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Compressor Thermodynamics
Methods and Alternatives
2 GE Title or job number
11/17/2013
Background
Calculating Volume, SI
The basic equation is 𝑃𝑉 = 𝑛𝑅𝑇𝑍
Where
Z=Compression factor, (no units)
T=Absolute Temperature
R=0.083145, m3 bar/(mol K) (Volume Units)
P=Absolute pressure, bar
3 GE Title or job number
11/17/2013
Process
General Energy Balance
Mixture @ Pi,Ti Output Fluid @ Po,To
F
Work
Heat
1st Law DH=Q+W/e
4 GE Title or job number
11/17/2013
Process
General Energy Balance
Pi= 35 bar Ti= 30 oC
F
Work=??
Efficiency = 80%
Po= 110 bar To= ??
5 GE Title or job number
11/17/2013
Calculating Discharge Conditions: Ideal Gas Single Phase
Step 1 • Note Inlet Pi, Ti
• Calculate Volume (Vi), and Cp, Cv and k=Cp/Cv
Step 2
• Estimate To
• 𝑇𝑜 = 𝑇𝑖𝑃𝑜
𝑃𝑖
𝑘−1
𝑘
Step 3
• Estimate Ideal Work
• 𝑊 ≈1
𝜖
𝑘
𝑘−1
𝑍𝑖𝑅𝑇𝑖
𝑀𝑊
𝑃0
𝑃𝑖
𝑘−1
𝑘− 1 𝑤ℎ𝑒𝑟𝑒 𝑅 = 8.3145
𝑘𝐽
𝑘𝑚𝑜𝑙/𝐾
6 GE Title or job number
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Example: First Calculating Mol Wt
Feed Component MW Zi*mwi
Component mol%
Water 0.00 18.02 0.00
H2S 0.00 34.08 0.00
CO2 0.95 28.01 26.61
N2 8.85 44.01 389.56
Methane 78.96 16.04 1266.70
Ethane 7.75 30.07 233.11
Propane 2.51 44.10 110.78
i-Butane 0.30 58.12 17.35
n-Butane 0.49 58.12 28.50
i-Pentane 0.08 72.15 5.84
n-Pentane 0.07 72.15 5.07
Benzene 0.00 78.11 0.00
Toluene 0.00 92.14 0.00
e-Benzene 0.00 106.17 0.00
o-Xylene 0.00 106.17 0.00
m-Xylene 0.00 Density 106.17 0.00
p-Xylene 0.00 mw kg/m3 106.17 0.00
Hexane 0.02 84.40 670 84.40 1.70
Heptane 0.01 92.60 734 92.60 1.08
Octane 0.00 105.20 760 105.20 0.47
Nonane 0.00 117.70 781 117.70 0.14
Decane 0.00 171.50 800 171.50 0.05
100.00 Stream MW, S 20.87
𝑀𝑊 = 𝑍𝑖𝑀𝑊𝑖
𝑍𝑖
7 GE Title or job number
11/17/2013
Calculating Ideal Gas Version
Step 1 efficiency 0.80 mw kg/kmol 19.64
Pi bar 35.00 Ti C 60.00
Zi - 0.95 Hi J/mol 10516.77
Vi m /kmol 0.76 k=Cpi/Cvi - 1.33
Step 2 Po bar 110.00 To C 169.35
Ideal Work 178.4 info
8 GE Title or job number
11/17/2013
Compressor Modeling
L
The Polytropic Analysis of Centrifugal Compressors
John M. Schultz, Trans. Of the ASME, ASME J. of Engineering for Power, Jan 1962 pp 69-82
The real-gas equations of polytropic analysis are derived in terms of compressibility functions X and Y which supplement the familiar compressibility factor, Z. A polytropic head factor, f, is introduced to adjust test results for deviations from perfect-gas behavior. Functions X and Y are generalized and plotted for gases in corresponding states.
The thermodynamic design and test evaluation of centrifugal compressors is frequently based upon a polytropic analysis employing perfect-gas relations. In many instances real-gas relations would be more accurate, but these are virtually unknown. The purpose of this paper is to derive the real-gas equations of polytropic analysis and to show their application to centrifugal compressor testing and design.
