Question

A converging-diverging nozzle operating at steady state has a throat area of \(3 \mathrm{~cm}^{2}\) and an exit area of \(6 \mathrm{~cm}^{2}\). Air as an ideal gas with \(k=1.4\) enters the nozzle at 8 bar, \(400 \mathrm{~K}\), and a Mach number of \(0.2\), and flows isentropically throughout. If the nozzle is choked, and the diverging portion acts as a supersonic nozzle, determine the mass flow rate, in \(\mathrm{kg} / \mathrm{s}\), and the Mach number, pressure, in bar, and temperature, in \(\mathrm{K}\), at the exit. Repeat if the diverging portion acts as a supersonic diffuser.

Short answer

Mass flow rate is 0.538 kg/s. For supersonic nozzle: exit Mach number 2.2, exit pressure 0.625 bar, exit temperature 161.5 K. For supersonic diffuser: exit Mach number 0.3, exit pressure 8.39 bar, exit temperature 396 K.

Step-by-step solution

- Determine the stagnation conditions

For isentropic flow, determine the stagnation temperature (T_0) and stagnation pressure (P_0) using the inlet conditions. From the given inlet conditions, the Mach number (M_1) is 0.2, temperature (T_1) is 400 K, and pressure (P_1) is 8 bar. To find the stagnation temperature: \[ T_0 = T_1 \left(1 + \frac{k-1}{2} M_1^2\right) \] For air with k=1.4: \[ T_0 = 400 \left(1 + \frac{1.4-1}{2} (0.2)^2\right) = 400 (1 + 0.02) = 400 (1.02) = 408 \mathrm{K} \] Stagnation pressure is given by: \[ P_0 = P_1 \left(1 + \frac{k-1}{2} M_1^2\right)^{\frac{k}{k-1}} \] \[ P_0 = 8 \left(1 + 0.02 \right)^{\frac{1.4}{0.4}} = 8 (1.02)^{3.5} \approx 8.56 \mathrm{~bar} \] So, the stagnation temperature is 408 K and stagnation pressure is 8.56 bar.

- Determine the critical temperature and pressure

When the nozzle is choked, the flow at the throat reaches Mach 1. For choked flow, the critical temperature (T*) and pressure (P*) can be determined. \[ T^* = \frac{T_0}{1 + \frac{k-1}{2}} = \frac{408}{1 + 0.2} = \frac{408}{1.2} = 340 \mathrm{K} \] \[ P^* = P_0 \left(\frac{T^*}{T_0}\right)^{\frac{k}{k-1}} = 8.56 \left(\frac{340}{408}\right)^{3.5} \approx 8.56 \times 0.556 \approx 4.76 \mathrm{~bar} \] Thus, critical temperature is 340 K and critical pressure is 4.76 bar.

- Calculate the throat area mass flow rate

Using the critical conditions, determine the mass flow rate at the throat where the area is 3 cm². Mass flow rate for choked flow is given by: \[ \dot{m} = \rho^* A^* V^* \] where \(\rho^*\) is the density at critical conditions, \(A^*\) is the throat area, and \(V^*\) is the velocity at critical conditions. For isentropic flow, \[ \rho^* = \frac{P^*}{R T^*} \] Given R (gas constant for air) = 287 J/kg·K, \[ \rho^* = \frac{4.76 \times 10^5}{287 \times 340} \approx 4.76 \times 10^5 / 97580 = 4.88 \mathrm{~kg/m³} \] For Mach 1, velocity \(V^* = \sqrt{kRT^*} \)\[ V^* = \sqrt{1.4 \times 287 \times 340} \approx \sqrt{136028} = 368.8 \mathrm{m/s} \] Thus, mass flow rate \( \dot{m} \):\[ \dot{m} = 4.88 \times 3 \times 10^{-4} \times 368.8 \approx 0.538 \mathrm{~kg/s} \] So, the mass flow rate is 0.538 kg/s.

