dynamic modeling and optimization of large-scale cryogenic...

Dynamic Modeling and Optimizationof Large-Scale Cryogenic Processes

Maria Soledad Diaz

Planta Piloto de Ingeniería QuímicaUniversidad Nacional del Sur -CONICET

Bahía Blanca, ARGENTINA

2

Outline

Objective

Natural Gas Processing Plants

Methodology

Mathematical Models

Discussion of Results

Conclusions and Current Work

3

Objective

Dynamic optimization model for natural gas processing plant units

• Dynamic energy and mass balances• Phase equilibrium calculations with cubic equation of state• Hydraulic correlations• Phase change in countercurrent heat exchanger• Carbon dioxide precipitation in column• Phase existence in column

Simultaneous dynamic optimization approach

• Discretization of state and control variables• Resolution of large-scale Nonlinear Programming problem

Analysis of main variables temporal and spatial profiles

4

Natural Gas in Argentina

Component Feed A Feed B

Nitrogen 1.44 1.37

Carbon dioxide 0.65 1.30

Methane 90.43 89.40

Ethane 4.61 4.43

Propane 1.76 2.04

Butanes 0.77 0.96

Pentanes+ 0.34 0.50

5


Provide ethane as raw material for olefin plants (ethylene) and petrochemicalHigh ethylene yieldMinimum by-products

EP II EP I PEA D

PEBDL PEBD

PEBDL

Pta. Fracc. de LGN

Pta. NH3 y Pta. UreaAlmacen.

de GN I

VCM PVC

Pta.CloroSoda

Posta de Inflamables

a Bahía Blanca a Bahía Blanca

Ingeniero White

Puerto Galván

EP II EP I PEA D

PEBDL PEBD

PEBDL

Pta. Fracc. de LGN


de GN I

VCM PVC

Pta.CloroSoda



Ingeniero White

Puerto Galván

EP II EP I PEA D

PEBDL PEBD

PEBDL

Pta. Fracc. de LGN


de GN I

VCM PVC

Pta.CloroSoda



Ingeniero White

Puerto Galván

GN I

a Bahía Blanca

acce

so a

Gen

eral

Cer

ri

General D. Cerri

Ruta Nº3 Sur

ferrocarril

ferro

carr

il

a Médanos

Esta

ción

D. C

err i

Esta

ción

Agu

ara

Polo Petr

oquímico B

ahía Blan

ca

GN I

a Bahía Blanca

acce

so a

Gen

eral

Cer

ri

General D. Cerri

Ruta Nº3 Sur

ferrocarril

ferro

carr

il

a Médanos

Esta

ción

D. C

err i

Esta

ción

Agu

ara

Polo Petr

oquímico B

ahía Blan

ca

NGL Plant

6

Natural Gas Plants at Bahia Blanca

LPG Gasoline

Residual GasResidual Gas

C3 + C3=

C4

C5+

EthaneEthane

EthyleneEthylene

Natural GasNatural Gas NGLNGL

LPG Gasoline533533355355

700 700 MtMt/y/y

24 24 MMsmMMsm33/d/d 5 5 MMsmMMsm33/d/d

7


Cryogenic sector (demethanizing), Conventional fractionation (deethanizing, depropanizing, debutanizing, gasoline stabilization), CO2 removalCurrent technology: Turboexpansion (Old technology: Absorption)High pressure Cryogenic conditions

8

NATURAL GAS PLANT

Pipelines

DEHYDRATION

TRAIN B

TRAIN A

TRAIN C

EthaneTreatment CO2

Propane

Butane

GasolineSeparation Fractionation

Ethane To Ethylene Plant

Absorption Plant

TRAIN A

TRAIN B

Compression/Recompression

DEHYDRATION

C2 C3 C4

C3

C3

C4

C4

To Recompression

9

Cryogenic Sector

Turboexpander

HighPressureSeparator

to deethanizer column

Cryogenic HeatExchangers

Demethanizer

10

Previous Work

Operating conditions optimization in natural gas processing plants (NLP) (Diaz, Serrani, Be Deistegui, Brignole, 1995)

Automatic design and debottlenecking of ethane extraction plants (MINLP) (Diaz, Serrani, Bandoni Brignole, 1997)

Thermodynamic model effect on the design and optimization of natural gas plants (NLP) (Diaz, Zabaloy, Brignole, 1999)

