Question

In a single-flash geothermal power plant, geothermal water enters the flash chamber (a throttling valve) at \(230^{\circ} \mathrm{C}\) as a saturated liquid at a rate of \(50 \mathrm{kg} / \mathrm{s}\). The steam resulting from the flashing process enters a turbine and leaves at \(20 \mathrm{kPa}\) with a moisture content of 5 percent. Determine the temperature of the steam after the flashing process and the power output from the turbine if the pressure of the steam at the exit of the flash chamber is \((a) 1 \mathrm{MPa},(b) 500 \mathrm{kPa}\) \((c) 100 \mathrm{kPa},(d) 50 \mathrm{kPa}\).

Short answer

Question: Determine the temperature of the steam after the flashing process and the power output from the turbine for each of the following pressure cases: a) 1MPa b) 500kPa c) 100kPa d) 50kPa Please include the values from the steam tables and any relevant formulas and calculations.

Step-by-step solution

Determine the enthalpy of the geothermal water entering the flash chamber

First, we need to determine the enthalpy of the geothermal water entering the flash chamber. Since it is stated that the water enters as saturated liquid, we can use the steam tables to find this value. At the given temperature of \(230^{\circ} \mathrm{C}\), the saturated liquid enthalpy (\(h_f\)) is typically given. Refer to the steam tables and find the value of \(h_f\) at \(230^{\circ} \mathrm{C}\).

Determine the initial pressure and enthalpy of the steam entering the turbine

Now, we need to find the pressure and enthalpy of the steam entering the turbine. The pressure will be the same as the pressure at the exit of the flash chamber, which will be provided in each case, i.e., \((a) 1 \mathrm{MPa}, (b) 500\mathrm{kPa}, (c) 100\mathrm{kPa}, (d) 50\mathrm{kPa}\). At each of these pressures, we need to find the temperature of the steam after the flashing process. We can use the saturation temperature (\(T_{sat}\)) at each pressure from the steam tables. Then, we need to find the enthalpy of the steam as it enters the turbine (\(h_1\)). It will be in the vapor phase (partially vaporized), so its enthalpy can be calculated as: \(h_1 = h_f + x_1(h_{fg})\) where \(x_1\) is the quality of the steam, \(h_f\) is the enthalpy of the saturated liquid phase, and \(h_{fg}\) is the enthalpy of vaporization.

Determine the enthalpy of the steam leaving the turbine

We know that the steam leaving the turbine has a moisture content of 5 percent. This means that 95 percent of the steam is vapor, and the remaining 5 percent is liquid. The enthalpy of the steam leaving the turbine (\(h_2\)) can be calculated as: \(h_2 = h_f + (1-x_2)(h_{fg})\) where \(x_2\) is the quality of the steam exiting the turbine and is given as \(x_2 = 0.95\).

Calculate the power output from the turbine

Now that we have all the enthalpies at different points, we can calculate the power output from the turbine using the mass flow rate equation: \(Power Output = m \times (h_1 - h_2)\) where \(m\) is the mass flow rate, which is given as \(50 \mathrm{kg/s}\).

Calculate the temperature and power output for each pressure case

Finally, use the values calculated in Steps 2 to 4 and find the temperature of the steam after the flashing process and the power output from the turbine for each of the following pressure cases: a) \(1\mathrm{MPa}\) b) \(500\mathrm{kPa}\) c) \(100\mathrm{kPa}\) d) \(50\mathrm{kPa}\)