S.A. Klein and G.F. Nellis Cambridge University Press, 2011

Transcription

1 16-1 A flow nozzle is to be used to determine the mass flow rate of air through a 1.5 inch internal diameter pipe. The air in the line upstream of the meters is at 70 F and 95 psig. The barometric pressure = 14.7 psia. The diameter at the throat of the flow nozzle is 0.25 in. Assume that the flow nozzle operates isentropically. a.) Prepare a plot that relates the mass flow rate of the air in lb m /min to the absolute pressure at the nozzle throat in psia. b.) What is the maximum flow rate that can be measured by this nozzle in lb m /min?

2 16-2 For the Formula Car challenge, the air inlet to the engine is required to have a minimum area of 0.25 in 2. You are requested to evaluate the two inlet air design options shown below. Design A uses a converging nozzle with a throat area of 0.25 in 2. Design B uses a converging-diverging nozzle with a throat area of 0.25 in 2 and an exit area of in 2. The nozzle in both of these designs exhausts to the inlet plenum for a turbo charger. Assume that both nozzles operate isentropically. Air Inlet P atm =15psi Inlet Plenum Turbo Charger DESIGN A: A throat =0.25in 2 Air Inlet P atm =15psi Inlet Plenum Turbo Charger DESIGN B: A throat =0.25in 2 A exit =0.415in 2 a.) Compare the maximum air flow rates for the nozzles in Design A and Design B. Which nozzle provides the largest mass flow rate? b.) Prepare a plot of the mass flow rate for the converging nozzle in Design A as a function of the ratio of the plenum pressure to the inlet pressure. c.) Prepare a plot of the mass flow rate for the converging nozzle in Design B as a function of the ratio of the plenum pressure to the inlet pressure, assuming that flow in the diverging section of the nozzle is subsonic. d.) The turbocharger raises the pressure of the air from the pressure in the inlet plenum pressure to the pressure at which air is charged into the cylinders. Explain which nozzle (Design A or Design B) you would you recommend for this purpose any why.

3 16-3 A converging-diverging nozzle has the geometry shown in Figure 16P-3. Figure 16P-3 Nozzle Geometry a.) Assuming isentropic flow conditions otherwise, determine the lowest and highest pressures at the exit plane for which a shock wave is expected to develop within the diverging part of the nozzle b.) The pressure is at the exit of the nozzle is measured to be 400 kpa. Determine the velocity and temperature at the exit.

4 16-4 A rocket engine steady generates combustion gas at a pressure of 682 psia, as measured by a pressure sensor attached to the body of the motor as shown in Figure 16P-4. The combustion gas consists of 40% carbon dioxide and 60% water vapor (molar basis) at 1245 F. The nozzle has a throat diameter of 18 in and it discharges to the atmosphere with a barometric pressure of 14.5 psia. Experimental data verify that the performance of this system can be predicted by one-dimensional isentropic flow theory. Assuming ideal gas behavior, determine: a.) The required exit diameter of the nozzle in order for it to operate at design conditions. b.) The mass flow rate of gas out of the engine. c.) The thrust produced by the engine in lbf Figure 16P-4: Rocket engine

5 16-5 A converging-diverging and a converging nozzle are connected in a parallel flow arrangement, as shown in Figure 16P-5. The inlet air conditions for both nozzles are 300 K, 10 bar. The back pressure is varied from 10 bar to 1 bar. Both nozzles have the same minimum throat area = m 2. In addition, the cross-sectional area at the inlet of the converging nozzle and the exit area of the converging-diverging nozzle are both m 2. Figure 16P-5: Converging diverging and converging nozzles a.) Plot the mass flow rate though the converging nozzle as a function of back pressure for back pressures between 10 bar and 1 bar. b.) Plot the mass flow rate though the converging-diverging nozzle as a function of back pressure for back pressures between 10 bar and 1 bar.

