Dynamic Modeling and Optimization of Large-Scale ... - CEPAC

Dynamic Modeling and Optimization of Large-Scale Cryogenic Processes Maria Soledad Diaz Planta Piloto de Ingeniería Química Universidad Nacional del Sur -CONICET Bahía Blanca, ARGENTINA

Outline ¾ Objective ¾ Natural Gas Processing Plants ¾ Methodology ¾ Mathematical Models ¾ Discussion of Results ¾ Conclusions and Current Work

2

Objective ¾ 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

¾ Simultaneous dynamic optimization approach • Discretization of state and control variables • Resolution of large-scale Nonlinear Programming problem

¾ Analysis of main variables temporal and spatial profiles

3

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

4

Natural Gas Processing Plants ¾ Provide ethane as raw material for olefin plants (ethylene)

and petrochemical ¾ High ethylene yield ¾ Minimum by-products

a Bahía Blanca

a Bahía Blanca Ingeniero White

EP II

EP I

PEA D Pta. Clo ro Soda

PEBDL PEBD

a Médanos

tr

General D. Cerri

l Ce r ri

Pta. NH3 y Pta. Urea Pta. Fracc. de LGN

ferrocarril

Pe

o

Estación Aguara

Estación D. Cerri

lo Po

ic m uí oq

Almacen. de GN I

Gen e ra

GN I

NGL Plant

a hí Ba

ca an Bl

so a

ro f er

ril car

ac ce

Su r Nº 3 Ru t a

VCM PVC

a Bahía Blanca

PEBDL Puerto Galván Posta de Inflamables

5

Natural Gas Plants at Bahia Blanca

Natural Gas 24 MMsm3/d

NGL

5 MMsm3/d

Residual Gas

LPG Gasoline

355

Ethane

LPG Gasoline

533 C3 + C3= C4

700 Mt/y

C5+

Ethylene 6

Natural Gas Processing Plants ¾ Cryogenic sector (demethanizing), Conventional

fractionation (deethanizing, depropanizing, debutanizing, gasoline stabilization), CO2 removal ¾ Current technology: Turboexpansion (Old technology: Absorption) ¾ High pressure ¾ Cryogenic conditions

7

NATURAL GAS PLANT D E H Y D R A T I O N

Ethane To Ethylene Plant Ethane Treatment

CO2

TRAIN A

TRAIN B

C2

C3

C4

Propane

Compression/ Recompression

D E H Y D R A T I O N

TRAIN C

Separation

Butane Fractionation Gasoline

To Recompression TRAIN A

C3

C4

C3

C4

TRAIN B

Pipelines

Absorption Plant

8

Cryogenic Sector

Turboexpander

Demethanizer

Cryogenic Heat Exchangers High Pressure Separator

to deethanizer 9 column

Previous Work ¾ Operating conditions optimization in natural gas

processing plants (NLP) (Diaz, Serrani, Be Deistegui, Brignole, 1995)

¾ Automatic design and debottlenecking of ethane extraction

plants (MINLP) (Diaz, Serrani, Bandoni Brignole, 1997)

¾ Thermodynamic model effect on the design and

optimization of natural gas plants (NLP) (Diaz, Zabaloy, Brignole, 1999)

¾ Flexibility of natural gas processing plants - Dual mode

operation (Bilevel NLP) (Diaz, Bandoni, Brignole, 2002)

10

Dynamic Optimization Problem General Formulation 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)=0 G(z(t) , y(t), u(t), p, t) = 0 zO = z(0) z(t) ∈ [zl, zu] u(t) ∈ [ul, uu] t, time z, differential variables y, algebraic variables

y(t) ∈ [xl, xu] , p ∈ [pl, pu] ,

tf, final time u, control variables p, time independent parameters

11

Dynamic Optimization Strategy Simultaneous Approach

Cervantes, Biegler (1998)

Nonlinear DAE optimization problem

Discretization of Control and State variables Collocation on finite elements

Large-Scale Nonlinear Programming Problem (NLP) Interior Point Algorithm

Biegler, Cervantes, Waechter (2002) 12

rSQP Algorithm Initialization Linearization Objective function, constraints and gradients at xk

