COMPARISON OF POLYTROPIC AND ISENTROPIC EFFICIENCY OF NATURAL GAS COMPRESSOR

CALCULATED USING ASPEN HYSYS AND USING MANUAL CALCULATIONS

MUHAMMAD WAQAS MANZOOR

PROCESS ENGINEER

Contact: engr.waqasmanzoor@gmail.com

• Isentropic Efficiency – Definition and mathematical derivation

• BHP and Isentropic head of compressor

• Polytropic Efficiency – Definition and mathematical derivation

• BHP and Polytropic head of compressor

• Relationship between Isentropic and Polytropic Efficiency

• Aspen HYSYS Simulation Results

• Comparison of Simulation Results with Manual Calculations(GPSA Method)

• Compressor Performance Calculations using manual calculationmethods

• Conclusion

SYNOPSIS

• Actual horse power required by compressor is always greater than Isentropichorsepower.

• Isentropic efficiency is the ratio of Isentropic horsepower to Actual horsepowerrequired by compressor.

• Or, 𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒

𝐴𝑐𝑡𝑢𝑎𝑙 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒

• Thermodynamically, Isentropic process is represented by the following relation,

𝑃𝑉𝑘 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Or, 𝑃1𝑉1𝑘 = 𝑃2𝑉2

𝑘

Or, 𝑃2 𝑃1= 𝑉1

𝑉2

𝑘Eq. 01

Or, 𝑃2 𝑃1=

𝑃2

𝑃1

𝑇1

𝑇2

𝑘

Or, 𝑇2𝑇1

𝑘= 𝑃2

𝑃1

𝑘−1

Or, 𝑇2𝑇1 = 𝑃2

𝑃1

𝑘−1𝑘 Eq. 02

• T2 in Eq. 02, is final ‘Isentropic’ temperature and can also be represented as T2,s.

ISENTROPIC EFFICIENCY OF COMPRESSOR

• Eq. 02, therefore, can also be written as under,

Or, 𝑇2,𝑠

𝑇1 = 𝑃2𝑃1

𝑘−1𝑘 Eq. 02

• Now, Isentropic efficiency of the compressor is given by,

𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒

𝐴𝑐𝑡𝑢𝑎𝑙 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒

Or, η𝑖𝑠𝑒𝑛,𝐶 =ℎ2,𝑠−ℎ1

ℎ2−ℎ1=

𝑇2,𝑠−𝑇1

𝑇2−𝑇1Eq. 03

Putting , 𝑇2,𝑠 = 𝑇1 𝑃2𝑃1

𝑘−1𝑘 in Eq. 03, we get,

η𝑖𝑠𝑒𝑛,𝐶 =𝑇1

𝑃2𝑃1

𝑘−1𝑘−𝑇1

𝑇2−𝑇1

Or, η𝑖𝑠𝑒𝑛,𝐶 =

𝑃2𝑃1

𝑘−1𝑘−1

𝑇2

𝑇1−1

Eq. 04

Or,η𝑖𝑠𝑒𝑛,𝐶

𝑃2

𝑃1

𝑘−1𝑘−1

=1

𝑇2

𝑇1−1

=𝑇1

𝑇2−𝑇1Eq. 05

ISENTROPIC EFFICIENCY OF COMPRESSOR

Contd.

• Now, Actual Work done of the compressor is given by,

𝐴𝑐𝑡𝑢𝑎𝑙 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝐶𝑝 × 𝑇2 − 𝑇1

Or, 𝐴𝑐𝑡𝑢𝑎𝑙 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝐶𝑝 × 𝑇2 − 𝑇1𝑇1

𝑇1

Or, 𝐴𝑐𝑡𝑢𝑎𝑙 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 =𝐶𝑝𝑇1𝑇1

𝑇2−𝑇1

Eq. 06

Putting,𝑇1

𝑇2−𝑇1=

η𝑖𝑠𝑒𝑛,𝐶

𝑃2

𝑃1

𝑘−1𝑘−1

in Eq. 06, we get,

𝐴𝑐𝑡𝑢𝑎𝑙 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 =𝐶𝑝𝑇1

η𝑖𝑠𝑒𝑛,𝐶

𝑃2

𝑃1

𝑘−1𝑘−1

Or, 𝐴𝑐𝑡𝑢𝑎𝑙 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 =𝐶𝑝𝑇1

η𝑖𝑠𝑒𝑛,𝐶 𝑃2𝑃1

𝑘−1𝑘 − 1

And, 𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝐶𝑝𝑇1 𝑃2𝑃1

𝑘−1𝑘 − 1

ISENTROPIC EFFICIENCY OF COMPRESSOR

Contd.

