Question

Air at \(900 \mathrm{kPa}\) and \(400 \mathrm{K}\) enters a converging nozzle with a negligible velocity. The throat area of the nozzle is \(10 \mathrm{cm}^{2} .\) Approximating the flow as isentropic, calculate and plot the exit pressure, the exit velocity, and the mass flow rate versus the back pressure \(P_{b}\) for \(0.9 \geq\) \(P_{b} \geq 0.1 \mathrm{MPa}\).

Short answer

Answer: The exit Mach number is calculated using the isentropic flow relation: \(\frac{P_{0}}{P_{b}} = \left(1 + \frac{\gamma - 1}{2} M_2^2\right)^{\frac{\gamma}{\gamma - 1}}\).

Step-by-step solution

Write down the given information

The given information is: - Inlet pressure: \(P_{1} = 900\,\text{kPa}\) - Inlet temperature: \(T_{1} = 400\,\text{K}\) - Throat area: \(A^{*} = 10\,\text{cm}^2\) - Back pressure range: \(0.1\,\text{MPa} \leq P_{b} \leq 0.9\,\text{MPa}\)

Calculate the initial Mach number

Using the isentropic flow equation for Mach number in terms of stagnation (total) pressure, we have $$ \frac{P_{0}}{P_{1}} = \left(1 + \frac{\gamma - 1}{2} M_1^2\right)^{\frac{\gamma}{\gamma - 1}} $$ Mach number, \(M = 0\) since velocity is negligible. Therefore the total pressure $$ P_{0} = P_{1} $$

Calculate mass flow rate and initialize variables

The mass flow rate can be calculated using the following equation: $$ \dot{m} = \text{Constant flow rate} = \rho A^{*} V^{*} $$ We will use the ideal gas law to find the density of air at the inlet $$ \rho_1 = \frac{P_1}{RT_1} $$ To find the constants required for the analysis, we use: - Specific gas constant for air: \(R = 287\, \text{J/(kg}\cdot\text{K)}\) - Ratio of specific heats for air: \(\gamma = 1.4\) Now we can calculate the density \(\rho_1\) and the mass flow rate \(\dot{m}\).

Calculate exit properties as a function of back pressure

For each given back pressure \(P_{b}\), we can calculate the following exit properties: 1. Exit Mach number: Using the isentropic flow relation between the total pressure and the back pressure, we have $$ \frac{P_{0}}{P_{b}} = \left(1 + \frac{\gamma - 1}{2} M_2^2\right)^{\frac{\gamma}{\gamma - 1}} \Rightarrow M_{2} = f(P_{b}, \gamma) $$ 2. Exit velocity: Using the relation between Mach number, speed of sound and velocity, we compute the exit velocity $$ V_{2} = M_{2}a_{2} = M_{2} \sqrt{\gamma R T_{2}} $$ where \(T_{2}\) can be calculated using the isentropic flow equation $$ T_{2} = T_{1}\left(\frac{P_{b}}{P_{1}}\right)^{\frac{\gamma - 1}{\gamma}} $$ 3. Mass flow rate: Mass flow rate remains constant, and can be found using Step 3. Once we calculate the above three properties for each back pressure value, we can plot them as required.

Plot the results

With the calculated exit pressure, velocity, and mass flow rate, we can create the following plots: 1. Exit pressure (\(P_{b}\)) vs back pressure (\(P_{2}\)) 2. Exit velocity (\(V_{2}\)) vs back pressure (\(P_{2}\)) 3. Mass flow rate (\(\dot{m}\)) vs back pressure (\(P_{2}\)). This will provide a comprehensive picture of how the flow properties change with different back pressures through the converging nozzle, as per the isentropic flow assumption.