dynamic modeling and optimization of large-scale cryogenic processes

Dynamic Modeling and Optimizationof Large-Scale Cryogenic Processes

Maria Soledad Diaz

Planta Piloto de Ingeniería Química Universidad 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• Boiler dynamic optimization for controllers parameters

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 Processing Plants

Provide ethane as raw material for olefin plants (ethylene) and petrochemical

Technology: Turboexpansion High pressure Cryogenic conditions

5

NATURAL GAS PLANT

Pipelines

DEHYDRATION

TRAIN B

TRAIN A

TRAIN C

Ethane Treatment CO2

Propane

Butane

Gasoline Separation Fractionation

Ethane To Ethylene Plant

Absorption Plant

TRAIN A

TRAIN B

Compression/Recompression

DEHYDRATION

C2 C3 C4

C3

C3

C4

C4

To Recompression

6

Natural gas composition

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

7

Cryogenic Sector

Turboexpander

HighPressureSeparator

to deethanizer column

Cryogenic Heat Exchangers

Demethanizer

8

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)

9

Dynamic Optimization Problem

min (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

10

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)

11

Simultaneous Dynamic Optimization

Discretization of differential equations (DAEs) and state and control variables

Large-scale nonlinear problem Profile constraints handled directly Direct incorporation of equipment design variables DAE model solved only once Converges for unstable systems

12

Nonlinear Programming Problem

Application of Barrier method

0)(s.t xc

0 x

n

jjslnxfmin

1)(

0)(s.t xc

0 xs

As 0, x*() x*

13

Ts2_i = 322 K

ht

V

ToRecompression

ToTurboexpander

Natural Gas

Residual Gas

To Demethanizing Column

Cryogenic Heat Exchangers and HP Separator

Phase Change

Horizontal High Pressure Separator

Rodriguez, Bandoni, Diaz (2004)

14

Energy Balances: Partial Differential Equations System

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

A*hz

)t,z(Ttvtt

)t,z(Tt

tt

supt

HE Model (no Phase Change)

Spatial Discretization: Finite Differences ODE System

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

A*hz

)t,z(Tsvst

)t,z(Ts

ss

sups

Tube side

Shell side

15

Energy Balance at cell i

iissi

supsii

ii TsTtL*A*Cp*s

A*hTsTs

zvs

dtdTs

1

HE Model (no Phase Change)

E-1

Ts1 Ts2 Ts3 Ts4 Ts5 Ts6

Ts1 Ts2 Ts3 Ts4 Ts5 Ts6

Tt1 Tt2 Tt3 Tt4 Tt5 Tt6

Cold streamat Tt0

Hot streamat Ts0

iitti

suptii

ii TtTsL*A*Cp*t

A*hTtTt

zvt

dtdTt

1Tube side

Shell side

Multicell model

16

i,jjji,j T*BAz

ji,j

ji,j A*

Fv

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

17

Energy Balance at cell i

Algebraic Equations at cell i

tiiiiiiiii

i + Q*H - V*h - L*Hp + Vp*hpLpdt

dE

HE Model (Partial Phase Change on Shell Side)

j,ij,ij,i xKy

iidealii HHH

iidealii hhh

Vj,i

Lj,i

j,iK

i,mli,t T*A*UQ 0

j,i

nc

ji

idealj,i

ideali x)T(hh

1

j,i

nc

ji

idealj,i

ideali y)T(HH

1

j

j,ij

j,i xy 0

Rodriguez, Bandoni, Diaz (2005)

18

Compressibility factor

Residual enthalpies

Fugacity coefficients


)x,Ps,Ts(zz

)y,Ps,Ts(zz

iiiLL

i

iiiVV

i

)x,Ps,Ts(

)y,Ps,Ts(

iiiLL

j,i

iiiVV

j,i

Soave-Redlich-Kwong EoS

Li

iLi

Li

i PmolsTs*R*z*sPs V

i

iVi

Vi

i PmolsTs*R*z*sPs

)x,Ps,Ts(hh

)y,Ps,Ts(HH

iiii

iiii

19

)(

)(**6.02)(

2

iimN

iPV

Tcww

432 )Re()Re()Re()Re()(

iE

iD

iC

iBAiKf

0max *)(*)(

)Re(DiVi

i V

)(**)(*)(*)( iPCRliPBiPAiP cwcs

2

2 )i(V)*i(*)i(Kf*NKa)i(P maxVcc


Bell-Delaware method Pressure drop Pi

20

- VLp+ Vp - Ldt

dM

L

LinV

MLongDVol

2

2

2222

ttt

L

LL hrh

rharccosrr

PmolLongM

VVV VolM LV MMM

r

ttsv

hPPCL

High Pressure Separator

Horizontal tank

21

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

1

e

s

e

s

PP

TT

22

iiiiii LVLVF

dtdM 11

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

dtdm

1111 j=1,…,ncomp

iFiiFiiiiiiiiiiii QsrhHFhLHVhLHV

dtdE 11111

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

Demethanizing Column

Diaz, Tonelli, Bandoni, Biegler (2003)

