Question

Air enters a converging-diverging nozzle of a supersonic wind tunnel at 150 psia and \(100^{\circ} \mathrm{F}\) with a low velocity. The flow area of the test section is equal to the exit area of the nozzle, which is \(5 \mathrm{ft}^{2}\). Calculate the pressure, temperature, velocity, and mass flow rate in the test section for a Mach number \(\mathrm{Ma}=2 .\) Explain why the air must be very dry for this application.

Short answer

Create a short answer based on the provided solution: To find the properties (pressure, temperature, velocity, and mass flow rate) in the test section of a converging-diverging nozzle when the Mach number is 2, we need to follow these steps: 1. Calculate the initial density of the dry air using the Ideal Gas Law. 2. Calculate the initial speed of sound in the dry air using the specific heat ratio. 3. Use the isentropic flow equations to find the final pressure, temperature, and density in the test section by plugging in the known values. 4. Calculate the final velocity in the test section using the Mach number and the speed of sound. 5. Determine the mass flow rate in the test section using the final density, velocity, and exit area of the nozzle. Air must be very dry in this application to ensure accurate and consistent results, as water vapor content could influence the flow properties, making it difficult to maintain a Mach number of 2 and causing potential condensation in the test section.

Step-by-step solution

Calculate the initial density of dry air

Using the Ideal Gas Law, we can find the initial air density (\(\rho_{1}\)) and convert the temperature to the absolute scale (Rankine): \(\rho_{1} = \frac{P_{1}}{R_{air} T_{1}}\) where, \(P_{1} = 150\,\text{psia}\) \(T_{1} = 100^{\circ} \mathrm{F} = 100 + 459.67 = 559.67\,\mathrm{R}\) \(R_{air} = 1716\,\frac{\text{ft.lb}}{\text{slug.R}}\) Now, plug in the values to get \(\rho_{1}\).

Calculate the initial speed of sound in dry air

The initial speed of sound (\(a_{1}\)) in dry air can be found using the following equation: \(a_{1} = \sqrt{\gamma_{1} R_{air} T_{1}}\) where, \(\gamma_{1}\) is the specific heat ratio for dry air, which is approximately \(1.4\). Now, plug in the specific heat ratio, and other known values to calculate \(a_{1}\).

Calculate the values of the ratios of specific heats

We need to calculate the values of the ratios of specific heats (\(\gamma\)) for dry air: \(\gamma = \frac{C_{p}}{C_{v}}\) where \(C_{p}\) and \(C_{v}\) are the specific heat capacities of dry air at constant pressure and constant volume, respectively. As dry air is mostly composed of diatomic molecules (approximately \(99\%\)), we can take \(\gamma \approx 1.4\).

Use isentropic flow equations to find final properties

Now, using the isentropic flow equations, we can find the final pressure (\(P_{2}\)), temperature (\(T_{2}\)), and density (\(\rho_{2}\)) in the test section: \(P_{2} = P_{1} \left[1 + \frac{\gamma - 1}{2} \mathrm{Ma}^2\right]^{-\frac{\gamma}{\gamma - 1}}\) \(T_{2} = T_{1} \left[1 + \frac{\gamma - 1}{2} \mathrm{Ma}^2\right]^{-1}\) \(\rho_{2} = \rho_{1} \left[1 + \frac{\gamma - 1}{2} \mathrm{Ma}^2\right]^{\frac{1}{\gamma - 1}}\) where \(\mathrm{Ma}=2\) is the Mach number at the test section. Plug in the known values and calculate the required properties.

Calculate mass flow rate and velocity in the test section

First, calculate the final velocity (\(v_{2}\)) in the test section using the Mach number: \(v_{2} = \mathrm{Ma} \cdot a_{2}\) where \(a_{2}\) is the speed of sound in the test section. It can be found using \(a_{2} = \sqrt{\gamma R_{air} T_{2}}\). Insert known values to find \(v_{2}\). Next, calculate the mass flow rate (\(\dot{m}\)) in the test section: \(\dot{m} = \rho_{2} \cdot v_{2} \cdot A_{2}\) where \(A_{2} = 5\,\mathrm{ft}^{2}\) is the exit area of the nozzle. Plug in the known values to find the mass flow rate.

