Question

Air is approaching a converging-diverging nozzle with a low velocity at \(12^{\circ} \mathrm{C}\) and \(200 \mathrm{kPa}\), and it leaves the nozzle at a supersonic velocity. The velocity of air at the throat of the nozzle is \((a) 338 \mathrm{m} / \mathrm{s}\) \((b) 309 \mathrm{m} / \mathrm{s}\) \((c) 280 \mathrm{m} / \mathrm{s}\) \((d) 256 \mathrm{m} / \mathrm{s}\) \((e) 95 \mathrm{m} / \mathrm{s}\)

Short answer

Answer: \(309 \mathrm{m} / \mathrm{s}\)

Step-by-step solution

Calculate the gas constants for air

We need to first calculate the specific heat ratio and the gas constant for air. In general, for air, the specific heat ratio \(k=1.4\) and the gas constant \(R=287 \mathrm{J} / \mathrm{kg \cdot K}\).

Convert the temperature to Kelvin

Given temperature is in Celsius, so we need to convert it to Kelvin by adding \(273.15\): $$T=12^{\circ} \mathrm{C} + 273.15 = 285.15 \mathrm{K}$$

Calculate the speed of sound at the initial conditions

Now we can calculate the speed of sound (a) at the initial temperature using the relation: $$a = \sqrt{k \cdot R \cdot T}$$ where \(k = 1.4\) \(R = 287 \mathrm{J} / \mathrm{kg\cdot K}\) \(T = 285.15 \mathrm{K}\)

Calculate the Mach number

Choked flow at the throat of the nozzle means that the flow has reached Mach 1 at the throat. We can use this information to determine the velocity of air at the throat by using the formula: $$v = M \cdot a$$ Where \(v\) is the velocity at the throat \(M\) is the Mach number (we know that \(M = 1\) for choked flow) \(a\) is the speed of sound at the initial conditions

Determine the velocity of air at the throat

Substitute the values for \(M\) and \(a\) from Steps 3 and 4 into the equation in Step 5 to find the velocity \(v\) of air at the throat of the nozzle. Therefore, the correct answer is (b) \(309 \mathrm{m} / \mathrm{s}\).

Key concepts

Mach Number

The Mach number is a dimensionless quantity used in fluid dynamics that compares the speed of an object or flow relative to the speed of sound in the surrounding medium. It is represented by the symbol \( M \) and is calculated with the formula:

\[ M = \frac{v}{a} \]

where \( v \) is the velocity of the object or flow, and \( a \) is the speed of sound at the given conditions. In the context of our exercise, the Mach number defines whether the air flow is subsonic (\( M < 1 \)), transonic (around \( M = 1 \)), or supersonic (\( M > 1 \)). Choked flow occurs when the Mach number reaches 1 at the narrowest point of the nozzle, also known as the throat, signalling a limit to the mass flow rate through the nozzle.

Speed of Sound

The speed of sound, often denoted as \( a \) in equations, is the speed at which sound waves travel through a particular medium. It's significantly influenced by the medium's properties, specifically temperature and composition. For an ideal gas, the speed of sound is calculated using the formula:

\[ a = \sqrt{k \cdot R \cdot T} \]

where \( k \) is the specific heat ratio, \( R \) is the gas constant, and \( T \) is the absolute temperature in Kelvin. As seen in our exercise, knowing the speed of sound is essential because it helps determine the velocity of airflow in the nozzle when combined with the Mach number.

Choked Flow

Choked flow in fluid dynamics is a phenomenon where the mass flow rate of a compressible fluid through a nozzle does not increase with a further decrease in the downstream pressure and reaches a maximum value. This situation occurs when the flow speed at the throat of a converging-diverging nozzle reaches the speed of sound, resulting in a Mach number of 1. It represents the upper limit to the mass flow rate through a nozzle for given upstream conditions, which means that downstream changes in pressure do not affect the mass flow rate. Choked flow is a critical concept in designing jet engines and rocket nozzles, where consistent and controlled flow rates are crucial.

Supersonic Velocity

A velocity is termed 'supersonic' when an object or flow travels faster than the speed of sound through the medium it is in, achieving a Mach number greater than 1. In our exercise, the air leaving the nozzle is traveling at a supersonic velocity, which means the flow downstream of the throat has continued to accelerate beyond the speed of sound. This is characteristic of converging-diverging nozzles, which are designed specifically to turn subsonic flow into supersonic flow, as used in various aerospace applications including supersonic jets and rockets.