Analysis of a Geothermal Rankine Cycle System

Question 1. A simple layout is

Question 2

A. Working pressure of the boiler (P_boiler) = 1200 kPa (absolute)

Initial temperature of the refrigerant (T_initial) = Temperature of hot geothermal water - 5°C

Specific heat capacity of geothermal water (c_water) = 4.2 kJ/kg·K

Isentropic efficiency of the screw expander (?_expander) = 80%

Isentropic efficiency of the boiler feed pump (?_pump) = 89%

Saturation pressure at the condenser (P_condenser) = 720 + Z kPa

1. Calculate specific enthalpy and specific entropy at different points of the Rankine cycle.
2. Inlet to the Generator (Point 1)

Using R134a tables at P_boiler and T_initial: h_1 = 309.7 kJ/kg s_1 = 1.275 kJ/kg·K

3. Outlet of the Boiler Feed Pump (Point 2)

Isentropic pump: h_2 = h_1 s_2 = s_1

4. Outlet of the Generator (Point 3

To calculate q_gen, you need to know the temperature at Point 3.

Let's assume T_3 = 100°C. h_3 = h_2 + q_gen

Using R134a tables at P_boiler and T_3: h_3 = 437.5 kJ/kg

5. Outlet of the Turbine/Expander (Point 4)

Isentropic expander: h_4 = h_3 * (1 - ?_expander) h_4 = 372.0 kJ/kg

6. Calculate the exit temperature of the turbine (T4) using R134a tables

At P_boiler and h_4: T4 = 94.9°C

B. Sketch the Rankine cycle on T-S, P-V, and h-s diagrams. Label the points and show the saturated liquid and saturated vapor lines.

C. Calculate for 1 kg/s of refrigerant circulation:
i. The necessary electric power input for the boiler feed pump motor (P_pump
P_pump = (h_2 - h_1) / ?_pump P_pump = (0 kJ/kg) / 0.89
P_pump = 0 kW (assuming the pump has minimal work input)
ii. The electric power output of the generator (P_generator)
P_generator = (h_3 - h_4) × ?_expander P_generator = (437.5 kJ/kg - 372.0 kJ/kg) × 0.80= 52.6 kW
iii. The net electrical output power (P_net): P_net = P_generator - P_pump P_net = 52.6 kW
iv. The rate of heat delivery to the fluid in the boiler (Q_boiler): Q_boiler = m_dot × (h_1 - h_2) Q_boiler = (1 kg/s) × (309.7 kJ/kg - 0 kJ/kg)=309.7 kW
v. The rate of heat rejection at the condenser (Q_condenser)
Q_condenser = m_dot × (h_3 - h_4) Q_condenser = (1 kg/s) × (437.5 kJ/kg - 372.0 kJ/kg)
Q_condenser =65.5 kW
vi. The efficiency of conversion of heat energy to net electrical energy (?_conversion): ?_conversion = (P_net / Q_boiler) ×100% = (52.6 kW / 309.7 kW) × 100% =17%

Question 3

• Efficiency of the pump (?_pump) = 0.89 (89?ficient)
• Efficiency of the expander (?_expander) = 0.80 (80?ficient)

Isentropic enthalpy values

• h_1 = 309.7 kJ/kg
• h_2 = h_1
• h_3 = 437.5 kJ/kg
• h_4 = 372.0 kJ/kg

1. Calculate Actual State Parameters

1. Exit of the Feed Pump (Point 2_actual): Use the efficiency of the pump (?_pump) to find h_2_actual
   • h_2_actual = h_1 + (h_2 - h_1) / ?_pump
   • h_2_actual = 309.7 kJ/kg + (309.7 kJ/kg - 309.7 kJ/kg) / 0.89
   • h_2_actual = 309.7 kJ/kg
2. Exit of the Turbine/Expander (Point 4_actual): Use the efficiency of the expander (?_expander) to find h_4_actual
   • h_4_actual = h_3 - ?_expander × (h_3 - h_4)
   • h_4_actual = 437.5 kJ/kg - 0.80 ×(437.5 kJ/kg - 372.0 kJ/kg)
   • h_4_actual = 376.6 kJ/kg

• T_2_actual = 100°C for the exit of the feed pump.
• T_4_actual = 90°C for the exit of the turbine/expander.
• Q_boiler = 10 kW
• Efficiency of the pump (?_pump) = 0.89 (89?ficient)
• Efficiency of the expander (?_expander) = 0.80 (80?ficient)

We have the actual state parameters:

• T_2_actual = 373.15 K
• T_4_actual = 363.15 K

B

C
i. Mass Flow Rate of Refrigerant (m_dot)
Using the new Q_boiler value
m_dot = Q_boiler / (h_1 - h_2_actual) m_dot = 100 kW / 0.89
m_dot = 100 kW / (0.89 )
m_dot = 112.36 kg/s
ii. Calculate the necessary electric power input for the boiler feed pump motor (P_pump_actual)
P_pump_actual = m_dot × (h_2_actual - h_1) / ?_pump P_pump_actual = 112.36 kg/s × (309.7 kJ/kg - 309.7 kJ/kg) / 0.89
P_pump_actual = 0 kW
iii. Calculate the electric power output of the generator (P_generator_actual):
Using the actual enthalpy values and ?_expander
P_generator_actual = m_dot× (h_3 - h_4_actual) × ?_expander
= 112.36 kg/s × (437.5 kJ/kg - 376.6 kJ/kg) × 0.80= 2251.75 kW
iv. Calculate the net electrical output power (P_net_actual)
P_net_actual = P_generator_actual - P_pump_actual
P_net_actual = 2251.75 kW
v. Calculate the rate of heat delivery to the fluid in the boiler (Q_boiler)
Q_boiler = m_dot × (h_1 - h_2_actual) Q_boiler = 112.36 kg/s × (309.7 kJ/kg - 309.7 kJ/kg)
Q_boiler = 0 kW
vi. Calculate the rate of heat rejection at the condenser (Q_condenser)
Q_condenser = m_dot × (h_3 - h_4_actual)= 112.36 kg/s × (437.5 kJ/kg - 376.6 kJ/kg = 6837.43 kW

Question 4

• Q_geothermal (Average Monthly Geothermal Heat Capacity)= 2000 kW
• P_demand (Monthly Average Energy Demand)=1500 kW
• h_1 (Specific Enthalpy at Inlet, Point 1)= 309.7 kJ/kg
• h_2_actual (Specific Enthalpy at Outlet of the Feed Pump, Point 2_actual)=309.7 kJ/kg

A. Monthly Average Mass Flow Rate of Refrigerant and Feed Pump Capacity:
1.Calculate the monthly required heat (Q_required): Q_required = Q_geothermal ×30 days × 24 hours × 3600 seconds
Q_required = 2000 kW × 30 days × 24 hours × 3600 seconds = 5,184,000,000 kJ
2.m_dot = Q_required / (h_1 - h_2_actual) m_dot = 5,184,000,000 kJ / (309.7 kJ/kg - 309.7 kJ/kg) = 0 kg/s
3.The flow capacity of the feed pump is not required since the calculated mass flow rate is zero.
B. Monthly Average Net Power Generation Capacity:
1.P_net_actual = m_dot × (h_3 - h_4_actual) × ?_expander
Since m_dot is zero, P_net_actual is also zero.
2.The yearly average net power generation capacity is also zero since P_net_actual is zero for all months.
C. Monthly Average Capacity in Percentage:
1.Monthly Average Capacity Percentage = (P_net_actual / P_demand) × 100%
2.Monthly Average Capacity Percentage = (0 kW / 1500 kW) * 100% = 0%

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