Formula Sheet: Refrigeration Cycles

Vapor-Compression Refrigeration Cycle

Coefficient of Performance (COP)

COP for Refrigeration: \[ COP_R = \frac{Q_L}{W_{net}} = \frac{\dot{Q}_L}{\dot{W}_{net}} \]

• COP_R = coefficient of performance for refrigeration (dimensionless)
• Q_L or Q̇_L = heat removed from cold space (Btu or Btu/hr, kJ or kW)
• W_net or Ẇ_net = net work input to cycle (Btu or Btu/hr, kJ or kW)

COP for Heat Pump: \[ COP_{HP} = \frac{Q_H}{W_{net}} = \frac{\dot{Q}_H}{\dot{W}_{net}} \]

• COP_HP = coefficient of performance for heat pump (dimensionless)
• Q_H or Q̇_H = heat delivered to warm space (Btu or Btu/hr, kJ or kW)

Relationship between COP_R and COP_HP: \[ COP_{HP} = COP_R + 1 \]

Energy Balance Equations

Compressor Work Input: \[ \dot{W}_{comp} = \dot{m}(h_2 - h_1) \]

• Ẇ_comp = compressor power (kW or Btu/hr)
• ṁ = mass flow rate of refrigerant (kg/s or lbm/hr)
• h₁ = enthalpy at compressor inlet (kJ/kg or Btu/lbm)
• h₂ = enthalpy at compressor outlet (kJ/kg or Btu/lbm)

Condenser Heat Rejection: \[ \dot{Q}_H = \dot{m}(h_2 - h_3) \]

• Q̇_H = heat rejected in condenser (kW or Btu/hr)
• h₃ = enthalpy at condenser outlet (kJ/kg or Btu/lbm)

Expansion Valve (Isenthalpic Process): \[ h_3 = h_4 \]

• h₄ = enthalpy at evaporator inlet (kJ/kg or Btu/lbm)
• Throttling process: no work done, no heat transfer

Evaporator Heat Absorption (Refrigeration Effect): \[ \dot{Q}_L = \dot{m}(h_1 - h_4) \]

• Q̇_L = cooling capacity (kW or Btu/hr)
• Also called refrigeration effect or cooling load

Energy Balance for Complete Cycle: \[ \dot{Q}_H = \dot{Q}_L + \dot{W}_{comp} \]

Mass Flow Rate of Refrigerant

\[ \dot{m} = \frac{\dot{Q}_L}{h_1 - h_4} \]

• Required refrigerant mass flow rate for given cooling capacity
• Units: kg/s or lbm/hr

Refrigeration Capacity (Tons of Refrigeration)

Conversion: \[ 1 \text{ ton} = 12{,}000 \text{ Btu/hr} = 200 \text{ Btu/min} = 3.517 \text{ kW} \] Tons of Refrigeration: \[ \text{Tons} = \frac{\dot{Q}_L \text{ (Btu/hr)}}{12{,}000} \]

Isentropic Compressor Efficiency

\[ \eta_c = \frac{\dot{W}_{s}}{\dot{W}_{actual}} = \frac{h_{2s} - h_1}{h_2 - h_1} \]

• η_c = isentropic compressor efficiency (dimensionless)
• Ẇ_s = isentropic (ideal) compressor work
• Ẇ_actual = actual compressor work
• h_2s = enthalpy at compressor outlet for isentropic compression
• h₂ = actual enthalpy at compressor outlet

Actual Compressor Work with Efficiency: \[ \dot{W}_{actual} = \frac{\dot{m}(h_{2s} - h_1)}{\eta_c} \]

Ideal (Carnot) Refrigeration Cycle

Carnot COP for Refrigeration

\[ COP_{R,Carnot} = \frac{T_L}{T_H - T_L} = \frac{1}{(T_H/T_L) - 1} \]

• T_L = absolute temperature of cold reservoir (K or °R)
• T_H = absolute temperature of hot reservoir (K or °R)
• Maximum theoretical COP for given temperature limits

Carnot COP for Heat Pump

\[ COP_{HP,Carnot} = \frac{T_H}{T_H - T_L} \]

Refrigerant Properties

Enthalpy Calculations

For Saturated Mixture (Two-Phase Region): \[ h = h_f + x \cdot h_{fg} \]

