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Introduction to Compressible Flow Review of thermodynamics 12.1 An air flow in a duct passes through a thick filter. What happens to the pressure, temperature, and density of the air as it does so? Hint: This is a throttling process Answer:

Introduction to Compressible Flow 12.3 A vendor claims that an adiabatic compressor takes in air at atmospheric pressure and 50 oF and delivers the air at 150 psig and 200 oF. Is this possible? Justify your answer by calculation. Sketch the process on a Ts diagram. Answer:  S = -0.1037 Btu/lbm oR so no feasible (violates 2nd law of thermodynamics)

Introduction to Compressible Flow

12.5 Air initially at 50 psia and 660 oR expands to atmospheric pressure. The process by which this expansion occurs is defined by the expression pV1.3 = constant. Calculate the final temperature and the change in entropy through the process Answer: T2 = 498 oR  S = 0.0161 Btu/lbm oR

Introduction to Compressible Flow

12.7 Air expands without heat transfer through a turbine from a pressure of 10 bars and a temperature of 1400 oK to a pressure of 1 bar. If the turbine has an efficiency of 80 percent, determine the exit temperature and the changes in enthalpy and entropy across the turbine, if the turbine is generating 1 MW of power, what is the mass flow rate of air through the turbine? Answer: T2 = 860 oK,  h = 542 kJ/kg,  S = 171.7 J/kg oK, m = 1.845 kg/s

Introduction to Compressible Flow

12.9 An automobile supercharger is a device that pressurizes the air that is used by the engine for combustion to increase the engine power (how does it differ from a turbocharger?). A supercharger takes in air at 70 oF and atmospheric pressure and boosts it to 200 psig, at an intake rate of 0.5 ft3/s. What are the pressure, temperature, and volume flow rate at the exit? (The relatively high exit temperature is the reason an intercooler is also used.) Assuming a 70 percent efficiency, what is the power drawn by the supercharger? Hint: the efficiency is defined as the ratio of the isentropic power to actual power. Answer: W=8.26 kW

Introduction to Compressible Flow

12.11 Air is contained in a piston-cylinder device. Temperature of the air is 100 oC. Using the fact that for a reversible process the heat transfer q = ∫Tds, compare the amount of heat (J/kg) required to raise the temperature of the air to 1200 oC at (a) constant pressure and (b) constant volume. Verify your results using the first law of thermodynamics. Plot the process on a Ts diagram. Answer - computation: dQ/dm = 1104 kJ/Kg at constant pressure; dQ/dm = 789 kJ/kg at constant volume

Introduction to Compressible Flow

12.12 The four-stroke Otto cycle of a typical automobile engine is sometimes modeled as an ideal airstandard closed system. In this simplified system the combustion process is modeled as a heating process, and the exhaust-intake process as a cooling process of the working fluid (air). The cycle consists of: isentropic compression from state 1 (p1=100 kPa (abs), T1=20 oC, V1 =500 cc) to state 2 (V2=V1/8.5); isometric (constant volume) heat addition to state 3 (T 3=2750 oC); isentropic expansion to state 4 (V4 =V1); and isometric cooling back to state 1. Plot the pV and Ts diagrams for this cycle, and find the efficiency, defined as the net work (the cycle area in pV space) divided by the heat added. Computation

Introduction to Compressible Flow

12.13 The four stroke cycle of a typical diesel engine is sometimes modeled as an ideal air standard closed system. In this simplified system the combustion process is modeled as a heating process, and the exhaust-intake process as a cooling process of the working fluid (air). The cycle consist of: isentropic compression from state 1 (p1 = 100 kPa abs, T1 = 20 oC, V1 = 500 cc) to state 2 (V 2 = V1/12.5); isometric heat addition to state 3 (T3 = 3000 oC); isobaric heat addition to state 4 (V4 = 1.75 V3); isentropic expansion to state 5; and isometric cooling back to state 1. Plot the pV and Ts diagrams for this cycle, and find the efficiency, defined as the net work (the cycle area in pV space) divided by the heat added. Answer - computation:  =58.8%

