Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness Calculator

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Air-Standard Cycles

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Fundamentals of IC Engine

✖Compression ratio is ratio of volume of cylinder to volume combustion chamber.ⓘ Compression Ratio [r]			+10% -10%
✖Final Temperature can be referred as the temperature reached after the combustion in the engine.ⓘ Final Temperature [T_f]			+10% -10%
✖Initial Temperature can be referred as the temperature after intake stroke in the engine.ⓘ Initial Temperature [T_i]			+10% -10%
✖Universal Gas Constant is a physical constant that appears in an equation defining the behavior of a gas under theoretically ideal conditions. Its unit is joulekelvin−1mole−1.ⓘ Universal Gas Constant [R]			+10% -10%
✖Molar Specific Heat Capacity at Constant Volume , Cv ( of a gas ) is the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.ⓘ Molar Specific Heat Capacity at Constant Volume [C_v]			+10% -10%
✖The effectiveness of heat exchanger is defined as the ratio of the actual heat transfer to the maximum possible heat transfer.ⓘ Effectiveness of Heat Exchanger [ε]			+10% -10%

✖The Thermal Efficiency of Stirling Cycle (in %) represents the fraction of heat converted into useful work in an engine working on the Stirling cycle.ⓘ Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness [η_stirling]

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Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness

Formula

`"η"_{"stirling"} = 100*(("[R]"*ln("r")*("T"_{"f"}-"T"_{"i"}))/("R"*"T"_{"f"}*ln("r")+"C"_{"v"}*(1-"ε")*("T"_{"f"}-"T"_{"i"})))`

Example

`"19.88603"=100*(("[R]"*ln("20")*("423K"-"283K"))/("8.314"*"423K"*ln("20")+"100J/K*mol"*(1-"0.5")*("423K"-"283K")))`

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Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness Solution

STEP 0: Pre-Calculation Summary

Formula Used

Thermal Efficiency of Stirling Cycle = 100*(([R]*ln(Compression Ratio)*(Final Temperature-Initial Temperature))/(Universal Gas Constant*Final Temperature*ln(Compression Ratio)+Molar Specific Heat Capacity at Constant Volume*(1-Effectiveness of Heat Exchanger)*(Final Temperature-Initial Temperature)))
η_stirling = 100*(([R]*ln(r)*(T_f-T_i))/(R*T_f*ln(r)+C_v*(1-ε)*(T_f-T_i)))
This formula uses 1 Constants, 1 Functions, 7 Variables

Constants Used

[R] - Universal gas constant Value Taken As 8.31446261815324

Functions Used

ln - The natural logarithm, also known as the logarithm to the base e, is the inverse function of the natural exponential function., ln(Number)

Variables Used

Thermal Efficiency of Stirling Cycle - The Thermal Efficiency of Stirling Cycle (in %) represents the fraction of heat converted into useful work in an engine working on the Stirling cycle.
Compression Ratio - Compression ratio is ratio of volume of cylinder to volume combustion chamber.
Final Temperature - (Measured in Kelvin) - Final Temperature can be referred as the temperature reached after the combustion in the engine.
Initial Temperature - (Measured in Kelvin) - Initial Temperature can be referred as the temperature after intake stroke in the engine.
Universal Gas Constant - Universal Gas Constant is a physical constant that appears in an equation defining the behavior of a gas under theoretically ideal conditions. Its unit is joule*kelvin−1*mole−1.
Molar Specific Heat Capacity at Constant Volume - (Measured in Joule Per Kelvin Per Mole) - Molar Specific Heat Capacity at Constant Volume , Cv ( of a gas ) is the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.
Effectiveness of Heat Exchanger - The effectiveness of heat exchanger is defined as the ratio of the actual heat transfer to the maximum possible heat transfer.

STEP 1: Convert Input(s) to Base Unit

Compression Ratio: 20 --> No Conversion Required
Final Temperature: 423 Kelvin --> 423 Kelvin No Conversion Required
Initial Temperature: 283 Kelvin --> 283 Kelvin No Conversion Required
Universal Gas Constant: 8.314 --> No Conversion Required
Molar Specific Heat Capacity at Constant Volume: 100 Joule Per Kelvin Per Mole --> 100 Joule Per Kelvin Per Mole No Conversion Required
Effectiveness of Heat Exchanger: 0.5 --> No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

η_stirling = 100*(([R]*ln(r)*(T_f-T_i))/(R*T_f*ln(r)+C_v*(1-ε)*(T_f-T_i))) --> 100*(([R]*ln(20)*(423-283))/(8.314*423*ln(20)+100*(1-0.5)*(423-283)))

Evaluating ... ...

