Consider the flow of air through the piping system shown in Fig. P 11.60 . If the system is chok

step 1 So for this question first we can find the speed of sound at air in section 1, which is 347 .2 m

Consider the flow of air through the piping system shown in...

Consider the flow of air through the piping system shown in Fig. $\mathrm{P} 11.60 .$ If the system is choked, determine the location where the Mach number is $1 .$ Calculate the pressure ratio $p_{B} / p_{0},$ that causes choking in this system.

Key Concepts

Choked Flow

Choked flow occurs in compressible fluids when the local flow velocity reaches the speed of sound (Mach 1) at a specific location, usually at the throat or minimum cross-sectional area of a duct or nozzle. In this state, the mass flow rate becomes independent of the downstream pressure, meaning that further reduction in exit pressure does not increase the mass flow. This phenomenon is crucial in both the design and analysis of high-speed gas flow systems.

Mach Number

The Mach number is a dimensionless quantity that represents the ratio of the flow velocity to the local speed of sound. It is pivotal in compressible flow dynamics because it determines whether the flow is subsonic (Mach < 1), sonic (Mach = 1), or supersonic (Mach > 1). In many applications, especially those involving nozzles or ducts, locating the position where the Mach number reaches exactly 1 is important for understanding flow behavior and ensuring stability.

Critical Pressure Ratio

The critical pressure ratio is the specific ratio of downstream to upstream pressure that causes the flow to become choked. It is derived from isentropic flow relations and serves as the threshold at which the flow velocity at the throat reaches Mach 1. This ratio is vital in determining the conditions needed to induce choked flow in a system.

Isentropic Flow Relations

Isentropic flow relations describe the behavior of a compressible fluid undergoing adiabatic and reversible processes. They provide analytic links between pressure, temperature, density, and Mach number in the flow. These relations are employed to derive expressions for the critical pressure ratio and are essential for analyzing and predicting the conditions under which choking occurs in ducts and nozzles.

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