Flexibility of natural gas processing plants - Dual mode operation (Bilevel NLP) (Diaz, Bandoni, Brignole, 2002)

11

Dynamic Optimization Problem

min Φ(z(tf),y(tf), u(tf), tf , p)z(t),y(t), u(t), tf , p

st

F(dz(t)/dt ,(z(t), y(t), u(t), p, t)=0G(z(t) , y(t), u(t), p, t) = 0

zO = z(0)

z(t) ∈ [zl, zu] , y(t) ∈ [xl, xu]u(t) ∈ [ul, uu] , p ∈ [pl, pu]

t, time tf, final timez, differential variables u, control variablesy, algebraic variables p, time independent parameters

General Formulation

12

Dynamic Optimization Strategy

Nonlinear DAE optimization problem

Discretization of Control and State variables

Collocation on finite elements

Large-Scale Nonlinear Programming Problem (NLP)

Interior Point Algorithm

Simultaneous Approach Cervantes, Biegler (1998)

Biegler, Cervantes, Waechter (2002)

13

rSQP AlgorithmInitialization

LinearizationObjective function, constraints

and gradients at xk

Step for dependent variables (dY)(Linear system)

Step for independent variables (dZ)QP in null space (Inequality constraints)

Line search or Trust region

Check ConvergenceOptimal Solution

dx = YdY + ZdZ

Basis selection

QP

14

Nonlinear Programming Problem

Application of Barrier method

)(xfmin

0)(s.t =xc

0 ≥x

∑−=

n

jjxlnxfmin

1)( µ

0)(s.t =xc

As µ 0, x*(µ) x*

Sequence of barrier problems for decreasing µ values

15

Barrier Method Algorithm: Primal-Dual ApproachInitialization

Step for dependent variables (dY)(Linear system)

Step for independent variables (dZ)Unconstrained QP in null space

Line search or Trust region

Check Barrier Problem Convergence

Check NLP Convergence

dx = YdY + ZdZ

Step for dual variables dv

Update µ, ε

Update B

B, µ,ε

No

Update B

OptimalSolution

QP

BP

16

TurboexpanderArea 203

Area 204 Area 501

High pressureseparator

HeatExchangers

Columm

Cryogenic Sector

17

Ts2_i = 322 K

ht

V

ToRecompression

ToTurboexpander

Natural Gas

Residual Gas

To DemethanizingColumn

Cryogenic Heat Exchangers and HP Separator

Phase Change Horizontal High

Pressure Separator

Rodriguez, Bandoni, Diaz (2004)

Baffled shell-and-tube single-pass countercurrent heat exchangers

Tubes: residual gas

Shell: natural gas feed

Partial condensation: Second heat exchanger

18

Energy Balances: Partial Differential Equations System

( ))t,z(Tt)t,z(TsL*A*Cp*t

A*hz

)t,z(Ttvtt

)t,z(Tttt

supt −=∂

∂+

∂∂

ρ

HE Model (no Phase Change)

( ))t,z(Ts)t,z(TtL*A*Cp*s

A*hz

)t,z(Tsvst

)t,z(Tsss

sups −=∂

∂−

∂∂

ρ

Tube side

Shell side

First-order hyperbolic PDE

Method of Lines

Spatial Discretization: Backward Finite Differences

for Convection Term ODE System

z z+∆z

Flowing Fluid

)z(*Ts),z(Ts)z(*Tt),z(Tt)t(Tso)t,L(Ts

)t(Tto)t,(Tt

====

00

0

19

Energy Balance at cell i

( ) ( )iissi

supsii

ii TsTtL*A*Cp*s

A*hTsTs

zvs

dtdTs

−+−∆

= − ρ1

HE Model (no Phase Change)

( ) ( )iitti

suptii

ii TtTsL*A*Cp*t

A*hTtTt

zvt

dtdTt

−+−∆

−= − ρ1Tube side

Shell side

Multicell model

20

ji,j

ji,j A*

Fv

ρ=

i = 1, …, N (cells)j = tube or shell side

Algebraic Equations (no Phase Change)

G

ToRecompression

Natural Gas

Residual Gas

HE model DAE System

i,ji,ji,j T*R*z

P*M=ρ

21

HE Model (Partial Phase Change on Shell Side)

