Understanding Air Mass Flow Rate Calculation for a Nozzle

This problem requires calculating the mass flow rate of air entering a nozzle under specific conditions. The mass flow rate is a crucial parameter in fluid dynamics, representing the mass of fluid passing through a given cross-section per unit time.

Given Parameters

We are provided with the following information about the air entering the nozzle:

• Inlet Temperature ($T_1$): 27 °C
• Inlet Pressure ($P_1$): 100 kPa
• Inlet Velocity ($V_1$): 100 m/s
• Inlet Area ($A_1$): 0.5 m²
• Fluid: Air (assumed to behave as an ideal gas)

Calculation Steps

Step 1: Convert Units

First, we need to convert the given temperature and pressure into standard SI units (Kelvin for temperature and Pascals for pressure) to use them in thermodynamic equations.

• Temperature Conversion: $T_1 = 27 \text{ °C} + 273.15 = 300.15 \text{ K}$
• Pressure Conversion: $P_1 = 100 \text{ kPa} = 100 \times 10^3 \text{ Pa}$

Step 2: Determine Air Density

We can calculate the density of air at the inlet using the ideal gas law, which relates pressure, density, specific gas constant, and temperature.

The ideal gas law is given by:

$$ P = \rho R T $$

Where:

• $P$ is the absolute pressure
• $\rho$ is the density
• $R$ is the specific gas constant for the substance
• $T$ is the absolute temperature

For air, the specific gas constant ($R_{air}$) is approximately 287 J/(kg·K).

Rearranging the formula to solve for density ($\rho_1$):

$$ \rho_1 = \frac{P_1}{R_{air} T_1} $$

Now, substitute the values:

$$ \rho_1 = \frac{100 \times 10^3 \text{ Pa}}{287 \text{ J/(kg·K)} \times 300.15 \text{ K}} $$

$$ \rho_1 \approx \frac{100000}{86143.05} \text{ kg/m}^3 $$

$$ \rho_1 \approx 1.1608 \text{ kg/m}^3 $$

Step 3: Calculate Mass Flow Rate

The mass flow rate ($\dot{m}$) for a steady flow system is calculated by multiplying the density, the cross-sectional area, and the velocity of the fluid.

The formula for mass flow rate is:

$$ \dot{m} = \rho \times A \times V $$

Using the inlet conditions:

$$ \dot{m} = \rho_1 \times A_1 \times V_1 $$

Substitute the calculated density and given area and velocity:

$$ \dot{m} = 1.1608 \text{ kg/m}^3 \times 0.5 \text{ m}^2 \times 100 \text{ m/s} $$

$$ \dot{m} = 1.1608 \times 50 \text{ kg/s} $$

$$ \dot{m} \approx 58.04 \text{ kg/s} $$

Conclusion

The calculated mass flow rate through the nozzle is approximately 58.04 kg/s. Comparing this value to the given options, it closely matches 58 kg/s.