• Another equation which is used to compute BHP of compressor based on IsentropicEfficiency is as follows,

𝐵𝐻𝑃 =𝑃1𝑉1

η𝑖𝑠𝑒𝑛,𝐶𝑘−1

𝑘

𝑃2𝑃1

𝑘−1𝑘 − 1 Eq. 07

• The units of BHP would be ‘Watts’, if P1 is in Pascals and V1 is in m3/sec, in aboveequation.

• 𝑉1 in Eq. 07 can also be replaced with the following relation,

𝑉1 =𝑊

𝜌1=

𝑛𝑀

𝑃1𝑀𝑍𝑅𝑇1

=𝑛𝑍𝑅𝑇1𝑃1

• The units of V1 would be m3/s, if W is in kg/s, 𝜌1 is in kg/m3, n is in kgmole/s, M is inkg/kgmole, T is in Kelvins, P1 is in Pascals, and R is equal to 8314.47 Pa-m3/kgmol/K

Putting 𝑉1 =𝑛𝑍𝑅𝑇

𝑃1in Eq. 07, we get,

𝐵𝐻𝑃 =𝑃1 ×

𝑛𝑍𝑅𝑇1𝑃1

η𝑖𝑠𝑒𝑛,𝐶𝑘 − 1𝑘

𝑃2

𝑃1

𝑘−1𝑘− 1

BHP AND ISENTROPIC HEAD OF COMPRESSOR

Contd.

Or, 𝐵𝐻𝑃 =𝑛𝑍𝑅𝑇1

η𝑖𝑠𝑒𝑛,𝐶𝑘−1

𝑘

𝑃2𝑃1

𝑘−1𝑘 − 1

In other words, 𝐵𝐻𝑃 ∝𝑛𝑇1𝑘−1

𝑘

𝑃2𝑃1

𝑘−1𝑘 − 1

• Isentropic Head can also be computed from above equation as under,

𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝐻𝑒𝑎𝑑 (𝑚𝑒𝑡𝑒𝑟𝑠) =𝐵𝐻𝑃 𝑊𝑎𝑡𝑡𝑠 × η𝑖𝑠𝑒𝑛,𝐶

𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑘𝑔/𝑠 × 9.8 𝑚/𝑠2

BHP AND ISENTROPIC HEAD OF COMPRESSOR

• There is another equation which is used to calculate Isentropic Head directly and ismentioned in GPSA standard.

• It can be derived from BHP equation (Eq. 07), directly. However, as we are going tocompute ‘Isentropic Head’, so the isentropic efficiency would not be used in the equation.

• We have,

𝐵𝐻𝑃 (𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐) =𝑃1𝑉1𝑘−1

𝑘

𝑃2𝑃1

𝑘−1𝑘 − 1

Or, 𝜌1𝑔ℎ𝑉1 =𝑃1𝑉1𝑘−1

𝑘

𝑃2𝑃1

𝑘−1𝑘 − 1

Or, ℎ𝑖𝑠𝑒𝑛, 𝐶 =𝑃1

𝜌1𝑔𝑘−1

𝑘

𝑃2𝑃1

𝑘−1𝑘 − 1 Eq. 08

Now, putting 𝜌 = 𝑃1𝑀𝑍𝑅𝑇1

in Eq. 08, we get,

ℎ𝑖𝑠𝑒𝑛, 𝐶 =𝑍𝑅𝑇1

𝑀𝑔𝑘−1

𝑘

𝑃2𝑃1

𝑘−1𝑘 − 1 Eq. 09

Where, g=9.8 m/s2 , M=Molecular Mass, R=Universal Gas Constant = 8314.47 Pa-m3/kgmol/K

• The unit of Isentropic Head ‘hisen,C’ would be ‘meter’ if T1 is in Kelvins, M is in kg/kgmol, gis in m/s2, and R is in Pa-m3/kgmol/K.

ISENTROPIC HEAD OF COMPRESSOR

• As the gas is compressed, its compression ratio increases along thecompression path, until it becomes maximum at compressor discharge.

• Due to varying compression ratio, the Isentropic efficiency also varies alongthe compression path.

• Therefore, even by using ‘constant’ Isentropic efficiency, we can not computereal / actual work done, as it is not expressing the compression processadequately.

• In order to incorporate the effect of varying the Isentropic efficiency, we candivide the compression path into ‘infinite’ number of small steps, so that theIsentropic efficiency is constant along each of these small steps.

