simone (equations and methods)

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SIMONE

Equations andMethods

ver. 5

rel. November 2002

© SIMONE Research Group s.r.o. & LIWACOM Informationstechnik GmbH



SIMONE Equations and methods

© SIMONE Research Group, s.r.o.

2

This issue has a special position within the SIMONE documentation library. While the

purpose of all the other issues is to support the user by information practical for real use, this

text offers the information about basic equations which the SIMONE system is based on:

while all other is-sues treat the fruits of the SIMONE tree shown bellow, this issue is

dedicated to its roots.

It is definitely out of the possibilities of this material to describe the trunk of the tree – the

theory behind SIMONE which is considerably complicated and abstract. SIMONE is based

on several original methods tailor-made for the pipeline networks. The main are the

integration method and the method for sophisticated exploitation of the sparseness of the

matrix.

Those who are interested only to know how it works but not why it works so well are

recommended to ignore this issue and to turn attention only to other sheets of the SIMONE

documentation library.

1

M o m e n t u

m

M a s s

S t a t e

H e a t

I n t e g r a

t i o n

D e c o

m p o s i t i o

n

O p t i m

i z a t i o

n

E q u a

t i o n

B a l l a

n c e

B a l l a n c e

E q u a t i o n

M e t h o d M

e t h .

M e t h .

Error

Identific. Leak

Detection

State

Reconstr-uction

Heat

Dynamics

Quality

Tracking

Compressor

Stations

Mixing

Stations

Logic

Results

Monito-

ring

Topo

Visuali-

zation

Set-Point

Optimization

Config.

Optimization

Financial

Optimization

Prefrencial

Optimiza-

tion

Automatic

Optimum

Design

Network

Adjustment

State to State

Optimization



SIMONE Equations and Methods

© LIWACOM Informationstechnik GmbH

3

Contents

1. Units .......................................................................................................................................7

1.1. Gauge pressure for low-pressure networks ................................................................7

2. Flow equations ....................................................................................................................... 8

2.1. Continuity equation .................................................................................................... 8

2.2. Momentum equation ..................................................................................................8

2.3. Friction factor .............................................................................................................9

2.3.1. Hofer formula..................................................................................................... 9

2.3.2. Nikuradze formula............................................................................................10

2.3.3. PMT-1025 formula........................................................................................... 10

2.3.4. Pipe efficiency.................................................................................................. 10

2.3.5. Comparison of Hofer and Nikuradze formulae................................................ 10

3. Equation of state for real gas ................................................................................................ 13

3.1. AGA (American Gas Association)........................................................................... 14

3.2. Papay formula .......................................................................................................... 15

3.3. Redlich-Kwong equation..........................................................................................15

3.4. BWR equation .......................................................................................................... 16

3.4.1. Simple fluid approach ...................................................................................... 16

3.4.2. Full mixing rules ..............................................................................................17

3.5. AGA8 DC92 equation.............................................................................................. 18

3.6. Thermodynamic properties of real gas.....................................................................18

3.6.1. Specific heat ..................................................................................................... 18

3.6.2. Isentropic exponent ..........................................................................................19

3.6.3. Joule-Thomson coefficient ............................................................................... 20

3.7. Calculation of gas properties using gas composition...............................................21





4

3.7.1. Calculation gas relative density and volumetric heating value using gas

composition ...................................................................................................................... 21

3.7.2. Conversion of molar to volumetric fractions and vice versa ........................... 23

3.8. Gas viscosity ............................................................................................................ 23

3.8.1. Equation Lee64 ................................................................................................ 24

3.8.2. Equation Lee66 ................................................................................................ 24

4. Quality Tracking ..................................................................................................................25

4.1. Solution procedure ................................................................................................... 25

4.2. Quality parameters ................................................................................................... 26

4.3. Mixing rules .............................................................................................................26

5. Heat Dynamics ..................................................................................................................... 29

5.1. Differential Equations ..............................................................................................29

5.2. Heat Transfer Coefficient in SIMONE ....................................................................31

5.3. Suppressing Joule-Thomson effect for pipes ........................................................... 33

6. Compressor Stations............................................................................................................. 34

6.1. Basic thermodynamics relations...............................................................................34

6.1.1. Pressure ratio .................................................................................................... 34

6.1.2. Adiabatic head.................................................................................................. 34

6.1.3. Adiabatic efficiency .........................................................................................34

6.1.4. Shaft power ...................................................................................................... 34

6.1.5. Discharge temperature...................................................................................... 35

6.1.6. Isentropic exponent ..........................................................................................36

6.1.7. Volumetric flow rate ........................................................................................ 37

6.2. Centrifugal compressor ............................................................................................37

6.3. Reciprocating compressor ........................................................................................38

6.4. Gas turbine ...............................................................................................................39





5

6.5. Gas engine ................................................................................................................39

6.6. Electro drive .............................................................................................................40

6.7. Steam drive (combined cycle).................................................................................. 41

6.8. Gas cooler................................................................................................................. 41

6.9. Local pressure loss ................................................................................................... 42

7. Controlled valves..................................................................................................................43

7.1. Preheating power...................................................................................................... 43

7.2. Valves with characteristic ........................................................................................43

7.2.1. Characteristic A (Mokveld).............................................................................. 44

7.2.2. Characteristic B (Argus)................................................................................... 45

7.3. Local pressure loss ................................................................................................... 45

8. Resistor................................................................................................................................. 47

9. Solution methods..................................................................................................................48

10. State reconstruction ............................................................................................................ 50

10.1. State reconstruction in SIMONE.......................................................................... 51

10.2. Outlying Measured Data ......................................................................................52

10.3. State Reconstruction Parameters .......................................................................... 52

10.4. Tools for Scanning State Reconstruction ............................................................. 53

11. Steady-State Optimization.................................................................................................. 55

11.1. Objective Function ............................................................................................... 55

11.2. Constraints............................................................................................................ 55

11.3. SIMONE Method of Steady-State Optimization ................................................. 55

12. Hydrate formation risk analysis .........................................................................................57

12.1. Equilibrium condition .......................................................................................... 57

12.2. Water dew point ................................................................................................... 58





6

12.2.1. Bukacek aprroximation .................................................................................... 58

12.3. Hydrate equilibrium .............................................................................................59

12.3.1. Motiee approximation ...................................................................................... 59

12.3.2. Carson-Katz method......................................................................................... 60

12.3.3. Ponomarev approximation ............................................................................... 60

12.3.4. Remarks on accuracy and choice of models .................................................... 61

12.4. Effect of inhibitors ............................................................................................... 63

12.5. Risk indicators...................................................................................................... 64





7

1. Units1.1. Gauge pressure for low-pressure networks

For low-pressure networks the local pressure is expressed in terms of overpressure in the

order of up to 5 kPa. The effect of lower gas density with respect to air density (i.e. lift force)

therefore leads to non-negligible changes of local overpressure with elevation.

SIMONE calculates all pressures as absolute ones. For correct simulation of low-pressure

systems the local air pressure depending on node elevation is used for conversion between

absolute and gauge value of pressure. This is done by selection of “kPah” pressure unit. This

unit can be used both for value visualization and for input of pressure conditions or set points.

