Let's break down the problem step-by-step: --- ## **Problem Recap** A combined cycle power plant consists of: - **Gas (combustion) turbine efficiency**: 40% - **HRSG (Heat Recovery Steam Generator)**: Loses 20% of the incoming thermal energy at the stack - **Steam turbine efficiency**: 40% - **Generator efficiency**: 98% - **Neglect auxiliaries** **Questions:** 1. **Overall plant efficiency?** 2. **Heat rate?** --- ## **Step 1: Define the Energy Flow** Let’s use a basis of **100 units of fuel energy input**. ### **Gas Turbine:** - **Output (to shaft):** \( 40\% \) of input \[ = 0.40 \times 100 = 40 \text{ units (gas turbine shaft output)} \] - **Exhaust to HRSG:** \[ = 100 - 40 = 60 \text{ units (exhaust energy to HRSG)} \] --- ### **HRSG (Heat Recovery Steam Generator):** - **Stack Losses:** \( 20\% \) of incoming (to HRSG) energy is lost \[ = 0.20 \times 60 = 12 \text{ units (lost to stack)} \] - **Energy to Steam Turbine:** \[ = 60 - 12 = 48 \text{ units (to steam turbine)} \] --- ### **Steam Turbine:** - **Cycle Efficiency:** \( 40\% \) \[ = 0.40 \times 48 = 19.2 \text{ units (steam turbine shaft output)} \] --- ### **Total Shaft Output (before Generator):** \[ \text{Total Shaft Output} = 40 \text{ (GT)} + 19.2 \text{ (ST)} = 59.2 \text{ units} \] --- ### **Generator Efficiency:** \( 98\% \) \[ \text{Electrical Output} = 0.98 \times 59.2 = 58.016 \text{ units} \] --- ## **Step 2: Calculate Overall Plant Efficiency** \[ \text{Overall Efficiency} = \frac{\text{Electrical Output}}{\text{Fuel Input}} \times 100\% \] \[ = \frac{58.016}{100} \times 100\% = 58.016\% \] --- ## **Step 3: Calculate the Heat Rate** **Heat Rate** is defined as the amount of fuel energy required to generate **1 kWh** of electricity (commonly in kJ/kWh or BTU/kWh). \[ \text{Heat Rate} = \frac{3600 \text{ kJ}}{\text{Efficiency (as decimal)}} \] \[ = \frac{3600}{0.58016} \] \[ \approx 6203.8 \text{ kJ/kWh} \] --- ## **Summary Table** | Step | Value | |---------------------------|-------------| | Overall Plant Efficiency | 58.0% | | Heat Rate | 6204 kJ/kWh | --- ## **Final Answers** ### **(a) What is the overall plant efficiency?** \[ \boxed{58.0\%} \] ### **(b) What is the heat rate?** \[ \boxed{6204 \text{ kJ/kWh}} \] If you need any clarifications or help understanding any step, let me know!