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K J Somaiya College of Engineering (K J Somaiya), Mumbai
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## COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ) Solution

STEP 0: Pre-Calculation Summary
Formula Used
theoretical_coefficient_of_performance = (Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion)/((Polytropic index/(Polytropic index-1))*((Heat Capacity Ratio-1)/Heat Capacity Ratio)*((Ideal temp at end of isentropic compression-Ideal temp at the end of isobaric cooling)-(Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion)))
COPtheoretical = (T1-T4)/((n/(n-1))*((γ-1)/γ)*((T2-T3)-(T1-T4)))
This formula uses 6 Variables
Variables Used
Temperature at the start of Isentropic Compression - The temperature at the start of Isentropic Compression is the temperature from which the cycle starts. (Measured in Kelvin)
Temperature at the end of Isentropic Expansion - Temperature at the end of Isentropic Expansion is the temperature from where isentropic expansion ends and isobaric expansion starts. (Measured in Kelvin)
Polytropic index- The polytropic index is that defined via a polytropic equation of state. The index dictates the type of thermodynamic process.
Heat Capacity Ratio- The heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure (Cp) to heat capacity at constant volume (Cv). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas.
Ideal temp at end of isentropic compression - Ideal temp at end of isentropic compression is the intermediate temperature from where isobaric cooling starts. (Measured in Kelvin)
Ideal temp at the end of isobaric cooling - Ideal temp at the end of isobaric cooling is the intermediate temperature in the cycle where isentropic expansion starts (Measured in Kelvin)
STEP 1: Convert Input(s) to Base Unit
Temperature at the start of Isentropic Compression: 300 Kelvin --> 300 Kelvin No Conversion Required
Temperature at the end of Isentropic Expansion: 273 Kelvin --> 273 Kelvin No Conversion Required
Polytropic index: 0.2 --> No Conversion Required
Heat Capacity Ratio: 10 --> No Conversion Required
Ideal temp at end of isentropic compression: 350 Kelvin --> 350 Kelvin No Conversion Required
Ideal temp at the end of isobaric cooling: 325 Kelvin --> 325 Kelvin No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
COPtheoretical = (T1-T4)/((n/(n-1))*((γ-1)/γ)*((T2-T3)-(T1-T4))) --> (300-273)/((0.2/(0.2-1))*((10-1)/10)*((350-325)-(300-273)))
Evaluating ... ...
COPtheoretical = 60
STEP 3: Convert Result to Output's Unit
60 --> No Conversion Required
60 <-- Theoretical Coefficient of Performance
(Calculation completed in 00.047 seconds)

## < 11 Other formulas that you can solve using the same Inputs

Power required to maintain pressure inside the cabin(excluding ram work)
power_input = ((Mass of air*Specific Heat Capacity at Constant Pressure*Actual temperature of Rammed Air)/(Compressor efficiency))*((Cabin Pressure/Pressure of rammed air)^((Heat Capacity Ratio-1)/Heat Capacity Ratio)-1) Go
Power required to maintain pressure inside the cabin(including ram work)
power_input = ((Mass of air*Specific Heat Capacity at Constant Pressure*Ambient air temperature)/(Compressor efficiency))*((Cabin Pressure/Atmospheric Pressure)^((Heat Capacity Ratio-1)/Heat Capacity Ratio)-1) Go
Mass of air to produce Q tonnes of refrigeration in terms of exit temperature of cooling turbine
mass_of_air = (210*Tonnage of Refrigeration)/(1000*Specific Heat Capacity at Constant Pressure*(Temperature at the end of Isentropic Expansion-Actual exit Temperature of cooling turbine)) Go
Polytropic work
polytropic_work = ((Final Pressure of System*Volume of gas 2)-(Initial Pressure of System*Volume of gas 1))/(1-Polytropic index) Go
Temperature ratio at the start and end of ramming process
temperature_ratio = 1+(((Velocity^2)*(Heat Capacity Ratio-1)))/(2*(Heat Capacity Ratio*[R]*Initial Temp.)) Go
Heat Absorbed during Constant pressure Expansion Process
heat_absorbed = Specific Heat Capacity at Constant Pressure*(Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion) Go
Heat Rejected during Constant pressure Cooling Process
heat_rejected = Specific Heat Capacity at Constant Pressure*(Ideal temp at end of isentropic compression-Ideal temp at the end of isobaric cooling) Go
work = (Mass of Gas*[R]*(Initial Temp.-Final Temp.))/(Heat Capacity Ratio-1) Go
COP of Bell-Coleman Cycle for given Compression ratio and adiabatic index(γ)
theoretical_coefficient_of_performance = 1/(Compression or Expansion Ratio^((Heat Capacity Ratio-1)/Heat Capacity Ratio)-1) Go
Specific heat capacity at constant pressure when Adiabatic Index is Given
constant_pressure_specific_heat_capacity = (Heat Capacity Ratio*[R])/(Heat Capacity Ratio-1) Go
Local Sonic or Acoustic velocity at Ambient air conditions
sonic_velocity = (Heat Capacity Ratio*[R]*Initial Temp.)^0.5 Go