N = 8; ncomp = 10

23

Algebraic Equations

Vj,i

Lj,i

j,iK

j,ij,ij,i xKy

iidealii HHH

iidealii hhh


Compressibility factor ( )

Residual enthalpies (Hi, hi)

Fugacity coefficients ( )

Li

Vi zz ,

Lji

Vji ,, ,

SRK

j

j,ij

j,i xy 0

24

Vapor volume


Li

Li

i,plateiV

iMHDVol

2

2

Vi

Vi

Vi VolM

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

Internal energy holdup

Algebraic Equations

25

Liquid holdup and hydraulic correlations


Liii

iLi HwoHwD.M

2

28960

Li

Li

Vi

Vi

iiVi

ii Pmol

PmolCoAhole

V..Ho

2410565

32

3201495030/

iLi

ii

/i Weirl

LFw.Hwo

iii HwoHwHldrop

2111 VKKPP V

TOP Pressure drop

Li

Liiiii PmolHldropHo..PP

5

1 10249170

26

Phase Equilibrium

Carbon dioxide solubility constraints

SCO,i

LCO,i

VCO,i fff 222


SCO,i

LCO,i

VCO,i 222

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

27

Solid phase

VCO,i

SCO,i

SCO,i Pf 222

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

Carbon dioxide solubility constraints SCO,i

VCO,i ff 22


28

Gibbs Free Energy minimization at stage i

Demethanizing Column: Phase Existence

Raghunathan, Diaz, Biegler (2004)

Karush Kuhn Tucker conditions

Raghunathan, Biegler (2003)

MPECs as additional constraints

IPOPT-C under AMPL

29


Relaxed Equilibrium Conditions

j,ij,ij,iiij,iij,i x)y,x,P,T(Ky

Li

Vii ss 1

0Li

Li Ms

0Vi

Vi Ms

00 Li

Vi s,s

,M,M

M,M

M,M

Vi

Li

Vi

Li

Vi

Li

001

001

001

30


Liquid Holdup and Hydraulic correlations

0 Li

Li MM

L

iLi

Lii

iLi MMHwD.M

2

28960

Lii

iLi HwoD.M

2

28960

32

3201495030/

iLi

ii

/i Weirl

LFw.Hwo

00 Li

Li M,M

2

3

)M(kL Lidi

31

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

System DAE

.stdtTTmin

out

tfSP

3063038058060

20

Alg. Eqns = 403Differ. Eqns = 37

Optimization Problem: HEs + HPS

G

ht

L

T

Tout

(10+8 cells in HEs)

32

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

)

33


Optimization variables: G, L

0 5 10 15 20 25 307071727374757677787980818283

Tout

G0

G0 (

Km

ol/m

in)

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)

34

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


35

Pressure profile, shell side Condensed fraction, 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

36

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

Alg. Eqns = 385Differ. Eqns = 97

Optimization Problem: Demethanizing Column

37

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)ss = 80.9 %

Feed A: Low CO2 content (0.65%)

38


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)

ss = 74.5 %

39


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



ss = 76.6 %

40


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

methane m

ole fraction

Feed A

Optimization variable:

Reboiler Heat Duty (QR)


ss = 77.8 % (PTOP=19bar)

Active constraint: Methane mole fraction in bottoms

41

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

42

Numerical Results: Demethanizer Start Up

MPECs


PTOP, QR

3rd time period17944 discretized variables576 complementarity constraints97 iterations

Time (min)

Reb

oile

r Hea

t Dut

y (M

J/m

in)

43

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

44


DAE Optimization model Differential 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

45


DAE Optimization model Optimization 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)

46

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 addition of MPECs for Gibbs free energy minimization

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

47

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, 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, 623-641, 2002

48

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, 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 - 18 May 2005b Giardini Naxos, Italy

49

Appendix: Soave Redlich Kwong Equation of State

)bV(TRa

bVVz

g

k

k,j*kjjjj

*j

*j

*j )k(xz;axx;zxa 1

j

cjcjgjj P/TR.b;bb 086640

505022

11427470

.

cjjj

.

cj

cjgj T

Tm;P

TR.a

217605741480 jjj ...m

Compressibility factor for a mixture

Soave modification (Temperature dependence)

)z,z( Vi

Li

50

Appendix: Soave Redlich Kwong Equation of State

zBln

bb

aza

BA)Bzln()z(

bb

ln j*jjjj

j 12

1

j j

j*j

*j

gg

mzx

zBln

TbRTRH

1

111

TRbPB;

TRPaA

gg

22

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

j,iV

j,i

Residual enthalpy in mixture )h,H( ii

dynamic modeling and optimization of large-scale cryogenic processes

Documents

bandoni brignole

p stfdztdt

algebraic variables

dynamic modeling

olefin plants ethylene

control variablesy

state variables collocation

differential variablesu