The conversion is based on the following equation

( )h p p p air −=l

Here

l p Local overpressure [kPah]

p Absolute gas pressure [kPa]

( )h pair Local air pressure (absolute) depending on elevation [kPa]

h Elevation [m]

The elevation change of air pressure is derived form the barometric equation using the

definition of International Standard Atmosphere:

( )

−⋅=

0

0 expT R

gh ph p

air

air

325.1010 = p kPa is air pressure at sea level

150 =t oC is air temperature at sea level (used for the barometric equation)

15.27300 += t T K

89.286=air R J.kg -1.K -1 is specific gas constant of air

9.80665= g m.s -2 is gravity acceleration





8

2. Flow equations

Simulation of dynamic processes running with gas transport and distribution is based on the

non-linear partial differential equations describing the dynamics – continuity equation and

momentum equation.

For a more sophisticated and detailed description of the heat dynamic processes running with

the gas flow, another partial differential equation must be linked to those mentioned above –

see sections 4 and 5.

2.1. Continuity equation

0=+t

S x

m

∂

∂ρ

∂

∂

Here

m Mass flow rate [kg.s-1]

S Pipe cross section [m2] ρ Gas density [kg.m-3] x Lengths coordinate [m]t Time [s]

2.2. Momentum equation

021 2

=+++−− R f dx

dh g x

P

xwt wt

m

S ρ ∂

∂

∂

∂ρ

∂

∂ρ

∂

∂

Here

p Pressure [Pa] g Gravity acceleration [m.s-2] ρ Gas density [kg.m

-3]

R f Pressure loss due to friction per unit of pipe length [Pa.m-1]

h Geodetic height [m]S Pipe cross section [m

2]

m Mass flow rate [kg.s-1]w Flow velocity [m.s-1] x Lengths coordinate [m]t Time [s]

The unit hydraulic resistance in a pipeline with a circular cross section is described by Darcy-Weisbach equation

ρ λ

D

ww f R

2

||=





9

Here

λ Friction factor [1]w Flow velocity [m.s-1] ρ Gas density [kg.m-3]

D Pipe internal diameter [m]

2.3. Friction factor

SIMONE is equipped with several formulae for the friction factor λ (being selected using the

LAMBDA scenario parameter):

• Hofer formula (default)

• Nikuradze formula

• PMT-1025 formula (Gazprom)

Any other formula can be built into the system on user’s request.

2.3.1. Hofer formula

Hofer formula is an explicit approximation of the general Colebrook-White formula.

2

71.37

Relog

Re

518.4log2

1

+

=

D

k λ

Here

k Pipe equivalent roughness [m]

D Pipe internal diameter [m]Re Reynolds number [1]

Reynolds number is defined

D

m

v

Dw

πµ

4Re ==

Here

D Pipe internal diameter [m]w Flow velocity [m.s-1]ν Gas kinematical viscosity [m

2 s

-1]


Gas dynamic viscosity [kg.m-1

.s-1

]

The value of dynamic viscosity for natural gas is approximately 10-5

kg.m-1

.s-1

.





10

2.3.2. Nikuradze formula

Nikuradze formula, valid for high Reynolds numbers, has the form

2

138.1log2

1

+

=k

Dλ

It is easy to see that as +∞→Re the Hofer formula is equal to Nikuradze.

2.3.3. PMT-1025 formula

2.02

Re

158067.0

+= D

k λ

Here the pipe equivalent roughness is taken constant

mm03.0=k

2.3.4. Pipe efficiency

The pipe efficiency pipeη is usually introduced as correction factor to λ 1 in the momentum

or Darcy-Weisbach equations, i.e. the friction term becomes

ρ η λ

Dww f

pipe

R2

||2

=

There is simple and straightforward link between flow rate and pipe efficiency for steady

flow: If one assumes the pipe efficiency 0.99 instead of 1.00, then the flow rate for constant

pressure drop is also reduced to 0.99 of its original value.

The pipe efficiency can be altered by setting CORLAM scenario parameter for subsystem or

individual pipe element (the default value is 1).

2.3.5. Comparison of Hofer and Nikuradze formulae

The following comparison of both formulas is presented to illustrate the differences between

them and their advantages and shortcomings.





12

To get rough figures about flow rate and Reynolds number see the following graph (drawn for

natural gas of normal density 0.73 kg/m3, i.e., relative density 0.565):

Reynolds number / flow rate

10

100

1000

10000

100000

1.00E+06 1.00E+07 1.00E+08 1.00E+09

Re

Q[1000m3/h]

D=1400 mm

1200 mm

900 mm

500 mm

300 mm

For D=1000 mm, k=0.01 mm, and usual load of about 106 m 3/h results Re=2.4×107 and

therefore the non-negligible difference between Hofer and Nikuradze can be seen!





13

3. Equation of state for real gas

The equation of state for real gas is usually written in the form:

RTz p ρ =

Here

p Pressure [Pa] ρ Gas density [kg.m-3]

R Gas constant [J.kg-1.K -1]

T Absolute temperature [K]

z Compressibility factor [1]

The gas constant is related to molar weight or relative density (specific gravity) of gas

M

R R

~

=

air

air air air

air M z

M z

R z

R z r

0

0,

0

0,

0,

0 === ρ

ρ

Here

R~

Universal gas constant [J.kmol-1.K -1]

Molar weight [kg.kmol-1]r Gas relative density [1] ρ Density [kg.m-3]


Subscript 0 Stands for standard condition (101.325 kPa and 0, 15 or 20ºC)

Subscript air Stands for air

The non-ideal behavior of the gas is expressed by the compressibility coefficient z . The

compressibility coefficient is function of pressure, temperature and composition of gas. The

following choices are available in SIMONE:

• 2-parametric equations of state:

( )r r T p z z ,=

Here

c

r p

p p = Dimensionless reduced pressure [1]

c

r

T

T T = Dimensionless reduced temperature [1]





14

∑=

= N

i

icic p x p1

, Pseudo critical pressure of gas mixture [Pa]

∑=

= N

i

icic T xT 1

, Pseudo critical temperature of gas mixture [K]

N x x K1 Molar fractions of gas components [1]

ic p , Critical pressure of component i [Pa]

icT , Critical pressure of component i [K]

• 3-parametric equations of state:

( ),, r r T p z z =

Here

∑=

= N

i

ii x1

ω ω Acentric factor of gas mixture [1]

iω Acentric factor of component i [1]

• General equation of state:

( ) N x xT p z z K1,,=

More complex mixing rules are used than the above mentioned.

The following equations of state are currently available in SIMONE (being selected by the

ZET scenario parameter):

• AGA formula

• Papay formula (default)

• Redlich-Kwong equation

• BWR in simplified form (Fasold et al.)

• BWR

• AGA8 DC92 (ISO 12213-2)

Other equations of state can be implemented on user’s demand.

3.1. AGA (American Gas Association)

r

r r

T

p p z 533.0257.01 −+=

For natural gases, this relationship is adequate for pressures up to 70 bars.





15

3.2. Papay formula

( ) ( )r r r r T pT p z 878.1exp274.0260.2exp52.31 2 −+−−=

For natural gases, this relationship is adequate for pressures up to 150 bars.

3.3. Redlich-Kwong equation

Redlich-Kowng equation of state is usually written in the following form

)~

(~~

~

bV V T

a

bV

T R p

+−

−=

Here

p Pressure [Pa]

T Temperature [K]

R~

Universal gas constant [J.mol-1.K -1]

V ~

Molar volume of gas [m3.mol-1]

c

ca

p

T Ra

5.22~

Ω=

c

cb

p

T Rb

~

Ω=

( )12913/1 −=Ωa

3

123/1 −

=Ωa

c Stands for pseudo critical values of gas mixture

For pure gases, Redlich-Kwong equation is the most precise 2-constant equation of state.