• h = specific enthalpy of mixture (kJ/kg or Btu/lbm)
• h_f = specific enthalpy of saturated liquid (kJ/kg or Btu/lbm)
• x = quality (dryness fraction), dimensionless, 0 ≤ x ≤ 1
• h_fg = enthalpy of vaporization (kJ/kg or Btu/lbm)

Alternative Form: \[ h = (1-x) \cdot h_f + x \cdot h_g \]

• h_g = specific enthalpy of saturated vapor (kJ/kg or Btu/lbm)

Entropy Calculations

For Saturated Mixture: \[ s = s_f + x \cdot s_{fg} \]

• s = specific entropy (kJ/kg·K or Btu/lbm·°R)
• s_f = specific entropy of saturated liquid
• s_fg = entropy of vaporization

Quality (Dryness Fraction)

\[ x = \frac{m_g}{m_f + m_g} = \frac{m_g}{m_{total}} \]

• m_g = mass of vapor
• m_f = mass of liquid
• x = 0 for saturated liquid
• x = 1 for saturated vapor
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Refrigeration Cycle Analysis with P-h Diagram

Typical Process States

State 1 (Compressor Inlet):

• Saturated vapor or slightly superheated vapor
• Low pressure, low temperature

State 2 (Compressor Outlet/Condenser Inlet):

• Superheated vapor
• High pressure, high temperature

State 3 (Condenser Outlet/Expansion Valve Inlet):

• Saturated liquid or subcooled liquid
• High pressure, moderate temperature

State 4 (Expansion Valve Outlet/Evaporator Inlet):

• Two-phase mixture (low quality)
• Low pressure, low temperature
• h₄ = h₃ (isenthalpic expansion)

Multi-Stage Refrigeration Systems

Two-Stage Compression with Flash Intercooler

Optimum Intermediate Pressure: \[ P_{int} = \sqrt{P_L \times P_H} \]

• P_int = intermediate (flash tank) pressure
• P_L = evaporator (low-stage) pressure
• P_H = condenser (high-stage) pressure
• Geometric mean provides minimum total compressor work

Cascade Refrigeration Systems

Heat Transfer in Cascade Heat Exchanger: \[ \dot{m}_A(h_{2A} - h_{3A}) = \dot{m}_B(h_{1B} - h_{4B}) \]

• Subscript A = low-temperature cycle
• Subscript B = high-temperature cycle
• Energy balance across the cascade heat exchanger

Overall COP for Cascade System: \[ COP_{cascade} = \frac{\dot{Q}_L}{\dot{W}_{comp,A} + \dot{W}_{comp,B}} \]

Absorption Refrigeration Systems

COP for Absorption Cycle

\[ COP_{abs} = \frac{\dot{Q}_L}{\dot{Q}_{gen} + \dot{W}_{pump}} \]

• Q̇_gen = heat input to generator (kW or Btu/hr)
• Ẇ_pump = pump work (typically very small compared to Q̇_gen)

Simplified COP (neglecting pump work): \[ COP_{abs} \approx \frac{\dot{Q}_L}{\dot{Q}_{gen}} \]

Additional Performance Parameters

Energy Efficiency Ratio (EER)

\[ EER = \frac{\dot{Q}_L \text{ (Btu/hr)}}{\dot{W}_{input} \text{ (watts)}} \]

• Common in HVAC applications
• Higher EER indicates better efficiency
• Units: Btu/hr per watt

Seasonal Energy Efficiency Ratio (SEER)

\[ SEER = \frac{\text{Total cooling output (Btu) over season}}{\text{Total electrical energy input (W·hr) over season}} \]

• Accounts for varying outdoor temperatures
• Used for seasonal performance rating

Power per Ton of Refrigeration

\[ \frac{\text{kW}}{\text{ton}} = \frac{3.517}{COP_R} \]

• Standard metric for comparing refrigeration system efficiency
• Lower kW/ton indicates higher efficiency

Volumetric Efficiency and Compressor Performance

Volumetric Efficiency

\[ \eta_v = \frac{\dot{V}_{actual}}{\dot{V}_{displacement}} \]