Basis selection

Step for dependent variables (dY) (Linear system) Step for independent variables (dZ) QP in null space (Inequality constraints)

QP

dx = YdY + ZdZ Line search or Trust region Check Convergence

Optimal Solution

13

Nonlinear Programming Problem Application of Barrier method n

min f (x)

min f ( x) − µ ∑ ln x j

s.t c( x) = 0

s.t c( x) = 0

j =1

x ≥0 As µ Î 0,

x*(µ) Î x*

Sequence of barrier problems for decreasing µ values

14

Barrier Method Algorithm: Primal-Dual Approach Initialization BP

B, µ,ε

Step for dependent variables (dY) (Linear system) Update B Step for independent variables (dZ) Unconstrained QP in null space

Update B

QP

dx = YdY + ZdZ Step for dual variables dv Update µ, ε Line search or Trust region No

Check Barrier Problem Convergence

Check NLP Convergence

Optimal Solution 15

Cryogenic Sector Turboexpander Area 203

High pressure separator

Columm

Heat Exchangers

Area 204

Area 501

16

Cryogenic Heat Exchangers and HP Separator To Recompression

Rodriguez, Bandoni, Diaz (2004)

Residual Gas

V

To Turboexpander

ht

Phase Change

Horizontal High Pressure Separator

To Demethanizing Column

Baffled shell-and-tube single-pass countercurrent heat exchangers Tubes: residual gas

Ts2_i = 322 K

Shell: natural gas feed Natural Gas

Partial condensation: Second heat exchanger

17

HE Model (no Phase Change) Energy Balances: Partial Differential Equations System Tube side Shell side

ht * Asup ∂Tt( z ,t ) ∂Tt( z ,t ) (Ts( z ,t ) − Tt( z ,t )) + vt = ∂t ∂z ρt * Cpt * At * L

hs * Asup ∂Ts( z ,t ) ∂Ts( z ,t ) − vs = (Tt( z ,t ) − Ts( z ,t )) ∂t ∂z ρs * Cps * As * L

Tt( 0 ,t ) = Tto( t ) Ts( L ,t ) = Tso( t ) Tt( z ,0 ) = Tt * ( z ) Ts( z ,0 ) = Ts * ( z )

Flowing Fluid

First-order hyperbolic PDE

z

z+∆z

Method of Lines Spatial Discretization: Backward Finite Differences for Convection Term Æ ODE System

18

HE Model (no Phase Change) Energy Balance at cell i Tube side

Shell side

ht * Asup dTti vti = − (Tti − Tti −1 ) + (Tsi − Tti ) dt ∆z ρti * Cpt * At * L

hs * Asup dTsi vsi = (Tsi − Tsi −1 ) + (Tti − Tsi ) ρsi * Cp s * As * L dt ∆z

Multicell model

19

Algebraic Equations (no Phase Change) ρ j ,i =

v j ,i =

M*P z j ,i * R * T j ,i

To Recompression

Residual Gas

Fj

ρ j ,i * A j G

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

HE model

DAE System Natural Gas

20

HE Model (Partial Phase Change on Shell Side) Mass Balance at cell i Vapor Phase

dM V ,i = Vi −1 - Vi - mVL ,i dt

Liquid Phase

dM L ,i = Li −1 - Li + mVL ,i dt

Component

dmij dt

Rodriguez, Bandoni, Diaz (2005)

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

= Vi −1 yi −1, j + Li −1 xi −1, j − Vi yi , j − Li xi , j

Energy Balance at cell i dEi = Li −1*hi −1 + Vi −1*H i −1 - Li*hi - Vi*H i + Qit dt

21

HE Model (Partial Phase Change on Shell Side) Momentum Balance at cell i Vapor Phase

d ( M V ,i vV ,i ) = Vi −1vV ,i −1 - Vi vV ,i + FP ,i − Fint er ,i − mVL ,i vL ,i dt

