Question

Using EES (or other) software, determine the shape of a converging-diverging nozzle for air for a mass flow rate of \(3 \mathrm{kg} / \mathrm{s}\) and inlet stagnation conditions of \(1400 \mathrm{kPa}\) and \(200^{\circ} \mathrm{C}\). Approximate the flow as isentropic. Repeat the calculations for 50 -kPa increments of pressure drop to an exit pressure of \(100 \mathrm{kPa}\). Plot the nozzle to scale. Also, calculate and plot the Mach number along the nozzle.

Short answer

Answer: To determine the shape of a converging-diverging nozzle for given conditions, follow these steps: 1. Find the inlet properties using the given inlet stagnation temperature and pressure. 2. Write down the mass flow rate equation with the given mass flow rate. 3. Repeat the nozzle design calculations for different exit pressure values. 4. Determine the Mach number along the whole nozzle using isentropic equations. 5. Calculate the nozzle shape and plot it to scale for each exit pressure value. 6. Calculate the Mach number along the nozzle for each exit pressure value and plot the Mach number profile.

Step-by-step solution

Find Inlet Properties

First, determine the inlet properties for the given inlet stagnation conditions. We know the inlet stagnation temperature \(T_0 = 200^{\circ}\mathrm{C}\) and the inlet stagnation pressure \(P_0 = 1400\mathrm{kPa}\).

Mass Flow Rate Equation

Write down the mass flow rate equation with the given mass flow rate, which is \(m = 3\mathrm{kg/s}\). $$ m = \dot{m} = \rho A V = \rho A \left( \frac{\gamma R T}{1 + \frac{\gamma-1}{2}M^2} \right)^{1/2} M $$ Where \(\rho\) is the density, \(A\) is the area of the nozzle, \(V\) is the velocity, \(M\) is the Mach number, \(\gamma\) is the specific heat ratio, and \(R\) is the specific gas constant for air.

Repeat Calculations for Different Pressure Drops

Set the exit pressure \(P_e\) to the highest value (1400 kPa) and then repeat the nozzle design calculations for 50 kPa decrements until the exit pressure reaches the lowest value of 100 kPa.

Determine Mach Number Along the Nozzle

For each exit pressure value, calculate the Mach number along the whole nozzle using the isentropic equations: $$ \frac{P}{P_0} = \left( \frac{1 + \frac{\gamma-1}{2}M^2}{\frac{\gamma+1}{2}} \right)^{\frac{\gamma}{\gamma-1}} $$

Calculate Nozzle Shape and Plot to Scale

Based on the Mach number and the other flow parameters, calculate the nozzle shape and plot it to scale. Make sure to use the same axis scaling for all the pressure drops to accurately compare the shape differences.

Mach Number Profile Plot

Calculate the Mach number along the nozzle for each exit pressure value and plot them on a graph with the distance from the inlet (\(x\) axis) and the Mach number (\(y\) axis). By following these steps, we can design the converging-diverging nozzle for the given air flow and inlet stagnation conditions for different pressure drops and plot it to scale. Additionally, we can calculate and plot the Mach number along the nozzle for a comprehensive understanding of the flow characteristics within the nozzle.

Key concepts

Isentropic Flow

Understanding isentropic flow is essential when dealing with nozzle design, particularly in aerospace applications. Isentropic flow is a theoretical concept where the flow is assumed to occur without any entropy change, which implies no heat transfer or friction losses, and can be considered as an idealized process. This simplification is immensely useful for calculating various properties in a flow, such as pressure, temperature, and density, especially when the process approaches a reversible adiabatic condition.

For a converging-diverging nozzle operating under isentropic conditions, we can apply the isentropic flow equations to determine the state of the gas at any point in the nozzle. The equation, involving the Mach number, temperature, and specific heat ratios, describes how the pressure ratio between any two points in the flow is related to the Mach number. When designing nozzles, we commonly approximate the flow as isentropic as long as the process takes place with minimal loss.

Mach Number Calculation

Calculation of the Mach number is fundamental in the study of nozzle flow dynamics. The Mach number is a dimensionless unit describing the ratio of the speed of the fluid to the speed of sound within the fluid. It is given by \( M = \frac{V}{a} \), where \( V \) is the fluid velocity and \( a \) is the speed of sound in the medium.

Within a converging-diverging nozzle, the Mach number progressively increases from subsonic levels to supersonic as the gas accelerates to the nozzle's throat and continues expanding afterwards. The precise calculation of Mach number along the nozzle length is vital, as it affects the nozzle's overall performance and is critical for proper design and operation. This requires an iterative process involving the isentropic relations and mass flow rate equations, giving engineers the necessary insight into the flow characteristics for optimization.

Nozzle Flow Characteristics

The nozzle flow characteristics are highly influenced by its design, especially in a converging-diverging nozzle. This type of nozzle has three distinct sections: a converging section, a throat, and a diverging section. The flow accelerates in the converging part, reaches sonic speed at the throat (Mach 1), and continues accelerating to supersonic speeds in the diverging section.

The shape and size of the nozzle determine how efficiently it converts thermal and pressure energy into kinetic energy. The area ratio between the throat and exit areas plays a key role in determining the exit Mach number for a given set of inlet conditions. Control over these characteristics is crucial to ensure the nozzle performs as intended, whether it's to achieve maximum thrust in rocket engines or to control the exhaust velocity in a jet engine.

Mass Flow Rate Equation

The mass flow rate equation is a cornerstone in nozzle design, providing the relationship between the flow rate and the physical properties of the fluid, including its density \(\rho\), cross-sectional area \(A\), and velocity \(V\). It's expressed as \( m = \rho A V \) for steady flow conditions. However, when dealing with compressible flows like those in nozzles, this equation becomes more complex because the fluid density changes with pressure and temperature.

In the context of converging-diverging nozzles, the mass flow rate equation integrates the Mach number and the thermodynamic properties of the gas, resulting in the expanded equation shown in the solution steps. This equation is key to analyzing the design and performance of nozzles, ensuring that the desired mass flow rate matches the operational requirements of the engine or system in which the nozzle is used.