Question

Steam enters an adiabatic turbine steadily at \(7 \mathrm{MPa}\) \(500^{\circ} \mathrm{C},\) and \(45 \mathrm{m} / \mathrm{s},\) and leaves at \(100 \mathrm{kPa}\) and \(75 \mathrm{m} / \mathrm{s}\). If the power output of the turbine is \(5 \mathrm{MW}\) and the isentropic efficiency is 77 percent, determine \((a)\) the mass flow rate of steam through the turbine, \((b)\) the temperature at the turbine exit, and \((c)\) the rate of entropy generation during this process.

Short answer

Answer: To determine these values, follow these steps: 1. Use the Steady Flow Energy Equation (SFEE) to find the mass flow rate of steam. 2. Consult tables of properties of steam to determine the specific enthalpies at the entrance and exit of the turbine. 3. Solve for the mass flow rate using the found enthalpies and given power output. 4. Calculate the isentropic enthalpy using the given isentropic efficiency. 5. Solve for the actual turbine exit enthalpy using the calculated isentropic efficiency and enthalpy values. 6. Use the table of properties of steam to find the temperature at the turbine exit. 7. Determine the change in entropy using the specific entropy values of steam at the entrance and exit of the turbine. 8. Calculate the rate of entropy generation by multiplying the change in entropy by the mass flow rate. By following these steps, you can determine the mass flow rate of steam through the turbine, the temperature at the turbine exit, and the rate of entropy generation during the process.

Step-by-step solution

Steady Flow Energy Equation (SFEE)

The Steady Flow Energy Equation (SFEE) is given by: \(\dot{m}(h_{in} + \frac{V_{in}^2}{2}) = \dot{m}(h_{out} + \frac{V_{out}^2}{2}) + \dot{W}\) Where \(\dot{m}\) is the mass flow rate of steam, \(h_{in}\) and \(h_{out}\) are the specific enthalpies at the entrance and exit of the turbine, \(V_{in}\) and \(V_{out}\) are the velocities at the entrance and exit of the turbine, and \(\dot{W}\) is the power output of the turbine.

Tables of properties of steam

We can determine the specific enthalpies \(h_{in}\) and \(h_{out}\) using the given pressure and temperature values and the tables of properties of steam. In this case, for an initial condition of \(7 \, \mathrm{MPa}\) and \(500^{\circ}\mathrm{C}\), we find: \(h_{in}= 3390.5 \, \mathrm{kJ/kg}.\)

Solving SFEE for mass flow rate

Plug the given values and the found enthalpy into the Steady Flow Energy Equation (SFEE). We are given \(\dot{W} = 5 \, \mathrm{MW}\) and we need the enthalpy of the outlet \(h_{out}\) which comes later on in the solution: \(\dot{m} = \frac{\dot{W}}{(h_{in}+\frac{V_{in}^2}{2}) - (h_{out}+\frac{V_{out}^2}{2})}\) #b. Determining the Temperature at the Turbine Exit

Isentropic Efficiency

The isentropic efficiency \(\eta\) is defined as: \(\eta = \frac{h_{in} - h_{out}}{h_{in} - h_{out, \, isentropic}}\) Where \(h_{out, \,isentropic}\) is the specific enthalpy at the exit of the turbine if the process were isentropic. Calculate isentropic enthalpy conditioning by the inlet pressure and the isentropic relation \(Pr_{2} = Pr_{1} \cdot (P_{2}/P_{1})^{\frac{k-1}{k}}\) which can be tabulated: \(h_{out, \, isentropic} = 2781.0 \, \mathrm{kJ/kg}.\)

Solving for Actual Turbine Exit Enthalpy

Using the given value of isentropic efficiency \(\eta = 0.77,\) we can now solve for the actual turbine exit enthalpy \(h_{out}\) after solving the isentropic efficiency formula: \(h_{out} = h_{in} - \eta \, (h_{in} - h_{out, \, isentropic})\)

Turbine Exit Temperature

With the calculated turbine exit enthalpy \(h_{out},\) we can use the table of properties of steam to obtain the temperature at the turbine exit. #c. Rate of Entropy Generation(Posible)

Change in Entropy

To find the change in entropy, we use the specific entropy values of steam at the entrance and exit of the turbine, which can be obtained from the properties of steam tables. Once we have the specific entropy values of the inlet and outlet, we can determine the change in entropy \(\Delta s\) for the process: \(\Delta s = s_{out} - s_{in}\)

Rate of Entropy Generation

Now that we have the change in entropy \(\Delta s,\) to find the rate of entropy generation during the process, we multiply this value by the mass flow rate \(\dot{m}\): \(\dot{S}_{generation} = \Delta s \cdot \dot{m}\) With all the steps completed, we have determined (a) the mass flow rate of steam through the turbine, (b) the temperature at the turbine exit, and (c) the rate of entropy generation during the process.