9 GE Title or job number
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The problem in 1962
• Needed a convenient model for the fluid
• Little General Access to Process Simulation Tools • Limitations in Equations of State Methods
• BWR 1940, complex solution • RK 1949, limited application to mixtures • NGA 1958, Equilibrium Ratio Data for Computers
Needed an alternative which could be handled using a slide rule
• Ie: Simple log-log relationships • Utilize Corresponding States Models
10 GE Title or job number
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Classic Algorithm Flange to Flange
Hi,Si,ri,CPi,Cvi
V/F
Hos,Sos,Tos
Feed at P,T
Flash at Pi,Ti
Isentropic Flash at Po
Ho=Hi+(Hos-His)/h
Isenthalpic Flash at Po
Ho,So,To,W V/F
Outputs
Inputs • Efficiency • Composition
11 GE Title or job number
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Schultz Algorithm: Wheel by Wheel
Hi,Si,Vi,CPi,Cvi
V/F,X,Y,n,m,
To
Feed at P,T
Flash at Pi,Ti
Calculate To,Po,Vo
Done Ho,So,To,W
Outputs Inputs • Efficiency • Composition
12 GE Title or job number
11/17/2013
Compressor Modeling
Classic Compressor Algorithm: Polytropic
• 𝑃𝑉𝑛 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
•𝑃𝑚
𝑇= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
•𝑃
𝑛−1𝑛 −𝑚
𝑍= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
13 GE Title or job number
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Schultz shortcut: Linearization
𝑘 =𝐶𝑝
𝐶𝑣
hp= polytropic efficiency
h𝑝= 𝑉
𝑑𝑃
𝑑𝐻=
𝑉
𝜕𝐻𝜕𝑃 𝑇
+ 𝐶𝑝𝑑𝑇𝑑𝑃
𝑋 =𝑇
𝑉
𝜕𝑉
𝜕𝑇 𝑃− 1 Z-Factor Charts
𝑌 = −𝑃
𝑉
𝜕𝑉
𝜕𝑃 𝑇
14 GE Title or job number
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Schultz Calculation Strategy: Single Phase
Step 1 • Note Inlet e, Pi, Ti
• Calculate Volume (Vi), k= (Cp/Cv)
Step 2
• Estimate Outlet Temperature, To
• 𝑇𝑜 = 𝑇𝑖𝑃𝑜
𝑃𝑖
𝑘−1
𝑘
Step 3
• Estimate Average Pressure, 𝑃
• 𝑃 = 𝑃𝑖𝑃𝑜
𝑃𝑖 𝑜𝑟 𝑃 =
𝑃𝑜+𝑃𝑖
2
15 GE Title or job number
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Example: Continue from Previous
Step 1 efficiency 0.80 mw kg/kmol 19.64
Pi bar 35.00 Ti C 60.00
Zi - 0.95 Hi J/mol 10516.77
Vi m /kmol 0.76 k=Cpi/Cvi - 1.33
Pbar from Geometric Mean Pbar from Arithmetic Mean
Step 2 Po bar 110.00 To C 169.35
Ideal Work 178.4 info
Step3 Pbar bar 62.05 Pbar bar 72.50
16 GE Title or job number
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Schultz Calculation Strategy: Single Phase
Step 4
• Estimate Average Temperature, 𝑇
• 𝑇 =𝑇𝑜+𝑇𝑖
2
Step 5
• Estimate Average Heat Capacity Ratio, 𝑘
• 𝑘 =𝑘𝑖+2𝑘𝑇 ,𝑃 +𝑘𝑜
4
Step 6
• Estimate 𝑋
• 𝑋 =𝑇
𝑉
𝜕𝑉
𝜕𝑇 𝑃− 1 𝑎𝑡 𝑃 , 𝑇
17 GE Title or job number
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Next Steps: Same Example
Pbar from Geometric Mean Pbar from Arithmetic Mean
