Summary

Choked flow is a phenomenon that limits the mass flow rate of a compressible fluid flowing through nozzles, orifices and sudden expansions. Generally speaking it is the mass flux after which a further reduction in downstream pressure will not result in an increase in mass flow rate.

Definitions

\(A\):Cross sectional area
\(C_{d}\):Coefficient of Discharge
\(G\):Mass flux
\(K\):Resistance coefficient
\(k\):Index of polytropic Change
\(M\):Mach number
\(P\):Pressure
\(u\):Velocity
\(w\):Mass flow rate
\(Y\):Expansion coefficient
\(\gamma\):Ratio of heat capacities, \(C_{p}/C_{v}\)
\(\sigma\):Specific gravity

Subscripts:

\(1\):Upstream conditions
\(2\):Downstream conditions
\(c\):Choked conditions
\(t\):Throat (minimum flow area) conditions


Introduction

Non-choked Flow

As a compressible fluid passes through a restriction there are changes in both velocity and pressure. The fluid starts upstream at a higher pressure, which falls as it increases velocity flowing through the restriction, and may continue to fall as the velocity increases through the vena contracta. After passing the vena contracta the fluid will begin expand to fill the cross sectional area of the pipe and as it does will slow down and regain pressure.

Vena Contracta after a Restriction Orifice

For normal non-choked flow with a given inlet pressure, reducing the outlet pressure will cause a greater differential pressure across the restriction and therefore increase the fluid flow rate and velocity. This holds true until the flow rate is increased to the point that the fluid reaches the local sonic velocity at the throat of the restriction and becomes choked.

Choked Flow

As a compressible fluid reaches the speed of sound (i.e. has a Mach number of 1), pressure changes can no longer be communicated upstream as the speed of which these pressure changes are propagated is limited by the speed of sound. In a nozzle or restriction this has the effect of isolating the upstream side from the downstream side at the throat. Because of this effect any reduction in downstream pressure will have no effect on the flow rate, as the increased pressure differential is not 'felt' upstream of the restriction.

The establishment of choked flow can be identified as the point at which the ratio of the minimum fluid pressure to inlet pressure \(\left( P_{min}/P_{1} \right)\) falls below the critical pressure ratio in the fluid.

Choked Flow

It should be noted that while downstream changes in pressure will not effect the mass flow rate when the flow is choked, changes in the upstream pressure may still have an effect as it will affect the local speed of sound at the throat, and thus change the mass flow rate at which the system becomes choked.

Supersonic Flow

Reducing the downstream pressure of a choked system will not result in increased mass flow rate but it will however, result in an increased velocity of the fluid after the restriction. After reaching the point of choked flow, further reductions in downstream pressure will result in the fluid accelerating away from the throat and in some cases achieving supersonic speeds (Mach number > 1).

Depending on the outlet nozzle design shock waves may form as the fluid returns to subsonic speeds.



Mass Flux and Velocity at Choked Flow Conditions

General equations

The mass flux and velocity at which choked conditions begin can be calculated using the following equations.

\[\displaystyle G_{c} = \sqrt{kP_{c}\rho_{c}}\]

\[\displaystyle u_{c} = \sqrt{\frac{kP_{c}}{\rho_{c}}}\]

Here \(k\) is the index of polytropic change and is dependent on the thermodynamic conditions of the fluid flow.

Adiabatic Flow

For adiabatic conditions the index of polytropic change is equal to ratio of specific heat capacities \(\left( k = \gamma \right)\).

Isothermal Flow

For isothermal flow the index of polytropic change is equal to unity \(\left( k = 1 \right)\).

The typically abrupt nature of constrictions which result in choked flow make it unlikely that the system will behave isothermally, and therefore adiabatic flow is typically a better assumption for practical use.

Expansion Factor

The Expansion Factor \(Y\) is an experimentally derived value that can be used to quantify the difference between and incompressible fluid and a compressible fluid for a nozzle or orifice. This allow for a simplification of the calculation provided the expansion factor value is available.

For the conditions at the choke point this value is known as the Critical Expansion Factor. And may be used to determine the mass flux or velocity at choked conditions.

\[\displaystyle Q_{1}=KYA_{t}\sqrt{2\cfrac{\Delta p}{\rho_{1}}}\]

\[\displaystyle Q_{1}=C_{d}YA_{t}\sqrt{2\cfrac{\Delta p}{\rho_{1}}}\left(\cfrac{1}{\sqrt{1-\beta^4}}\right)\]



Further Reading

  1. Albright's Chemical Engineering Handbook, L. Albright
  2. Flow Measurement Engineering Handbook, R. W. Miller
  3. Marks' Standard Handbook for Mechanical Engineers 11th Edition