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):
- ( Pb > P^ ):* Flow is not choked; Mach number < 1 at the exit, ( P_e = P_b ).
- ( Pb = P^ ):* Flow is barely choked; Mach number = 1 at the exit, ( P_e = P^* = P_b ).
- ( 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.