Second Law of Thermodynamics - It is impossible to construct a device that operates in a cycle and produces no effect other than the removal of heat from a body at one temperature and the absorption of an equal quantity of heat by a body at a higher temperature. Clausius, 1850

Notes on the examples used in the VibroNurse turbine efficiency calculation software:

Example 1 Simple Turbine – Based on a 1MW Turbine with no heat exchanger or waste heat recovery unit.

Example 2 Heat Exchanged – Based on a 25MW Turbine with a heat exchanger.

Example 3 Combined Heat Recovery – Based on a 50MW Turbine with combined heat recovery. The stored model includes 10% compressor air-cooling bypass.

Example 4 Heat Exchanged + Combined Heat Recovery – Based on a 50MW Turbine with combined heat recovery and heat exchanger. The stored model includes 10% compressor air-cooling bypass.

All the preset examples are based on turbines burning natural gas.

Guideline for Turbine Performance

Turbine Model	Manufacturer	Efficiency (%)
Saturn 20	Solar	25
Centaur 40	Solar	28
Centaur 50	Solar	29
Taurus 60	Solar	30
Taurus 70	Solar	34
Mars 90	Solar	32
Mars 100	Solar	32
Titan 120	Solar	35
Avon	Rolls Royce	28
RB211 24G	Rolls Royce	36
RB 211-T 24GT	Rolls Royce	39
Trent 50	Rolls Royce	42
Trent 60	Rolls Royce	40
LM2500+	GE	40
LM6000	GE	43
TB5000	Ruston / European	25

All the above models in the table assume a simple turbine set-up i.e. no heat exchanger or combined heat recovery

 
Turbine Efficiency Calculations

Note on Calculation Method Several calculations have been split over multiple lines, as this is how they are entered in the computer code. Where this has been done the following convention / style applies. X = 1 X = X + 2 Results in X = 3

Calculate the Compressor Ratio

Compressor_Pressure_Ratio = Compressor_Discharge_Pressure / Compressor_Inlet_Pressure

Calculate Exit Pressures

Compressor inlet pressure is used for the air pressure outside the system

Heat_Exchanger_Exit_High_Pressure = Compressor_Pressure_Ratio - Exchanger_High_Pressure_Loss

Combustion_Chamber_Exit_Pressure = Heat_Exchanger_Exit_High_Pressure - Combustion_Pressure_Loss

Heat_Exchanger_Exit_Low_Pressure = Compressor_Inlet_Pressure

Turbine_Exit_Pressure = Heat_Exchanger_Exit_Low_Pressure + Exchanger_Low_Pressure_Loss

Calculation of Specific Heats

Constant_Pressure_Process = Gas_Constant × Heat_Ratio / (Heat_Ratio - 1)

Compressor_Specific_Heat = Gas_Constant × Compressor_Polytropic_Exponent / (Compressor_Polytropic_Exponent - 1)

Turbine_Specific_Heat = Gas_Constant × Turbine_Polytropic_Exponent / (Turbine_Polytropic_Exponent - 1)

Compressor Physical Efficiency

Compressor_Exergy_Efficiency = Nominal_Compressor_Efficiency - (Compressor_Pressure_Ratio - 1)

Calculated Compressor Exit Temperature

Compressor_Exponentnent = ((Compressor_Polytropic_Exponent - 1) / Compressor_Polytropic_Exponent) × (1 / Compressor_Exergy_Efficiency)
Compressor_Exit_Temperature = (Compressor_Pressure_Ratio ^ Compressor_Exponentnent) × Compressor_Inlet_Temperature


Turbine Physical Parameters

Turbine_Cooling_Loss = Exhaust_Temperature × (1 - Air_Cool_Compressor_Bypass)
Turbine_Cooling_Loss = Turbine_Cooling_Loss + Compressor_Exit_Temperature × Air_Cool_Compressor_Bypass

Calculated Turbine Exit Temperature

Turbine_Exit_Temperature = Turbine_Exit_Pressure / Combustion_Chamber_Exit_Pressure

Turbine_Exponent = ((Turbine_Polytropic_Exponent - 1) / Turbine_Polytropic_Exponent) × Compressor_Exergy_Efficiency
Turbine_Exit_Temperature = Turbine_Exit_Temperature ^ Turbine_Exponent
Turbine_Exit_Temperature = Turbine_Exit_Temperature × Turbine_Cooling_Loss

Heat Exchanger

Adjust the Exchanger Parity Flow Effectiveness Value to take into account any air bypass.