Redich-Kwong equation is currently used in simplified form with the simple definition of

pseudo critical pressure and temperature:

∑

∑

=

=

=

=

N

i

icic

N

i

icic

T xT

p x p

1

,

1

,

In such a case, Redlich-Kwong equation is used as 2-parametric equation of the

form ( )r r T p z z ,= .

The precise more complex mixing rules or other modifications of Redlich-Kwong equation

can be implemented on user’s demand.





16

3.4. BWR equation

Benedict-Webb-Rubin equation of state is usually written in the following form (and

traditional physical units!)

( ) ( ) ( )22

2

36320

00~exp~1

~~~~~~~~ ρ γ ρ γ

ρ ρ α ρ ρ ρ −+++−−

−−+=

T

caaT Rb

T

C AT R BT R p

Here

p Pressure [atm]

T Temperature [K]

R~

Universal gas constant [atm.dm3 mol

-1.K

-1]

ρ ~ Molar density of gas [mol.dm-3]

0 A , 0 B , 0C , a ,b , c ,α ,γ BWR equation constants (traditional units)

3.4.1. Simple fluid approach

Following Fasold et al., the constants for BWR equation can for natural gases be linked to

three parameters of the gas mixture:

c

c

p

RT A A

22

00

~

′= 6900000.006000000.00 +−=′ c A α

c

c

p

RT B B

~

00

′= 0781383.000797872.00 +=′ c B α

c

c

p

RT C C

24

00

~

′= 0340000.002000000.00 +=′ cC α

2

33 ~

c

c

p

RT aa ′= 2128330.004083330.0 −=′ ca α

2

22 ~

c

c

p

RT bb ′= 0706000.001700000.0 −=′ ca α

2

35 ~

c

c

p

RT cc ′= 0980000.002400000.0 −=′ ca α

3

33 ~

c

c

p

RT α α ′= 0138333.000183333.0 +−=′ cα α

2

22 ~

c

c

p

RT γ γ ′= 1332000.001400000.0 +−=′ cα γ

∑=

= N

i

icic p x p1

, Pseudo critical pressure of gas mixture [atm]

∑=

= N

i

icic T xT

1

, Pseudo critical temperature of gas mixture [K]





17

∑=

= N

i

icic x1

,α α Critical Riedel factor for gas mixture [1]

ic,α Critical Riedel factor for component i

N x x K1 Molar fractions of gas components

Critical Riedel factor is calculated using the acentric factor by equation of Edmister

ω α 8763.47839.5 +=c

As a result, this simplification leads to 3-parametric equation of the form ( )ω ,, r r T p z z = .

3.4.2. Full mixing rules

The general mixing rules for BWR equation of state are

2

1

,00

= ∑

=

N

i

ii A x A

∑=

= N

i

ii B x B1

,00

2

1

,00

= ∑

=

N

i

ii C xC

3

1

3

= ∑

=

N

i

ii a xa

3

1

3

= ∑

=

N

i

ii b xb

3

1

3

= ∑

=

N

i

ii c xc

3

1

3

= ∑

=

N

i

ii x α α

2

1

=

∑= N

i

ii x γ γ

Here

N x x K1 Molar fractions of gas components [1]

Subscript i Stands for BWR equation constants for component i

The constants of BWR equation for basic 21 gas components listed in ISO 12213 are predefined in SIMONE. The user has the possibility to add new gas component and enter the

constants for it.





18

3.5. AGA8 DC92 equation

The expansion of compressibility factor according to AGA8 DC92 (ISO 12213-2) equation is

( ) ( )nnn k

r n

b

r

k

r nnnn

nn

nr ck cbC C B z ρ ρ ρ ρ ρ −−+−+= ∑∑ =

∗

=

∗

exp~

1

58

13

18

13

Here


B Second virial coefficient (function of temperature and

composition)[dm

3.mol

-1]

ρ ~ Molar gas density [mol.dm-3

]

r ρ Dimensionless reduced density [1]

nb , nc , nk Constants [1]∗nC Coefficients dependent on temperature and composition [1]

The reduced density is related to molar density by equation

ρ ρ ~3 K r =

Here

3 K Mixture size parameter (function of composition) [dm3.mol

-1]

The procedure for calculation of composition and temperature-dependent coefficients is rather

complicated. For further details please refer to ISO 12213-2.

3.6. Thermodynamic properties of real gas

All important thermodynamic properties of gas can be derived from the equation of state,

namely

• Specific heat

• Isentropic exponent

• Joule-Thomson coefficient

3.6.1. Specific heat

Molar gas specific heat of real gas is calculated by

( ) ( ) N p p p x xT pcT cc ,,,,~~~1

0K∆+=

Here

pc~ Molar specific heat (for constant pressure) [J.mol-1.K -1]





19

( )T c p0~ Molar specific heat of ideal gas [J.mol

-1.K

-1]

( ) N p x xT pc ,,,,~1 K∆ Correction to real gas behavior [J.mol

-1.K

-1]

Specific heat per unit of mass is then

M

cc

p

p

~=

pc Specific heat of real gas [J.kg-1

.K -1

]

Molar weight of gas [kg.mol-1]

The ideal gas specific heat is a function of temperature and in SIMONE is approximated by

second-order parabola:

( )20~

T C T B AT c p ⋅+⋅+=

For gas mixture the coefficients are simply mixed form pure component values

∑=

=

N

i

i

i

i

i

C

B

A

x

C

B

A

1

Here

N x x K1 Molar fractions of gas components

Subscript i Stands for mixture component i

The real gas correction can be expressed by equation

p

dp

T

z T

T

z T Rc

p

p p

p ∫

∂

∂+

∂∂

−=∆0

2

222

~~

Here

R~

Universal gas constant [J.mol-1

.K -1

]


3.6.2. Isentropic exponent

Isentropic exponent κ is defined by relations

.~

const V p =κ or

S V

p

p

V

∂

∂−= ~

~

κ

and can be calculated by the formula





20

2

~

~

T

p

p Z c

R Z

z

−

=κ

Here

p Pressure [Pa]

V ~

Molar volume [m3.mol-1]

S Entropy [J.mol-1

.K -1

]


N x xT

p p

z p z Z

K1,

∂∂

−= Dimensionless derivative of compressibility factor [1]

N x x p

T T

z T z Z

K1,

∂∂

+= Dimensionless derivative of compressibility factor [1]

R~


pc~ Molar specific heat of real gas (for constant pressure) [J.mol-1

.K -1

]

3.6.3. Joule-Thomson coefficient

The (differential) Joule-Thomson coefficient is defined as

H

JT p

T

∂∂

=µ

and can be calculated from the equation

( ) z Z c

R

p

T T

p

JT −= ~

~

µ

Here

JT Joule-Thomson coefficient [K.Pa-1] p

Pressure [Pa]T Temperature [K]

H Enthalpy [J.mol-1.K -1]


N x x p

T T

z T z Z

K1,

∂∂

+= Dimensionless derivative of compressibility factor [1]

R~


pc~ Molar specific heat of real gas (for constant pressure) [J.mol-1.K -1]

The integral Joule-Thomson effect, i.e. temperature change caused by pressure drop during

isenthalpic process, can be described by the initial-value problem





21

( )

( )

( )22

11

,

pT T

T pdp

dT

T pT

JT

=

=

=

µ

Here

1 p Starting pressure [Pa]

1T Starting temperature [K]

2 p Final pressure [Pa]

( ) pT Solution of the initial-value problem

2T Final temperature [K]

3.7. Calculation of gas properties using gas compositionIn gas transport & distribution, volumetric quantities and units are widely used:

• Flow rates and line pack are expressed in standard volumetric units (defined for

standard condition – pressure 101.325 kPa and temperature typically 0, 15 or 20ºC)

• Volumetric heating value – per unit volume in standard condition; thermal billing is

based on volumetric flow rate and volumetric heating value

• Relative density is used to convert volume of gas to mass.

• The composition of gas can be expressed in volumetric fractions (defined for standard

condition) rather then in molar or mass fractions.

On the other hand, all equations mentioned above (flow equation, equations of state) are

based on mass or molar quantities.

SIMONE supports the conversion of molar quantities to volumetric and vice versa using

formulae consistent, whenever possible, with ISO 6976 norm.

3.7.1. Calculation gas relative density and volumetric heating valueusing gas composition

Gas relative density r is related to gas molar mass

air

air

M z

M z r

0

0,=

Here

0,air z Compressibility factor of air at standard condition [1]

( ) 99941.0C0,0 =° p z air

( ) 99958.0C15,0 =° p z air