Introduction to Compressible Flow

12.15 A tank of volume V= 10 m 3 contains compressed air at 15 oC. The gage pressure in the tank is 4.50 MPa. Evaluate the work required to fill the tank by compressing air from standard atmosphere conditions for: a) Isothermal compression b) Isentropic compression followed by cooling at constant pressure. What is the peak temperature of the isentropic compression process? Calculate the energy removed during cooling for process b). Assume ideal gas behavior and reversible processes. Label state points on a Ts diagram a Pv diagram for each process Answer: W = 176MJ

Ws =228 MJ

Ts (max) = 858 oK

m = 1.845 kg/s

Introduction to Compressible Flow

12.17 Natural gas, with the thermodynamic properties of methane, flows in an underground pipeline of 0.6 m. The gage pressure at the inlet to a compressor station is 0.5 MPa; outlet pressure is 8.0 MPa (gage). The gas temperature and speed at the inlet are 13 oC and 32 m/s, respectively. The compressor efficiency is  = 0.85. Calculate the mass flow rate of natural gas through the pipeline. Label state points on a Ts diagram for compressor inlet and outlet. Evaluate the gas temperature and speed at the compressor outlet and the power required to drive the compressor. Answer: m = 36.7 kg/s

T2 = 572 oK

V2 = 4.75 m/s

W = 23 MW

Introduction to Compressible Flow

12.18 Over time the efficiency of the compressor of Problem 12.17 drops. At what efficiency will the power required to attain 8.0 MPa (gage) exceed 30 MW? Plot the required power and the gas exit temperature as functions of efficiency Answer - computation

Introduction to Compressible Flow

12.19 Improper maintenance on the turbine of problem 12.7 has resulted in a gradual decrease in its efficiency over time. Assuming that the efficiency drops by 1 percent per year, how long would it take for the power output of the turbine to drop 950 kW, assuming that the entrance conditions, flow rate, and exhaust pressure were all kept constant? Answer:  T = 4 years

Introduction to Compressible Flow

12.20 In an isothermal process, 0.1 cubic feet of standard air per minute (SCFM) is pumped into a balloon. Tension in the rubber skin of the balloon is given by σ = kA, where k=200 lbf/ft3, and A is the surface area of the balloon in ft2 Compute the time required to increase the balloon radius from 5 to 7 in

Introduction to Compressible Flow

12.21 For the balloon process of Problem 12.20 we could define a “volumetric ratio” as the ratio of the volume of standard air supplied to the volume increase of the balloon, per unit time. Plot this ratio over time as the balloon radius is increased from 5 to 7 in. Answer - computation

Introduction to Compressible Flow

Propagation of sound waves 12.23 a) b) c) d)

Calculate the speed of sound at 20 oC for, Hydrogen Helium Nitrogen Carbon dioxide Answer: H2 = 1305 m/s

He = 1005 m/s

N2 = 349 m/s

CO2 = 267 m/s

Introduction to Compressible Flow

12.25 You have designed a device for determining the bulk modulus, Ev, of a material. It works by measuring the time delay between sending a sound wave into a sample of the material and receiving the wave after it travels through the sample and bounces back. As a test, you use a 1-m rod of steel (Ev = 200 GN/m2). What time delay should your device indicate? You now test a 1-m rod (1 cm diameter) of an unknown material and find a time delay of 0.5 ms. The mass of the rod is measured to be 0.25 kg. What is this material’s bulk modulus? Answer: T = 198  s

Ev = 12.7 GN/m2

Introduction to Compressible Flow

12.27 A submarine sends a sonar signal to detect the enemy. The reflected wave returns after 3.25 s. Estimate the separation between the submarines. (As an approximation assume the seawater is at 20 oC.) Answer: x = 2.5 km

Introduction to Compressible Flow

12.29 Next-generation missiles will use scramjet engines to travel at Mach numbers as high as 7. If a scramjet-powered missile travels at Mach 7 at an altitude of 85,000 ft, how long will it take for the missile to travel 600 nautical miles? Assume standard atmospheric conditions. (Note: This is the range for the Tomahawk missile, which uses a conventional propulsion system, but it takes 90 min to cover that same distance.) Answer: T = 531 seconds