η_stirling = 19.8860316408311

STEP 3: Convert Result to Output's Unit

19.8860316408311 --> No Conversion Required

FINAL ANSWER

19.8860316408311 ≈ 19.88603 <-- Thermal Efficiency of Stirling Cycle

(Calculation completed in 00.004 seconds)

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Indian Institute of Technology (IIT (ISM) ), Dhanbad, Jharkhand

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< 18 Air-Standard Cycles Calculators

Mean Effective Pressure in Dual Cycle

Go Mean Effective Pressure of Dual Cycle = Pressure at Start of Isentropic Compression*(Compression Ratio^Heat Capacity Ratio*((Pressure Ratio in Dual Cycle-1)+Heat Capacity Ratio*Pressure Ratio in Dual Cycle*(Cut-off Ratio-1))-Compression Ratio*(Pressure Ratio in Dual Cycle*Cut-off Ratio^Heat Capacity Ratio-1))/((Heat Capacity Ratio-1)*(Compression Ratio-1))

Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness

Go Thermal Efficiency of Stirling Cycle = 100*(([R]*ln(Compression Ratio)*(Final Temperature-Initial Temperature))/(Universal Gas Constant*Final Temperature*ln(Compression Ratio)+Molar Specific Heat Capacity at Constant Volume*(1-Effectiveness of Heat Exchanger)*(Final Temperature-Initial Temperature)))

Work Output for Dual Cycle

Go Work Output of Dual Cycle = Pressure at Start of Isentropic Compression*Volume at Start of Isentropic Compression*(Compression Ratio^(Heat Capacity Ratio-1)*(Heat Capacity Ratio*Pressure Ratio*(Cut-off Ratio-1)+(Pressure Ratio-1))-(Pressure Ratio*Cut-off Ratio^(Heat Capacity Ratio)-1))/(Heat Capacity Ratio-1)

Work Output for Diesel Cycle

Go Work Output of Diesel Cycle = Pressure at Start of Isentropic Compression*Volume at Start of Isentropic Compression*(Compression Ratio^(Heat Capacity Ratio-1)*(Heat Capacity Ratio*(Cut-off Ratio-1)-Compression Ratio^(1-Heat Capacity Ratio)*(Cut-off Ratio^(Heat Capacity Ratio)-1)))/(Heat Capacity Ratio-1)

Mean Effective Pressure in Diesel Cycle

Go Mean Effective Pressure of Diesel Cycle = Pressure at Start of Isentropic Compression*(Heat Capacity Ratio*Compression Ratio^Heat Capacity Ratio*(Cut-off Ratio-1)-Compression Ratio*(Cut-off Ratio^Heat Capacity Ratio-1))/((Heat Capacity Ratio-1)*(Compression Ratio-1))

Thermal Efficiency of Dual Cycle

Go Thermal Efficiency of Dual Cycle = 100*(1-1/(Compression Ratio^(Heat Capacity Ratio-1))*((Pressure Ratio in Dual Cycle*Cut-off Ratio^Heat Capacity Ratio-1)/(Pressure Ratio in Dual Cycle-1+Pressure Ratio in Dual Cycle*Heat Capacity Ratio*(Cut-off Ratio-1))))

Mean Effective Pressure in Otto Cycle

Go Mean Effective Pressure of Otto Cycle = Pressure at Start of Isentropic Compression*Compression Ratio*(((Compression Ratio^(Heat Capacity Ratio-1)-1)*(Pressure Ratio-1))/((Compression Ratio-1)*(Heat Capacity Ratio-1)))

Thermal Efficiency of Atkinson Cycle

Go Thermal Efficiency of Atkinson Cycle = 100*(1-Heat Capacity Ratio*((Expansion Ratio-Compression Ratio)/(Expansion Ratio^(Heat Capacity Ratio)-Compression Ratio^(Heat Capacity Ratio))))

Work Output for Otto Cycle

Go Work Output of Otto Cycle = Pressure at Start of Isentropic Compression*Volume at Start of Isentropic Compression*((Pressure Ratio-1)*(Compression Ratio^(Heat Capacity Ratio-1)-1))/(Heat Capacity Ratio-1)

Air Standard Efficiency for Diesel Engines

Go Air Standard Efficiency of Diesel Cycle = 100*(1-1/(Compression Ratio^(Heat Capacity Ratio-1))*(Cut-off Ratio^(Heat Capacity Ratio)-1)/(Heat Capacity Ratio*(Cut-off Ratio-1)))

Thermal Efficiency of Diesel Cycle

Go Thermal Efficiency of Diesel Cycle = 100*(1-1/Compression Ratio^(Heat Capacity Ratio-1)*(Cut-off Ratio^Heat Capacity Ratio-1)/(Heat Capacity Ratio*(Cut-off Ratio-1)))

Thermal Efficiency of Lenoir Cycle

Go Thermal Efficiency of Lenoir Cycle = 100*(1-Heat Capacity Ratio*((Pressure Ratio^(1/Heat Capacity Ratio)-1)/(Pressure Ratio-1)))

