2.4 Thermodynamic Cycles

2.4 Thermodynamic Cycles

1. Power and Refrigeration Cycles

Power Cycles

• Cycles that convert heat into work (net work output).

• Objective: Maximize thermal efficiency ($\eta_{th}$).

• $\eta_{th} = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$

Refrigeration Cycles

• Cycles that transfer heat from a low-temperature space to a high-temperature space by consuming work.

• Objective: Maximize Coefficient of Performance ($COP$).

• $COP_R = \frac{Q_L}{W_{net}}$

Heat Pump Cycles

• Similar to refrigeration cycles but aim to supply heat to a high-temperature space.

• $COP_{HP} = \frac{Q_H}{W_{net}} = COP_R + 1$

2. Vapor Compression Refrigeration Cycle

Components

1. Compressor: Compresses refrigerant vapor (1→2).

2. Condenser: Condenses refrigerant vapor to liquid (2→3).

3. Expansion Valve: Throttles liquid refrigerant (3→4).

4. Evaporator: Evaporates refrigerant liquid to vapor (4→1).

Processes

1. 1→2: Isentropic compression (ideal) or actual compression.

2. 2→3: Constant-pressure heat rejection (condensation).

3. 3→4: Isenthalpic (constant enthalpy) throttling.

4. 4→1: Constant-pressure heat absorption (evaporation).

Performance Parameters

• Refrigerating Effect: $q_L = h_1 - h_4$ (kJ/kg)

• Work Input: $w_{in} = h_2 - h_1$ (kJ/kg)

• Heat Rejected: $q_H = h_2 - h_3$ (kJ/kg)

• Coefficient of Performance: $COP_R = \frac{q_L}{w_{in}} = \frac{h_1 - h_4}{h_2 - h_1}$

3. Vapor Absorption Refrigeration Cycle

Basic Principle

• Uses a heat source (e.g., waste heat, solar energy) instead of mechanical work to drive the refrigeration cycle.

• Uses a refrigerant-absorbent pair (e.g., Ammonia-Water, Water-LiBr).

Main Components

1. Generator: Heat input separates refrigerant from absorbent.

2. Condenser, Expansion Valve, Evaporator: Same as vapor compression cycle.

3. Absorber: Absorbs refrigerant vapor into weak solution.

4. Pump: Pumps strong solution from absorber to generator.

5. Heat Exchanger: Improves efficiency.

Advantages

• Can use low-grade heat (solar, waste heat).

• Quiet operation (few moving parts).

• Lower maintenance.

Disadvantages

• Lower COP compared to vapor compression.

• Larger size and higher initial cost.

• Potentially toxic working fluids.

4. Rankine Cycle (Vapor Power Cycle)

Ideal Rankine Cycle Processes

1. 1→2: Isentropic compression in pump (water: compressed liquid).

2. 2→3: Constant-pressure heat addition in boiler (water heated to steam).

3. 3→4: Isentropic expansion in turbine (steam produces work).

4. 4→1: Constant-pressure heat rejection in condenser (steam condensed to water).

Performance Parameters

• Pump Work: $w_p = h_2 - h_1 \approx v_1 (P_2 - P_1)$

• Turbine Work: $w_t = h_3 - h_4$

• Heat Added: $q_{in} = h_3 - h_2$

• Heat Rejected: $q_{out} = h_4 - h_1$

• Net Work: $w_{net} = w_t - w_p$

• Thermal Efficiency: $\eta_{th} = \frac{w_{net}}{q_{in}} = \frac{(h_3 - h_4) - (h_2 - h_1)}{h_3 - h_2}$

Improvements to Rankine Cycle

• Superheating: Increases $h_3$, increases efficiency, reduces moisture at turbine exit.

• Reheating: Steam is expanded in high-pressure turbine, reheated, then expanded in low-pressure turbine.

• Regeneration: Using extracted steam from turbine to preheat feedwater.

5. Brayton Cycle (Gas Turbine Cycle)

Ideal Brayton Cycle Processes

1. 1→2: Isentropic compression (in compressor).

2. 2→3: Constant-pressure heat addition (in combustion chamber).

3. 3→4: Isentropic expansion (in turbine).

4. 4→1: Constant-pressure heat rejection.

Performance Parameters

• Pressure Ratio: $r_p = \frac{P_2}{P_1}$

• Compressor Work: $w_c = h_2 - h_1 = c_p (T_2 - T_1)$

• Turbine Work: $w_t = h_3 - h_4 = c_p (T_3 - T_4)$

• Net Work: $w_{net} = w_t - w_c$

• Heat Added: $q_{in} = h_3 - h_2 = c_p (T_3 - T_2)$

• Thermal Efficiency (for ideal gas with constant $c_p$): $\eta_{th} = 1 - \frac{1}{r_p^{(k-1)/k}}$

6. Otto Cycle (Spark-Ignition Engine Cycle)

Ideal Otto Cycle Processes

1. 1→2: Isentropic compression.

2. 2→3: Constant-volume heat addition (combustion).

3. 3→4: Isentropic expansion (power stroke).

4. 4→1: Constant-volume heat rejection (exhaust).

Performance Parameters

• Compression Ratio: $r = \frac{V_1}{V_2} = \frac{v_1}{v_2}$

• Thermal Efficiency (for ideal gas with constant $c_v$): $\eta_{th} = 1 - \frac{1}{r^{k-1}}$

7. Diesel Cycle (Compression-Ignition Engine Cycle)

Ideal Diesel Cycle Processes

1. 1→2: Isentropic compression.

2. 2→3: Constant-pressure heat addition (combustion).

3. 3→4: Isentropic expansion.

4. 4→1: Constant-volume heat rejection.

Performance Parameters

• Compression Ratio: $r = \frac{V_1}{V_2}$

• Cut-off Ratio: $r_c = \frac{V_3}{V_2}$

• Thermal Efficiency (for ideal gas with constant $c_p, c_v$): $\eta_{th} = 1 - \frac{1}{r^{k-1}} \left[ \frac{r_c^k - 1}{k(r_c - 1)} \right]$

8. Efficiency and COP Comparison

Maximum Theoretical Efficiencies

• Carnot Efficiency (all cycles): $\eta_{Carnot} = 1 - \frac{T_L}{T_H}$

• Otto Cycle Efficiency: Depends only on compression ratio $r$ and $k$.

• Diesel Cycle Efficiency: Depends on compression ratio $r$, cut-off ratio $r_c$, and $k$.

• Brayton Cycle Efficiency: Depends only on pressure ratio $r_p$ and $k$.

Typical Efficiency Ranges

• Rankine Cycle (power plants): 30-45%

• Brayton Cycle (gas turbines): 25-40%

• Otto Cycle (gasoline engines): 25-30%

• Diesel Cycle (diesel engines): 30-40%

COP Values

• Vapor Compression Refrigerators: $COP_R$ typically 2-4

• Carnot Refrigerator: $COP_{R,Carnot} = \frac{T_L}{T_H - T_L}$ (theoretical maximum)

• Heat Pumps: $COP_{HP}$ typically 3-5

Important Relationships

• For any reversible heat engine operating between $T_H$ and $T_L$: $\frac{Q_H}{T_H} = \frac{Q_L}{T_L}$ and $\eta_{max} = 1 - \frac{T_L}{T_H}$

• For any reversible refrigerator or heat pump: $COP_{R,max} = \frac{T_L}{T_H - T_L}$ $COP_{HP,max} = \frac{T_H}{T_H - T_L}$