( ) 99963.0C20,0 =° p z air





22

kPa325.1010 = p

air M Molar weight of air [kg.kmol-1]

9626.28=air M kg.kmol-1

0 z Compressibility factor of gas at standard condition [1]

Molar mass of gas mixture [kg.kmol-1]

∑=

= N

i

ii M x M 1

i x Molar fractions of mixture components [1]

i M Molar weights of mixture components [kg.kmol-1]

Molar gas heating value is the amount of heat released by burning 1 kmol of gas at constant

pressure kPa325.1010 = p while the temperature of gas and air before combustion and the

temperature of combustion products are the same:

[ ] H U T H ~

[MJ.kmol-1]

H T Temperature of combustion products (0, 15, 20 or 25oC)

SIMONE uses the upper (superior) heating value (i.e. all water in combustion products is

assumed to condense) at the temperature C25°= H T .

Molar heating value for gas mixture is given by simple weighted sum of molar heating values

of individual mixture components:

∑=

= N

i

iU iU H x H 1

,~~

Volumetric gas heating value is the amount of heat released by burning 1 m3 std. of gas at

constant pressure kPa325.1010 = p (while the temperature of gas and air before combustion

and the temperature of combustion products are the same):

[ ]0,T T H H H U U = [MJ/m3 std.]

0T Temperature at standard condition (typically 0, 15 or 20ºC) used to define

1 m3 std.

The relation of molar and volumetric heating value is given by the formula

[ ] [ ]

0

0

0

0 ~

~

,T R

p

z

T H T T H H U

H U =

For the above listed conversion formulae, the compressibility factor of gas at standard

condition 0 z is necessary.

For known composition of gas mixture, the compressibility factor at standard condition can be

calculated by the ISO 6976 formula





23

2

1

0 1

−= ∑

=

N

i

ii b x z

ib Summation factor for component i [1]

The data for individual components (heating values iU H ,

~; summation factors ib for 0, 15 and

20ºC) are taken from ISO 6976:1995(E), Table 1-3.

3.7.2. Conversion of molar to volumetric fractions and vice versa

Volumetric fractions i y are defined for standard condition kPa325.1010 = p , 0T (typically 0,

15 or 20ºC). The conversion formulae are

N i

z

y

z

y

x N

j j

j

i

i

i ,,1,

1 ,0

,0K==

∑=

N i

z x

z x y

N

j

j j

ii

i ,,1,

1

,0

,0K==

∑=

Here

i z ,0 Compressibility factor of component i at standard condition [1]

The values of i z ,0 are taken from ISO 6976:1995(E), Table 2.

3.8. Gas viscosity

By default, the gas viscosity is taken as constant entered within the network editor. However

for the case of e.g. high-pressure systems with coated pipes the effect of varying viscosity can

be included.

SIMONE supports currently two classic correlations for gas viscosity:

• Lee64 – Lee, Starling, Dolan, Elington (1964)

• Lee66 – Lee, Gonzalez, Eakin (1966)

Any user-preferred equation of the form

( ) xT ,, ρ = or ( ) xT p ,,µ =

can be easily included as well. Here





24

Dynamic viscosity [Pa.s] ρ Gas density (mean value for pipe element) [kg.m-3] p Pressure (mean value for pipe element) [bar]

T Gas temperature (mean value from temperature profile for pipe element) [K] x Gas composition – vector of molar fractions [1]

3.8.1. Equation Lee64

This correlation for viscosity of natural gases was published in 1964 by Lee, Starling, Dolan,

Elington (AIChE Journal, Sept. 1964, 10(5), pp. 694-697). Gas viscosity is expressed in terms

of density, temperature and molar mass:

( )( )

X Y

M T

X

T M

T M K

X K Y

04.011.1

0095.05.191457.2

9.124.122

0063.077.7

exp

5.1

+=

++=

+++

=

⋅⋅= ρ µ

Here

Gas dynamic viscosity [µPoise=10-7Pa.s] ρ Gas density [g.cm-3]

T Gas temperature [°R]

Gas molar mass [g.mol-1]

3.8.2. Equation Lee66

The correlation by Lee, Starling, Dolan, Elington (1964) has been modified by Lee, Gonzalez,

Eakin (JPT August 1966, pp. 997-1000) for better fit with density calculated by simple

methods (Theorem of Corresponding States). Later on, the coefficients were mentioned in

slightly modified form by Danesh (Danesh: PVT and Phase Behavior of Petroleum Reservoir

Fluids, Elsevier, 1998):

( )

( )

X Y

M T

X

T M T M K

X K Y

2.04.2

01009.04.986

448.3

26.192.2090160.0379.9

exp

5.1

−=

++=

+++=

⋅⋅= ρ µ

Here again

Gas dynamic viscosity [µPoise=10-7Pa.s] ρ Gas density [g.cm-3]

T Gas temperature [°R]Gas molar mass [g.mol-1]





25

4. Quality Tracking

To describe the dynamics of particular components of a gas mixture within the pipeline

system in an exact way, the mathematical model must contain the mass balance equation for

each component:

N it

cS

x

mc ii K1,0 ==+∂

∂ρ

∂

∂

Here


S Pipe cross section [m2] ρ Gas density [kg.m-3]

N i cc K Mass fractions of gas components [1] x Lengths coordinate [m]t Time [s]

The momentum equation remains untouched.

4.1. Solution procedure

The number of the above mentioned additional partial differential equations would complicate

the solution essentially. Therefore, an approximate solution has been adopted, based on the

following presumptions:

• The quality signal is discrete, i.e., its value is known only for the discrete time of

sampling and no information exists within the sampling period;

• Neither the time step of sampling need not be equal nor to the time step of calculation

nor to the sampling period of other variables.

The value of the sample of quality parameter is related to an infinitesimal volume of gas just

present at the sampling time in the place of sampling – quality flag . The quality flag is

characterized by its position and by a set of values corresponding to the respective tracked

quality parameters.

The quality tracking is then formulated as a task to

• Follow the movement of all quality flags entering the network in the supply points

until they leave the system

• Simulate the mixing of quality values in all nodes of a network to which more than

one gas stream is linked to.

The moving of quality flags is simulated by a two steps procedure:

1. In each time step of simulation or state reconstruction, all the state variables for each pipe element are calculated. Being the calculation finished, the flow-speed values are

used by the quality tracking procedure to calculate the new positions of all quality





26

flags, the mixing of quality parameters in the crossings of the network, and the

interpolated quality values in nodes.

2. As some parameters of the simulation model itself depend on the gas parameters, (gas

composition, relative density, heating value, critical temperature and critical pressure

etc.), all these values are used in the next step of simulation to calculate the new

values of corresponding coefficients.

4.2. Quality parameters

The quality parameters are generally divided into three groups:

1.

Gas physical properties

• Gas relative density

• Gas volumetric heating value

• Pseudocritical temperature of gas mixture

• Pseudocritical pressure of gas mixture• Acentric factor of gas mixture

• Three coefficients of ideal gas specific heat temperature expansion

2. User-defined parameters

• Any user-defined property of gas (per 1 m3 std. of gas)

3.

Gas composition

• Molar fractions of gas components

Such partitioning corresponds to the quality tracking options being set for the individual

network in topoeditor. The available options are:

• Uniform gas quality – the network is filled by gas of one composition only.• Gas physical properties tracking – various gases at individual supplies can enter the

