Calculating Air Mass Flow Rate in a Pipe

The question asks for the mass flow rate of air entering a pipe under steady flow conditions. To find the mass flow rate ($\dot{m}$), we need the density of the air at the inlet ($\rho_1$), the cross-sectional area of the pipe ($A$), and the inlet velocity ($V_1$). The formula for mass flow rate is:

First, let's list the given information and convert units to be consistent with SI units (meters, kilograms, seconds, Kelvin, Pascals, Joules):

• Pipe diameter, $D = 28 \text{ cm} = 0.28 \text{ m}$
• Inlet pressure, $P_1 = 200 \text{ kPa} = 200 \times 10^3 \text{ Pa}$
• Inlet temperature, $T_1 = 20^\circ\text{C}$. We need to convert this to Kelvin: \(T_1 = 20 + 273.15 = 293.15 \text{ K}\).
• Inlet velocity, $V_1 = 5 \text{ m/s}$
• Gas constant for air, $R = 0.287 \text{ kJ/kg.K} = 0.287 \times 10^3 \text{ J/kg.K}$

Note that the outlet conditions ($P_2$ and $T_2$) are given, but for calculating the mass flow rate *at the inlet*, we only need the inlet conditions and the gas constant, since the mass flow rate is constant in steady flow.

Next, we need to calculate the cross-sectional area of the pipe. Since the pipe is circular, the area is given by:

Now, we need to find the density of air at the inlet conditions ($\rho_1$). We can use the ideal gas law, which states \(P V = m R T\). Dividing by volume \(V\) and rearranging gives \(P = \frac{m}{V} R T\). Since density \(\rho = \frac{m}{V}\), we have \(P = \rho R T\). We can solve for density:

Finally, we can calculate the mass flow rate using the formula \(\dot{m} = \rho_1 \times A \times V_1\):

The calculated mass flow rate of air is approximately 0.7318 kg/s.

Let's compare this result with the given options:

• 7.318 kg/s
• 0.7318 kg/s
• 0.3079 kg/s
• 10.727 kg/s

Our calculated value matches one of the options.