Reasoning for the use of very dry air

The air must be very dry for this application for multiple reasons: 1. Water vapor content in the air would influence the flow properties, such as specific heat ratio, speed of sound, and density. This would affect the obtained results and make it difficult to maintain a Mach number of 2 in the test section. 2. The presence of water vapor in the air could lead to condensation in the test section, particularly when the air expands and cools down. This could negatively affect experiments, as well as the performance and accuracy of the supersonic wind tunnel. Therefore, it is essential to use very dry air to ensure accurate and consistent results in the supersonic wind tunnel application.

Key concepts

Mach Number

The Mach number (Ma) is a dimensionless quantity used in aerodynamics to characterize the speed of an object or flow relative to the speed of sound in the medium through which it travels. It is named after the Austrian physicist Ernst Mach. In technical terms, the Mach number is defined as the ratio of the object's speed to the speed of sound in the fluid:

\[\begin{equation}Ma = \frac{v}{a}\end{equation}\]
where \(v\) is the velocity of the object and \(a\) is the speed of sound in the fluid. The speed of sound, in turn, depends on the properties of the fluid, such as its temperature and composition.

In the given exercise, a Mach number of 2 signifies that the flow speed in the test section of the supersonic wind tunnel is twice the speed of sound at those conditions. This high-speed flow is essential for testing and analyzing aerodynamic properties and behaviors that only occur at supersonic speeds, including shock waves and boundary layer effects.

Mach numbers are grouped into regimes: subsonic (Ma < 1), transonic (around Ma=1), supersonic (1 < Ma < 5), and hypersonic (Ma > 5). Each regime brings unique aerodynamic challenges and characteristics. Thus, knowing the Mach number helps engineers and scientists to anticipate and control the flow behavior around aerodynamic models placed inside the wind tunnel.

Isentropic Flow Equations

Isentropic flow equations are a set of mathematical formulas describing the behavior of a compressible fluid passing through a nozzle when the flow is isentropic, which means it is reversible and adiabatic, with no heat exchange or friction. This idealizes how a gas behaves when it is compressed or expanded without losses, and these equations are vital in determining the properties of the flow at various stages within devices like supersonic wind tunnels.

The core isentropic relations are derived from the conservation of mass, momentum, and energy principles. They relate the pressure (P), temperature (T), density (\(\rho\)), and velocity (v) at any two points in the flow when the specific heat ratio (\(\gamma\)) is constant:

\begin{align*} \frac{P_{2}}{P_{1}} &= \left[1 + \frac{\gamma - 1}{2} Ma^2\right]^{-\frac{\gamma}{\gamma - 1}} \ \frac{T_{2}}{T_{1}} &= \left[1 + \frac{\gamma - 1}{2} Ma^2\right]^{-1} \ \frac{\rho_{2}}{\rho_{1}} &= \left[1 + \frac{\gamma - 1}{2} Ma^2\right]^{\frac{1}{\gamma - 1}} \ \end{align*}
These equations allow us to connect the initial and final states of the flow through a nozzle by only knowing the Mach number and specific heat ratio. In the exercise, these equations are used after determining initial conditions of air flow to calculate the pressure, temperature, and density in the test section of the wind tunnel. Isentropic flow equations are pivotal because they provide a means of predicting conditions for supersonic flows without conducting an experiment for every new scenario.

Ideal Gas Law

The ideal gas law is an equation of state that relates the pressure (P), volume (V), temperature (T), and amount of gas (n), to provide a very simple model of the behavior of many gases under a variety of conditions. While real gases have complex interactions that can deviate from this model, the ideal gas law is a helpful approximation for many practical situations, including engineering problems and initial state calculations in wind tunnel testing.

The ideal gas law is represented as:\[\begin{equation}PV = nRT\end{equation}\]
Here, \(R\) is the universal gas constant, which takes on different values depending on the units used for pressure, volume, and temperature. In applications like the supersonic wind tunnel, the equation is often expressed in terms of gas density (\(\rho\)), as follows:

\[\begin{equation}P = \rho RT\end{equation}\]
where \(R\) is specific to the gas in question – for air, it's typically measured in ft·lb/slug·R, and density is in slugs/ft³.

In the context of the exercise, the ideal gas law is crucial for calculating initial air density and speed of sound, which are the starting points for predicting the behavior of air traveling through the nozzle and reaching supersonic speeds. For accurate predictions and the highest fidelity in experimental settings, it is crucial that the air is dry to maintain the validity of the ideal gas law's assumptions, as moisture can substantially change the gas properties.