Question

There is a convergent-divergent duct inside a wind tunnel. Both the duct and the tunnel are horizontal. When operating, air reaches Mach 1 at the throat of the duct and Mach 2 at the exit of the duct. The air flows into the duct at the standard condition of 101,325 Pa and 298.15 K. Given that the specific heat ratio is 1.4 and the universal gas constant is 287 J/kg/K, (a) Determine the pressure and the density at the throat of the duct. (b) Determine the temperature and the velocity at the exit of the duct.

Answer 1

To find the pressure and density at the throat, use the isentropic flow relations. To find the temperature and velocity at the exit, use the area-Mach number relation for a convergent-divergent duct.

Step-by-step explanation:

To determine the pressure and density at the throat of the convergent-divergent duct, we can use the isentropic flow relations.

At the throat, the Mach number is 1, which means the flow is sonic. We can use the equation:

Mach number = (velocity at throat) / (speed of sound)

Since the Mach number is 1, we can solve for the velocity at the throat:

Velocity at throat = Mach number * speed of sound

We can then use the ideal gas law to determine the density at the throat:

Density at throat = pressure at throat / (specific gas constant * temperature at throat)

By plugging in the given values, we can find the pressure and density at the throat of the duct.

To determine the temperature and velocity at the exit of the duct, we can use the area-Mach number relation for a convergent-divergent duct. The equation is:

Area ratio = (1 / Mach number) * ((2 / (specific heat ratio + 1)) * (1 + ((specific heat ratio - 1) / 2) * Mach number^2))^((specific heat ratio + 1) / (2 * (specific heat ratio - 1)))

By plugging in the given values, we can calculate the area ratio, and then use it to find the Mach number and velocity at the exit of the duct. Lastly, we can use the ideal gas law to determine the temperature at the exit.