Steam enters an adiabatic turbine operating at steady state at 2MPa, 400 C with velocity of 50 m/s and exit at 15 KPasquality of 90% with velocity or 180 m/s) The elevation of inlet is47m higher than at the exit and the mass flow rte is 5 kg/s. Let g = 9.81 m/s. determine the powerdeveloped by the turbine.

To determine the power developed by an adiabatic turbine, we will use the first law of thermodynamics for a steady flow process, which states that the rate of change of energy in the system is equal to the rate of heat transfer minus the rate of work done by the system on the surroundings. For an adiabatic process, heat transfer ( Q̇ ) is zero. The energy change includes enthalpy change, kinetic energy change, and potential energy change.

The basic formula for power developed by the turbine is:

\[ P = \dot{m} \cdot (h_1 - h_2 + \frac{1}{2} \cdot (v_1^2 - v_2^2) + g \cdot (z_1 - z_2) ) \]

Where: - \( P \) is the power developed by the turbine - \( \dot{m} \) is the mass flow rate (5 kg/s) - \( h_1 \) and \( h_2 \) are the specific enthalpies at inlet and exit, respectively - \( v_1 \) and \( v_2 \) are the velocities at inlet and exit, respectively - \( g \) is the acceleration due to gravity (9.81 \( m/s^2 \)) - \( z_1 \) and \( z_2 \) are the elevations at inlet and exit, respectively

Step 1: Determining \( h_1 \) and \( h_2 \)

Using steam tables, we can find the specific enthalpy \( h_1 \) at the inlet condition (2 MPa, 400°C), as the state is superheated steam.

For the exit, since the steam has a quality (x) of 90%, we must find enthalpy \( h_2 \) using the saturation temperature at 15 kPa and the quality. We use the following formula for the specific enthalpy at exit:

\[ h_2 = h_{f, exit} + x \cdot (h_{fg, exit}) \]

Where \( h_{f, exit} \) is the enthalpy of saturated liquid and \( h_{fg, exit} \) is the enthalpy of vaporization at the exit pressure.

Step 2: Applying the energy equation

\[ P = 5 \cdot (h_1 - h_2 + \frac{1}{2} \cdot (50^2 - 180^2) + 9.81 \cdot (47 - 0) ) \]

Notice that you must ensure all units are consistent, specifically that enthalpy is in kJ/kg and velocity in m/s.

Step 3: Calculate the power output

Once you have the values for \( h_1 \) and \( h_2 \), substitute them into the equation along with the other known values:

\[ P = 5 \cdot (h_1 - h_2 + \frac{1}{2} \cdot (2500 - 32400) + 9.81 \cdot 47 ) \]

Finally, calculate the value of \( P \) to determine the power developed by the turbine in kilowatts (since mass flow rate and enthalpy are usually given in kg/s and kJ/kg, respectively).

Extra: An adiabatic turbine is a type of turbine in which there is no heat exchange with the surroundings; this means the system is thermally isolated, and all the work done is due to the change in energy of the steam flowing through it. The concept of specific enthalpy is essential as it represents the total energy of a substance per unit mass, composed of internal energy plus the energy related to pressure and volume.

The term "quality" of steam refers to the proportion of the steam that is in vapor form in a saturated mixture; a quality of 90% means that 90% of the mass is in vapor form, while the remaining 10% is in liquid form. The kinetic energy and potential energy terms in the energy equation are typically small compared to the enthalpy change and often can be neglected unless high velocities or significant height differences are involved, like in this case.