Air is cooled as it flows through a 20 -cm-diameter duct....

Air is cooled as it flows through a 20 -cm-diameter duct. The inlet conditions are $\mathrm{Ma}_1=1.2, T_{01}=350 \mathrm{~K}$, and $P_{01}$ $=240 \mathrm{kPa}$ and the exit Mach number is $\mathrm{Ma}_2=2.0$. Disregarding frictional effects, determine the rate of cooling of air.

Key Concepts

Static Temperature Reduction in Accelerating Flows

In a duct where the flow accelerates, part of the thermal energy is converted to kinetic energy, causing a drop in the static temperature while the stagnation temperature remains constant in an ideal isentropic process. Understanding this conversion is crucial when determining cooling effects in high-speed flows.

Isentropic Flow Relations

Isentropic flow relations describe the relationships between pressure, temperature, density, and velocity in a fluid undergoing an adiabatic and reversible process. These relations allow the conversion between stagnation (total) properties and static properties using the Mach number, and they are key for analyzing cooling or heating effects in ducts with varying flow speeds.

Mass Flow Rate in Duct Flow

The determination of the mass flow rate in a duct is essential as it links the area of the duct, the fluid density, and the velocity of the flow. Knowing the mass flow rate allows one to quantify the overall energy change per unit time, which is necessary for computing the rate of cooling or heating in the duct.

Mach Number

Mach number is the ratio of the flow speed to the local speed of sound and is a fundamental parameter in compressible fluid dynamics. It determines whether the flow is subsonic, transonic, or supersonic, and it directly affects how pressure, temperature, and density change in a gas during variations in flow speed, particularly within ducts or nozzles.

Total (Stagnation) Temperature

Stagnation or total temperature represents the temperature a fluid attains when it is brought to rest isentropically. It is an important concept in high-speed flows because it remains constant along a streamline in an ideal, frictionless, adiabatic (isentropic) process, even while static temperature may vary due to changes in flow speed.

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