Lesson 15b: Compressible Flow and Choking in Converging Ducts

Key Topics

• Compressible Flow in Converging Ducts
• Critical Conditions and Mass Flow Rate Calculation
• Definition and Implications of Choking
• Example Problem

Compressible Flow in a Converging Duct

• Scenario Considered: Flow from a pressurized tank into a converging duct.
• Initial Conditions: Stagnation conditions are in the tank; interested in Mach number and pressure at the exit.
• Flow Characteristics:
  • Starts at zero speed, accelerates in the duct.
  • Subsonic Flow: Speed and Mach number increase, pressure and temperature decrease.
  • Supersonic Flow in a Converging Duct: Not possible, as it would require reversing these trends.
  • Thus, the flow remains subsonic with the Mach number approaching 1 but not exceeding it.

Choking in Flow

• Choking Definition: When Mach number equals 1 at the exit plane, and flow is sonic.
• Critical Conditions: Denoted by an asterisk (e.g., ( T^* ), ( P^* ), ( \rho^* )).
• Choked Flow: Further decrease in back pressure (Pb) does not affect flow.*

Determining If Flow is Choked

• Cases Based on Back Pressure (Pb):
  1. ( Pb > P^ ):* Flow is not choked; Mach number < 1 at the exit, ( P_e = P_b ).
  2. ( Pb = P^ ):* Flow is barely choked; Mach number = 1 at the exit, ( P_e = P^* = P_b ).
  3. ( Pb < P^ ):* Flow is choked; Mach number = 1 at the exit, ( P_e = P^* ) but ( P^* > P_b ).

Example Problem

• Conditions: Given ( P_0 ), ( T_0 ), back pressure, and exit area.
• Objective: Determine if the flow is subsonic, sonic, or supersonic.
• Findings:
  • Flow cannot be supersonic.
  • Subsonic or Sonic Decision: Compare ( P_b ) to ( P^* ).
  • Critical Back Pressure: ( P^* = 0.5283 \times P_0 = 83.47 \text{kPa} ).
  • Given ( P_b > P^ ):* Flow is subsonic.*

Calculation of Conditions

• Mach Number Calculation at Exit Plane:
  • Used stagnation equations.
  • Mach number ( M_a = 0.823 ).
• Temperature at Exit Plane:
  • Calculated as ( T_e = 458K ).

Mass Flow Rate Calculation

• Equation Derived:
  • ( \dot{m} = \frac{P M_a}{\sqrt{RT}} A )
  • Applied ideal gas law and stagnation conditions.
• For Choked Flow:
  • ( M_a = 1 ) and ( A = A^* ).
• Example Calculation:
  • ( m \dot{} = 3.54 \text{ kg/s} ) for given conditions.
  • ( m_{\dot{max}} = 3.64 \text{ kg/s} ) if flow was choked.*_

This lecture covers essential concepts and equations for analyzing compressible flow in converging ducts, particularly the conditions under which flow becomes choked and how to calculate associated flow properties.