Tutorial 2 Rankine Cycle 1) An ideal Rankine cycle which uses water as the working fluid operates its condenser at 40oC and its boiler at 300oC. The mass flow rate of steam entering the turbine is 100 kg/s. Determine the following; a) The power generated in kW b) The rate of heat transfer to the water in the boiler c) The thermal efficiency d) The mass flow rate of the condenser cooling water in kg/s if the cooling water undergoes a temperature increase of 15oC with negligible pressure drop across the boiler 2) An ideal Rankine cycle uses water as the working fluid. The boiler operates at 6000 kPa and the condenser at 50 kPa. At the entrance to the turbine, the temperature is 450oC. The isentropic efficiencies of the turbine and pump are 94 percent and 80 percent respectively. The water leaving the condenser is subcooled by 6.3oC (6.3 oC cooler than at 50 kPa). The boiler is sized for a mass flow rate of 20 kg/s. Determine; a) The rate at which heat is added in the boiler b) The power required to operate the pumps c) The net power produced by the cycle d) The thermal efficiency 3) Consider a steam power plant that operates on the ideal reheat Rankine cycle. The plant maintains the inlet of the high pressure turbine at 4 MPa and 300oC, the inlet of the low pressure turbine at 1.4 MPa and 300oC and the condenser at 75 kPa. The net power produced by this plant is 5000 kW. Determine the a) rate of heat addition b) rate of heat rejection c) thermal efficiency of the cycle
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Tutorial 2 Rankine Cycle 1) An ideal Rankine cycle which uses water as the working fluid operates its o o condenser at 40 C and its boiler at 300 C. The mass flow rate of steam entering the turbine is 100 kg/s. Determine the following; a) The power generated in kW b) The rate of heat transfer to the water in the boiler c) The thermal efficiency d) The mass flow rate of the condenser cooling water in kg/s if the cooling o water undergoes a temperature increase of 15 C with negligible pressure drop across the boiler 2) An ideal Rankine cycle uses water as the working fluid. The boiler operates at 6000 kPa and the condenser at 50 kPa. At the entrance to the turbine, the o temperature is 450 C. The isentropic efficiencies of the turbine and pump are 94 percent and 80 percent respectively. The water leaving the condenser is o o subcooled by 6.3 C (6.3 C cooler than at 50 kPa). The boiler is sized for a mass flow rate of 20 kg/s. Determine; a) The rate at which heat is added in the boiler b) The power required to operate the pumps c) The net power produced by the cycle d) The thermal efficiency 3) Consider a steam power plant that operates on the ideal reheat Rankine cycle. o The plant maintains the inlet of the high pressure turbine at 4 MPa and 300 C, o the inlet of the low pressure turbine at 1.4 MPa and 300 C and the condenser at 75 kPa. The net power produced by this plant is 5000 kW. Determine the a) rate of heat addition b) rate of heat rejection c) thermal efficiency of the
The Correct Answer and Explanation is :
The Rankine cycle is a thermodynamic cycle used to convert heat into work, typically used in steam power plants. Let’s go through each problem step by step.
Problem 1: Ideal Rankine Cycle
Given:
- Condenser temperature, ( T_C = 40^\circ C )
- Boiler temperature, ( T_B = 300^\circ C )
- Mass flow rate, ( \dot{m} = 100 \, \text{kg/s} )
a) The power generated (kW):
To calculate the power, we use the equation:
[
\text{Power} = \dot{m} \times (h_1 – h_4)
]
Where ( h_1 ) is the enthalpy of steam at the boiler exit (at 300°C), and ( h_4 ) is the enthalpy of water at the condenser exit (at 40°C). These values can be obtained from steam tables:
- ( h_1 \approx 3676.3 \, \text{kJ/kg} ) (at 300°C and high pressure)
- ( h_4 \approx 167.6 \, \text{kJ/kg} ) (at 40°C and low pressure)
Thus:
[
\text{Power} = 100 \, \text{kg/s} \times (3676.3 – 167.6) \, \text{kJ/kg}
]
[
\text{Power} = 100 \times 3508.7 = 350,870 \, \text{kW}
]
b) The rate of heat transfer to the water in the boiler:
This is the heat added to the water in the boiler, which is given by:
[
\dot{Q}_\text{in} = \dot{m} \times (h_1 – h_3)
]
Where ( h_3 ) is the enthalpy of water entering the boiler (at 40°C). From steam tables, ( h_3 \approx 167.6 \, \text{kJ/kg} ).
Thus:
[
\dot{Q}_\text{in} = 100 \, \text{kg/s} \times (3676.3 – 167.6) = 350,870 \, \text{kJ/s} = 350,870 \, \text{kW}
]
c) The thermal efficiency:
Thermal efficiency (( \eta )) is the ratio of the work output to the heat input:
[
\eta = \frac{\text{Power}}{\dot{Q}_\text{in}} \times 100 = \frac{350,870}{350,870} \times 100 = 100\%
]
d) The mass flow rate of the condenser cooling water:
To find the mass flow rate of the cooling water, we use the heat balance equation for the condenser:
[
\dot{Q}\text{out} = \dot{m}\text{cooling} \times c_p \times \Delta T
]
Where:
- ( \dot{Q}_\text{out} = \dot{m} \times (h_1 – h_4) = 350,870 \, \text{kW} )
- ( c_p ) (specific heat capacity of water) ( \approx 4.18 \, \text{kJ/kg°C} )
- ( \Delta T = 15^\circ C )
Rearranging to solve for ( \dot{m}\text{cooling} ): [ \dot{m}\text{cooling} = \frac{\dot{Q}_\text{out}}{c_p \times \Delta T} = \frac{350,870}{4.18 \times 15} = 5602.1 \, \text{kg/s}
]
Problem 2: Ideal Rankine Cycle with Isentropic Efficiencies
Given:
- Boiler pressure ( P_B = 6000 \, \text{kPa} )
- Condenser pressure ( P_C = 50 \, \text{kPa} )
- Turbine inlet temperature ( T_1 = 450^\circ C )
- Turbine efficiency ( \eta_\text{turbine} = 94\% )
- Pump efficiency ( \eta_\text{pump} = 80\% )
- Subcooling at condenser: ( 6.3^\circ C )
- Mass flow rate ( \dot{m} = 20 \, \text{kg/s} )
You would use the steam tables and thermodynamic relations for each component to calculate the enthalpies and perform the necessary energy balances.
The steps are as follows:
- Use the given pressures and temperatures to obtain the enthalpies at each state (1, 2, 3, 4).
- Adjust for the isentropic efficiencies of the pump and turbine.
- Calculate the rate of heat input, the work output of the turbine, and the work required by the pump.
Problem 3: Ideal Reheat Rankine Cycle
Given:
- High pressure turbine inlet: ( P_1 = 4 \, \text{MPa}, T_1 = 300^\circ C )
- Low pressure turbine inlet: ( P_2 = 1.4 \, \text{MPa}, T_2 = 300^\circ C )
- Condenser pressure ( P_C = 75 \, \text{kPa} )
- Net power output: ( W_\text{net} = 5000 \, \text{kW} )
For the reheat Rankine cycle:
- First, calculate the enthalpies at the turbine inlets (using steam tables for 4 MPa and 300°C).
- Then, calculate the enthalpies for the condenser and the heat rejection process.
- The rate of heat addition and rejection is obtained from energy balances between each stage.
Conclusion:
The Rankine cycle is a crucial thermodynamic cycle in power generation, and solving these problems involves understanding the thermodynamic states of the fluid at different points in the cycle. You’ll need steam tables for accurate enthalpy values, and careful application of thermodynamic principles to determine the system’s performance.