Problems
Speed of Sound
Determine the speed of sound in helium (based on constant specific heats) and air (based on variable specific heats, tables) at 300 K and 1500 K. Compute the corresponding Mach numbers for a velocity of 290 m .
Speed of Sound in R-134a
Determine the speed of sound in refrigerant R-134a at 1 MPa, 60 ◦C. Use table data!
Security Valve
A steam boiler produces saturated vapor at 17.5 bar. The security valve has a smallest free area of 20 cm2. Determine the maximum mass flow that can be produced so that no pressure is building up when the valve is open.
Laval Nozzle
A Laval nozzle is to be designed such that it delivers 4 kg of air at 10 ◦C and 1 bar at twice the speed of sound. The air that expands in the nozzle is delivered by an isentropic compressor that draws air at 1 bar and 10 ◦C.
Consider air as ideal gas with constant specific heats, R = 0.287 kJ , cp =1.004 kJ .
1. Determine the cross section at the end of the nozzle.
2. Determine temperature, pressure, mass density and velocity in the throat.
3. Determine stagnation pressure and stagnation temperature for the nozzle flow.
4. Determine the power consumed by the compressor.
5. Determine the heat that must be withdrawn from the flow between compressor and nozzle.
Nozzle Flow
Consider a Laval nozzle for rocket propulsion. Pressure and temperature in the combustion chamber are 10 bar and 2500 K, respectively. The mass flow through the nozzle is 30 kg of combustion product (ideal gas, constant specific heats, R = 0.287 kJ , k = 1.4), the cross section at the end of the nozzle is Ae = 700 cm2 and the flow is isentropic throughout the nozzle. The environmental pressure is 0.9 bar.
1. Do you expect supersonic or subsonic flow at the outlet? Why?
2. Compute the area of the throat of the nozzle.
3. Find pressures and gas velocities at throat and end.
4. Discuss the flow behind the nozzle
Rocket Engine
A converging-diverging nozzle is fed from a combustion chamber at temperature T0 = 2200 K. The flow through the nozzle is isentropic, and the outflow is supersonic with the velocity v = 1400 m . The pressure in the throat is measured as p∗ = 4 bar.
The gas flowing through the nozzle can be considered as an ideal gas with constant specific heats: cp = 0.98 kJ , k = cp/cv = 1.4.
1. Determine the pressure in the combustion chamber.
2. Determine temperature and pressure at the nozzle exit.
3. Determine the speed of sound at the nozzle exit.