Question

Argon gas is approaching a converging-diverging nozzle with a low velocity at \(20^{\circ} \mathrm{C}\) and \(120 \mathrm{kPa}\), and it leaves the nozzle at a supersonic velocity. If the cross-sectional area of the throat is \(0.015 \mathrm{m}^{2},\) the mass flow rate of argon through the nozzle is \((a) 0.41 \mathrm{kg} / \mathrm{s}\) (b) \(3.4 \mathrm{kg} / \mathrm{s}\) \((c) 5.3 \mathrm{kg} / \mathrm{s}\) \((d) 17 \mathrm{kg} / \mathrm{s}\) \((e) 22 \mathrm{kg} / \mathrm{s}\)

Short answer

a) 0.41 kg/s b) 3.4 kg/s c) 5.3 kg/s d) 17 kg/s e) 22 kg/s

Step-by-step solution

Determine the Ideal Gas Properties of Argon

At \(20^{\circ}\mathrm{C}\) or \(293\,\mathrm{K}\) and \(120\,\mathrm{kPa}\), retrieve the properties of argon gas, which are the specific gas constant \((R)\) and the specific heat ratio \((k)\). Look up values for argon are: \(R = 208.13\,\mathrm{J}(\mathrm{kg}\cdot\mathrm{K})^{-1}\) and \(k = 1.67\).

Calculate the Mach Number and Velocity at the Throat

To find the mass flow rate, we need to determine the velocity of the argon gas at the throat of the nozzle. Mach number \((M_{throat})\) is defined as the ratio of the flow velocity (\((V_{throat})\)) to the local speed of sound (\((a_{throat})\)). In a converging-diverging nozzle, the throat condition is choked with a Mach number equal to 1. First, we determine the speed of sound at the inlet: \(a_{inlet} = \sqrt{k \cdot R \cdot T_{inlet}}\). Next, we calculate the choked throat condition (Mach number 1) and velocity: \(V_{throat} = M_{throat} \cdot a_{inlet}\).

Calculate the Mass Flow Rate

We can now determine the mass flow rate (\({m'}\)) of the argon gas through the nozzle using the following formula: \({m'} = \rho_{inlet} \cdot A_{throat} \cdot V_{throat}\), where \(\rho_{inlet}\) is the density of argon gas at the inlet and \(A_{throat}\) is the cross-sectional area of the throat. To find the density at the inlet, \(\rho_{inlet} = \dfrac{P_{inlet}}{R \cdot T_{inlet}}\). Finally, we can calculate the mass flow rate of argon gas, and compare it with the given answer choices.

Determine the Correct Answer

After calculating the mass flow rate, match it with one of the options given in the exercise: (a) \(0.41\,\mathrm{kg/s}\), (b) \(3.4\,\mathrm{kg/s}\), (c) \(5.3\,\mathrm{kg/s}\), (d) \(17\,\mathrm{kg/s}\), or (e) \(22\,\mathrm{kg/s}\).

Key concepts

Converging-Diverging Nozzle

A converging-diverging nozzle, often found in supersonic jet engines, is a device designed to accelerate a fluid to high velocities. It works on the principle of gas dynamics, beginning with a narrowing section (converging), followed by a minimum cross-sectional area point known as the throat, and then widening (diverging).

As the gas flows through the converging section, its velocity increases while the pressure and temperature decrease. When the gas reaches the throat, it achieves the speed of sound; this condition is called 'choked flow.' Beyond the throat, in the diverging section, the gas expands and can exceed the speed of sound (become supersonic), further increasing in velocity as pressure continues to drop. This process is closely related to the mass flow rate of the gas, determined by the throat's area and the gas properties at that point.

Mach Number

The Mach number is a dimensionless quantity in fluid dynamics that represents the ratio of the speed of an object or gas flow to the speed of sound in the surrounding medium. It is named after the Austrian physicist Ernst Mach.

Mach numbers are used to characterize the flow regime: subsonic (Mach number < 1), transonic (Mach number around 1), supersonic (Mach number > 1), and hypersonic (Mach number > 5). Particularly in a converging-diverging nozzle, knowing the Mach number at the throat enables engineers to predict whether the flow is choked and to calculate the subsequent behavior of the gas as it exits the nozzle at potentially supersonic speeds.

Specific Gas Constant

The specific gas constant, denoted by 'R', is a unique value for each gas that relates the gas temperature, pressure, and density in the equation of state for an ideal gas, given by PV = nRT, where P is pressure, V is volume, n is the amount of substance, and T is temperature. This constant is crucial for solving various thermodynamic problems.

In particular, it allows us to find the density of the gas when calculating the mass flow rate through a nozzle. Since the specific gas constant varies between different gases, this ensures that calculations account for the inherent properties of the working fluid, such as argon in the given exercise.

Specific Heat Ratio

The specific heat ratio (also known as the adiabatic index or gamma, denoted as 'k') is the ratio of the specific heat at constant pressure (Cp) to the specific heat at constant volume (Cv). It is a crucial parameter in thermodynamics and fluid mechanics as it affects the speed of sound in a medium and the thermodynamic processes that occur during the flow of gases.

For monatomic gases like argon, the specific heat ratio is higher than for diatomic gases like oxygen or nitrogen. This property influences how gases expand or compress in different situations, such as when flowing through a converging-diverging nozzle. When calculating the mass flow rate, the value of 'k' affects the temperature and velocity of the gas, which in turn impacts the overall performance of the nozzle.