Understanding Axial Turbine Stage Performance

In this problem, we are asked to find the degree of reaction for an axial turbine stage. We are given key parameters like the relative velocities at the rotor inlet and outlet, the mean rotor peripheral speed, and the specific work output of the stage.

Given Parameters:

• Relative velocity at rotor inlet, \(V_{r1} = 80\) m/s
• Relative velocity at rotor outlet, \(V_{r2} = 150\) m/s
• Mean rotor peripheral speed, \(U = 68.4\) m/s (This parameter is not directly needed for the calculation of the degree of reaction using the enthalpy method with relative velocities, but it is provided in the question context).
• Work output per unit mass in the stage, \(W = 13500\) J/Kg

What is Degree of Reaction?

The degree of reaction (\(R\)) in a turbine stage is a measure of how much of the total enthalpy drop across the stage occurs in the rotor. It is defined as the ratio of the enthalpy drop in the rotor to the total enthalpy drop in the stage.

The total enthalpy drop in the stage is equal to the work output per unit mass, assuming an adiabatic process and negligible kinetic energy change between the stage inlet and outlet (or considering the net change). So, \(\Delta h_{stage} = W\).

For an adiabatic flow through the rotor passages, and assuming constant mean blade speed (which is implied by a single value of \(U\) given), the enthalpy drop in the rotor (\(\Delta h_r\)) is given by the change in the square of the relative velocities:

\[ \Delta h_r = \frac{V_{r2}^2 - V_{r1}^2}{2} \] where \(V_{r1}\) is the relative velocity at the rotor inlet and \(V_{r2}\) is the relative velocity at the rotor outlet.

Calculating the Degree of Reaction

The formula for the degree of reaction using enthalpy drops is:

\[ R = \frac{\Delta h_r}{\Delta h_{stage}} \] Substituting the expressions for \(\Delta h_r\) and \(\Delta h_{stage}\) (which is equal to the work output \(W\)), we get:

\[ R = \frac{\frac{V_{r2}^2 - V_{r1}^2}{2}}{W} \]

Step-by-Step Calculation:

First, calculate the enthalpy drop in the rotor:

\[ \Delta h_r = \frac{(150 \, \text{m/s})^2 - (80 \, \text{m/s})^2}{2} \] \[ \Delta h_r = \frac{22500 \, (\text{m/s})^2 - 6400 \, (\text{m/s})^2}{2} \] \[ \Delta h_r = \frac{16100 \, (\text{m/s})^2}{2} \]

Since \(1 \, (\text{m/s})^2 = 1 \, \text{J/Kg}\), the enthalpy drop is:

\[ \Delta h_r = 8050 \, \text{J/Kg} \] Now, calculate the degree of reaction using the work output \(W = 13500\) J/Kg:

\[ R = \frac{\Delta h_r}{W} \] \[ R = \frac{8050 \, \text{J/Kg}}{13500 \, \text{J/Kg}} \] \[ R \approx 0.596 \]

Comparing with Options:

The calculated value for the degree of reaction is approximately 0.596. Let's compare this with the given options:

• Option 1: 0.6
• Option 2: 0.7
• Option 3: 0.8
• Option 4: 0.9

The value 0.596 is nearest to 0.6.

Thus, the nearest value of the degree of reaction for the given axial turbine stage is 0.6.