pipeline network. To describe the gas behavior, only the set of gas physical properties

is used and tracked over the network. This implies that only 2- or 3-parametric

equations of state can be used.

• Gas composition tracking – various gases at individual supplies can enter the pipeline

network and both gas physical properties and gas composition are tracked over the

network.

4.3. Mixing rules

Let us consider the mixing of k streams of gas in one node:

(1)

(2)

(k )

…





27

The mixing of different gas streams is described by the following balance equations:

• Mass balance:

( )

∑==

k

j

j

mm 1

• Molar balance (consider no chemical reactions):

( )

( )∑=

=k

j j

j

M

m

M

m

1

Here

m Mass flux [kg.s-1]

Gas molar weight [kg.kmol-1]

Superscript )( j Gas stream j

Therefore the mixing rule for all gas quality parameters can be written in a compact form:

( )

( )

( )

∑∑ =

=

=

k

j

j

j

N

j

j

j

j

j

j

c

j

c

j

U

j

j

j

k

j j

j

N

c

c

U

x

x

C

B

A

p

T

H

M

M

m

M

m

x

x

C

B

A

p

T

H

M

1

)(

)(

)(

1

)(

)(

)(

)

)(

)(

)(

)(

)(

1

1

~

~

1

~

~

ψ

ω

ψ

ω

MM

Here

Molar mass of gas [kg.kmol-1]

U H ~

Molar heating value of gas [MJ.kmol-1]

cT Pseudo critical pressure of gas [K]

c p Pseudo critical temperature of gas [bar]

ω Acentric factor of gas [1]

C B A ,, Coefficients of ideal gas specific heat expansion ( )20~

T C T B AT c p ⋅+⋅+=

i x Molar fractions of mixture components [1]





28

~ Any (user-defined) parameter per 1 kmol of gas [x.kmol-1]





29

5. Heat Dynamics

In the standard SIMONE package, heat dynamics are respected by calculating the temperature

changes in a pipeline behind a compressor station or pressure reducer using a fixed

exponential model. The temperature of each node can be entered individually for a particulartime period.

In most situations, this concept is quite satisfactory. Nevertheless, there are occasions when

the description of heat dynamics is required – pipe sections downstream of compressor

stations, underwater lines, and long sections with high-pressure drop. In all of these cases, the

omission of heat balance results in a loss of accuracy.

Therefore the heat dynamics model is available in SIMONE.

5.1. Differential Equations

Heat dynamics consists of two components:

• Heat transients in the gas resulting from the Joule-Thompson effect –”longitudinal”

dynamics.

• Heat transients resulting from the heat exchange between the flowing gas and the

enveloping tube together with its further environment –”axial” dynamics.

The following form of energy equation is used in SIMONE:

01 =++

−

+−

∂∂

+∂∂

E

p p

P Qdx

dhSwg

x

p

T

z

z

T Sw

t

p

T

z

z

T S

x

T w

t

T cS ρ

∂

∂

∂

∂

∂

∂

∂

∂ ρ

Here

S Pipe cross section [m 2] ρ Gas density [kg.m-3]

pc Gas specific heat [J.kg-1.K -1]

T Gas temperature [K]

t Time [s]w Flow velocity [m.s-1] x Lengths coordinate [m] p Pressure [Pa]

z Compressibility factor [1] g Gravity acceleration [m.s-2]

h Geodetic height [m]

E Q Heat flux from gas through the inner surface of

pipe to the surrounding soil per unit of length

[J.s-1.m-1]

The description of the heat flux E Q should describe the effect of heat capacity of pipe-

surrounding soil, e.g. if gas cooler than near soil enters suddenly the pipe, the back heat flux

from soil to gas (until the near soil is cooled) occurs and should be modeled properly.





30

The character of the dynamic processes in the pipe and the surrounding soil is „stiff” – it

consists of a quick component corresponding to the near soil and a very slow one

corresponding to the remote mass of soil. Therefore the process needs to be modeled by a

minimum of two capacitors with substantially different time constants:

( ) ( )

( ) ( ) soil s s s s

s s s s

T T T T t d

T d C

T T T T t d

T d C

−−−=

−−−=

232122

2

212111

1

α α

α α

Here

T Gas temperature [K]

1 sT Near soil temperature at the surface of first layer (capacitor) [K]

2 sT Near soil temperature at the surface of second layer (capacitor) [K] soil T Far soil temperature being not influenced by temperature changes of

gas; this value is referred in SIMONE scenario as Ground Temperature

(GT)

[K]

1C Heat capacity of first layer (per unit of pipe length) [J.m-1.K -1]

2C Heat capacity of second layer (per unit of pipe length) [J.m-1.K -1]

1α Heat transfer coefficient through first layer (per unit of pipe length) [J.m-1.K -1.s-1]

2α Heat transfer coefficient through second layer (per unit of pipe length) [J.m-1.K -1.s-1]

3α Heat transfer coefficient between the surface of second layer and the

surface of constant far soil temperature (per unit of pipe length)

[J.m-1.K -1.s-1]

The heat flux E Q is then given by the first right-hand side term in the equation for first layer

temperature:

( )11 s E T T Q −=α

The steady-state solution of axial heat dynamic leads to formula

( ) ( ) soil E T T Dk t Q −=+∞→ π

Here

D Internal pipe diameter [m]

k Overall heat transfer coefficient between gas and far soil (per unit of

internal pipe surface); this value is referred in SIMONE scenario as

Heat Transfer Coefficient (HTC)

[J.m-2.K -1.s-1]

The parameters of 2-layer model of axial heat dynamics must therefore obey the steady-state

condition

Dk π α α α

1111

321=++





31

The dynamics of 2-layer model can be fully described by the following 4 parameters:

32

22

21

11

33

11

α α τ

α α τ

π

α

π

α

+=

+=

=

=

C

C

Dk k

Dk k

Here

1k , 3k Dimensionless parameters describing the splitting of overall heat transfer

coefficient into 3 resistors. They can be altered using SIMONE parameters LEEKK1, LEEKK3

[1]

1τ , 2τ Time constants of the 2 capacitors. They can be altered using SIMONE

parameters LEEKT1, LEEKT2.

[s]

The default settings for axial model dynamics ( 101 =k , 25.13 =k , h81 =τ , h1002 =τ )

correspond well for the behavior of fully-buried pipes.

SIMONE allows both models of axial heat dynamic to be used within scenarios:

• SIMPLE – only the steady-state heat flux between pipe and soil is used. This model isfor use in steady-state scenarios.

• FULL – the full axial dynamic is respected. However, the initial condition is to be

carefully prepared – the initial gas temperature distribution should correspond to

steady-state profile in order to initialize reasonably the axial model.

5.2. Heat Transfer Coefficient in SIMONE

In the following paragraph, the basic recommendations for the value of heat transfer

coefficient are summarized.

Heat flow between gas and surrounding environment (in a steady-state situation) is describedin SIMONE by equation

( ) soil E T T DQ −= HTCπ

Here

E Q Heat flux from gas through the inner surface of pipe to the surrounding

soil per unit of length

[W.m-1]

D Internal pipe diameter [m]

T Gas temperature [K]HTC Heat transfer coefficient between gas and far soil (per unit of internal [W.m-2.K -1]





32

pipe surface); this value is referred in SIMONE scenario as Heat

Transfer Coefficient (HTC)

soil T Soil temperature being not influenced by temperature changes of gas;

this value is referred in SIMONE scenario as Ground Temperature

(GT)

[K]

Let we suppose cylindrical model of n layers of different materials where the last layer has