Mass Balance at cell i

i,VLiii,V - m - VV

dtdM

1−=Vapor Phase

i,VLiii,L m - LL

dtdM

+= −1

j,iij,iij,iij,iiij xLyVxLyV

dtdm

−−+= −−−− 1111

Liquid Phase

Component

Energy Balance at cell i

tiiiiiiiii

i + Q*H - V*h - L*H + V*hLdt

dE1111 −−−−=

mVL,i: interfacial mass-transport rate from vapor to liquid phase

Rodriguez, Bandoni, Diaz (2005)

22


Momentum Balance at cell i

i,Li,VLi,erinti,Pi,Vii,Vii,Vi,V vmF F v - VvV

dt)vM(d

−−+= −− 11Vapor Phase

i,wi,Li,VLi,erinti,Pi,Lii,Lii,Li,L FvmF F v - LvL

dt)vM(d

−+++= −− 11Liquid Phase

Algebraic Equations at cell i

Void fraction based on cross-section area

i,ViiTi,V LeAM ρθ=

T

i,Vi A

A=θ

i,LiiTi,L )(LeAM ρθ−= 1

i,Vi,V

ii,V A

Vvρ

=

i,Li,V

ii,L )A(

Lvρ−

=1

23


Driving Force ( ) Tiiiii,P APsPsF θθ −= −− 11

Interfacial shear stress

( )22

4i,Li,V

i,Li,f

ii,erint vvf

DF −=

ρθ

Interfacial friction factor

+=

Dff

/

i,V

i,Lvi,f

δρρ

31

241

>

≤≤=

−

−

300000460

3000040000790200

250

Re,Re.

Re,Re.f

.

.

v

Friction resistance on wall surface

wi,w DF τ4

=

24


j,ij,ij,i xKy =

,HHH iidealii ∆−=

,hhh iidealii ∆−=

Vj,i

Lj,i

j,iKφφ

=

j,inc

ji

idealj,i

ideali x)T(hh ∑

==

1

j,inc

ji

idealj,i

ideali y)T(HH ∑

==

1

∑∑ =−j

j,ij

j,i xy 0

Equilibrium ratio

Summation Eqns

Internal Energy ii,Lii,Vi hMHME +=

25

Compressibility factor

Residual enthalpies

Fugacity coefficients


)x,Ps,Ts(zz

)y,Ps,Ts(zz

j,iiiLL

i

j,iiiVV

i

=

=

)x,Ps,Ts(

)y,Ps,Ts(

j,iiiLL

j,i

j,iiiVV

j,i

φφ

φφ

=

=

Soave-Redlich-Kwong EoS

Li

iLii,L

i PmolsTs*R*z*

Psρ

= Vi

iVii,V

i PmolsTs*R*z*

Psρ

=

)x,Ps,Ts(hh

)y,Ps,Ts(HH

j,iiii

j,iiii

∆=∆

∆=∆

26

- VLp+ Vp - Ldt

dM=

L

LV

MLongrVolρ

π −= 2

−+

−= 2222

ttt

L

LL hrh

rharccosrr

PmolLongM πρ

VVV VolM ρ= LV MMM +=

( )wL

tLstv /

)rh(gPPxCLρρρ ++−

=

High Pressure Separator

Horizontal tank

27

Turboexpander

Feed: vapor from High Pressure Separator

Polytropic expansion (η = 1.26, natural gas)

Assumption: Static mass and energy balances

Thermodynamic predictions: Soave-Redlich-Kwong

Algebraic equations: Same as Flash

Outlet partially condensed stream: to Demethanizing column

ηη 1−

=

e

s

e

s

PP

TT

28

iiiiii LVLVF

dtdM

−−++= −+ 11

j,iij,iij,iij,iij,iiij xLyVxLyVzF

dtdm

−−++= −−++ 1111 j=1,…,ncomp

( )[ ] iFiiFiiiiiiiiiiii QsrhHFhLHVhLHV

dtdE

+−+−−−+= −−++ ϕϕ 11111

Differential Equations at stage i (1 ≤ i ≤ N)

Demethanizing Column

Diaz, Tonelli, Bandoni, Biegler (2003)

N = 8; ncomp = 10

29

Algebraic Equations

Vj,i

Lj,i

j,iKφφ

= j,ij,ij,i xKy =

iidealii HHH ∆−=

iidealii hhh ∆−=


Compressibility factor ( )

Residual enthalpies (∆Hi, ∆hi)

Fugacity coefficients ( )