Introduction to Compressible Flow

12.31 The Boeing 727 aircraft of Example 9.8 cruises at 520 mph at 33,000 ft altitude on a standard day. Calculate the cruise Mach number of the aircraft. If the maximum allowable operating Mach number for the aircraft is 0.9, what is the corresponding flight speed? Answer: M = 0.776

V = 269 m/s

Introduction to Compressible Flow

12.32 Investigate the effect of altitude on Mach number by plotting the Mach number of a 500 mph airplane as it flies at altitudes ranging from sea level to 10 km. Answer – computation

Introduction to Compressible Flow

12.33 You are watching a July 4th fireworks display from a distance of one mile. How long after you see an explosion do you hear it? You also watch New Year’s fireworks (same place and distance). How long after you see an explosion do you hear it? Assume it’s 75 oF in July and 5 oF in January). Answer: T = 4.66 seconds (July)

 T = 5 seconds (January)

Introduction to Compressible Flow

12.35 You need to estimate the speed of a hypersonic aircraft traveling at Mach 7 and 120,000 ft. Not having a table of atmospheric tables handy, you remember that through the stratosphere (approximately 36,000 ft to 72,000 ft) the temperature of air is nearly constant at 390 oR, and you assume this temperature for your calculation. Later, you obtain the appropriate data and recalculate the speed. What was the percentage error? What would the percentage error have been if you used the air temperature at sea level?

Introduction to Compressible Flow Answer: - 5.42% (assuming stratospheric temperature)

9.08% (assuming sea level temperature)

12.37 While working on a pier on a mountain lake, you notice that the sounds of your hammering are echoing from the mountains in the distance. If the temperature is 25 oC and the echoes reach you 3 seconds after the hammer strike. How far away are the mountains? Answer: x= 519 m

Introduction to Compressible Flow

12.38 Use data for specific volume to calculate and plot the speed of sound in saturated liquid water over the temperature range from 0 to 200 oC. Answer-computation

Introduction to Compressible Flow

12.39 Compute the speed of sound at sea level in standard air. By scanning data from Table A.3 into your PC (or using Fig. 3.3), evaluate the speed of sound and plot for altitudes to 90 km Answer-computation

Introduction to Compressible Flow

12.41 The temperature varies linearly from sea level to approximately 11 km altitude in the standard atmosphere Evaluate the lapse rate—the rate of decrease of temperature with altitude—in the

Introduction to Compressible Flow standard atmosphere. Derive an expression for the rate of change of sonic speed with altitude in an ideal gas under standard atmospheric conditions. Evaluate and plot from sea level to 10 km altitude. Answer - computation

Introduction to Compressible Flow 12.43 Consider the hypersonic aircraft of Problem 12.35. How long would it take for an observer to hear the aircraft after it flies over the observer? In that elapsed time, how far did the aircraft travel? Answer: T = 116.1 seconds

Introduction to Compressible Flow 12.45 A photograph of a bullet shows a Mach angle of 32o. Determine the speed of the bullet for standard air. Answer: V = 642 m/s

Introduction to Compressible Flow 12.47 An F-4 aircraft makes a high-speed pass over an airfield on a day when T = 35 oC. The aircraft flies at M =1.4 and 200 m altitude. Calculate the speed of the aircraft. How long after it passes directly over point A on the ground does its Mach cone pass over point A? Answer: V = 493 m/s

 T = 0.398 sec

Introduction to Compressible Flow

12.49 An aircraft passes overhead at 3 km altitude. The aircraft flies at M=1.5; assume air temperature is constant at 20 oC. Find the air speed of the aircraft. A headwind blows at 30 m/s. How long after the aircraft passes directly overhead does its sound reach a point on the ground? Answer: V – 515 m/s

t = 6.92 sec

Introduction to Compressible Flow

12.51 A supersonic aircraft flies at 3 km altitude at a speed of 1000 m/s on a standard day. How long after passing directly above a ground observer is the sound of the aircraft heard by the ground observer? Answer:  x = 1043 – 1064 m

