91Ó°ÊÓ

To find the Mach number, we first need to calculate the speed of sound in Helium at a given temperature. This is important because Mach number is defined as the ratio of the flow velocity to the speed of sound. The speed of sound in an ideal gas can be found using the formula: \[ a = \sqrt{\gamma R T } \]Where:

• \( \gamma \) is the specific heat ratio, which for Helium is 1.66,
• \( R \) is the specific gas constant for Helium, equal to 2077 J/kg·K.
• \( T \) is the temperature in Kelvin, given as 300 K.

Substituting the values into the formula gives us:\[ a = \sqrt{1.66 \times 2077 \times 300} \approx 1006.4 \text{m/s} \]Next, we use the calculated speed of sound to find the Mach number with the velocity of the flow:\[ M = \frac{v}{a} \]Where:

z
• \( v \) is the local velocity of the flow, which is 280 m/s.

Substituting the values, we get:\[ M = \frac{280}{1006.4} \approx 0.278 \]This tells us that the flow velocity is approximately 27.8% of the speed of sound.

The stagnation temperature is the temperature of the gas when it is brought to rest adiabatically. It accounts for the kinetic energy of the flowing gas.The formula to calculate the stagnation temperature is:\[ T_0 = T \(1 + \frac{\gamma - 1}{2} M^2 \) \]Where:

• \( T \) is the static temperature, which is 300 K.
• \( \gamma \) is the specific heat ratio, which is 1.66.
• \( M \) is the Mach number, which we have calculated as 0.278.

Substituting the values, we get:\[ T_0 = 300 \(1 + \frac{1.66 - 1}{2} \times 0.278^2 \) \approx 302.3 \text{K} \]This means that if the gas was brought to rest isentropically, its temperature would increase slightly to 302.3 K.

Stagnation pressure is the pressure a fluid attains when it is brought to rest isentropically. It represents the maximum possible pressure of the gas.The formula to calculate the stagnation pressure is:\[ P_0 = P \(1 + \frac{\gamma - 1}{2} M^2 \)^{\frac{\gamma}{\gamma - 1}} \]Where:

• \( P \) is the static pressure, given as 150 kPa.
• \( \gamma \) is the specific heat ratio, which is 1.66.
• \( M \) is the Mach number, which we calculated as 0.278.

Substituting the values, we get:\[ P_0 = 150 \(1 + \frac{1.66 - 1}{2} \times 0.278^2 \)^{\frac{1.66}{1.66 - 1}} \approx 152.4 \text{kPa} \]This indicates that the pressure would increase slightly to 152.4 kPa if the flow were brought to rest isentropically.

The ideal gas law is a fundamental equation that describes the relationship between the pressure, volume, temperature, and amount of gas. Although it is not explicitly needed for the specific calculations in the exercise, it provides a fundamental understanding of gas behavior.The ideal gas law is given by the equation:\[ PV = nRT \]Where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the number of moles of the gas.
• \( R \) is the universal gas constant, 8.314 J/mol·K.
• \( T \) is the temperature of the gas in Kelvin.

Rearranging for specific gas constants and moles for practical use, it can also be written as:\[ P = \rho RT \]Where \( \rho \) is the density of the gas. This form makes it easier to apply directly when dealing with fluid dynamics, such as calculating parameters in a flowing gas, where density, pressure, and temperature are often the variables of interest.