Question

A gas mixture at a pressure of \(800\) kPa and a density of \(5\) kg/m\(^3\) enters a turbine stage. The temperature of the gas at the nozzle exit and the stage exit are \(790\) K and \(750\) K, respectively. Assume the specific heats are constant for the gas mixture in the range of temperatures considered. The specific heat at constant pressure is \(0.72\) kJ/kg-K and the ratio of specific heats is \(1.33\). The value of the degree of reaction of the turbine stage is _______ (rounded off to 2 decimal places).

Hint:

Degree of reaction is the fraction of the stage's static enthalpy drop that happens across the rotor: \(\Lambda = (T_2-T_3)/(T_1-T_3)\). You need the stage inlet temperature \(T_1\) from the ideal gas law, using \(R = c_p(\gamma-1)/\gamma\).

Answer 1

Correct Answer: 0.27

Step 1: Recall the definition of degree of reaction.
A turbine stage has a fixed row of nozzles followed by a moving row (the rotor). The degree of reaction \(\Lambda\) is the fraction of the stage's total static enthalpy drop that occurs in the rotor:
\[ \Lambda = \frac{h_2 - h_3}{h_1 - h_3} \]
where 1 = stage inlet, 2 = nozzle exit (rotor inlet), 3 = stage exit (rotor exit). Since \(c_p\) is constant here, enthalpy differences are just \(c_p\) times temperature differences, so the \(c_p\) cancels and
\[ \Lambda = \frac{T_2 - T_3}{T_1 - T_3} \]
We already know \(T_2 = 790\) K and \(T_3 = 750\) K, but \(T_1\) (stage inlet temperature) is not given directly, only \(p_1\) and \(\rho_1\).

Step 2: Find the gas constant R from cp and gamma.
\[ R = c_p\left(\frac{\gamma - 1}{\gamma}\right) = 0.72\left(\frac{0.33}{1.33}\right) = 0.72(0.24812) = 0.17865 \text{ kJ/kg-K} \]

Step 3: Find T1 from the ideal gas law.
\[ p_1 = \rho_1 R T_1 \implies T_1 = \frac{p_1}{\rho_1 R} \]
\[ T_1 = \frac{800}{(5)(0.17865)} = \frac{800}{0.89324} = 895.7 \text{ K} \]

Step 4: Compute the degree of reaction.
\[ \Lambda = \frac{T_2 - T_3}{T_1 - T_3} = \frac{790 - 750}{895.7 - 750} = \frac{40}{145.7} \]

Final Answer:
\[ \boxed{\Lambda \approx 0.27} \]