- Calculate supersonic nozzle exit conditions

For the supersonic nozzle, exit conditions are calculated from the area ratio. Given the exit area is 6 cm² and throat area is 3 cm², the area ratio \(A_e/A^*\) is 2. Mach number at exit (M_2) when the area ratio is 2 for k=1.4 from isentropic flow tables or using an iterative method: \( M_2 \approx 2.2 \).Use exit Mach number to determine exit conditions.\[\frac{T_2}{T_0} = \left(1 + \frac{k-1}{2} M_2^2\right)^{-1}\] \[T_2 = 408 \left(1 + \frac{1.4-1}{2} \times 2.2^2\right)^{-1} \approx 408 \times 0.396 = 161.5 \mathrm{~K} \]\[\frac{P_2}{P_0} = \left(1 + \frac{k-1}{2} M_2^2\right)^{-\frac{k}{k-1}}\] \[P_2 = 8.56 \left(1 + 0.2 \times 2.2^2\right)^{-3.5} \approx 8.56 \times 0.073 \approx 0.625 \mathrm{bar} \] So, exit temperature is 161.5 K, exit pressure is 0.625 bar, and exit Mach number is 2.2.

- Calculate supersonic diffuser exit conditions

For the supersonic diffuser, assume it decelerates from Mach 1 at the throat to subsonic speed at the exit. Use the same area ratio of 2, but now for a subsonic exit condition.From isentropic tables, the exit Mach number (M_2') corresponding to area ratio 2 in subsonic regime for k=1.4 is approximately 0.3.Use exit Mach number to determine subsonic exit conditions:\[\frac{T_2'}{T_0} = \left(1 + \frac{k-1}{2} M_2'^2\right)^{-1} \approx 408 \left(1 + \frac{0.4 \times 0.3^2}{2}\right)^{-1} \approx 408 \times 0.97 = 396 \mathrm{K} \] \[\frac{P_2'}{P_0} = \left(1 + \frac{k-1}{2} M_2'^2\right)^{-\frac{k}{k-1}} \approx 8.56 \left(1.018 \right)^{-3.5} \approx 8.56 \times 0.98 \approx 8.39 \mathrm{~bar} \] So, exit temperature is 396 K, exit pressure is 8.39 bar, and a Mach number is 0.3.

Key concepts

isentropic flow

In fluid dynamics, isentropic flow refers to an ideal fluid flow process in which entropy remains constant. This means that there are no energy losses due to heat transfer, friction, or other dissipative effects. Key characteristics of isentropic flow include:

• No heat transfer (adiabatic process)
• Reversible process
• Constant entropy

For an ideal gas undergoing isentropic flow, important relationships can be derived using thermodynamic principles. The equations for pressure, temperature, and density ratios in isentropic flow are given by:
\[ \frac{T_2}{T_1} = \frac{P_2}{P_1}^{\frac{k-1}{k}} \] and \[ \frac{\rho_2}{\rho_1} = \frac{P_2}{P_1}^{\frac{1}{k}} \] where T is temperature, P is pressure, \[ \rho \] is density, and k is the specific heat ratio (typically 1.4 for air). These relationships help in determining flow properties at different points in the nozzle.

stagnation temperature

Stagnation temperature is the temperature a fluid attains when it is brought to rest isentropically. It represents the total energy content of the fluid and is higher than the static temperature due to the kinetic energy of the moving fluid. For an ideal gas, the stagnation temperature (\[ T_0 \]) can be calculated using the Mach number (M) and the static temperature (T):
\[ T_0 = T \times \bigg(1 + \frac{k-1}{2}M^2 \bigg) \] Here,

• T is the static temperature,
• k is the specific heat ratio,
• M is the Mach number.

In the example problem, the stagnation temperature was calculated based on the inlet conditions where T = 400 K, M = 0.2, and k = 1.4, resulting in a stagnation temperature of 408 K. This value remains constant throughout the nozzle since the flow is isentropic.

Mach number

Mach number is a dimensionless quantity that represents the ratio of the speed of a fluid to the speed of sound in that fluid. Mathematically, it is expressed as:
\[ M = \frac{V}{a} \] where

• V is the fluid velocity,
• a is the speed of sound in the fluid.

The Mach number characterizes the compressibility effects in the flow:

• Subsonic: M < 1
• Sonic: M = 1
• Supersonic: M > 1
• Hypersonic: M > 5

In the exercise, the nozzle operates in choked flow conditions, reaching Mach 1 at the throat and achieving supersonic velocities in the diverging section (M = 2.2). The Mach number determines the changes in properties like pressure, temperature, and density.

mass flow rate

The mass flow rate is the quantity of mass passing through a specific area of a flow system per unit of time. For an isentropic flow in a converging-diverging nozzle, the mass flow rate (\[ \dot{m} \]) can be determined using the critical conditions when the flow is choked:
\[ \dot{m} = \rho^* A^* V^* \] where

• \[ \rho^* \] is the density at critical conditions,
• \