Liquid Phase

d ( M L ,i vL ,i ) = Li −1vL ,i −1 - Li vL ,i + FP ,i + Fint er ,i + mVL ,i vL ,i − Fw ,i dt

Algebraic Equations at cell i AV ,i θi = AT

Void fraction based on cross-section area

M V ,i = AT Leiθi ρV ,i

vV ,i =

M L ,i = AT Lei ( 1 − θi )ρ L ,i

vL ,i

Vi

AV ,i ρV ,i Li = ( 1 − AV ,i )ρ L ,i

22

HE Model (Partial Phase Change on Shell Side) Driving Force

FP ,i = ( Psi −1θ i −1 − Psiθ i ) AT

Interfacial shear stress

Fint er ,i =

ρ 4 θi 2 f f ,i L ,i (vV ,i − vL ,i ) D 2

1/ 3   ρ L ,i  δ    = f v 1 + 24    ρV ,i  D 

Interfacial friction factor

f f ,i

Friction resistance on wall surface

Fw ,i =

4 τw D

0.079 Re −0.25 ,4000 ≤ Re ≤ 30000 fv =  −0.20 0 . 046 Re , Re > 30000 

23

HE Model (Partial Phase Change on Shell Side) Internal Energy

Ei = M V ,i H i + M L ,i hi

Summation Eqns

∑ yi , j − ∑ xi , j = 0 j

Equilibrium ratio

hi =

hiideal

Hi =

H iideal

− ∆hi ,

− ∆H i ,

j

Ki,j

φ iL, j = V φi , j

h iideal H

ideal i

yi , j = K i , j xi , j nc

= ∑ h iideal ( T i )x i , j ,j j =1

nc

= ∑ H j =1

ideal i,j

( Ti )y i , j 24

HE Model (Partial Phase Change on Shell Side) Compressibility factor

z iV = z V ( Ts i , Ps i , y i , j ) z iL = z L ( Ts i , Ps i , x i , j ) Residual enthalpies

∆H

i

= ∆ H ( Ts i , Ps i , y i , j )

Soave-Redlich-Kwong EoS

∆ h i = ∆ h ( Ts i , Ps i , x i , j ) Fugacity coefficients

φ iV, j = φ V ( Ts i , Ps i , y i , j ) φ i L, j = φ L ( Ts i , Ps i , x i , j ) Psi =

ρ L ,i * ziL * R * Tsi PmolsiL

Psi =

ρV ,i * ziV * R * Tsi PmolsiV

25

High Pressure Separator dM = Lp+ Vp - L - V dt 2

VolV = πr Long − L = Cv x

ML

ρL

(Pt − Ps ) + g ρ L ( ht + r ) ρL / ρw

M V = ρV VolV ML =

Horizontal tank

M = MV + M L

ρ L Long 

2   ht  2 2 2 + − π r r arccos h r h −    t t 

Pmol L 

r

 26

Turboexpander ¾ Feed: vapor from High Pressure Separator ¾ Polytropic expansion (η = 1.26, natural gas) η −1 η

Ts  Ps =   Te  Pe 

¾ Assumption: Static mass and energy balances ¾ Thermodynamic predictions: Soave-Redlich-Kwong ¾ Algebraic equations: Same as Flash ¾ Outlet partially condensed stream: to Demethanizing column

27

Demethanizing Column Diaz, Tonelli, Bandoni, Biegler (2003)

Differential Equations at stage i (1 ≤ i ≤ N) dM i = Fi + Vi +1 + Li −1 − Vi − Li dt dmij dt

= Fi zi , j + Vi +1 yi +1, j + Li −1 xi −1, j − Vi yi , j − Li xi , j

j=1,…,ncomp

dEi = Vi +1 H i +1 + Li −1hi −1 − Vi H i − Li hi − Fi [ϕi H Fi + (1 − ϕi )hFi ] + Qsri dt N = 8; ncomp = 10

28

Demethanizing Column Algebraic Equations Compressibility factor ( z iV , z iL) Residual enthalpies (∆Hi, ∆hi) V L Fugacity coefficients (φ i , j , φ i , j)

