Question

Atmospheric air enters the air compressor of a simple combined gas-steam power system at 14.7 psia and \(80^{\circ} \mathrm{F}\). The air compressor's compression ratio is \(10 ;\) the gas cycle's maximum temperature is \(2100^{\circ} \mathrm{F} ;\) and the air compressor and turbine have an isentropic efficiency of 90 percent. The gas leaves the heat exchanger \(50^{\circ} \mathrm{F}\) hotter than the saturation temperature of the steam in the heat exchanger. The steam pressure in the heat exchanger is 800 psia, and the steam leaves the heat exchanger at \(600^{\circ} \mathrm{F}\). The steam- condenser pressure is 5 psia and the isentropic efficiency of the steam turbine is 95 percent. Determine the overall thermal efficiency of this combined cycle. For air, use constant specific heats at room temperature

Short answer

Answer: The overall thermal efficiency of a combined gas-steam power system is the ratio of the net work output to the total heat input in the system. It can be determined by analyzing the gas cycle (air compressor and turbine) and the steam cycle (steam turbine and heat exchanger), calculating the net work and heat input for both cycles, and then combining the results to find the overall thermal efficiency as follows: Overall Thermal Efficiency = Net Work / Heat Input.

Step-by-step solution

1. Analyze the Gas Cycle (Air Compressor and Turbine)

To analyze the gas cycle, we need to find the work and heat input/output of the air compressor and turbine. In both cases, we will use the isentropic efficiency formulas, which relate the actual and ideal work using isentropic efficiency. For the air compressor, the actual and ideal work can be found using the following formulas: Actual Work: \(W_{comp,actual} = \frac{c_p(T_2 - T_1)}{\eta_{comp}}\) Ideal Work: \(W_{comp,ideal} = c_p(T_{2s} - T_1)\) Using given constant specific heat capacity at room temperature, compression ratio, and isentropic efficiency, the actual work of the air compressor can be calculated. For the turbine, the actual and ideal work can be found using the following formulas: Actual Work: \(W_{turb,actual} = \eta_{turb} \cdot c_p(T_3 - T_{4s})\) Ideal Work: \(W_{turb,ideal} = c_p(T_3 - T_4)\) Using given gas cycles maximum temperature, isentropic efficiency, and constant specific heat capacity at room temperature, the actual work of the turbine can be calculated.

2. Analyze the Steam Cycle (Steam Turbine and Heat Exchanger)

To analyze the steam cycle, we need to find the work and heat input/output of the steam turbine and heat exchanger. For the steam turbine, we can calculate the actual and ideal work using the isentropic efficiency formulas: Actual Work: \(W_{st,actual} = \eta_{st} \cdot (h_5 - h_{6s})\) Ideal Work: \(W_{st,ideal} = h_5 - h_6\) By using the steam table with given information, we can find all the enthalpies (\(h_5\), \(h_{6s}\), and \(h_6\)) and then calculate the actual work of the steam turbine. In the heat exchanger, we can calculate the heat input as follows: Heat Input: \(Q_{H,steam} = \sum{m_i \cdot (h_i - h_{i+1})}\), where \(m_i\) is the mass flow rate of the steam and \((h_i - h_{i+1})\) is the enthalpy change in the heat exchanger. Using the steam table and given steam parameters, we can calculate the heat input in the heat exchanger.

3. Calculate the Net Work and Heat Input for the Combined Cycle

Now that we have the work and heat input for the gas and steam cycles, we can calculate the net work and heat input of the combined cycle. Net Work: \(W_{net} = W_{turb,actual} + W_{st,actual} - W_{comp,actual}\) Heat Input: \(Q_{in} = Q_{H,steam} + c_p(T_3 - T_2)\)

4. Determine the Overall Thermal Efficiency of the Combined Cycle

Finally, using the net work and heat input, the overall thermal efficiency of the combined cycle can be calculated as follows: Overall Thermal Efficiency: \(\eta = \frac{W_{net}}{Q_{in}}\) Following these steps with the given parameters, one can determine the overall thermal efficiency of the combined gas-steam power system.