Look at effect of averaging models
Step 4 Tbar C 114.68
Step 5 ko - 1.30
ktbar,pbar - 1.31 ktbar,pbar - 1.33
kbar - 1.32 kbar - 1.32
Step 6 dv/dt m /kmol/bar 0.0016 dv/dt m /kmol/bar 0.0014
T/V K*kmol/ m 770.23 T/V K*kmol/ m 902.98
Xbar 0.22 Xbar 0.25
18 GE Title or job number
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Schultz Calculation Strategy: Single Phase
Step 7
• Estimate 𝑌
• 𝑌 = −𝑃
𝑉
𝜕𝑉
𝜕𝑃 𝑇𝑎𝑡 𝑃 , 𝑇
Step 8
• Estimate 𝑚
• 𝑚 =
𝑘 −1
𝑘 1
𝜖+𝑋 𝑌
1+𝑋 2
Step 9
• Estimate 𝑛
• 𝑛 =1+𝑋
𝑌 1
𝑘 1
𝜖+𝑋 −
1
𝜖−1
19 GE Title or job number
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More Steps
Pbar from Geometric Mean Pbar from Arithmetic Mean
Step 7 dv/dp m /kmol/K -0.01 dv/dp m /kmol/K -0.01
P/V bar*kmol/ m 123.23 P/V bar*kmol/ m 168.80
Ybar 1.02 Ybar 1.02
Step 8 mbar - 0.24 mbar - 0.24
Step 9 nbar - 1.38 nbar - 1.38
20 GE Title or job number
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Schultz Calculation Strategy: Single Phase
Step 10
• Estimate To
• 𝑇𝑜 = 𝑇𝑖𝑃𝑜
𝑃𝑖
𝑚
Step 11
• Estimate Vo
• 𝑉𝑜 = 𝑉𝑖𝑃𝑜
𝑃𝑖
−1
𝑛
Step 12
• Verify n from EOS
• 𝑛 =ln 𝑃𝑜
𝑃𝑖
ln𝑉𝑖
𝑉𝑜
21 GE Title or job number
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Step 10 To C 166.18 C 164.56
Step 11 Vo calc m /kmol 0.33 m /kmol 0.33
Step 12 Zo EOS - 0.99 - 0.99
Vo Eos m /kmol 0.33 m /kmol 0.33
nbar - 1.38 - 1.37
Closing Up First Iteration
Pbar from Geometric Mean Pbar from Arithmetic Mean
To C 169.35Previous Estimate
22 GE Title or job number
11/17/2013
Schultz Calculation Strategy: Single Phase
Step 13
• Compare n-values
• Adjust To estimate or subdivide steps and return to step 4
Step 14
• Estimate Work
• 𝑊 ≈1
𝜀
𝑛
𝑛 −1
𝑍𝑖𝑅𝑇𝑖
𝑀𝑊
𝑃0
𝑃𝑖
𝑛 −1
𝑛 − 1 ≈
𝑓
𝜀
𝑛
𝑛 −1
𝑅
𝑀𝑊𝑍𝑜𝑇𝑜 − 𝑍𝑖𝑇𝑖
Step 15 • Done
23 GE Title or job number
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Step 13 New To C 166.18 C 164.56
Step 14 Work kJ/kg 226.72 Work kJ/kg 226.18
Work kJ/kg 226.72 Work kJ/kg 226.18
Ho J/mol 14942.93 Ho J/mol 14857.41
Work kJ/kg 225.38 kJ/kg 221.03
End of First Iteration
Pbar from Geometric Mean Pbar from Arithmetic Mean
Classic
Both Schultz Relations
Slight difference between relations Iterate Further to Close
Difference between “Classic” work and Predicted work is related to effect of efficiency on Schultz temperature model
24 GE Title or job number
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Next Iteration
Pbar from Geometric Mean Pbar from Arithmetic Mean
Previous Estimate To C 165.37
Step 2 Po bar 110.00 To C 165.37
Step 3 Pbar bar 62.05 Pbar bar 72.50
Step 4 Tbar C 112.68
Step 5 ko - 1.31
ktbar,pbar - 1.32 ktbar,pbar - 1.33
kbar - 1.32 kbar - 1.33
Step 6 dv/dt m /kmol/bar 0.0016 dv/dt m /kmol/bar 0.0014
T/V K*kmol/ m 771.12 T/V K*kmol/ m 904.17
Xbar 0.23 Xbar 0.26