Ki , j

φiL, j = V φi , j

SRK

yi , j = K i , j xi , j

∑ yi , j − ∑ xi , j = 0 j

j

H i = H iideal − ∆H i hi = hiideal − ∆hi

29

Demethanizing Column Algebraic Equations 2

Vapor volume

M iL  Di  V Voli = π   H plate ,i − L ρi  2

Vapor and liquid holdup

M iV = ρ iV VoliV M i = M iV + M iL

Component holdup

mi , j = M iV yi , j + M iL xi , j

Internal energy holdup

Ei = M iV ( H i −

pi

ρ

V i

) + M iL ( hi −

pi

ρ

L i

) 30

Demethanizing Column Hwo Hw

Liquid holdup and hydraulic correlations i

2

M iL

 Di  ( = 0.896π   Hwi + Hwoi )ρiL  2

Hwoi = 30(0.01495)

2/ 3

  Li  Fwi  L  ρi Weirli 

2/ 3

Hldropi = β ( Hwi + Hwoi )

  Vi  Hoi = 5.56.10−4  V  ρi AholeiCoi  Pressure drop

2

i+1

Equivalent height of vapor free liquid Pressure drop through aerated liquid

 ρiV PmoliV    L L   ρi Pmoli 

Dry hole pressure drop

P1 = PTOP − KKρ1V V12 L Pi = Pi −1 + 0.24917.10 −5 ( Hoi + Hldropi )ρiL Pmoli31

Demethanizing Column Carbon dioxide solubility constraints Phase Equilibrium

µiV,CO 2 = µiL,CO 2 = µiS,CO 2 f iV,CO 2 = f i ,LCO 2 = f i S,CO 2

Assumption No hydrocarbon in solid phase To avoid CO2 precipitation

f i S,CO 2 = f i S,CO 2

f iV,CO 2 ≤ f i ,SCO 2 f i ,LCO 2 ≤ f i S,CO 2

But

f i ,LCO 2 = f iV,CO 2

from VLE calculations at each stage

32

Demethanizing Column Carbon dioxide solubility constraints

f iV,CO 2 ≤ f i ,SCO 2

Solid phase

f i ,SCO 2 = Pi ,SCO 2φiV,CO 2 t  Pi S,CO 2   TCO  T   T  2  − 14.480 ln t i  + 65.356 t i − 1 ln t  = 14.5681 − Ti   TCO 2   TCO 2    PCO 2 

 T  2   T 3  − 47.146  t i  − 1 + 14.540 t i  − 1  TCO 2   TCO 2   

Vapor phase

f iV,CO 2 = yi ,CO 2 Pi φiV,CO 2

33

Demethanizing Column: Phase Existence Raghunathan, Diaz, Biegler (2004)

Gibbs Free Energy minimization at stage i

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)

34

Demethanizing Column: Phase Existence Relaxed Equilibrium Conditions

yi , j = γ i K i , j ( Ti , Pi , xi , j , yi , j )xi , j ∑ yi , j − ∑ xi , j = 0 j

γ i −1 =

γ = 1 ⇒ M iL > 0 , M iV > 0

j

siV

γ < 1 ⇒ M iL = 0 , M iV > 0

− siL

0 ≤ M iL ⊥ siL ≥ 0 0 ≤ M iV ⊥ siV ≥ 0

γ > 1 ⇒ M iL > 0 , M iV = 0 , Complementarity Constraints

35

Demethanizing Column: Phase Existence Liquid Holdup and Hydraulic correlations 2

D  M iL = 0.896π  i  Hwi ρiL + M iL+ − M iL−  2

Hwo Hw

i

i+1 2

D  M iL+ = 0.896π  i  Hwoi ρiL  2 Hwoi = 30(0.01495)