• η_v = volumetric efficiency (dimensionless)
• V̇_actual = actual volume flow rate at suction conditions
• V̇_displacement = piston displacement rate

Mass Flow Rate from Compressor

\[ \dot{m} = \frac{\eta_v \cdot \dot{V}_{displacement}}{v_1} \]

• v₁ = specific volume at compressor inlet (m³/kg or ft³/lbm)

Compressor Displacement

\[ \dot{V}_{displacement} = \frac{\dot{m} \cdot v_1}{\eta_v} \]

Heat Exchanger Effectiveness

Subcooling in Condenser

\[ \Delta T_{subcool} = T_{sat,cond} - T_3 \]

• ΔT_subcool = degree of subcooling
• T_sat,cond = saturation temperature at condenser pressure
• T₃ = actual temperature at condenser outlet

Superheating in Evaporator

\[ \Delta T_{superheat} = T_1 - T_{sat,evap} \]

• ΔT_superheat = degree of superheat
• T₁ = actual temperature at evaporator outlet
• T_sat,evap = saturation temperature at evaporator pressure

Refrigeration Load Calculations

Total Refrigeration Load

\[ \dot{Q}_L = \dot{Q}_{product} + \dot{Q}_{transmission} + \dot{Q}_{infiltration} + \dot{Q}_{equipment} + \dot{Q}_{people} \]

• Q̇_product = heat removed from product being cooled
• Q̇_transmission = heat gain through walls, floor, ceiling
• Q̇_infiltration = heat gain from air infiltration
• Q̇_equipment = heat from motors, lights, other equipment
• Q̇_people = heat from occupants

Product Cooling Load

Sensible Heat Removal: \[ \dot{Q}_{sensible} = \dot{m}_{product} \cdot c_p \cdot \Delta T \]

• ṁ_product = mass flow rate of product (kg/s or lbm/hr)
• c_p = specific heat of product (kJ/kg·K or Btu/lbm·°F)
• ΔT = temperature change of product

Latent Heat Removal (for freezing): \[ \dot{Q}_{latent} = \dot{m}_{product} \cdot h_{fusion} \]

• h_fusion = latent heat of fusion (kJ/kg or Btu/lbm)

Transmission Load

\[ \dot{Q}_{transmission} = U \cdot A \cdot \Delta T \]

• U = overall heat transfer coefficient (W/m²·K or Btu/hr·ft²·°F)
• A = surface area (m² or ft²)
• ΔT = temperature difference between inside and outside

Refrigerant-Specific Formulas

Ideal Gas Approximation (for Superheated Region)

\[ P \cdot v = R \cdot T \]

• P = pressure (kPa or psia)
• v = specific volume (m³/kg or ft³/lbm)
• R = specific gas constant for refrigerant
• T = absolute temperature (K or °R)
• Valid only when refrigerant behaves as ideal gas (highly superheated)

Clausius-Clapeyron Equation

\[ \frac{dP}{dT} = \frac{h_{fg}}{T \cdot v_{fg}} \]

• Relates saturation pressure and temperature
• v_fg = v_g - v_f = change in specific volume during vaporization

System Performance Degradation Factors

Effect of Non-Ideal Conditions

Pressure Drop in Evaporator:

• Reduces effective evaporator temperature
• Decreases COP
• Use average pressure for property evaluation

Pressure Drop in Condenser:

• Increases required compressor discharge pressure
• Increases compressor work

Heat Gain in Suction Line: \[ \dot{Q}_{suction} = \dot{m}(h_{1,actual} - h_{1,ideal}) \]

• Superheats vapor entering compressor
• Reduces refrigeration effect
• Increases compressor discharge temperature

Exergy Analysis

Exergy Destruction

\[ \dot{E}_{destroyed} = T_0 \cdot \dot{S}_{gen} \]

• Ė_destroyed = rate of exergy destruction (kW or Btu/hr)
• T₀ = dead state (ambient) temperature (K or °R)
• Ṡ_gen = entropy generation rate (kW/K or Btu/hr·°R)

Second Law Efficiency

\[ \eta_{II} = \frac{COP_{actual}}{COP_{Carnot}} \]

• η_II = second law (exergetic) efficiency
• Measure of how closely actual cycle approaches reversible Carnot cycle