## < 5 Other formulas that calculate the same Output

Coefficient of Performance (for given h4)
theoretical_coefficient_of_performance = (Enthalpy of the vapour refrigerant at T1-Enthalpy of the vapour refrigerant at T4)/(Enthalpy of the vapour refrigerant at T2-Enthalpy of the vapour refrigerant at T1) Go
Coefficient of Performance (for given hf3)
theoretical_coefficient_of_performance = (Enthalpy of the vapour refrigerant at T1-Sensible heat at temperature T3)/(Enthalpy of the vapour refrigerant at T2-Enthalpy of the vapour refrigerant at T1) Go
COP of Bell-Coleman Cycle for given Compression ratio and adiabatic index(γ)
theoretical_coefficient_of_performance = 1/(Compression or Expansion Ratio^((Heat Capacity Ratio-1)/Heat Capacity Ratio)-1) Go
Theoretical Coefficient of Performance of a refrigerator
theoretical_coefficient_of_performance = Heat Extracted from Refrigerator/Work Go
Energy Performance Ratio of Heat Pump
theoretical_coefficient_of_performance = Heat Delivered to Body/Work Go

### COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ) Formula

theoretical_coefficient_of_performance = (Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion)/((Polytropic index/(Polytropic index-1))*((Heat Capacity Ratio-1)/Heat Capacity Ratio)*((Ideal temp at end of isentropic compression-Ideal temp at the end of isobaric cooling)-(Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion)))
COPtheoretical = (T1-T4)/((n/(n-1))*((γ-1)/γ)*((T2-T3)-(T1-T4)))

## What is Bell Coleman cycle?

The Bell Coleman Cycle (also called as the Joule or "reverse" Brayton cycle) is a refrigeration cycle where the working fluid is a gas that is compressed and expanded but does not change phase

## How to Calculate COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ)?

COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ) calculator uses theoretical_coefficient_of_performance = (Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion)/((Polytropic index/(Polytropic index-1))*((Heat Capacity Ratio-1)/Heat Capacity Ratio)*((Ideal temp at end of isentropic compression-Ideal temp at the end of isobaric cooling)-(Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion))) to calculate the Theoretical Coefficient of Performance, COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ)= (t1-t4) / ((Polytropic Index / (Polytropic Index-1)) * ((Adiabatic_Index-1)/Adiabatic_index) * ((t2-t3)-(t1-t4))). Theoretical Coefficient of Performance and is denoted by COPtheoretical symbol.