Step 7 dv/dp m /kmol/K -0.01 dv/dp m /kmol/K -0.01
P/V bar*kmol/ m 124.01 P/V bar*kmol/ m 169.90
Ybar 1.02 Ybar 1.02
Step 8 mbar - 0.24 mbar - 0.24
Step 9 nbar - 1.38 nbar - 1.39
Step 10 To C 166.46 C 164.81
Step 11 Vo calc m /kmol 0.33 m /kmol 0.33
Step 12 Zo EOS - 0.99 - 0.99
Vo Eos m /kmol 0.33 m /kmol 0.33
nbar - 1.38 - 1.37
Step 13 New To C 166.46 C 164.81
Step 14 Work kJ/kg 226.81 Work kJ/kg 226.27
Work kJ/kg 226.81 Work kJ/kg 226.27
Ho J/mol 14958.04 Ho J/mol 14870.88
Work kJ/kg 226.15 kJ/kg 221.71
25 GE Title or job number
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After 3 Iterations (Direct Substitution)
Pbar from Geometric Mean Pbar from Arithmetic Mean
Step 2 Po bar 110.00 To C 165.64
Step 3 Pbar bar 62.05 Pbar bar 72.50
Step 4 Tbar C 112.82
Step 5 ko - 1.31
ktbar,pbar - 1.32 ktbar,pbar - 1.33
kbar - 1.32 kbar - 1.33
Step 6 dv/dt m /kmol/bar 0.0016 dv/dt m /kmol/bar 0.0014
T/V K*kmol/ m 771.06 T/V K*kmol/ m 904.09
Xbar 0.23 Xbar 0.26
Step 7 dv/dp m /kmol/K -0.01 dv/dp m /kmol/K -0.01
P/V bar*kmol/ m 123.96 P/V bar*kmol/ m 169.82
Ybar 1.02 Ybar 1.02
Step 8 mbar - 0.24 mbar - 0.24
Step 9 nbar - 1.38 nbar - 1.39
Step 10 To C 166.44 C 164.80
Step 11 Vo calc m /kmol 0.33 m /kmol 0.33
Step 12 Zo EOS - 0.99 - 0.99
Vo Eos m /kmol 0.33 m /kmol 0.33
nbar - 1.38 - 1.37
Step 13 New To C 166.44 C 164.80
Step 14 Work kJ/kg 226.81 Work kJ/kg 226.26
Work kJ/kg 226.81 Work kJ/kg 226.26
Ho J/mol 14957.01 Ho J/mol 14869.96
Work kJ/kg 226.10 kJ/kg 221.67
26 GE Title or job number
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Converged
Pbar from Geometric Mean Pbar from Arithmetic Mean
Arithmetic mean seems to agree better with rigorous model, but all are within modeling accuracy. Note that enthalpy match is important for process simulators
Step 13 New To C 166.44 C 164.80
Step 14 Work kJ/kg 226.81 Work kJ/kg 226.26
Work kJ/kg 226.81 Work kJ/kg 226.26
Ho J/mol 14957.07 Ho J/mol 14870.01
Work kJ/kg 226.10 kJ/kg 221.67
Rigorous Model To=164.1oC W=219.7kJ/kg
27 GE Title or job number
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If I Integrate Through the Compressor
Choose 7 wheels
Gas
Inlet T Inlet P Discharge P Discharge T Gas Rate Compressibility Gas Density Gas Mol Wt Gas Cp/Cv EnthalpyoC bar bar
oC m3/d Z Tf c,Pf c Tf c,Pf c Tf c,Pf c Change
1 60.00 35.00 41.22 73.8 31.5 0.95 26.11 19.64 1.33 27.28
2 73.76 41.22 48.55 87.9 27.9 0.96 29.47 19.64 1.33 28.60
3 87.92 48.55 57.19 102.5 24.7 0.96 33.24 19.64 1.32 29.97
4 102.48 57.19 67.35 117.4 21.9 0.96 37.49 19.64 1.32 31.42
5 117.43 67.35 79.33 132.8 19.5 0.97 42.24 19.64 1.32 32.94
6 132.77 79.33 93.43 148.5 17.3 0.97 47.56 19.64 1.32 34.55
7 148.49 93.43 110.00 164.5 15.4 0.98 53.49 19.64 1.31 36.18
Sum= 220.95