Thermal Efficiency of Ericsson Cycle

Go Thermal Efficiency of Ericsson Cycle = (Higher Temperature-Lower Temperature)/(Higher Temperature)

Air Standard Efficiency for Petrol engines

Go Air Standard Efficiency of Otto Cycle = 100*(1-1/(Compression Ratio^(Heat Capacity Ratio-1)))

Relative Air-Fuel Ratio

Go Relative Air Fuel Ratio = Actual Air Fuel Ratio/Stoichiometric Air Fuel Ratio

Air Standard Efficiency given Relative Efficiency

Go Air Standard Efficiency = Indicated Thermal Efficiency/Relative Efficiency

Thermal Efficiency of Otto Cycle

Go OTE = 1-1/Compression Ratio^(Heat Capacity Ratio-1)

Actual Air Fuel Ratio

Go Actual Air Fuel Ratio = Mass of Air/Mass of Fuel

Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness Formula

What is Stirling cycle ?

The Carnot cycle has a low mean effective pressure because of its very low work output. Hence, one of the modified forms of the cycle to produce higher mean effective pressure whilst theoretically achieving full Carnot cycle efficiency is the 'Stirling cycle'. The thermal efficiency of Stirling engines is 40% while the efficiency of similar Otto and Diesel engines are 25 and 35%, respectively.

How to Calculate Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness?

Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness calculator uses Thermal Efficiency of Stirling Cycle = 100*(([R]*ln(Compression Ratio)*(Final Temperature-Initial Temperature))/(Universal Gas Constant*Final Temperature*ln(Compression Ratio)+Molar Specific Heat Capacity at Constant Volume*(1-Effectiveness of Heat Exchanger)*(Final Temperature-Initial Temperature))) to calculate the Thermal Efficiency of Stirling Cycle, The Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness formula is defined as the fraction of heat converted into useful work in an engine working on the Stirling cycle. It takes into account the compression ratio as well as heat exchanger effectiveness. Thermal Efficiency of Stirling Cycle is denoted by η_stirling symbol.

How to calculate Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness using this online calculator? To use this online calculator for Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness, enter Compression Ratio (r), Final Temperature (T_f), Initial Temperature (T_i), Universal Gas Constant (R), Molar Specific Heat Capacity at Constant Volume (C_v) & Effectiveness of Heat Exchanger (ε) and hit the calculate button. Here is how the Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness calculation can be explained with given input values -> 19.88603 = 100*(([R]*ln(20)*(423-283))/(8.314*423*ln(20)+100*(1-0.5)*(423-283))).

FAQ

What is Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness?

The Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness formula is defined as the fraction of heat converted into useful work in an engine working on the Stirling cycle. It takes into account the compression ratio as well as heat exchanger effectiveness and is represented as η_stirling = 100*(([R]*ln(r)*(T_f-T_i))/(R*T_f*ln(r)+C_v*(1-ε)*(T_f-T_i))) or Thermal Efficiency of Stirling Cycle = 100*(([R]*ln(Compression Ratio)*(Final Temperature-Initial Temperature))/(Universal Gas Constant*Final Temperature*ln(Compression Ratio)+Molar Specific Heat Capacity at Constant Volume*(1-Effectiveness of Heat Exchanger)*(Final Temperature-Initial Temperature))). Compression ratio is ratio of volume of cylinder to volume combustion chamber, Final Temperature can be referred as the temperature reached after the combustion in the engine, Initial Temperature can be referred as the temperature after intake stroke in the engine, Universal Gas Constant is a physical constant that appears in an equation defining the behavior of a gas under theoretically ideal conditions. Its unit is joule*kelvin−1*mole−1, Molar Specific Heat Capacity at Constant Volume , Cv ( of a gas ) is the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume & The effectiveness of heat exchanger is defined as the ratio of the actual heat transfer to the maximum possible heat transfer.

How to calculate Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness?

The Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness formula is defined as the fraction of heat converted into useful work in an engine working on the Stirling cycle. It takes into account the compression ratio as well as heat exchanger effectiveness is calculated using Thermal Efficiency of Stirling Cycle = 100*(([R]*ln(Compression Ratio)*(Final Temperature-Initial Temperature))/(Universal Gas Constant*Final Temperature*ln(Compression Ratio)+Molar Specific Heat Capacity at Constant Volume*(1-Effectiveness of Heat Exchanger)*(Final Temperature-Initial Temperature))). To calculate Thermal Efficiency of Stirling Cycle given Heat Exchanger Effectiveness, you need Compression Ratio (r), Final Temperature (T_f), Initial Temperature (T_i), Universal Gas Constant (R), Molar Specific Heat Capacity at Constant Volume (C_v) & Effectiveness of Heat Exchanger (ε). With our tool, you need to enter the respective value for Compression Ratio, Final Temperature, Initial Temperature, Universal Gas Constant, Molar Specific Heat Capacity at Constant Volume & Effectiveness of Heat Exchanger and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.

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