energy engineering: efficiency

Polytropic efficiency is a measure of the efficiency of a compressor. It is defined as the ratio of the actual work done by the compressor to the work that would be done if the compression process were isentropic (reversible and adiabatic). The polytropic efficiency of a compressor is always less than its isentropic efficiency.

Formula of Compressor polytropic efficiency

The polytropic efficiency of a compressor can be calculated using the following formula:

Po/Pi = (To/Ti)^(n/(n-1))
n/(n-1) = log(Po/Pi)/log(To/Ti)
η = (n/(n-1))/(k/(k-1))
η = log(Po/Pi)/log(To/Ti)/(k/(k-1))

where η is the polytropic efficiency, Pi and Ti are the inlet pressure and temperature, Po and To are the outlet pressure and temperature, and k is the specific heat ratio of the gas.

η : polytropic efficiency
Po : outlet pressure (psig)
Ti : inlet temperature (degF)
To : outlet temperature (degF)
n : compression ratio
k : specific heat ratio

Python code of Compressor polytropic efficiency

The following Python code is an example of calculating the polytropic efficiency of a compressor when gas introduced at 30 psig, 140 degF is compressed to 125 psig, 350 degF. (specific heat ratio k = 1.243).

Run Python code below : https://www.mycompiler.io/view/JMIFkEueEHW

import math
def compressorpolytropiceff(Pi, Po, Ti, To, k):

Pi : inlet Pressure (psig)

Po = Po + 14.696
Pi = Pi + 14.696
To = To + 460
Ti = Ti + 460
if k == 1 or Pi == 0 or Ti == 0: return -1
return math.log(Po/Pi)/math.log(To/Ti) / (k/(k-1)) * 100

polyeff = compressorpolytropiceff(30, 125, 140, 350, 1.243)
print("compressor polytropic efficiency = ", polyeff)

When run the code, you will receive the following results.

compressor polytropic efficiency = 74.2%

In thermodynamics, an isentropic process is an ideal thermodynamic process that is adiabatic and reversible. In this system, the transfer of work is friction less and there is no net transfer of heat or mass.

Formula of Isentropic efficiency of steam turbine

The isentropic efficiency is the ratio of actual power to the isentropic power and this isentropic efficiency of a steam turbine can be calculated using the following formula:

η = (hi - ho) / (hi - hos)

Here,
hi    : enthalpy of the steam at the inlet of the turbine (btu/lb)
ho    : enthalpy of the steam at the outlet of the turbine (btu/lb)
hos   : isentropic enthalpy of the steam at the outlet of the turbine (btu/lb)

Python code

The following is a Python example that calculates the isentropic efficiency of a steam turbine when superheated steam at 600 psig, 700 degF operates the turbine and is extracted at 60 psig, 350 degF.

from pyXSteam.XSteam import XSteam

steamTable = XSteam(XSteam.UNIT_SYSTEM_FLS) # ft/lb/sec/°F/psi/btu

def isentropicefficiency(Pi, Ti, Po, To):
hi = steamTable.h_pt(Pi+14.696, Ti)
ho = steamTable.h_pt(Po+14.696, To)
si = steamTable.s_ph(Pi+14.696, hi)
hos= steamTable.h_ps(Po+14.696, si)

return (hi - ho) / (hi - hos) * 100

# from 600 psig + 700 degF to 60 psig + 350 degF

When run the code, you get the results below.

hi : enthalpy of steam Pi, Ti (btu/lb) = 1350.1
si : entropy of steam Pi, Ti (btu/lb-F) = 1.584
ho : enthalpy of steam Pi, Ti (btu/lb) = 1205.5
so : entropy of steam Po, To (btu/lb-F) = 1.657
hos : isentropic enthalpy of steam Po, si (btu/lb) = 1149.6
isentropicefficiency = (1350.1 - 1205.5) / (1350.1 - 1149.6) * 100 = 71.1
isentropicefficiency = 72.1

print("isentropicefficiency = ", isentropicefficiency(600, 700, 60, 350))