Nozzles and Diffusers

Nozzles and diffusers are steady-flow aerodynamic devices that convert between pressure (enthalpy) and kinetic energy. A nozzle accelerates a fluid producing higher exit velocity by converting stagnation enthalpy into kinetic energy; a diffuser decelerates a fluid increasing static pressure (recovering enthalpy). Ideal (isentropic) devices are adiabatic and reversible so total (stagnation) enthalpy and total temperature remain constant. In real devices irreversibilities produce total-pressure loss. Compressibility introduces Mach-number-dependent behavior: subsonic flows accelerate in converging passages, while supersonic flows require diverging passages to accelerate. For compressible isentropic flow, stagnation-to-static relations and area–Mach relations govern performance; choked flow occurs when Mach = 1 at the throat, limiting mass flow.

Governing FormulaKey relations (ideal gas, perfect gas constant R, heat capacities cp, cv, ratio gamma = cp/cv): - Steady-flow energy (adiabatic, no work): h0 = h + V^2/2 (stagnation enthalpy constant) - Stagnation temperature: T0 = T(1 + (gamma-1)/2 * M^2) - Stagnation pressure: P0 = P(1 + (gamma-1)/2 * M^2)^{gamma/(gamma-1)} - Isentropic area–Mach: A/A* = (1/M) * [ (2/(gamma+1))*(1 + (gamma-1)/2 * M^2) ]^{(gamma+1)/(2(gamma-1))} - Critical (choked) static-to-stagnation pressure at M=1: P*/P0 = (2/(gamma+1))^{gamma/(gamma-1)} - Choked (M=1) mass flow per unit area: (m_dot/A) = (P0/ sqrt(T0)) * sqrt(gamma/R) * (2/(gamma+1))^{(gamma+1)/(2(gamma-1))} - Normal-shock downstream Mach: M2^2 = (1 + (gamma-1)/2 * M1^2) / (gamma*M1^2 - (gamma-1)/2) (Assume steady flow, negligible potential energy changes, SI units.)

Knowledge Check

10 Questions

1.Which statement correctly distinguishes a nozzle from a diffuser (ideal, subsonic regime)?

2.For an adiabatic, steady, no-work nozzle operating with a perfect gas, which quantity is constant between inlet and exit (neglecting losses)?

3.A flow of a perfect gas (gamma = 1.4) has stagnation temperature T0 = 600 K and Mach number M = 2. What is the static temperature T (assume ideal gas)?

4.For air (gamma = 1.4) flowing from a large reservoir, the critical (choked) static-to-stagnation pressure ratio P*/P0 at Mach = 1 equals:

5.Air (gamma = 1.4, R = 287 J/kg·K) in a reservoir at total pressure P0 = 500 kPa and total temperature T0 = 400 K flows through a choked (M=1 at throat) nozzle with throat area A = 0.0020 m^2. What is the mass flow rate m_dot (kg/s)?

6.Which statement about how cross-sectional area must change to accelerate the flow is correct (compressible, 1-D, isentropic)?

7.In a practical adiabatic diffuser with friction (irreversible), which of the following statements is true between inlet and outlet?

8.A normal shock stands in a duct. Upstream Mach number is M1 = 2.5. For air (gamma = 1.4), what is the downstream Mach number M2 (normal-shock relation)?

9.Air (gamma = 1.4, R = 287 J/kg·K) has stagnation temperature T0 = 500 K and exits a nozzle at Mach M = 0.8. What is the exit velocity V (m/s)?

10.Air (γ = 1.4, cp = 1005 J/kg·K) has stagnation temperature T0 = 300 K at diffuser inlet where V1 = 200 m/s and static pressure p1 = 80 kPa. If the flow is decelerated isentropically to V2 = 50 m/s, what is the final static pressure p2 (kPa)?