to Mach-number relations which are very convenient (and tabulated). This is not so for a real gas such as steam. To illustrate, let steam at T0 = 500°C and p0 = 2 MPa expand isentropically through a converging nozzle whose exit area is 10 cm2. Using the
A double-tank system in Fig P9.75 has two identical converging nozzles of 1-in2 throat area. Tank 1 is very large, and tank 2 is small enough to be in steady-flow equilibrium with the jet from tank 1. Nozzle flow is isentropic, but entropy changes between 1 and 3 due to jet dissipation in tank 2.
A large reservoir at 20°C and 800 kPa is used to fill a small insulated tank through a converging-diverging nozzle with 1-cm2 throat area and 1.66-cm2 exit area. The small tank has a volume of 1 m3 and is initially at 20°C and 100 kPa. Estimate the elapsed time when (a) Shock waves begin to
The orientation of a hole can make a difference. Consider holes A and B in Fig. P9.78 which are identical but reversed, for the given air properties on either side, compute the mass flow through each hole and explain the difference.
A perfect gas (not air) expands isentropically through a supersonic nozzle with an exit area 5 times its throat area. The exit Mach number is 3.8. What is the specific heat ratio of the gas? What might this gas be? If po = 300 kPa, what is the exit pressure of the gas?
A large reservoir at 600 K supplies air flow through a converging-diverging nozzle with a throat area of 2 cm2, A normal shock wave forms at a section of area 6 cm2. Just downstream of this shock, the pressure is 150 kPa. Calculate (a) The pressure in the throat; (b) The mass flow; and (c)
A sea-level automobile tire is initially at 32 lbf/in2 gage pressure and 75°F. When it is punctured with a hole which resembles a converging nozzle, its pressure drops to 15 lbf/in2 gage in 12 min. Estimate the size of the hole, in thousandths of an inch?
Helium, in a large tank at 100°C and 400 kPa, discharges to a receiver through a converging-diverging nozzle designed to exit at Ma = 2.5 with exit area 1.2 cm2. Compute (a) The receiver pressure and (b) The mass flow at design conditions. (c) Also estimate the range of receiver pressures
Air at 500 K flows through a converging-diverging nozzle with throat area of 1 cm2 and exit area of 2.7 cm2. When the mass flow is 182.2 kg/h, a Pitot-static probe placed in the exit plane reads p0 = 250.6 kPa and p = 240.1 kPa. Estimate the exit velocity. Is there a normal shock wave in the duct?
When operating at design conditions (smooth exit to sea-level pressure), a rocket engine has a thrust of 1 million lbf. The chamber pressure and temperature are 600 lbf/in2 absolute and 4000°R, respectively. The exhaust gases approximate k = 1.38 with a molecular weight of 26. Estimate (a) The
Air flows through a duct as in Fig P9.84, where A1 = 24 cm2, A2 = 18 cm2, and A3 = 32 cm2. A normal shock stands at section 2. Compute(a) The mass flow,(b) The Mach number, and(c) The stagnation pressure at section 3.
A large tank at 300 kPa delivers air through a nozzle of 1-cm2 throat area and 2.2-cm2 exit area. A normal shock wave stands in the exit plane. The temperature just downstream of this shock is 473 K. Calculate (a) The temperature in the large tank; (b) The receiver pressure; and (c) The mass
Air enters a 3-cm diameter pipe 15 m long at V1 73 ms, p1 550 kPa, and T1 60C. The friction factor is 0.018. Compute V2, p2, T2, and p02 at the end of the pipe. How much additional pipe length would cause the exit flow to be sonic?
Air enters an adiabatic duct of L/D = 40 at V1 = 170 m/s and T1 = 300 K. The flow at the exit is choked. What is the average friction factor in the duct?
Air enters a 5- by 5-cm square duct at V1 900 ms and T1 300 K. The friction factor is 0.02. For what length duct will the flow exactly decelerate to Ma 1.0? If the duct length is 2 m, will there be a normal shock in the duct? If so, at what
Carbon dioxide flows through an insulated pipe 25 m long and 8 cm in diameter. The friction factor is 0.025. At the entrance, p = 300 kPa and T = 400 K. The mass flow is 1.5 kg/s. Estimate the pressure drop by (a) Compressible; and (b) Incompressible (Sect. 6.6) flow theory. (c) For what pipe
Air, supplied at p0 = 700 kPa and T0 = 330 K, flows through a converging nozzle into a pipe of 2.5-cm diameter which exits to a near vacuum. If f = 0.022, what will be the mass flow through the pipe if its length is (a) 0 m, (b) 1 m, and (c) 10 m?
Air flows steadily from a tank through the pipe in Fig. P9.91. There is a converging nozzle on the end. If the mass flow is 3 kg/s and the flow is choked, estimate (a) The Mach number at section 1; and (b) The pressure in the tank.
Modify Prob. 9.91 as follows: Let the tank pressure be 700 kPa, and let the nozzle be choked. Determine (a) Ma2; and (b) The mass flow. Keep To = 100°C.
Air flows adiabatically in a 3-cm-diameter duct with f = 0.015. At the entrance, V = 950 m/s and T = 250 K. How far down the tube will (a) The Mach number be 1.8; and (b) The flow be choked?
Compressible pipe flow with friction, Sec. 9.7, assumes constant stagnation enthalpy and mass flow but variable momentum. Such a flow is often called Fanno flow, and a line representing all possible property changes on a temperature-entropy chart is called a Fanno line. Assuming an ideal gas with k
Helium (Table A.4) enters a 5-cm-diameter pipe at p1 = 550 kPa, V1 = 312 m/s, and T1 = 40°C. The friction factor is 0.025. If the flow is choked, determine (a) The length of the duct and (b) The exit pressure.
Methane (CH4) flows through an insulated 15-cm-diameter pipe with f = 0.023. Entrance conditions are 600 kPa, 100°C, and a mass flow of 5 kg/s. What lengths of pipe will? (a) Choke the flow; (b) Raise the velocity by 50%; (c) Decrease the pressure by 50%?

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