Standard atmospheric air \(\left(T_{0}=59^{\circ} \mathrm{F}, p_{0}=14.7 \mathrm{psia}\right)\) is drawn steadily through a frictionless and adiabatic converging nozzle into an adiabatic, constant cross-sectional area duct. The duct is \(10 \mathrm{ft}\) long and has an inside diameter of \(0.5 \mathrm{ft}\). The average friction factor for the duct may be estimated as being equal to \(0.03 .\) What is the maximum mass flowrate in slugs/s through the duct? For this maximum flowrate, determine the valies of static temperature, static pressure, stagnation temperature, stagnation pressure, and velocity at the inlet [section (1)] and exit [section (2)] of the constant area duct. Sketch a temperature-entropy diagram for this flow.

Step by step solution

01

Calculate the initial conditions

Given the initial temperature and pressure, we can calculate the initial density using the ideal gas law. The equation for ideal gas law is \( P = ρRT \), where \( P \) is pressure, \( ρ \) is density, \( R \) is gas constant, and \( T \) is temperature.

02

Determine the maximum mass flow rate

For an adiabatic variable area duct, the maximum mass flow rate occurs when the flow is choked. The mass flow rate is given by the equation \( \dot{m} = ρAV \), where \( \dot{m} \) is mass flow rate, \( ρ \) is density, \( A \) is cross sectional area, and \( V \) is velocity.

03

Calculate the static temperature and pressure at the exit and the entrance

These values are determined from the isentropic flow relations. Static temperature and pressure at the exit can be calculated using the formulas:For temperature: \( T = T_0 - \frac{V^2}{2C_p} \),where \( T_0 \) is the total temperature, \( V \) is the velocity at the exit, \( C_p \) is the specific heat at constant pressure.For Pressure: \( P = P_0 \left( \frac{T}{T_0} \right)^{\gamma / (\gamma -1 )} \),where \( \gamma \) is the ratio of specific heats.

04

Determine Stagnation temperature and pressure

In an adiabatic flow, the stagnation temperature remains constant throughout the flow. It’s equal to the initial temperature. The stagnation pressure can be determined using the Bernoulli's equation:\( P_0 = P + \frac{1}{2} ρV^2 \)

05

Plot temperature-entropy diagram

The temperature-entropy (T-s) diagram represents the thermal state changes during the flow. The T-s diagram for the adiabatic process is a vertical line. It starts from the initial temperature at the inlet and ends at the final temperature at the exit.

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