• This Isentropic efficiency which is constant at each of the small steps, is knownas ‘Polytropic Efficiency’, and the resulting process is called ‘Polytropicprocess’.

POLYTROPIC EFFICIENCY OF COMPRESSOR

• In a Polytropic compression process, the compression path is divided intoinfinite number of small steps so that Isentropic efficiency is constant alongeach of these small steps.

• Whereas, an Isentropic compression process is that process in which ‘Entropy’remains constant throughout the process.

• Mathematically, the Polytropic process is expressed using a polytropicexponent ‘n’ instead of ‘k’ as in Isentropic process,

i.e. 𝑃𝑉𝑛 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Or, 𝑃1𝑉1𝑛 = 𝑃2𝑉2

𝑛

Or,𝑃2

𝑃1=

𝑉1

𝑉2

𝑛Eq. 10

Or,𝑇2

𝑇1=

𝑃2

𝑃1

𝑛−1

𝑛Eq. 11

• The Polytropic exponent ‘n’ is related to the Isentropic exponent ‘k’ in thefollowing form; where η𝑝,𝐶 is the Polytropic Efficiency of the compressor,

𝑛−1

𝑛=

1

η𝑝,𝐶

𝑘−1

𝑘Eq. 12

POLYTROPIC PROCESS

Putting𝑛−1

𝑛=

1

η𝑝,𝐶

𝑘−1

𝑘in Eq. 11, we get,

𝑇2

𝑇1=

𝑃2

𝑃1

1

η𝑝,𝐶

𝑘−1

𝑘Eq. 13

Taking natural log. of both sides we get,

𝑙𝑛𝑇2

𝑇1=

1

η𝑝,𝐶

𝑘−1

𝑘𝑙𝑛

𝑃2

𝑃1

Or, η𝑝,𝐶 =𝑙𝑛

𝑃2𝑃1

𝑘−1𝑘

𝑙𝑛𝑇2𝑇1

Eq. 14

Contd.

POLYTROPIC PROCESS

Replacing ‘k’ with ‘n’ in the Isentropic Head relation (Eq. 09), we get,

ℎ𝑝, 𝐶 =𝑍𝑅𝑇1

𝑀𝑔𝑛−1

𝑛

𝑃2𝑃1

𝑛−1𝑛− 1 Eq. 15

• The variable ‘hp,C’ above equation is known as Polytropic Head.

• The unit of Polytropic Head ‘hp,C’ would be ‘meter’ if T1 is in Kelvins, M is inkg/kgmol, g is in m/s2, and R is in Pa-m3/kgmol/K.

Putting𝑛−1

𝑛=

1

η𝑝,𝐶

𝑘−1

𝑘in Eq. 15, we get,

ℎ𝑝, 𝐶 =𝑍𝑅𝑇1η𝑝,𝐶

𝑀𝑔𝑘−1

𝑘

𝑃2𝑃1

𝑘−1

𝑘

1

η𝑝,𝐶 − 1 Eq. 16

• The BHP of compressor based on Polytropic Efficiency is given by,

𝐵𝐻𝑃 𝑊𝑎𝑡𝑡𝑠 =𝑃𝑜𝑙𝑦𝑡𝑟𝑜𝑝𝑖𝑐 𝐻𝑒𝑎𝑑 (𝑚𝑒𝑡𝑒𝑟𝑠)×𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑘𝑔/𝑠 × 9.8 𝑚/𝑠2

η𝑝,𝐶

POLYTROPIC HEAD

Putting 𝑃𝑜𝑙𝑦𝑡𝑟𝑜𝑝𝑖𝑐 𝐻𝑒𝑎𝑑 =𝑍𝑅𝑇1η𝑝,𝐶

𝑀𝑔𝑘−1

𝑘

𝑃2𝑃1

𝑘−1

𝑘

1

η𝑝,𝐶 − 1 in the equation of BHP, we

get,

𝐵𝐻𝑃 =𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 × 𝑍𝑅𝑇1

𝑀𝑘−1

𝑘

𝑃2𝑃1

𝑘−1

𝑘

1

η𝑝,𝐶 − 1 Eq. 17

• The unit of BHP would be ‘Watts’ if T1 is in ‘Kelvins’, M is in kg/kgmol, Mass flow rateis in ‘kg/s’, and R is in Pa-m3/kgmol/K.

POLYTROPIC HEAD

Contd.