Li

Vi zz ,

Lji

Vji ,, , φφ

SRK

∑∑ =−j

j,ij

j,i xy 0

30

Vapor volume


Li

Li

i,plateiV

iMHDVolρ

π −

=

2

2

Vi

Vi

Vi VolM ρ=

Li

Vii MMM +=

j,iLij,i

Vij,i xMyMm +=

Vapor and liquid holdup

Component holdup

)ph(M)pH(ME Li

ii

LiV

i

ii

Vii ρρ

−+−=Internal energy holdup

Algebraic Equations

31

Liquid holdup and hydraulic correlations


( ) Liii

iLi HwoHwD.M ρπ +

=

2

28960

= −

Li

Li

Vi

Vi

iiVi

ii Pmol

PmolCoAhole

V..Hoρρ

ρ

2410565

( )32

3201495030/

iLi

ii

/i Weirl

LFw.Hwo

=

ρ

( )iii HwoHwHldrop += β

2111 VKKPP V

TOP ρ−=Pressure drop

( ) Li

Liiiii PmolHldropHo..PP ρ++= −

−5

1 10249170

Equivalent height of vapor free liquid

Pressure drop through aerated liquid

Dry hole pressure drop

i

i+1

HwHwo

32

Phase Equilibrium

Carbon dioxide solubility constraints

SCO,i

LCO,i

VCO,i fff 222 ==


SCO,i

LCO,i

VCO,i 222 µµµ ==

Assumption

No hydrocarbon in solid phase

To avoid CO2 precipitation

But from VLE calculations at each stage

SCO,i

SCO,i ff 22 =

SCO,i

VCO,i ff 22 ≤

SCO,i

LCO,i ff 22 ≤

VCO,i

LCO,i ff 22 =

33

Solid phase

VCO,i

SCO,i

SCO,i Pf 222 φ=

−

+

−

−

−+

−

−=

154014114647

13566548014156814

3

2

2

2

22

2

2

2

tCO

it

CO

i

tCO

it

CO

i

i

tCO

tCO

SCO,i

TT.

TT.

TT.

TTln.

TT.

PP

ln

Vapor phase

VCO,iiCO,i

VCO,i Pyf 222 φ=

Carbon dioxide solubility constraints SCO,i

VCO,i ff 22 ≤


34

Gibbs Free Energy minimization at stage i

Demethanizing Column: Phase Existence

Raghunathan, Diaz, Biegler (2004)

Karush Kuhn Tucker conditions

Raghunathan, Biegler (2003)

MPECs (Mathematical Program with Equilibrium Constraints)

as additional constraints ( x.y = 0; x,y≥0)

IPOPT-C under AMPL (x.y ≤ δµ)

35


Relaxed Equilibrium Conditions

j,ij,ij,iiij,iij,i x)y,x,P,T(Ky γ=

Li

Vii ss −=−1γ

00 ≥⊥≤ Li

Li sM

,M,M

M,M

M,M

Vi

Li

Vi

Li

Vi

Li

001

001

001

=>⇒>

>=⇒<

>>⇒=

γ

γ

γ

00 ≥⊥≤ Vi

Vi sM

ComplementarityConstraints

∑∑ =−j

j,ij

j,i xy 0

36


Liquid Holdup and Hydraulic correlations

00 ≥⊥≤ −+ Li

Li MM

−+ −+

= L

iLi

Lii

iLi MMHwD.M ρπ

2

28960

Lii

iLi HwoD.M ρπ

2

28960

=+

( )32

3201495030/

iLi

ii

/i Weirl

LFw.Hwo

=

ρ

= + 2

3

)M(kL Lidi

HwHwo

i

i+1

Numerical Results

38

( )

{ }

)K(Tmin)/kmol(Gmin)/kmol(L

System DAE

.stdtTTmin

out

tfSP

3063038058060

20

≤≤≤≤≤≤

−∫

Optimization Problem: HEs + HPS

G

ht

L

T

Tout

(8+6 cells in HEs)

39

Optimization variables: G, L

20 elements

2 collocation points

18274 disc. variables

18 iter. (3 barrier problems)

Numerical Results: HEs + HPS

0 5 10 15 20 25 3065.300

65.305

65.310

65.315

65.320

65.325

65.330

h

Lt

Lt (K

mol

/min

)

time (min)

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

h (m

)

40


Optimization variables: G, L

0 5 10 15 20 25 307071727374757677787980818283

Tout

G0

G0 (

Kmol

/min

)

time (min)

300

301

302

303

304

305

306

307

Tt (K

)