How to calculate COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ) using this online calculator? To use this online calculator for COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ), enter Temperature at the start of Isentropic Compression (T1), Temperature at the end of Isentropic Expansion (T4), Polytropic index (n), Heat Capacity Ratio (γ), Ideal temp at end of isentropic compression (T2) and Ideal temp at the end of isobaric cooling (T3) and hit the calculate button. Here is how the COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ) calculation can be explained with given input values -> 60 = (300-273)/((0.2/(0.2-1))*((10-1)/10)*((350-325)-(300-273))).

### FAQ

What is COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ)?
COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ)= (t1-t4) / ((Polytropic Index / (Polytropic Index-1)) * ((Adiabatic_Index-1)/Adiabatic_index) * ((t2-t3)-(t1-t4))) and is represented as COPtheoretical = (T1-T4)/((n/(n-1))*((γ-1)/γ)*((T2-T3)-(T1-T4))) or theoretical_coefficient_of_performance = (Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion)/((Polytropic index/(Polytropic index-1))*((Heat Capacity Ratio-1)/Heat Capacity Ratio)*((Ideal temp at end of isentropic compression-Ideal temp at the end of isobaric cooling)-(Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion))). The temperature at the start of Isentropic Compression is the temperature from which the cycle starts, Temperature at the end of Isentropic Expansion is the temperature from where isentropic expansion ends and isobaric expansion starts, The polytropic index is that defined via a polytropic equation of state. The index dictates the type of thermodynamic process, The heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure (Cp) to heat capacity at constant volume (Cv). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas, Ideal temp at end of isentropic compression is the intermediate temperature from where isobaric cooling starts and Ideal temp at the end of isobaric cooling is the intermediate temperature in the cycle where isentropic expansion starts.
How to calculate COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ)?
COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ)= (t1-t4) / ((Polytropic Index / (Polytropic Index-1)) * ((Adiabatic_Index-1)/Adiabatic_index) * ((t2-t3)-(t1-t4))) is calculated using theoretical_coefficient_of_performance = (Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion)/((Polytropic index/(Polytropic index-1))*((Heat Capacity Ratio-1)/Heat Capacity Ratio)*((Ideal temp at end of isentropic compression-Ideal temp at the end of isobaric cooling)-(Temperature at the start of Isentropic Compression-Temperature at the end of Isentropic Expansion))). To calculate COP of Bell-Coleman Cycle for given temperatures, polytropic index(n) and adiabatic index(γ), you need Temperature at the start of Isentropic Compression (T1), Temperature at the end of Isentropic Expansion (T4), Polytropic index (n), Heat Capacity Ratio (γ), Ideal temp at end of isentropic compression (T2) and Ideal temp at the end of isobaric cooling (T3). With our tool, you need to enter the respective value for Temperature at the start of Isentropic Compression, Temperature at the end of Isentropic Expansion, Polytropic index, Heat Capacity Ratio, Ideal temp at end of isentropic compression and Ideal temp at the end of isobaric cooling and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Theoretical Coefficient of Performance?
In this formula, Theoretical Coefficient of Performance uses Temperature at the start of Isentropic Compression, Temperature at the end of Isentropic Expansion, Polytropic index, Heat Capacity Ratio, Ideal temp at end of isentropic compression and Ideal temp at the end of isobaric cooling. We can use 5 other way(s) to calculate the same, which is/are as follows -
• theoretical_coefficient_of_performance = Heat Extracted from Refrigerator/Work
• theoretical_coefficient_of_performance = Heat Delivered to Body/Work
• theoretical_coefficient_of_performance = 1/(Compression or Expansion Ratio^((Heat Capacity Ratio-1)/Heat Capacity Ratio)-1)
• theoretical_coefficient_of_performance = (Enthalpy of the vapour refrigerant at T1-Enthalpy of the vapour refrigerant at T4)/(Enthalpy of the vapour refrigerant at T2-Enthalpy of the vapour refrigerant at T1)
• theoretical_coefficient_of_performance = (Enthalpy of the vapour refrigerant at T1-Sensible heat at temperature T3)/(Enthalpy of the vapour refrigerant at T2-Enthalpy of the vapour refrigerant at T1)
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