6 16.6 An engineer is studying the leakage that occurs from a labyrinth seal in a super-critical carbon dioxide Brayton cycle. He has decided to model the leakage that occurs through the seal as a nozzle, in order to determine a limit on leakage rate and the temperature that may be experienced at the seal. Carbon dioxide enters the seal at 55 C and 75 bar. The exit area is estimated to be 4.5E-6 m 2. Assume that the process can be modeled as isentropic process and that choked flow occurs at the seal exit. a.) Calculate the leakage rate and the pressure and temperature at the seal exit assuming carbon dioxide can be modeled as an ideal gas. b.) The inlet pressure is above the critical pressure. Repeat part a, but do not assume that carbon dioxide obeys the ideal gas law.

7 16.7 A nozzle is used to determine the mass flow rate of methane (CH 4 ) as it flows through a pipeline. The stagnation conditions of the methane are 30 C, 10 bar. The exit diameter of the nozzle is 5.0 cm. The pressure downstream of the nozzle is 9 bar. The nozzle is adiabatic and, except for a normal shock (if one is present) the flow may be assumed to be reversible. a.) If a converging nozzle is used to make this measurement, what is the mass flow rate of methane? b.) If a converging-diverging nozzle with a throat diameter of 3.5 cm and an exit diameter of 5.0 cm is used to make this measurement, what is the mass flow rate of methane?

8 16-8 Air is flowing through a CD nozzle for which the cross-sectional area as a function of position is shown in Table 16P-8. The pressure at the inlet plenum is 550 kpa and the temperature is 42 C. A shock is observed to occur at position 8. Table 16P.8: Area at various cross-sectional areas in a nozzle Position Area (cm 2 ) 0 (inlet) (exit plane) 54 a) Determine the mass flow rate of air through this nozzle. b) Determine the pressures just upstream and just downstream of the shock wave, assuming the nozzle behaves isentropically. c) Determine the temperature, pressure and velocity of air at the nozzle exit plane.

9 16-9 A gas cylinder having a volume of 0.24 m 3 contains air at 20ºC, 24 bar. The cylinder is equipped with a valve. The valve is opened and the air is rapidly vented to the atmosphere at kpa through a valve with an exit diameter of cm. It is not clear how much thermal interaction there is between the air remaining in the cylinder and the cylinder walls. At one extreme, the venting process can be considered to be adiabatic. At the other extreme, the heat transfer coefficient between the air and the cylinder walls may be high so that the air remains at 20 C throughout the venting process. The valve can be modeled as a converging nozzle, which may or may not be choked, depending on the pressure in the cylinder. Assume air to behave as an ideal gas. a.) Assuming that the venting process is adiabatic, calculate and plot the pressure in the tank as a function of time. How much time is needed to reduce the pressure to 110 kpa? b.) Repeat part a, but assume the venting process is isothermal at 20 C.

10 16-10 Hydrogen has been proposed as a fuel for vehicles. This problem is concerned with the time it will take to fill the tank in the vehicle with high pressure hydrogen. The tank is a carbon-fiber reinforced shell with a volume of 150 liters. The filling stations supplies compressed hydrogen at 400 bar and 25 C. A vehicle enters a filling station with its hydrogen fuel tank at 25 C and 20 bar and the tank is filled to a pressure of 375 bar through a converging nozzle that has an exit diameter of 1 mm. (The nozzle is also used for measuring the mass flow rate.) In the actual filling process, the flow through the nozzle will initially be choked and at a later time, as the pressure in the tank increases, the flow at nozzle exit will be subsonic. Assume ideal gas behavior in your analysis. a.) Prepare a plot of the pressure in the tank as a function of time assuming that choked flow always occurs in the nozzle during the filling process. Based on your plot, determine the time required to fill the tank to 375 bar and the mass of hydrogen delivered. b.) Repeat the calculations for part a, but in this case, account for subsonic flow when it occurs. Compare the time and mass of hydrogen delivered to the tank with the results from part a.