0 5 10 15 20 25 30

211.4

211.5

211.6

211.7

211.8

211.9

212.0

Temperature vs time

Ts (K

)

time (min)

41

Temp. profile, tubes side Temp. profile, shell side(no phase change)


42

Pressure profile, shell side Liquid flowrate, shell side


0 211.5 21

30.540 1

23

45

6

55.5

55.8

56.1

56.4

56.7

57.0

57.3

57.6

57.9

58.2

58.5

Ps (bar)

time (min) cells 0 2 11.5 21 30.5 40

12

34

56 0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

L (Kmol/min)

time (min)cells

43

( )

{ }

SCO,i

VCO,i

CH,B

REB

TOP

tfSPethane

f.f

.xmin)/kJ(Q

)bar(P

System DAE

.stdtmin

22

4

20

900

0080020099

2215

≤

≤≤≤≤≤≤

−∫ ηη

Alg. Eqns = 385Differ. Eqns = 97

Optimization Problem: Demethanizing Column

44

Numerical Results: Demethanizer

0 10 20 30 40 50

16

18

20

P Ethane recovery

Time (min)

Pto

p (b

ar)

70

72

74

76

78

80

82

Ethane recovery (%)

Optimization variable: PTOP

20 finite elements2 collocation points21357 discretized variables40 iter. (3 barrier problems)ηss = 80.9 %

Feed A: Low CO2content (0.65%)

45


0 10 20 30 4016

18

20

P Ethane Recovery

Time (min)

Pto

p (b

ar)

72

76

80

84

Ethane recovery (%

)

Feed B: High CO2 content (2%)


20 finite elements2 collocation points43 iter. (3 barrier problems)

ηss = 74.5 %

46


0 10 20 30 40 5016

18

20 P Ethane recovery

Time (min)

Pto

p (b

ar)

74

76

78

80

82

84

Ethane recovery (%

)

Feed B (high CO2 content)

No solubility constraints



ηss = 76.6 %

47


0 5 10 15 20 25 30 35 40162

164

166

168

170

172

174

176

178

QREBOILER XCH4

Time (min)

Reb

oile

r Hea

t Dut

y (M

J/m

in)

0.003

0.004

0.005

0.006

0.007

0.008

Bottom m

ethane mole fraction

Feed A

Optimization variable:

Reboiler Heat Duty (QR)


ηss = 77.8 % (PTOP=19bar)

Active constraint: Methane mole fraction in bottoms

48

Numerical Results: Demethanizer Start Up

MPECs


PTOP, QR

Three time periods

Ideal Gas, Ideal solution

Solved in AMPL with IPOPT-C (Raghunathan, Biegler, 2003)

Time (min)

Etha

ne R

ecov

ery

49

Numerical Results: Demethanizer Start Up

MPECs


PTOP, QR

3rd time period17944 discretizedvariables576 complementarityconstraints97 iterations

Time (min)

Reb

oile

rHea

t Dut

y (M

J/m

in)

50

Boiler Dynamic Optimization

Rodriguez, Bandoni, Diaz (2005b)

E-1

CombustionChamber

Natural Gas Furnace Gas

Air

Superheater RISER 1 RISER 2

High Pressure Steam

Flue Gas

Feed Water

Dynamic optimization model to determine controller parameters

Boiler

Combustion chamber

Risers

Superheater

Steam drum

Gas path

PI Controller

(Liquid level in drum)

Manipulated: Feed water flowrate

51


DAE Optimization modelDifferential equations• Momentum and energy balances in risers and superheater• Mass and energy balances in drum• Integral part of controller and valve equation

Algebraic equations• Friction loss• Mixture properties• Vapor holdup in drum• Gas – temperature distribution equations in furnace, risers ans

superheaters (6 eqns.)• Air / fuel ratio • Drum geometric relations (5 eqns.)• Steam tables correlations for saturated and high pressure steam

density, pressure and enthalpy

52


DAE Optimization modelOptimization variables• PI controller parameters

Step change in high pressure steam demand

Additional PI controllers (current work)• Outlet steam pressure (fuel gas flowrate)• Superheated steam temperature (air excess)

0 200 400 600 800 1000

22

24

26

28

30

32

Feed

Wat

er F

low

rate

(lb/

s)

Time (s)0 200 400 600 800 1000

-4

-2

0

2

4

6

Liqu

id L

evel

Dru

m (f

t)

Time (s)