constant temperature equal to soil T . The heat flow between gas and surrounding environment is

described then as

( ) soil n

i i

i

i

E T T

D

DQ −=

∑=

+

1

1ln1

2

λ

π

Here

i D Diameter of layer i , for 1=i it is internal diameter of tube [m]

iλ Heat conductivity of material in the layer between i D and 1+i D [W.m-1.K -1]

Comparing of both equations one obtains the equation for the Heat Transfer Coefficient

∑=

+

=n

i i

i

i D

D D

1

11 ln

1

2HTC

λ

Due to high conductivity and small thickness of pipe wall the layer of pipe itself can be

neglected. Typical situation can be represented by two layers, bitumen and clay. Because λ

for bitumen is roughly the same as for wet clay only one layer is sufficient and equation for

HTC can be rearranged as

+=

D D

ξ

λ

21ln

2HTC

Here

ξ Thickness of the clay layer; at the outer surface of clay the temperature is assumed

to be equal to soil T

[m]





33

To show how thickness of clay layer ξ and pipe diameter D influence the relation between

HTC and λ see the following picture. The typical values for λ are summarized below:

Material λ [W.m -1.K -1]

Concrete 1.28 – 1.5

Dry clay 0.14Wet clay 0.65

Bitumen 0.6

HTC/lambda

0

1

2

3

4

5

6

200 400 600 800 1000 1200 1400

Diameter [mm]

H T C / l a m b d a

clay thickness 500 mm clay thickness 1000 mm

5.3. Suppressing Joule-Thomson effect for pipes

A user requirement to suppress Joule-Thomson effect in the longitudinal temperature profile

has been met. Therefore the heat dynamics model in SIMONE has been adapted to support

this requirement, if it is stated explicitly in the scenario using the JTEP parameter.

If one rearranges the energy equation to point out Joule-Thomson coefficient

( ) 01 =++−+−

∂∂

+∂∂

E JT p JT p P Qdx

dhSwg

x

pcSw

t

pcS

x

T w

t

T cS ρ

∂

∂ µ ρ

∂

∂ µ ρ ρ

then neglecting Joule-Thomson effect leads to

0=++−

∂∂

+∂∂

E P Qdx

dhSwg

t

pS

x

T w

t

T cS ρ

∂

∂ ρ





34

6. Compressor Stations6.1. Basic thermodynamics relations

6.1.1. Pressure ratio

i p p0=ε

Here

ε Pressure ratio [1]

i p Inlet (suction) pressure [Pa]

o p Outlet (discharge) pressure [Pa]

6.1.2. Adiabatic head

−

−=

−

11

1

κ

κ

ε κ

κ iiad z RT H

Here

ad H Adiabatic head [kJ.kg-1]

κ Isentropic exponent (mean value) [1]

R Gas constant [kJ.kg-1.K -1]

iT Inlet gas temperature [K]( )iii T p z z ,= Compressibility factor at inlet condition [1]

ε Pressure ratio [1]

6.1.3. Adiabatic efficiency

io

ad ad

H H

H

−=η

Here

ad η Adiabatic efficiency [1]


i H Gas enthalpy at inlet condition [kJ.kg-1]

o H Gas enthalpy at outlet condition [kJ.kg-1]

6.1.4. Shaft power

( )ad

ad io

H m H H m P

η

=−=





35

Here




i H Gas enthalpy at inlet condition [kJ.kg-1]

o H Gas enthalpy at outlet condition [kJ.kg-1]

6.1.5. Discharge temperature

The calculation of discharge temperature consists of two steps:

1. Temperature change due to compression of ideal gas. The formula is selected using

the THETAEQ (Theta equation) scenario parameter:

• Basic (default) equation taking into account

ad

i x T T κη

κ

ε

1−

=

• RG1991 (Fasold et al)

+

−=

−

11

1

ad

i x T T η

ε κ κ

• Isentropic equation

κ

κ

ε

1−

= i x T T

2. The temperature xT obtained in previous step is corrected to the real gas behavior

using formula selected by the THETACOR (Theta correction) scenario parameter:

• Basic (default)

( ) xo

i xo

T p z

z T T

,=

• RG1991 – iterate oT starting form xT (Fasold et al)

( )oo

i xo

T p z

z T T

,=

• No correction

xo T T =





36

Here

iT Inlet gas temperature [K]

κ Isentropic exponent (mean value) [1]ε Pressure ratio [1]


xT Ideal gas outlet temperature after compression [K]


oT Outlet gas temperature [K]

6.1.6. Isentropic exponent

The mean value of isentropic exponent is calculated using one of three methods according to

the KAPPA (Isentropic exponent calculation) scenario parameter:

1. Constant (default) – isentropic exponent is constant and equal the value being set in

topoeditor

2. RG Equation – temperature-dependent relation is used

( ) ( )15.273108824.5290.1 4 −×−= −T T k

Here

2

oi T T T

+= Mean gas temperature during compression [K]

3. Equation of state – isentropic exponent is calculated from currently used equation of

state

2

T

p

p Z c

R Z

z

−=κ

Here


N x xT

p p

z p z Z

K1,

∂∂

−= Dimensionless derivative of compressibility

factor[1]

N x x p

T T

z T z Z

K1,

∂∂

+= Dimensionless derivative of compressibility

factor[1]


pc Real gas specific heat (for constant pressure) [J.kg-1.K -1]

The mean value of isentropic exponent is used:





37

2

oi κ κ κ

+=

6.1.7. Volumetric flow rate

i

ii

i

ivol p

z RT m

mQ ==

ρ ,

Here

ivol Q , Volumetric flow rate at inlet condition [m3.s-1]


i ρ Gas density at inlet condition [kg.m-3]


iT Inlet gas temperature [K]

i z Compressibility factor at inlet condition [1]

6.2. Centrifugal compressor

SIMONE uses the working envelope (wheel map) of centrifugal compressor in the

coordinates (inlet volumetric flow rate ivol Q . – adiabatic head ad H ).

The envelope is approximated by set of curves expressing adiabatic head and adiabatic

efficiency as biquadratic polynomials in compressor speed and volumetric flow rate:

( )

( )

=

=

2

,

,

963

852

741

2

2

,

,

963

852

741

2

1

1

1

1

ivol

ivol ad

ivol

ivol ad

Q

Q

bbb

bbb

bbb

nn

Q

Q

aaa

aaa

aaa

nn H

η

Here



n Compressor revolutions (speed in r.p.m.) [min-1]


91 aa K , 91 bb K Coefficients





38

6.3. Reciprocating compressor

The working envelope of reciprocating compressor is drawn in coordinates (inlet volumetric

flow ivol Q . – shaft torque T M ).

The inlet volumetric flow rate corresponds to compressor speed

60.

nV Q W ivol =

Here


W V Working volume of compressor (per 1 shaft revolution) [m3]

n Revolutions [min

-1

]

The torque corresponds to shaft power

T S M n

P 60

2π =

Here

S P Shaft power [kW]

n Revolutions [min-1]

T M Torque [kNm]

Combining the above equations with the general formula for compressor shaft power, one

obtains the relation between adiabatic head and torque:

iad

ad

W T H

V M ρ

πη 2=

Here

W V Working volume of compressor (per 1 shaft revolution) [m3]

ad η Adiabatic efficiency (assumed constant within the whole

operation range of reciprocating compressor)

[1]


i ρ Gas density at inlet condition [kg.m-3]





39

6.4. Gas turbine

Gas turbine is described by two graphs:

1. Turbine maximum performance as function of turbine speed and ambient temperature,

approximated by biquadratic polynomial:

( )

=2

963

852

741

2

max

1

1

amb

amb

t

t

ccc

ccc

ccc

nn P

Here

max P Maximum turbine performance [kW]

n Revolutions (speed in r.p.m.) – turbine outputor compressor shaft revolutions

[min-1

]

ambt Ambient temperature [ºC]

91 cc K Coefficients

2. Fuel gas consumption as function of shaft power, approximated by parabola:

2

321 S S F P d P d d P ++=

Here

F P Fuel gas consumption in energy units [kWh.h-1]

S P Turbine performance [kW]

31 d d K Coefficients

6.5. Gas engine

Gas engine is described by two curves:

1. Shaft performance as a function of speed, approximated by parabola:

2

321max ncncc P ++=

Here

max P Maximum engine performance [kW]

n Engine revolutions (speed in r.p.m.) [min-1]






40

2. Fuel gas consumption as function of shaft power, approximated by parabola:

2

321 S S F P d P d d P ++=

Here

F P Fuel gas consumption in energy units [kWh.h-1]

E P Engine performance [kW]

n Engine revolutions (speed in r.p.m.) [min-1]


6.6. Electro drive

The electro drive is described by two characteristics:

1. Shaft performance curve:

a. For asynchronous drives and drives with continuous speed control the

maximum shaft performance is a function of speed (r.p.m.) and ambient

temperature (influence of cooling), described in similar way as that for gas

turbines:

( )

=2

963

852

741

2

max

1

1

amb

amb

t

t

ccc

ccc

ccc

nn P

Here

max P Maximum drive performance [kW]

n Drive revolutions (speed in r.p.m.) [min-1]

ambt Ambient temperature [ºC]


b. For synchronous drives the maximum shaft power is a function of revolutions,

described in similar way as that for gas engines:

2

321max ncncc P ++=

Here

max P Maximum drive performance [kW]

n Drive revolutions (speed in r.p.m.) [min-1]






41

2. Energy consumption curve is function of shaft power:

2

321 S S E P d P d d P ++=

E P Energy consumption [kW]

S P Drive performance [kW]


6.7. Steam drive (combined cycle)

For modeling of combined-cycle drives (primary gas turbine, steam boiler utilizing exhaust

heat and secondary steam turbine) the correlation of total energy consumption of primary

turbines and shaft power of secondary turbine is used:

2

321 SS SS FP P a P aa F ++= or2

321 FP FP SS F b F bb P ++=

Here

FP F Fuel gas consumption of primary gas turbine

(sum for all primary turbines) in energy units

[kWh.h-1]

SS P Shaft power of secondary steam turbine [kW]

31 aa K , 31 bb K Coefficients

To prevent on/off cycling, the combined cycle can be

• Started only if the fuel gas consumption of primary turbines is sufficiently high

start FP FP F F ,>

• Stopped only if the fuel gas consumption is lower than prescribed switch-off limit

start FT stop FP FP F F F ,, <<

6.8. Gas cooler

The gas cooler is described by an approximate characteristic equation

( )

−−+=m

k T T T T iciico exp,,

Here

iT Gas temperature at cooler inlet [K]

oT Gas temperature at cooler outlet [K]

icT , Coolant inlet temperature





42

m Gas mass flow rate [kg.s-1]k Constant coefficient [kg.s-1]

6.9. Local pressure loss

For the compressor station element, the pressure loss in inlet and outlet parts of piping yard ofthe compressor station can be modeled. In addition, the local pressure drop between stages

can be taken into account.

Local pressure loss is calculated using local pressure loss coefficient

2

2

1iioi w p p ξρ =−

Here

i p Pressure at resistor inlet [Pa]

o p Pressure at resistor outlet [Pa]

ξ Pressure loss coefficient [1]

i ρ Gas density at resistor inlet [kg.m-3]

iw Velocity at resistor inlet [m.s-1]

The gas velocity is calculated using representative internal diameter of compressor station D ,

entered in network element data:

i

i D

mw

ρ π 2

4=

Here

m Mass flow rate through element [kg.s-1]

D Element diameter [m]





43

7. Controlled valves7.1. Preheating power

Due to Joule-Thomson effect, the pressure drop in control valve is accompanied with

temperature drop. To avoid dangerously low temperatures at the outlet of controlled valve, the

gas is being preheated before it enters the valve.

The temperature drop is calculated from integral Joule-Thomson effect:

( )

( )

( )oo

JT

ii

pT T

T pdp

dT

T pT

=

=

=

,µ

Here

i p Inlet pressure [Pa]

iT Inlet temperature [K]

o p Outlet pressure [Pa]

JT Joule-Thomson coefficient [K.bar -1]


oT Outlet temperature [K]

The preheating power is calculated from the formula

( ) ( ) ( )

2

, 221,1

12

T pcT pcT T m P

p p +−=

Here

P Preheating power [kW]m Mass flow rate of gas [kg.s-1]

pc Gas specific heat [kJ.kg-1.K -1]

p Pressure [Pa]T Temperature [K]

Subscript 1 Exchanger inlet condition

Subscript 2 Exchanger outlet condition

7.2. Valves with characteristic

The valve in an intermediate position can be modeled as controlled valve element with

particular mode and setpoint equal to the opening percentage (using the SCVO scenario

parameter).

The flow rate through the valve in an intermediate position is a function of inlet pressure,

outlet pressure and the position of the valve. The valve position is expressed by opening





44

percentage (closed 0% and fully open 100%). An equation individual for each valve producer

& type describes this function:

( )K,,, o p pmm oi=

Here




o Opening percentage, 100,0∈o ; 0=closed; 100=fully open [1]

K Stands for other parameters, namely gas quality (relative

density) and inlet temperature

Two such formulae are currently implemented in SIMONE; other valve characteristics can beimplemented upon request.

7.2.1. Characteristic A (Mokveld)

The flow rate through valve is given by the formulae

( )

−

=

−=

1

21

1

11

3

1100

63.1

;5.1min

148.0241

p

p p

F Y

z rT

Y Y p F C C V VRV

Here

0V Volumetric flow rate at standard condition (101.325 kPa and 0ºC) [m3.h-1]

0V C Sizing coefficient [-]

o Opening percentage, 100,0∈o [1]

( )oC VR Characteristic coefficient depending on opening percentage [1]

( ) 00 =VR

C ; ( ) 1100 =VR

C

( )o F 1 Characteristic coefficient depending on opening percentage [1]

( ) 101 = F ; 1 F decreasing function

1 p Inlet pressure (absolute) [bar]

2 p Outlet pressure (absolute) [bar]

1T Inlet gas temperature [K]

1 z Compressibility factor at inlet condition [1]

r Gas relative density [1]

The valve behavior is therefore described by the sizing coefficient 0V C and functions ( )oC VR ,

( )o F 1 (being defined by table values).





45

7.2.2. Characteristic B (Argus)

The flow rate through valve is given by the formulae

• For sub critical flow

( )( ) 221

100

2

1

514

2

p p p

T Qo K K

p

p

VRVS −=

<

ρ

• For supercritical flow

( ) 10

1

0

2

1

257

2

T p

Qo K K

p

p

VRVS ρ =

≥

Here

VS K Sizing coefficient [m3.h-1]

o Opening percentage, 100,0∈o [1]

( )o K VR Function of opening percentage [1]

( ) 00 =VR K ; ( ) 1100 =VR K

0Q Volumetric flow rate at standard condition (101.325 kPa

and 0ºC)

[m3.h-1]

1 p Inlet pressure (absolute) [bar]

2 p Outlet pressure (absolute) [bar]

1T Inlet gas temperature [K]

0 ρ Gas density at standard condition (101.325 kPa, 0ºC) [kg.m-3]

The valve behavior is therefore described by the sizing coefficient VS K and function ( )o K VR

(being defined by table values).

7.3. Local pressure loss

For the control valve element, the pressure loss in inlet and outlet parts of piping yard of the

control station can be modeled.

Local pressure loss is calculated using local pressure loss coefficient

2

2






46

Here




i ρ Gas density at resistor inlet [kg.m-3]

iw Velocity at resistor inlet [m.s-1]

The gas velocity is calculated using representative internal diameter of controlled valve D ,

entered in network element data:

i

i D

mw

ρ π 2

4=

Here

m Mass flow rate through element [kg.s-1]

D Element diameter [m]





47

8. Resistor

A resistor is an element representing the lumped resistance of an armature, partially closed

valve, filtering equipment etc. The hydraulic resistance is characterized by the local pressure

loss coefficient. The resulting pressure drop is described by the equation

2

2


Here




i ρ Gas density at resistor inlet [kg.m

-3

]iw Velocity at resistor inlet [m.s-1]

