Assignment 5: EMH 222 Fluid Dynamics (Answer scheme)
Compressible flow Dr Mohd Sharizal
1. Air enters a compressor with a stagnation pressure of 135 kPa and a stagnation temperature of 32°C, and it is compressed to a stagnation pressure of 900 kPa. Calculate the power input to the compressor for a mass flow rate of 0.018 kg/s. Given for air, c p = 1.005 kJ/kg K and k = 1.4. (Assume the compression process to be isentropic) Solution
The inlet stagnation temperature and pressure and the exit stagnation pressure of air flowing through compressor are specified. The power input to the compressor is to be determined.
2.
Explain the conditions must be satisfied for a steady compressible flow to be isentropic Solution
An isentropic flow is a flow that is both adiabatic and reversible. That is, no heat is added to the flow, and no energy transformations occur due to friction or dissipative effects. Compressible flow flow in which e ntropy remains constant throughout the flow flow field. Page 1 of 10
If the entropy of a fluid element moving along a streamline in a flow remains constant, t he flow is isentropic along a streamline. However, the value of the entropy may be different along different streamlines, thus allowing entropy changes normal to the streamlines.
An example is the flow field behind a curved shock wave; here, streamlines that pass through different locations along the curved shock wave experience different increases in entropy. Hence, downstream from this shock, the entropy can be constant along a given streamline but differs from one streamline to another.
3. Air, with stagnation conditions of 800 kPa and 100°C, expands isentropically to a section of a duct where A 1 = 20 cm2 and p1 = 47 kPa. Compute ( a) Ma1, (b) the throat 2 ̇ . At section 2 between the throat and section 1, the area is 9 cm . ( d area, and ( c ) ) Estimate the Mach number at section 2. Solution
(Isentropic flow with area change)
k = 1.4, air
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4. A bicycle tire is filled with air at an absolute pressure of 169.12 kPa, and the temperature inside is 30.0°C. Suppose the valve breaks, and air starts to exhaust out of the tire into the atmosphere ( pa =100 kPa absolute and T a = 20.0°C). The valve exit is 2.00 mm in diameter and is the smallest cross-sectional area of the entire system. Frictional losses can be ignored here, i.e., one-dimensional isentropic flow is a reasonable assumption. ( a) Find the Mach number, velocity, and temperature at the exit plane of the valve (initially). ( b) Find the initial mass-flow rate out of the tire. ( c ) Estimate the velocity at the exit plane using the incompressible Bernoulli equation. How well does this estimate agree with the “exact” answer of part ( a)? Explain.
Solution
(Isentropic flow with area changes)
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Pabsolute or P o = Pgage + Patm = 169.12 kPa Critical pressure P*= 0.5283Po=(0.5283)(169.12 kPa) = 89.35 kPa < 100 kPa Therefore the flow is not choked
(291-267)/291x 100 ~ 8.25%
5. Air, at stagnation conditions of 500 K and 200 kPa, flows through a nozzle. At section 1, where the area is 12 cm 2, the density is 0.32 kg/m 3. Assuming isentropic flow, ( a) find the mass flow. ( b) Is the flow choked? If so, estimate A *. Also estimate ( c ) p1 and (d ) Ma1. Page 4 of 10
SOLUTION
Isentropic flow with area changes.
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6. Nitrogen enters a converging – diverging nozzle at 700 kPa and 400 K with a negligible velocity. Calculate the critical velocity, pressure, temperature, and density in the nozzle.
SOLUTION
(Isentropic flow through nozzles) Nitrogen enters a converging-diverging nozzle at a given pressure. The critical velocity, pressure, temperature, and density in the nozzle are to be determined.
From Table A-2- Ideal-gas specific heats of various common gases (At various temperature)
Table A-32
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7. Air enters a converging – diverging nozzle of a supersonic wind tunnel at 1 MPa and 300 K with a low velocity. If a normal shock wave occurs at the exit plane of the nozzle at Ma = 2, calculate the pressure, temperature, Mach number, velocity, and stagnation pressure after the shock wave.
SOLUTION
(Shock Waves and Expansion Waves)
Air flowing through a converging-diverging nozzle experiences a normal shock at exit. The effect of the shock wave on various properties is to be determined. Assumptions 1. Air is an ideal gas with constant specific heats. 2. Flow through t he nozzle is steady, one dimensional and isentropic before the shocks occurs. 3. The shock wave occurs at the exit plane Table A-2
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The fluid properties after the shock (denoted by subscript 2) are related to those before the shock through the functions listed in Table A-14. For Ma1= 2.0 we read. Table A-33
8. Air flowing steadily in a nozzle experiences a normal shock at a Mach number of Ma = 2.5. If the pressure and temperature of air are 61.64 kPa and 262.15 K, respectively, upstream of the shock, Calculate:
a) Pressure, temperature, velocity, Mach number, and stagnation pressure downstream of the shock b) Entropy change of air across the normal shock wave
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SOLUTION
(Shock Waves and Expansion Waves)
Air flowing through a nozzle experiences a no rmal shock. The effect of the shock wave on various properties is to be determined. Assumptions 1. Air is ideal gas with constant specific heat. 2. Flow through the nozzle is steady, one dimensional and isentropic before the shock occurs. Properties The properties of air are k = 1.4 and R= 0.287 kJ/ kg.K
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Air flowing through a nozzle experiences a normal shock. The entropy change of air across the normal shock is to be determined
Assumption 1. Air is an ideal gas with constant specific heat. 2. Flow through the nozzle is steady, one dimensional and isentropic before the shock occurs.
The entropy change across the shock is determined to be.
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