2/ 3

  Li   Fwi  L   ρi Weirli 

2/ 3

3   L + 2  Li = kd ( M i )     

0 ≤ M iL + ⊥ M iL − ≥ 0 36

Numerical Results

Optimization Problem: HEs + HPS min

T

tf ∫0

(T − TSP ) dt 2

st . h t

G

Tout

L

{DAE System} 60 ≤ L ≤ 80( kmol / min) 5 ≤ G ≤ 80( kmol / min) 303 ≤ Tout ≤ 306( K )

(8+6 cells in HEs) 38

Numerical Results: HEs + HPS Optimization variables: G, L 20 elements 2 collocation points

65.325

Lt

0.95

h

0.90 0.85 0.80

Lt (Kmol/min)

65.320

0.75 0.70

65.315

0.65 0.60

65.310

0.55 0.50

65.305

0.45 65.300 0

5

10

15

20

25

0.40 30

time (min) 39

h (m)

18274 disc. variables 18 iter. (3 barrier problems)

1.00

65.330

Numerical Results: HEs + HPS Optimization variables: G, L 83

307

82

G0

81

306

Tout

80

304

77 76

303

75 74

Temperature vs time 212.0

302

211.9

73 72

301

211.8

71 70 0

5

10

15

time (min)

20

25

300 30

Ts (K)

G0 (Kmol/min)

78

Tt (K)

305

79

211.7 211.6 211.5 211.4 0

5

10

15

time (min)

20

25

40

30

Numerical Results: HEs + HPS

Temp. profile, tubes side

Temp. profile, shell side (no phase change) 41

Numerical Results: HEs + HPS

58.5

35.0

58.2

30.0

57.9 57.6

25.0

57.3

Ps (bar)

20.0

57.0

L (Kmol/min)

56.7

15.0

56.4

10.0

56.1 55.8

5.0

55.5 0

2

11.5

time (min)

21

30.5

40 1

2

3

4

5

6

cells

Pressure profile, shell side

6

cells

0.0 5

4

3

2

1 0

2

11.5

21

30.5

40

time (min)

Liquid flowrate, shell side 42

Optimization Problem: Demethanizing Column min

tf ∫0

(ηethane − η SP ) dt 2

st .

{DAE System} 15 ≤ PTOP ≤ 22( bar ) 99 ≤ QREB ≤ 200( kJ / min) 0 ≤ xB ,CH 4 ≤ 0.008 f iV,CO 2 ≤ 0.90 f i ,SCO 2

Alg. Eqns = 385 Differ. Eqns = 97

43

Numerical Results: Demethanizer Feed A: Low CO2 content (0.65%)

82

20

Optimization variable: PTOP

80

2 collocation points

P Ethane recovery

18

Ptop (bar)

20 finite elements

76

74

21357 discretized variables 40 iter. (3 barrier problems)

16

72

ηss = 80.9 %

70 0

10

20

30

40

50

Time (min)

44

Ethane recovery (%)

78

Numerical Results: Demethanizer Feed B: High CO2 content (2%) Optimization variable: PTOP Ptop (bar)

2 collocation points

80

P Ethane Recovery

18

43 iter. (3 barrier problems)

76

ηss = 74.5 % 72

16 0

10

20

30

40

Time (min)

45

Ethane recovery (%)

20 finite elements

84

20

Numerical Results: Demethanizer Feed B (high CO2 content) 84

No solubility constraints

P Ethane recovery

20

20 finite elements

Ptop (bar)

80

78 18

2 collocation points

76

59 iter. (4 barrier problems) ηss = 76.6 %

74 16 0

10

20

30

40

50

Time (min)

46

Ethane recovery (%)

Optimization variable: PTOP

82

Numerical Results: Demethanizer Feed A Optimization variable: 178

Reboiler Heat Duty (QR)

57 iter. (5 barrier problems) ηss = 77.8 % (PTOP=19bar)

Reboiler Heat Duty (MJ/min)

2 collocation points

176

Active constraint: Methane mole fraction in bottoms

174

0.007

QREBOILER XCH4

172

0.006

170 0.005

168 166

0.004 164 162

0.003 0

5

10

15

20

25

30

35

40

Time (min)

