Question

Air flowing at \(32 \mathrm{kPa}, 240 \mathrm{K},\) and \(\mathrm{Ma}_{1}=3.6 \mathrm{is}\) forced to undergo an expansion turn of \(15^{\circ} .\) Determine the Mach number, pressure, and temperature of air after the expansion.

Short answer

Answer: The three main parameters we need to find at the exit of the expansion are the final Mach number (Ma₂), the pressure (P₂), and the temperature (T₂).

Step-by-step solution

Calculate the initial flow properties

first, we find the initial ratio of specific heats, the adiabatic index \(\gamma\). Assuming air is a diatomic gas (like Nitrogen and Oxygen), we can use \(\gamma = 1.4\). Given the initial Mach number \(\mathrm{Ma}_{1}=3.6\), we will compute the Prandtl-Meyer angle \(\nu_{1}\) and the total pressure and temperature ratios. We can express the Prandtl-Meyer angle (\(\nu\)) as follows: $$\nu(\mathrm{Ma}) = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \cdot \tan^{-1}\left(\sqrt{\frac{\mathrm{Ma}^2 - 1}{\gamma+1}}\right) - \tan^{-1}\left(\sqrt{\mathrm{Ma}^2 - 1}\right)$$ Using \(\gamma = 1.4\) and \(\mathrm{Ma}_{1}=3.6\), we can find \(\nu_{1}\).

Determine the flow parameters after expansion

Now, we compute the Prandtl-Meyer angle \(\nu_{2}\) after the expansion: $$\nu_{2} = \nu_{1} + 15^{\circ}$$ Then, we will calculate the final Mach number \(\mathrm{Ma}_{2}\), knowing the Prandtl-Meyer angle \(\nu_{2}\), which can be obtained using the inverse Prandtl-Meyer function.

Calculate the pressure and temperature ratios

With the final Mach number \(\mathrm{Ma}_{2}\), we can compute the pressure and temperature ratios. This can be done using the isentropic flow relations: $$\frac{T_2}{T_1} = \left(\frac{1+\frac{\gamma-1}{2}Ma_1^2}{1+\frac{\gamma-1}{2}Ma_2^2}\right)$$ $$\frac{P_2}{P_1} = \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}}$$

Determine the final pressure and temperature

Now, we have the pressure and temperature ratios, so we can calculate the final pressure \(P_{2}\) and temperature \(T_{2}\) as follows: $$P_{2} = P_{1} \cdot \frac{P_2}{P_1}$$ $$T_{2} = T_{1} \cdot \frac{T_2}{T_1}$$ In conclusion, by following these steps, we can determine the Mach number, pressure, and temperature of the air after the expansion turn.

Key concepts

Adiabatic Process

An adiabatic process is a thermodynamic process in which there is no heat transfer into or out of the system. In an adiabatic process, any changes in the energy of a gas are the result of work done by or on the gas. For instance, as air undergoes an adiabatic expansion, like in the problem above, it does work on the surrounding environment causing its temperature and pressure to drop.

In the context of Prandtl-Meyer expansion, this type of flow is assumed to be adiabatic since it happens very rapidly, and there's not enough time for significant heat exchange with the surroundings. Consequently, an adiabatic assumption permits the application of isentropic flow relations in the analysis of the flow patterns and resulting properties.

Mach Number

The Mach number is an important dimensionless quantity in aerodynamics, representing the ratio of the speed of an object moving through a fluid to the local speed of sound.

Mathematically, it is expressed as \( \mathrm{Ma} = \frac{v}{c} \) where \( v \) is the velocity of the object and \( c \) is the speed of sound in the fluid. When \( \mathrm{Ma} < 1 \), the flow is subsonic, while flows with \( \mathrm{Ma} > 1 \) are supersonic. As the Mach number increases beyond 1, the flow regime significantly alters and shock waves can form, which are essential elements for the exercise we are considering.

Isentropic Flow Relations

In the study of fluid dynamics, isentropic flow relations are tools that describe the changes in a flow when it moves through a variable area conduit in a reversible and adiabatic manner. During isentropic processes, the entropy of the system remains constant.

The relations are a set of equations that describe how the pressure, temperature, density, and velocity of a gas change in response to changes in area. They are derived from conservation laws and rely on the assumption that the flow is both isentropic and adiabatic. These relations are invaluable when solving for the properties of air after it undergoes an expansion turn, exemplified in the exercise.

Specific Heats Ratio

Specific heats ratio, also known as the adiabatic index or \( \gamma \), is the ratio of the specific heat at constant pressure (\(C_p\)) to the specific heat at constant volume (\(C_v\)). It plays a crucial role in thermodynamics and fluid dynamics, particularly in the analyses involving isentropic processes and adiabatic expansions.

For air, which is considered an ideal diatomic gas for many practical purposes, the value of \( \gamma \) is approximately 1.4. This value affects how the temperature, pressure, and volume of air change relative to each other during thermodynamic processes, such as the expansion turn described in the exercise. Understanding the specific heats ratio is essential for accurately determining the new state of an expanding gas.