Adjusted_Exchanger_Parity_Flow_Eff = Exchanger_Parity_Flow_Eff × (1 - Air_Cool_Compressor_Bypass)

Temperature_To_Combustion = (Turbine_Exit_Temperature - Compressor_Exit_Temperature) × Exchanger_Parity_Flow_Eff
Temperature_To_Combustion = Temperature_To_Combustion + Compressor_Exit_Temperature

Exchanger_Temperature_Out = Turbine_Exit_Temperature - (Turbine_Exit_Temperature - Compressor_Exit_Temperature) × Exchanger_Parity_Flow_Eff

Exergy Energy of Compressor and Turbine

Exergy_Energy_To_Compressor = Compressor_Specific_Heat × (Compressor_Exit_Temperature - Compressor_Inlet_Temperature)

Exergy_Energy_In_Turbine = Turbine_Specific_Heat × (Turbine_Exit_Temperature - Turbine_Cooling_Loss)

Compressor Friction Helmholtz Ratio

Compressor_Friction_Ar = Compressor_Exit_Temperature / Compressor_Inlet_Temperature
Compressor_Friction_Ar = ln (Compressor_Friction_Ar)
Compressor_Friction_Ar = 1 - (Compressor_Friction_Ar × (Compressor_Inlet_Temperature / (Compressor_Exit_Temperature - Compressor_Inlet_Temperature)))

Turbine Friction Helmholtz Ratio

Turbine_Friction_Ar = Turbine_Cooling_Loss / Turbine_Exit_Temperature
Turbine_Friction_Ar = ln (Turbine_Friction_Ar)
Turbine_Friction_Ar = 1 - (Turbine_Friction_Ar × (Compressor_Inlet_Temperature / (Turbine_Cooling_Loss - Turbine_Exit_Temperature)))

Caloric Energy Transferred

Compressor_Exergy_To_Caloric_Value = Exergy_Energy_To_Compressor × (1 - Compressor_Exergy_Efficiency)

Compressor_Intercooler_Flow_Out = Constant_Pressure_Process × (Compressor_Exit_Temperature - Compressor_Inlet_Temperature) - Exergy_Energy_To_Compressor

Turbine_To_Caloric_Value = Constant_Pressure_Process × (Turbine_Exit_Temperature - Turbine_Cooling_Loss) - Exergy_Energy_In_Turbine

Turbine_Work_To_Heat = Exergy_Energy_In_Turbine × (1 - (1 /Compressor_Exergy_Efficiency))

Heat_Exchanger_Caloric_Value_To_Air = Constant_Pressure_Process × (Temp_To_Combustion - Compressor_Exit_Temperature)

Caloric_Value_From_Combustion = Constant_Pressure_Process × (Turbine_Cooling_Loss - Temperature_To_Combustion)

Caloric_Value_In_Exhaust = Constant_Pressure_Process × Compressor_Inlet_Temperature

Caloric_Value_From_Gas = Turbine_To_Caloric_Value + Caloric_Value_From_Combustion
Caloric_Value_From_Gas = Caloric_Value_From_Gas
Caloric_Value_From_Gas = Exergy_Energy_To_Compressor × Caloric_Value_From_Gas

Total_Caloric_Value = Caloric_Value_From_Combustion + Turbine_To_Caloric_Value + Caloric_Value_From_Gas

Work Produced

Cycle_Exergy_Energy = (Conversion_Exhaust_Energy × Compressor_Friction_Ar × Compressor_Intercooler_Flow_Out)
Cycle_Exergy_Energy = Cycle_Exergy_Energy + (Conversion_Exhaust_Energy × Caloric_Value_Exhaust_To_Cooler)

Total_Turbine_Exergy_Energy = Exergy_Energy_to_Compressor + Exergy_Energy_In_Turbine + Cycle_Exergy_Energy - Caloric_Value_From_Gas

Key Deliverables

Compressor Friction
Compressor Friction = Compressor_Exergy_To_Caloric_Value × (1 – Compressor_Friction_Ar) / Total_Caloric_Value

Turbine Friction
Turbine_Friction = Turbine_Work_To_Heat × (1 - Turbine_Friction_Ar) / Total_Caloric_Value

Fuel to Mechanical Energy in Turbine Shaft
Mechanical_Efficiency = Total_Turbine_Exergy_Energy / ((-1) × Total_Caloric_Value)

Fuel to net mechanical-energy from turbine shaft

Turbine to Compressor Work Ratio
Turbine_To_Compressor_Work_Ratio = ((-1) × Exergy_Energy_In_Turbine) / Exergy_Energy_To_Compressor
 

 

Special Notes:

About Exergy...

The VibroNurse turbine calculation is based around the concept of Exergy which is the amount of useful work that can be drawn from this energy based on a certain ambient temperature of the environment. Exergy is not considered a form of energy but a designation of the quality of energy.

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About Caloric Energy...

Caloric energy is used in our equations to represent the amount of internal energy that will flow between two reservoirs.
 

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About Helmholtz Ratios...

The Helmholtz energy is the part of caloric energy that could be converted into external energy in a future process. The bound energy is the caloric energy that cannot be converted into external energy. The Helmholtz Ratio (Ar) is the ratio between the Helmholtz energy and the bound energy.
 

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