47

Bottom methane mole fraction

20 finite elements

0.008

Numerical Results: Demethanizer Start Up MPECs Optimization variable:

Three time periods Ideal Gas, Ideal solution

Ethane Recovery

PTOP, QR

Solved in AMPL with IPOPT-C (Raghunathan, Biegler, 2003) Time (min) 48

Numerical Results: Demethanizer Start Up MPECs

PTOP, QR 3rd time period 17944 discretized variables 576 complementarity constraints

Reboiler Heat Duty (MJ/min)

Optimization variable:

97 iterations Time (min) 49

Boiler Dynamic Optimization Rodriguez, Bandoni, Diaz (2005b)

Dynamic optimization model to determine controller parameters Boiler

Flue Gas

Combustion chamber

High Pressure Steam Feed Water

Risers Superheater

E-1

Air

Steam drum

Combustion Chamber

Superheater

RISER 1

RISER 2

Gas path Natural Gas

Furnace Gas

PI Controller (Liquid level in drum) Manipulated: Feed water flowrate

50

Boiler Dynamic Optimization DAE Optimization model ¾ Differential equations • Momentum and energy balances in risers and superheater • Mass and energy balances in drum • Integral part of controller and valve equation ¾ 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

51

Boiler Dynamic Optimization DAE Optimization model ¾ Optimization variables • PI controller parameters ¾ Step change in high pressure steam demand 6 32

4

Liquid Level Drum (ft)

Feed Water Flowrate (lb/s)

30

28

26

24

2

0

-2 22

-4 0

200

400

600

800

1000

0

200

400

Time (s)

600

800

1000

Time (s)

¾ Additional PI controllers (current work) • Outlet steam pressure (fuel gas flowrate) • Superheated steam temperature (air excess)

52

Conclusions ¾ Rigorous dynamic optimization models for cryogenic train in

Natural Gas Processing plant within a simultaneous approach ¾ Thermodynamic models with cubic equation of state ¾ Carbon dioxide solubility addressed in both liquid and vapor phases

as path constraints, at each column stage ¾ Phase detection through the first order optimality conditions for

Gibbs free energy minimization (MPECs) ¾ Boiler dynamic model for controller parameters ¾ Simultaneous approach provides efficient framework for the

formulation and resolution of DAE optimization problems for large-scale real plants

53

References ¾ Biegler, L. T., A. Cervantes, A. Waechter, “Advances in Simultaneous

¾

¾

¾

¾

¾

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, 1997 S. 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, 623641, 2002 54

References ¾ Diaz, S., S. Tonelli, A. Bandoni, L.T. Biegler, “Dynamic optimization for

¾

¾

¾

¾

¾

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, AIChE Annual Meeting, San Francisco, Nov. 16-19, 2003. Advances in Optimization Raghunathan, 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, 2004 Rodríguez, 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 55 18 May 2005b Giardini Naxos, Italy

Appendix: Soave Redlich Kwong Equation of State Compressibility factor L V for a mixture ( zi , zi )

z=

V aα − V − b RgT ( V + b )

aα = ∑ x*j z*j ; x*j =x j a jα j ; z j = ∑ x*k ( 1 − k j ,k ) k

b = ∑ b j ; b j = 0.08664 RgTcj / Pcj j

 0.42747 Rg2Tcj2

aj =   

Pcj

   

0. 5

  T 0.5  ;α j = 1 + m j 1 −      Tcj  

Soave modification (Temperature dependence)

m j = 0.48 + 1.574ω j − 0.176ω 2j 56

Appendix: Soave Redlich Kwong Equation of State Fugacity Coeff. for component j in mixture ( φiV, j ,φiL, j )

*  a z bj   B  2 α A j j j −  ln1 +  ln φ j = ( z − 1 ) − ln( z − B ) −  b B  aα b  z

bj

1 ∆H Residual enthalpy  B  * *  1 + m j  ln1 + ∑ x j z j  − = −1 −  in mixture ( ∆H i , ∆hi ) RgT bRg T  zj α j  

A=

bP aαP = ; B Rg2T 2 RgT 57