53

Conclusions

Rigorous dynamic optimization models for cryogenic train in Natural Gas Processing plant within a simultaneous approach

Thermodynamic models with cubic equation of state

Carbon dioxide solubility addressed in both liquid and vapor phases as path constraints, at each column stage

Phase detection through the first order optimality conditions for Gibbs free energy minimization (MPECs)

Boiler dynamic model for controller parameters

Simultaneous approach provides efficient framework for the formulation and resolution of DAE optimization problems for large-scale real plants

54

References

Biegler, L. T., A. Cervantes, A. Waechter, “Advances in Simultaneous Strategies for Dynamic Process Optimization,” Chem. Eng. Sci., 57, 575-593, 2002 Cervantes, A., L. T. Biegler, “Large-Scale DAE Optimization using Simultaneous Nonlinear Programming Formulations”, AIChE Journal, 44,1038, 1998 Diaz, S., A. Serrani, R. de Beistegui, E. Brignole, "A MINLP Strategy for the Debottlenecking problem in an Ethane Extraction Plant"; Computers and Chemical Engineering, 19s, 175-178, 1995 Diaz, S., A.Serrani, A. Bandoni, E. A. Brignole, “Automatic Design and Optimization of Natural Gas Plants", Industrial and Engineering Chemistry Research, 36, 2715-2724, 1997S. Diaz, M. Zabaloy, E. A. Brignole, “Thermodynamic Model Effect on the Design And Optimization of Natural Gas Plants”, Proceedings 78th Gas Processors Association Annual Convention, 38-45, March 1999, Nashville, Tennessee, USA Diaz, S., A. Bandoni, E.A. Brignole “Flexibility study on a dual mode natural gas plant in operation”, Chemical Engineering Communications, 189, 5, 623-641, 2002

55

References

Diaz, S., S. Tonelli, A. Bandoni, L.T. Biegler, “Dynamic optimization for switching between steady states in cryogenic plants”, Foundations of Computer Aided Process Operations, 4, 601-604, 2003 Diaz, S., A. Raghunathan , L. Biegler, “Dynamic Optimization of a Cryogenic Distillation Column Using Complementarity Constraints”, 449d, AIChEAnnual Meeting, San Francisco, Nov. 16-19, 2003. Advances in OptimizationRaghunathan, A., M.S. Diaz, L.T. Biegler, “An MPEC Formulation for Dynamic Optimization of Distillation Operations”, Computers and Chemical Engineering, 28, 2037-2052, 2004 M. Rodríguez, J. A. Bandoni, M. S. Diaz, “Optimal dynamic responses of a heat exchanger/separator tank system with phase change”, submitted to Energy Conversion and Management, 2004Rodríguez, M., J. A. Bandoni, M. S. Diaz, “Dynamic Modelling and Optimisation of Large-Scale Cryogenic Separation Processes”, IChEAP-7, The Seventh Italian Conference on Chemical and Process Engineering, 15 - 18 May 2005a Giardini Naxos, Italy Rodríguez, M., J. A. Bandoni, M. S. Diaz, “Boiler Controller Design Using Dynamic Optimisation”, PRES05, 8th Conference on Process Integration, Modelling and Optimisation for Energy Saving and Pollution Reduction, 15 -18 May 2005b Giardini Naxos, Italy

56

Appendix: Soave Redlich Kwong Equation of State

)bV(TRa

bVVz

g +−

−=

α

∑∑ −===k

k,j*kjjjj

*j

*j

*j )k(xz;axx;zxa 1αα

∑ ==j

cjcjgjj P/TR.b;bb 086640

−+=

=

505022

11427470

.

cjjj

.

cj

cjgj T

Tm;P

TR.a α

217605741480 jjj ...m ωω −+=

Compressibility factor for a mixture

Soave modification (Temperature dependence)

)z,z( Vi

Li

57

Appendix: Soave Redlich Kwong Equation of State

+

−−−−−=

zBln

bb

aza

BA)Bzln()z(

bb

ln j*jjjj

j 12

1αα

φ

∑

+

+−−=

∆−

j j

j*j

*j

gg

mzx

zBln

TbRTRH

α1

111

TRbPB;

TRPaA

gg== 22

α

Fugacity Coeff. for component j in mixture ),( L

j,iV

j,i φφ

Residual enthalpy in mixture )h,H( ii ∆∆

dynamic modeling and optimization of large-scale cryogenic...

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