Question

Air enters a rectangular duct at \(T_{1}=300 \mathrm{K}, P_{1}=\) \(420 \mathrm{kPa},\) and \(\mathrm{Ma}_{1}=2 .\) Heat is transferred to the air in the amount of \(55 \mathrm{kJ} / \mathrm{kg}\) as it flows through the duct. Disregarding frictional losses, determine the temperature and Mach number at the duct exit.

Short answer

Answer: To obtain the temperature and Mach number at the duct exit, follow these steps: 1. Determine the initial conditions at the duct inlet (given) 2. Calculate heat addition per unit mass (using energy equation) 3. Determine the exit temperature using specific enthalpy 4. Calculate the ratio of specific heat (for air) 5. Calculate the isentropic temperature ratio (using inlet Mach number) 6. Calculate the flow temperature ratio (using exit temperature) 7. Calculate the exit Mach number (using flow temperature ratio and ratio of specific heat) After performing these calculations, the temperature and Mach number at the duct exit are obtained.

Step-by-step solution

Determine the initial conditions at the duct inlet

Given the inlet temperature \(T_1 = 300\,\text{K}\), pressure \(P_1 = 420\,\text{kPa}\) and Mach number \(Ma_1 = 2\). We know the specific heat at constant pressure for air, \(c_p \approx 1005\,\text{J/kgK}\). We will now calculate the specific enthalpy at point 1. Using the formula, \(h_1 = c_p \times T_1\)

Calculate heat addition per unit mass

Given the heat transferred to the air, \(q = 55\,\text{kJ/kg}\). We need to determine the final specific enthalpy at the duct exit (point 2). We can account for the heat addition using the energy equation as follows, \(h_2 = h_1 + q\)

Determine the exit temperature using specific enthalpy

We have determined the specific enthalpy at the duct exit, and now we will use the specific heat to determine the temperature at the exit, \(T_2\). Using the formula, \(T_2 = \frac{h_2}{c_p}\)

Calculate the ratio of specific heat

To find the Mach number at the exit, we will use the isentropic flow relations. In order to do that, we need to determine the ratio of specific heat, \(\gamma\). For air, we have \(\gamma \approx 1.4\).

Calculate the isentropic temperature ratio

Now we will calculate the isentropic temperature ratio at the inlet (point 1) using the given Mach number, \(Ma_1\), and the ratio of specific heat \(\gamma\). The formula for the isentropic temperature ratio is: \(T_{1^*} = \frac{T_1}{1 + \frac{\gamma - 1}{2} Ma_1^2}\)

Calculate the flow temperature ratio

To find the Mach number at the exit, we need to determine the flow temperature ratio between the exit temperature, \(T_2\), and the isentropic temperature at the exit, \(T_{2^*}\). Using the isentropic conditions, we can calculate the flow temperature ratio as follows: \(\frac{T_2}{T_{2^*}} = \frac{T_2}{T_{1^*}}\)

Calculate the exit Mach number

We can now find the exit Mach number, \(Ma_2\), by using the flow temperature ratio and the ratio of specific heat, \(\gamma\). \(Ma_2 = \sqrt{\frac{2}{\gamma - 1} \left[ \left(\frac{T_2}{T_{2^*}}\right) - 1\right]}\) After performing these calculations, the temperature and Mach number at the duct exit are obtained.