Question

Helium enters a nozzle at \(0.6 \mathrm{MPa}, 560 \mathrm{K},\) and a velocity of \(120 \mathrm{m} / \mathrm{s}\). Assuming isentropic flow, determine the pressure and temperature of helium at a location where the velocity equals the speed of sound. What is the ratio of the area at this location to the entrance area?

Short answer

1. Calculate the initial properties of helium for the given conditions (0.6 MPa, 560 K, and 120 m/s). 2. Determine the Mach number at the entrance. 3. Use isentropic flow relations to find the conditions at the location where helium has a Mach number of 1 (speed of sound). 4. Calculate the pressure, temperature, and area ratio at the location where the velocity equals the speed of sound.

Step-by-step solution

Find the initial properties of helium

Using the initial conditions given (\(0.6 \mathrm{MPa}, 560 \mathrm{K}\), and \(120 \mathrm{m} / \mathrm{s}\)), we can find the properties of helium by looking up the values in a table for helium properties or using the gas constant and specific heat ratio. For helium, the gas constant (\(R\)) is \(2.0769 \mathrm{kJ} / (\mathrm{kg} \cdot \mathrm{K})\) and the specific heat ratio (\(\gamma\)) is \(1.66\). So, we can find the specific volume (\(v\)) using the ideal gas equation: \(v = \frac{RT}{P} = \frac{2.0769 \times 560}{600} = 1.932 \times 10^{-3} \mathrm{m}^3 / \mathrm{kg}\)

Calculate the Mach number at the entrance

The Mach number is the ratio of the flow velocity to the local speed of sound. The local speed of sound (\(c\)) can be calculated as: \(c = \sqrt{\gamma RT} = \sqrt{1.66 \times 2.0769 \times 560} = 1053.3 \mathrm{m} / \mathrm{s}\) Now, we can calculate the Mach number (\(M\)) at the entrance as: \(M = \frac{v}{c} = \frac{120}{1053.3} = 0.1139\)

Use isentropic flow relations to find the conditions when \(M = 1\)

Since the flow is assumed to be isentropic, we can use the isentropic flow relations to find the conditions at the location where helium has a Mach number of 1 (the speed of sound). Using the isentropic relation for pressure ratio: \(\frac{P_2}{P_1} = \left( \frac{1 + (\gamma - 1) / 2 \times M^2}{1 + (\gamma - 1) / 2} \right) ^{\frac{-\gamma}{\gamma - 1}}\) Substitute the known values: \(\frac{P_2}{0.6 \times 10^3} = \left(\frac{1 + (1.66 - 1) / 2}{1 + (1.66 - 1) / 2 \times 0.1139^2} \right) ^ {\frac{-1.66}{1.66 - 1}}\) Solve for \(P_2\): \(P_2 = 338.013 \mathrm{kPa}\) Similarly, we can find the temperature ratio using the isentropic relation: \(\frac{T_2}{T_1} = \left( \frac{1 + (\gamma - 1) / 2 \times M^2}{1 + (\gamma - 1) / 2} \right) ^ {\frac{-1}{\gamma - 1}}\) Substitute the known values and solve for \(T_2\): \(\frac{T_2}{560} = \left(\frac{1 + (1.66 - 1) / 2}{1 + (1.66 - 1) / 2 \times 0.1139^2} \right)^{\frac{-1}{1.66 - 1}}\) \(T_2 = 496.25 \mathrm{ K}\)

Find the area ratio

We can use the isentropic relation for the ratios of the flow area to find the ratio of the area at the location where \(M=1\) to the entrance area: \(\frac{A_2}{A_1} = \frac{M_1}{M_2} \left[ \frac{1+(\gamma - 1) / 2 \times M_2^2}{1+(\gamma - 1) / 2 \times M_1^2} \right] ^{\frac{\gamma+1}{2(\gamma - 1)}}\) Since \(M_1 = 0.1139\) and \(M_2 = 1\), substitute the known values: \(\frac{A_2}{A_1} = \frac{0.1139}{1} \left[ \frac{1+(1.66 - 1) / 2 \times 1^2}{1+(1.66 - 1) / 2 \times 0.1139^2} \right]^{\frac{3.32}{1.32}}\) Solve for the area ratio: \(\frac{A_2}{A_1} = 0.2469\) Thus, the pressure and temperature at the location where the velocity equals the speed of sound are \(338.013 \mathrm{kPa}\) and \(496.25 \mathrm{K}\), respectively. The ratio of the area at this location to the entrance area is approximately \(0.2469\).

Key concepts

Mach Number

The Mach number is a dimensionless parameter that represents the ratio of an object's speed to the local speed of sound in the fluid through which it is traveling. In the context of isentropic flow, such as the exercise scenario, it is essential to understand how the Mach number changes throughout the flow. The exercise starts by calculating the Mach number at the nozzle entrance, which is relatively low, indicating subsonic flow conditions.

Isentropic flow relations are employed to find conditions when the Mach number reaches one, a critical point known as sonic conditions. At this point, the flow velocity equals the local speed of sound, and distinct changes in pressure, temperature, and flow area occur. Students should recognize that this Mach number transition is pivotal in nozzle flow analysis, especially in applications like jet engines and rockets where achieving and surpassing sonic conditions is crucial.

Speed of Sound in Gases

The speed of sound in gases is calculated based on thermodynamic properties of the gas, notably its temperature and specific heat ratio. It is an essential variable in determining the Mach number. In the exercise, the speed of sound is calculated using the formula
\( c = \sqrt{\gamma RT} \)
where
\( \gamma \) is the specific heat ratio,
\( R \) is the gas constant, and
\( T \) is the temperature.

The speed of sound is integral to understanding compressible flow behavior in nozzles, as it establishes a benchmark for categorizing flow regimes as subsonic (Mach number < 1), sonic (Mach number = 1), and supersonic (Mach number > 1).

Understanding this concept helps students grasp why and how the flow properties change through the nozzle.

Nozzle Flow Analysis

Nozzle flow analysis involves studying how a fluid's velocity, pressure, and other properties transform as it passes through a nozzle. The exercise explores an isentropic flow, which is an idealized process where the flow is reversible and adiabatic, meaning there is no heat transfer or loss in total mechanical energy.

The nozzle's design also plays a significant role as it can accelerate the flow to reach sonic speeds at the nozzle’s throat, where the cross-sectional area is minimum. Physically, this correlates to the Mach number transitioning to one. The area ratio becomes critical since it determines the flow acceleration and the conditions under which the exiting flow can be maximized for thrust.

Importance of Area Ratio

The exercise concludes with finding the area's ratio at the sonic point to the entrance area, a parameter instrumental in nozzle design and performance evaluation.

Thermodynamic Properties of Helium

Helium, being a noble gas with unique properties, often requires special consideration in thermodynamic analysis. In the exercise, its thermodynamic properties such as specific heat ratio (\(\gamma\)) and gas constant (\(R\)) are used within the isentropic flow equations to determine the flow conditions at the point where flow velocity equals the speed of sound.

The specific heat ratio, which is relatively high for helium, significantly influences the speed of sound and hence impacts the Mach number and flow characteristics through the nozzle. The exercise demonstrates the application of these properties in real-world problems and highlights the importance of accurate property data for gases in engineering calculations.