Question

Steam enters a steady-flow turbine with a mass flow rate of \(13 \mathrm{kg} / \mathrm{s}\) at \(600^{\circ} \mathrm{C}, 8 \mathrm{MPa}\), and a negligible velocity. The steam expands in the turbine to a saturated vapor at \(300 \mathrm{kPa}\) where 10 percent of the steam is removed for some other use. The remainder of the steam continues to expand to the turbine exit where the pressure is \(10 \mathrm{kPa}\) and quality is 85 percent. If the turbine is adiabatic, determine the rate of work done by the steam during this process.

Short answer

Based on the steady-flow turbine process described above, the steam does work on its surroundings at a rate of approximately 17,096 kW.

Step-by-step solution

Identify the known and unknown parameters

We are given the mass flow rate (13 kg/s), temperature and pressure of the steam at the inlet (600°C, 8 MPa), the pressure at the point where 10% of the steam is removed (300 kPa), and the pressure and quality of the steam at the turbine exit (10 kPa, with 85% quality). The steam is assumed to be adiabatic, meaning no heat transfer occurs between the steam and its surroundings. We need to determine the rate of work done by the steam during this process.

Find the specific enthalpy at each stage of the process

To perform the energy balance, we need to find the specific enthalpy at the turbine inlet, at the pressure where 10% of the steam is removed, and at the turbine exit. 1. Inlet: Using the steam tables, look up the specific enthalpy at the given temperature and pressure (600°C, 8 MPa). Note that the values are in absolute temperature and pressure units. For the inlet, we find \(h_1 = 3590\,\mathrm{kJ/kg}\). 2. 10% removal: The pressure is given (300 kPa), and because it's a saturated vapor, we find its specific enthalpy using the steam tables as \(h_2 = 2750\,\mathrm{kJ/kg}\). 3. Exit: We are given the pressure (10 kPa) and the quality (85%). We can find the specific enthalpy of the saturated liquid (\(h_f\)) and the specific enthalpy of the saturated vapor (\(h_g\)) at this pressure, and then calculate the specific enthalpy of the mixture using the quality: \(h_3 = h_f + x(h_g - h_f)\). From the steam tables, \(h_f = 191.8\,\mathrm{kJ/kg}\) and \(h_g = 2584.6\,\mathrm{kJ/kg}\). Thus, \(h_3 = 191.8 + 0.85(2584.6 - 191.8) = 2220.1\,\mathrm{kJ/kg}\).

Apply the first law of thermodynamics and conservation of mass for the process

The mass flow rate at the inlet is \(m_1 = 13\,\mathrm{kg/s}\). After 10% removal of the steam, the mass flow rate of the remaining steam is \(m_2 = 0.9\cdot m_1 = 11.7\,\mathrm{kg/s}\). Then, apply the first law of thermodynamics for a steady-flow, adiabatic process: \(\dot{W} = \dot{m}_1 (h_1 - h_2) + \dot{m}_2 (h_2 - h_3)\). Plug in the known specific enthalpy values and mass flow rates: \(\dot{W} = 13(3590 - 2750) + 11.7(2750 - 2220.1)\).

Calculate the work rate

Evaluating the expression above, we find the rate of work done by the steam during the process: \(\dot{W} = 13(840) + 11.7(529.9) \approx 10902 + 6194 = 17,096\,\mathrm{kJ/s}\). This is the rate at which work is performed during the turbine operation. The positive value of the work rate indicates that the steam is doing work on its surroundings (as expected in a turbine). Note that the units are in kilojoules per second, which is equivalent to kilowatts. Therefore, the rate of work done by the steam during this process is approximately equal to \(17,096\,\mathrm{kW}\).