• The following equation relates the Isentropic Efficiency with PolytropicEfficiency,

η𝑖𝑠𝑒𝑛,𝐶 =

𝑃2𝑃1

𝑘−1𝑘 −1

𝑃2

𝑃1

𝑘−1𝑘

1η𝑝,𝐶 −1

Eq. 18

Or, 𝑃2𝑃1

𝑘−1

𝑘

1

η𝑝,𝐶 − 1 =

𝑃2𝑃1

𝑘−1𝑘 −1

η𝑖𝑠𝑒𝑛,𝐶

Or, 𝑃2𝑃1

𝑘−1

𝑘

1

η𝑝,𝐶 =

𝑃2𝑃1

𝑘−1𝑘 −1

η𝑖𝑠𝑒𝑛,𝐶+ 1

Or, 𝑙𝑛 𝑃2𝑃1

𝑘−1

𝑘

1

η𝑝,𝐶 = 𝑙𝑛

𝑃2𝑃1

𝑘−1𝑘 −1

η𝑖𝑠𝑒𝑛,𝐶+ 1

Or,𝑘−1

𝑘

1

η𝑝,𝐶𝑙𝑛 𝑃2

𝑃1 = 𝑙𝑛

𝑃2𝑃1

𝑘−1𝑘 −1

η𝑖𝑠𝑒𝑛,𝐶+ 1

ISENTROPIC AND POLYTROPIC EFFICIENCY

Contd.

Or,1

η𝑝,𝐶=

𝑙𝑛

𝑃2𝑃1

𝑘−1𝑘 −1

η𝑖𝑠𝑒𝑛,𝐶+1

𝑘−1

𝑘𝑙𝑛

𝑃2𝑃1

Or,1

η𝑝,𝐶=

𝑙𝑛

𝑃2𝑃1

𝑘−1𝑘 −1

η𝑖𝑠𝑒𝑛,𝐶+1

𝑙𝑛 𝑃2

𝑃1

𝑘−1𝑘

Or, η𝑝,𝐶 =𝑙𝑛

𝑃2𝑃1

𝑘−1𝑘

𝑙𝑛

𝑃2𝑃1

𝑘−1𝑘 −1

η𝑖𝑠𝑒𝑛,𝐶+1

ISENTROPIC AND POLYTROPIC EFFICIENCY

Contd.

Eq. 19

PROCESS FLOW DIAGRAMAspen HYSYS Simulation Snapshot

PROCESS PARAMETERS OF GAS STREAM AT COMPRESSOR SUCTION

COMPOSITION OF GAS AT COMPRESSOR SUCTION

COMPRESSOR SPECIFICATIONSAspen HYSYS Simulation

COMPRESSOR PERFORMANCE

MANUAL CALCULATION OF POLYTROPIC EFFICIENCY FROM POLYTROPIC HEAD AND BHP

• Polytropic Head (HYSYS) = 9833 m

• BHP (HYSYS) = 100 hp = 74.6 kW = 74600 W = 74600 kg-m2/s3

• Mass flow rate of gas = 4707 lb/h = 2136.978 kg/h = 0.593 kg/sec

• Polytropic Efficiency = Mass flow rate x Polytropic Head x 9.8 m/s2 / BHP

= 0.593kg/s x 9833m x 9.8m/s2 / 74600kg-m2/s3

= 0.76599

= 76.599 %

• Polytropic Efficiency calculated by HYSYS = 76.696 %

• The slight difference may be attributed to difference in calculation procedure in Aspen HYSYS, which is based on numerical integration.

MANUAL CALCULATION OF ISENTROPIC EFFICIENCY FROM ISENTROPIC HEAD AND BHP

• Isentropic Head = 9616 m

• BHP = 100 hp = 74.6 kW = 74600 W = 74600 kg-m2/s3

• Mass flow rate of gas = 4707 lb/h = 2136.978 kg/h = 0.593 kg/sec

• Isentropic Efficiency = Mass flow rate x Polytropic Head x 9.8 m/s2 / BHP

= 0.593kg/s x 9616m x 9.8m/s2 / 74600kg-m2/s3

= 0.74909

= 74.909 %

• Isentropic Efficiency specified in HYSYS = 75.0 %

• The slight difference may be attributed to difference in calculation procedure in Aspen HYSYS, which is based on numerical integration.

• The software Aspen HYSYS uses the similar equations as mentioned in GPSA standards.

• However, the slight difference observed in the manually calculated values and Aspen HYSYS simulation, may be attributed to the calculation method of the software which is based on numerical integration.

CONCLUSION