Introduction to Compressible Flow

12.53 The airflow around an automobile is assumed to be incompressible. Investigate the validity of this assumption for an automobile traveling at 60 mph. (Relative to the automobile the minimum air velocity is zero, and the maximum is approximately 120 mph.) Answer: Density change < 1.21%, so incompressible

Introduction to Compressible Flow

Local Isentropic Stagnation Properties 12.55 Plot the percentage discrepancy between the density at the stagnation point and the density at a location where the Mach number is M, of a compressible flow, for Mach numbers ranging from 0.05 to 0.95. Find the Mach numbers at which the discrepancy is 1 percent, 5 percent, and 10 percent. Answer- computation: M= 0.142 (1%)

M = 0.322 (5%)

M =0.464 (10%)

Introduction to Compressible Flow

12.59 Find the dynamic and stagnation pressures for the missile described in Problem 12.29 Answer: po = 1336 psia

pdyn = 1106 psia

Introduction to Compressible Flow

12.61 An aircraft flies at 250 m/s in air at 28 kPa and -50 oC . Find the stagnation pressure at the nose of the aircraft Answer: po 44.2 kPa

Introduction to Compressible Flow

12.63 For an aircraft traveling at M=2 at an elevation of 12 km, find the dynamic and stagnation pressures. Answer: pdyn = 54.3 kpa po =152 kPa

Introduction to Compressible Flow

12.65 Consider flow of standard air at 600 m/s. What is the local isentropic stagnation pressure? The stagnation enthalpy? The stagnation temperature? Answer: po=546 kPa

ho – h = 178 kJ/kg

To =466 oK

Introduction to Compressible Flow

12.67 An aircraft cruises at M=0.65 at 10 km altitude on a standard day. The aircraft speed is deduced from measurement of the difference between the stagnation and static pressures. What is the value of this difference? Compute the air speed from this actual difference assuming (a) compressibility and (b) incompressibility. Is the discrepancy in air speed computations significant in this case? Answer: po – p = 8.67 kPa

V = 195 m/s

V = 205 m/s

error using Bernoulli = 5.13%

Introduction to Compressible Flow

12.69 Modern high-speed aircraft use “air data computers” to compute air speed from measurement of the difference between the stagnation and static pressures. Plot, as a function of actual Mach number M, for M=0.1 to M=0.9, the percentage error in computing the Mach number assuming incompressibility (i.e., using the Bernoulli equation), from this pressure difference. Plot the percentage error in speed, as a function of speed, of an aircraft cruising at 12 km altitude for a range of speeds corresponding to the actual Mach number ranging from M=0.1 to M=0.9 Answer-computation:

Introduction to Compressible Flow

12.71 Air flows steadily through a length (1) denotes inlet and 2 denotes exit of insulated constantarea duct. Properties change along the duct as a result of friction a) Beginning with the control volume form of the first law of thermodynamics, show that the equation can be reduced to

h1  b)

V12 V2  h2  2  K 2 2

Denoting the constant by h0 (the stagnation enthalpy), show that for adiabatic flow of an ideal gas with friction

T0 k 1 2  1 M T 2 c)

For this flow does T01 = T02? p01 = p02? Explain these results

Introduction to Compressible Flow

12.73 For aircraft flying at supersonic speeds, lift and drag coefficients are functions of Mach number only. A supersonic transport with wingspan of 75 m is to fly at 780 m/s at 20 km altitude on a standard day. Performance of the aircraft is to be measured from tests of a model with 0.9 m wingspan in a supersonic wind tunnel. The wind tunnel is to be supplied from a large reservoir of compressed air, which can be heated if desired. The static temperature of air in the test section is to be 10 oC to avoid freezing of moisture. At what air speed should the wind tunnel tests be run to duplicate the Mach number of the prototype? What must be the stagnation temperature in the reservoir? What pressure is required in the reservoir if the test section pressure is to be 10 kPa (abs)? Answer: V= 890 m/s

To = 677 oK

po = 212 kPa

Introduction to Compressible Flow

12.75 The NASA X-43A Hyper-X experimental vehicle traveled at M= 9.68 at an altitude of 110,000 ft. Calculate the flight speed for these conditions. Determine the local stagnation pressure. Because the aircraft speed is supersonic, a normal shock wave occurs in front of a total-head tube. However, the shock wave results in a stagnation pressure decrease of 99.6 percent. Evaluate the stagnation pressure sensed by a probe on the aircraft. What is the maximum air temperature at stagnation points on the aircraft structure? Answer: po = 119.7 kPa

