Question

Air enters a converging-diverging nozzle at a pressure of \(1200 \mathrm{kPa}\) with negligible velocity. What is the lowest pressure that can be obtained at the throat of the nozzle?

Short answer

Answer: The lowest pressure that can be obtained at the throat of the converging-diverging nozzle is approximately \(816.78\,\mathrm{kPa}\).

Step-by-step solution

Identify given variables and relevant equations

The initial pressure (\(P_1\)) is given as \(1200\,\mathrm{kPa}\). We will assume isentropic flow, meaning constant entropy in this case, and that the air behaves as an ideal gas. For ideal gas isentropic flow, the pressure ratio across the nozzle can be related to the Mach number: \( \frac{P_2}{P_1} = \left( \frac{1+(\gamma-1)\frac{M^2}{2}}{1+(\gamma -1)\frac{M_1^2}{2}} \right)^{-\frac{\gamma}{\gamma - 1}} \) Here, \(P_2\) is the pressure at the throat of the nozzle, \(M\) is the Mach number, \(M_1\) is the initial Mach number, and \(\gamma\) is the ratio of specific heats. For air, \(\gamma \approx 1.4\). As \(M_1\) is negligible, the equation can be simplified to: \( \frac{P_2}{P_1} = \left( 1+(\gamma-1)\frac{M^2}{2} \right)^{-\frac{\gamma}{\gamma - 1}} \)

Determine the Mach number for choked flow

The throat of the nozzle experiences the choked flow condition, which occurs when the Mach number reaches 1. Therefore, \(M = 1\). Substitute this into the simplified equation: \( \frac{P_2}{P_1} = \left( 1+(\gamma-1)\frac{1^2}{2} \right)^{-\frac{\gamma}{\gamma - 1}} \)

Calculate the lowest pressure at the throat of the nozzle

Now that we have our equation for the pressure ratio and all the necessary variable values, we can find the lowest pressure at the throat of the nozzle by solving for \(P_2\): \(P_2 = P_1 \cdot \left( 1+(\gamma-1)\frac{1^2}{2} \right)^{-\frac{\gamma}{\gamma - 1}} \) \(P_2 = 1200\,\mathrm{kPa} \cdot \left(1+\frac{0.4 \cdot 1}{2} \right)^{-\frac{1.4}{0.4}} \) \(P_2 \approx 816.78\,\mathrm{kPa} \) Thus, the lowest pressure that can be obtained at the throat of the converging-diverging nozzle is approximately \(816.78\,\mathrm{kPa}\).

Key concepts

Isentropic Flow

Isentropic flow is an ideal flow process in which entropy remains constant. In the context of aerodynamics and thermodynamics, this simplification is significant when it comes to analyzing the behavior of fluids, and in particular, gases as they pass through a converging-diverging nozzle. The assumption of isentropic flow allows for the derivations of certain relationships between pressure, temperature, and density, which would otherwise require complex calculations.

When we consider isentropic flow through a nozzle, it implies that the process is both adiabatic (no heat transfer) and reversible. This theoretical approach makes it possible to use a simplified version of the continuity equation, energy conservation, and the ideal gas law to describe the gas properties at different stages within the nozzle. Specifically, the pressure and temperature within a nozzle will decrease as the velocity of the gas increases, a principle fundamental to understanding nozzle behavior.

One real-world manifestation of isentropic flow can be seen in jet engines and rockets, where the control of pressure and velocity in nozzles are crucial for efficient operation. Although true isentropic flow is an idealization (real flows have some level of friction and heat transfer), for many engineering calculations, the assumption of isentropic flow provides a sufficiently accurate model, particularly at supersonic speeds where dissipative effects are less consequential relative to inertial effects.

Mach Number

The Mach number is a dimensionless quantity in fluid dynamics used to compare the speed of an object or flow relative to the speed of sound in the surrounding medium. It's calculated by dividing the object's speed by the speed of sound in the given fluid:
\[ Mach\text{ }number (M) = \frac{speed\text{ }of\text{ }object}{speed\text{ }of\text{ }sound} \]
For gases, the speed of sound is dependent on the temperature and composition of the gas, and the Mach number is often used to categorize the regime of flow:

• Mach < 1: subsonic flow
• Mach = 1: sonic flow (the sound barrier)
• Mach > 1: supersonic flow
• Mach > 5: hypersonic flow

In the exercise, choked flow is a critical concept, which refers to the condition where the Mach number equals 1 at the throat of the nozzle. This signifies the maximum mass flow rate through the nozzle and is a fundamental limit in fluid dynamics, particularly for the design of jet engines and various types of turbines.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, and temperature of an ideal gas with the amount of that gas. The formula is expressed as:
\[ PV = nRT \]
where:

• \(P\) is the pressure,
• \(V\) is the volume,
• \(n\) is the number of moles,
• \(R\) is the universal gas constant, and
• \(T\) is the temperature in Kelvins.

Assuming the air behaves as an ideal gas – which is a valid assumption for most aerodynamic applications within reasonable ranges of temperature and pressure – the ideal gas law serves as a crucial tool for predicting how gases will respond when subjected to changes in conditions. For example, in the exercise involving the nozzle, the initial and final pressures and temperatures can be correlated using the ideal gas law, in combination with concepts such as isentropic flow and Mach number, to predict the gas behavior accurately.

Ratio of Specific Heats

The ratio of specific heats, often denoted as \(\gamma\), is a vital thermodynamic property in fluid dynamics and gas theory. It's the ratio of the specific heat capacity at constant pressure (\(C_p\)) to the specific heat capacity at constant volume (\(C_v\)):\[ \gamma = \frac{C_p}{C_v} \]
This ratio is crucial because it determines how a gas will react under different thermodynamic processes. For example, it affects the speed of sound in the gas (important when calculating Mach number), the shape of shock waves, and the thermodynamic efficiency of engines and nozzles.

In the case of air, which is treated as an ideal diatomic gas under standard conditions, the ratio of specific heats is approximately 1.4. This value is used in various calculations involving compressible flow, such as determining the pressure and temperature changes across a nozzle or when predicting the behavior of shock waves. A higher \(\gamma\) value indicates a gas with molecules having more degrees of freedom (such as diatomic vs. monatomic gases) and generally correlates with a stronger resistance to changes in temperature and pressure during adiabatic processes.