Question

Helium gas enters an adiabatic nozzle steadily at \(500^{\circ} \mathrm{C}\) and \(600 \mathrm{kPa}\) with a low velocity, and exits at a pressure of \(90 \mathrm{kPa}\). The highest possible velocity of helium gas at the nozzle exit is \((a) 1475 \mathrm{m} / \mathrm{s}\) \((b) 1662 \mathrm{m} / \mathrm{s}\) \((c) 1839 \mathrm{m} / \mathrm{s}\) \((d) 2066 \mathrm{m} / \mathrm{s}\) \((e) 3040 \mathrm{m} / \mathrm{s}\)

Short answer

Answer: The highest possible exit velocity of the helium gas is 1839 m/s.

Step-by-step solution

Recall the ideal gas law and adiabatic condition equations

For an ideal gas, the relationships between temperature, pressure, and specific volume can be represented using the following equation: \(PV = mRT\) Where: \(P\) = pressure, \(V\) = volume, \(m\) = mass, \(R\) = specific gas constant, and \(T\) = temperature. For an adiabatic process, we can relate the initial and final states using the relationship: \(\frac{P_1 V_1^{\gamma}}{T_1^{\gamma - 1}} = \frac{P_2 V_2^{\gamma}}{T_2^{\gamma - 1}}\) Where: \(P_1\), \(V_1\), and \(T_1\) are the initial pressure, specific volume, and temperature, \(P_2\), \(V_2\), and \(T_2\) are the final pressure, specific volume, and temperature, \(\gamma\) is the specific heat ratio of the gas.

Find the specific heat ratio and specific gas constant for helium

Helium is a monoatomic gas, thus \(\gamma = 1.66\) and \(R = 2077 \frac{\mathrm{J}}{\mathrm{kg}\cdot\mathrm{K}}\).

Determine the initial temperature in Kelvin

We know the initial temperature is \(500^{\circ}\mathrm{C}\). To convert it to Kelvin, we add 273.15: \(T_1 = 500 + 273.15 = 773.15 \mathrm{K}\)

Find the initial specific volume using the ideal gas law

Rewrite the ideal gas law, solving for the specific volume at initial conditions: \(V_1 = \frac{mRT_1}{P_1} = \frac{2077 \mathrm{\frac{J}{kg\cdot K}} \cdot 773.15 \mathrm{K}}{600000 \mathrm{Pa}} = 2.66 \mathrm{\frac{m^3}{kg}}\)

Find the final specific volume using the adiabatic equation

Rewrite the adiabatic equation, solving for the final specific volume (\(V_2\)): \(V_2 = \left(\frac{P_1 V_1^{\gamma}}{P_2}\right)^{\frac{1}{\gamma}} = \left(\frac{600000 \mathrm{Pa} \cdot (2.66 \mathrm{\frac{m^3}{kg}})^{1.66}}{90000 \mathrm{Pa}}\right)^{\frac{1}{1.66}} = 10.92 \mathrm{\frac{m^3}{kg}}\)

Find the final temperature using the adiabatic equation

Rewrite the adiabatic equation, replacing \(V_2\) with the value calculated in Step 5: \(T_2 = \frac{P_2 V_2^{\gamma - 1}}{P_1 V_1^{\gamma - 1}}T_1 = \frac{90,000 \mathrm{Pa} \cdot (10.92 \mathrm{\frac{m^3}{kg}})^{0.66}}{600,000 \mathrm{Pa} \cdot (2.66 \mathrm{\frac{m^3}{kg}})^{0.66}} \cdot 773.15 \mathrm{K} = 249.2 \mathrm{K}\)

Apply the conservation of energy equation

The conservation of energy equation with no heat transfer is: \(h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2}\) But we know \(V_1\) is very low and can be considered negligible. Thus, the equation becomes: \(h_1 = h_2 + \frac{V_2^2}{2}\)

Apply the ideal gas relationship between enthalpy and temperature

The relationship between enthalpy and temperature for an ideal gas is: \(h = C_p T\) Where \(C_p\) is the specific heat at constant pressure. For monoatomic gases like helium, \(C_p = \frac{5}{2}R\).

Calculate the final velocity

Rewrite the conservation of energy equation, replacing enthalpy with temperature, and solve for \(V_2\): \(V_2 = \sqrt{2 \cdot \left(C_p T_1 - C_p T_2\right)} = \sqrt{2 \cdot \frac{5}{2}R \cdot (T_1 - T_2)} = \sqrt{2 \cdot \frac{5}{2} \cdot 2077 \frac{\mathrm{J}}{\mathrm{kg}\cdot\mathrm{K}} \cdot (773.15 \mathrm{K} - 249.2 \mathrm{K})} = 1839 \mathrm{\frac{m}{s}}\) The highest possible velocity of helium gas at the nozzle exit is \(\boxed{1839 \mathrm{\frac{m}{s}}}\) (Option c).

Key concepts

Ideal Gas Law

At the heart of many thermodynamic processes lies the ideal gas law, expressed as the equation PV = mRT, where P stands for pressure, V is volume, m is mass of the gas, R is the specific gas constant, and T represents temperature.
This fundamental equation assumes that the gas molecules do not interact with each other except for perfectly elastic collisions and that the volume occupied by the gas molecules themselves is negligible compared to the volume of the container. In practical scenarios like flow through a nozzle, these assumptions allow us to predict how a gas will behave when subjected to changes in pressure, temperature, and volume. For instance, when helium gas expands through a nozzle, its volume increases, which according to the ideal gas law occurs with simultaneous changes in pressure and temperature that can ultimately determine the velocity of the gas at the exit of the nozzle.

Adiabatic Process

An adiabatic process is a thermodynamic transformation where no heat is exchanged between the system and its surroundings. For such a process in a gas, the heat content of the gas remains constant even as work is done on or by the gas, leading to changes in pressure and volume.
Adiabatic processes are characterized by alterations in the gas's temperature, which arises purely from work done on or by the gas, entirely independent of any heat transfer. This can be represented mathematically by the relationship \(\frac{P_1 V_1^{\gamma}}{T_1^{\gamma - 1}} = \frac{P_2 V_2^{\gamma}}{T_2^{\gamma - 1}}\), connecting the initial and final states of the process with \(\gamma\) being the specific heat ratio. In the context of a nozzle, the adiabatic expansion of a gas like helium can lead to a significant increase in velocity as the gas emerges from the nozzle exit.

Specific Heat Ratio

The specific heat ratio, denoted as \(\gamma\), is the ratio of the specific heat at constant pressure, \(C_p\), to the specific heat at constant volume, \(C_v\). Monoatomic gases like helium have \(\gamma = 1.66\), one of the highest among gas species, making these gases particularly amenable to adiabatic processes.
This ratio is crucial for calculating the changes in temperature and pressure during adiabatic processes because it embodies the thermodynamic properties of the gas. In our nozzle scenario, the specific heat ratio influences the expansion and cooling of helium as it moves through the nozzle and profoundly affects the final velocity attainable at the nozzle exit. Understanding the specific heat ratio helps us grasp the energy transformations involved in compression or expansion of gases without heat exchange with the surroundings.