To = 8350 oR

Introduction to Compressible Flow

12.77 Air is cooled as it flows without friction at a rate of 0.05 kg/s in a duct. At point 1 the conditions are M1=0.5, T1=500 oC, and p1=500 kPa (abs). Downstream, at point 2, the conditions are M 2=0.2, T2= - 18.57oC, and p2=639.2 kPa (abs). (Four significant figures are given to minimize round off errors.) Compare the stagnation temperatures at points 1 and 2 , and explain the result. Compute the rate of cooling. Compute the stagnation pressures at points 1 and 2. Should this process be isentropic or not? Justify your answer by computing the change in entropy between points 1 and 2. Plot static and stagnation state points on a Ts diagram. Answer: T01 = 812 oK

T02 = 257 oK

Q = - 27.9 kW

 s = - 1186 J/kg oK

Introduction to Compressible Flow

12.79 Air flows steadily through a constant-area duct. At section 1, the air is at 400 kPa (abs), 325 K, and 150 m/s. As a result of heat transfer and friction, the air at section 2 downstream is at 275 kPa (abs), 450 K. Calculate the heat transfer per kilogram of air between sections 1 and 2, and the stagnation pressure at section 2 . Answer: dQ/dm = 160 kJ/kg

p02 = 385 kPa

Introduction to Compressible Flow

12.81 Let us revisit the ramjet combustor in Problem 12.80. To more accurately model the flow, we now include the effects of friction in the duct. Once the effects of friction have been included, we find that the conditions at state 2 are now M 2=0.9, T2=1660 oF, and p2=1.6 psia. Recalculate the heat transfer per pound of air between sections 1 and 2 , and the stagnation pressure at section 2 . Answer: Q = 33.5 Btu/s

po2 = 2.71 psia

Introduction to Compressible Flow

12.83 Air enters a turbine at M1=0.4, T1=1250 oC, and p1=625 kPa (abs). Conditions leaving the turbine are M2=0.8, T2=650 oC, and p2=20 kPa (abs). Evaluate local isentropic stagnation conditions (a) at the turbine inlet and (b) at the turbine outlet. Calculate the change in specific entropy across the turbine. Plot static and stagnation state points on a Ts diagram. Answer: p01 = 698 kPa

T01 = 1575 oK

p02 = 30 kPa

T02 = 1041 oK

 s = 485 J/kg oK

Introduction to Compressible Flow

Critical Conditions 12.85 If a window of the cockpit in Problem 12.84 develops a tiny leak the air will start to rush out at critical speed. Find the mass flow rate if the leak area is 1mm2 Answer: m = 1.83 x 10 -4 kg/s

Introduction to Compressible Flow

12.87 A CO2 cartridge is used to propel a toy rocket. Gas in the cartridge is pressurized to 45 MPa (gage) and is at 25 oC. Calculate the critical conditions (temperature, pressure, and flow speed) that correspond to these stagnation conditions. Answer: T*= 260 oK

p* = 24.7 MPa

V* = 252 m/s

Introduction to Compressible Flow

12.89 Stagnation conditions in a solid propellant rocket motor are T 0=3000 K and p0=45 MPa (gage). Critical conditions occur in the throat of the rocket nozzle where the Mach number is equal to one. Evaluate the temperature, pressure, and flow speed at the throat. Assume ideal gas behavior with R=323 J/(kg_K) and k=1.2. Answer: Tt = 2730 oK

pt = 25.5 MPa

Vt = 1030 m/s

Introduction to Compressible Flow

12.91 Certain high-speed wind tunnels use combustion air heaters to generate the extreme pressures and temperatures required to accurately simulate flow at high Mach numbers. In one set of tests, a combustion air heater supplied stagnation conditions of 1.7 MPa and 1010 K. Calculate the critical pressure and temperature corresponding to these stagnation conditions.

Introduction to Compressible Flow