The gas velocity is calculated using representative internal diameter of resistor D , entered in

network element data:

i

i D

mw

ρ π 2

4=

The resistor element works in both directions of gas flow.

The temperature drop caused by Joule-Thomson effect is included in the resistor description:

( )

( )

( )oo

JT

ii

pT T

T pdp

dT

T pT

=

=

=

,µ

Here


iT Inlet temperature [K]


JT Joule-Thomson coefficient [K.bar -1]


oT Outlet temperature [K]





48

9. Solution methods

The set of original methods developed within the SIMONE project is what makes the power

and quality of SIMONE system.

As a rule, the description of a pipeline network consists of thousands of non-linear partial

differential equations and of hundreds of inequalities and constraints. In principle, the

requirements on a dynamic simulation model result from the great extent of reality being

described, and from the great complexity of its mathematical description:

1. The transients in network elements must be described with high accuracy. Otherwise,

due to the accumulation of errors in a great complex network, an unacceptable

inaccuracy may result.

2. High speed of simulation is required even for very extensive networks. This is

important to get high value of the ratio (real time)/(simulation time). The greater this

ratio, the wider the operation possibilities in practical, particularly on-line use.

Another essential benefit of the high speed of simulation will be appreciated in

building the (optimum) control system.

3. To bring down the memory demands is required, because due to the great number of

network elements and variables, they may exceed the possibilities of even powerful

computers.

4. In the network, both extremely slow transients and quick ones (after a rupture of a

tube or opening a valve) may occur. The model must simulate both with high accuracy

being simultaneously safe against any oscillations of the numeric solution.

5. The simulation model must copy the reality in all aspects. The model must therefore

simulate, besides the transients in pipes, the behavior of all other elements, namelycompressor stations, in full variety of their technological realization.

Besides all the technical reasons, there exists one good reason more why all these

requirements must be respected: the dispatcher will quickly abandon the system which will

not be quick, exact, user friendly, and will not allow him to do everything what can be done

with the real network and in a most similar manner.

To satisfy the above-mentioned requirements, SIMONE simulation kernel is based on the

following keystones:

• Modified implicit integration method was developed to ensure safe numerical stabilityof solution (item 4) in all transient conditions. It is safe against parametric oscillations,

which could rise due to the stepwise linearization of nonlinear equations, and exactly

maintains the mass balance condition.

• The demands on small memory consumption and high speed of calculation (items 2

and 3) are satisfied by means of a proprietary decomposition method that the

simulation system is based on. This method takes advantage of sparseness of the

system matrix and of its specific structure.

• Proprietary algorithmic procedure has been developed which allows the solution of

combined set of nonlinear partial differential equations (pipes) together with the

characteristics and constraints of non-pipe elements (compressor stations and

controlled valves). It satisfies the demand of realistic description of all aspects of





49

simulated reality (item 5) by an easy and flexible linkage of non-pipe elements into the

model of a pipeline network.





50

10. State reconstruction

The state of network is described by a complete set of values for all process variables, namely

pressures and flow rates, for the entire network. Naturally, because of the perpetual transients

in the network, the state changes continually.

The task to calculate the most probable estimate of the actual state of the network on the basis

of a restricted number of measured variables and considering the individual accuracy of

measurements is called state reconstruction.

In control theory the concept of State Observer is used for the state reconstruction task.

Nevertheless, the solution of this problem can be found in literature only for zero mean

random errors and zero systematic errors of measured variables.

The ability to find the state of given dynamic system on the basis of a set of values measured

over a time interval is called observability. Consider all non by-passed compressor stations,

control valves, and closed valves removed from the network. This divides the network in

separated sub-networks. If all supplies and off-takes are measured, set points of non-pipe

elements are known, and at minimum one pressure measurement exists in each sub-network,

the entire system is observable. The sub-network in which the pressure measurement is for a

longer time not available is not observable. The reason is the following. The errors in flow

rate measurements result in uncertainty of the in-take/off-take balance of the sub-network.

Without the information about pressure the system is unable to compensate the drift of

balance.

State reconstruction is build as Observer, which is using closed loop system consisting of

dynamic Model of the System and Controller.

System

err_1

measurement

Model

Input

err_2

Controller

-

estimated

state

real

state

What is above dashed line is for us unknown. We have only Model of the System and input

and restricted number of measurements that are corrupted by errors. If Model is ‘good’





51

approximation of System and Controller makes the closed loop stable then Observer (laying

bellow dashed line) is able to fulfill the task of state reconstruction.

The output of the dynamic model is compared with measured variables. The difference is

used by the controller that actuates on the model to minimize the difference i.e., to get the

state of the simulated system close to reality. The task is to find such a gain of the controller,which corresponds to an „optimum solution“. If the „optimum solution“is defined by the

minimum variance of state reconstruction error, the solution is known as Kalman filter.

10.1. State reconstruction in SIMONE

In the case of pipeline networks, such a solution is not practical due to the high dimension of

the system. Instead of an optimum, a sub-optimum solution combining the mathematical

approach with engineering aspects has been elaborated. The result is the real-time state

reconstruction.

The state reconstruction is performed in two subsequent steps:

1. In the first step, the simulation is calculated. Within the step, the corrections of in-

takes and off-takes are done in the nodes with pressure measurements. The correction

is done by the „additional artificial off-takes“, the quantity of which are stated on the

basis of pressure measurements considering the accuracy of pressure and flow rate

data.

Input pressure, output pressure, and flow rate are used to be measured for the

controlled elements (compressor stations and control valve). From these values the

special control mode is constructed to minimize for each controlled element theobjective

( ) ( ) ( ) ( )2

2

2

2

2

2

2

2

s

s

m po

oo

pi

ii mmmm p p p p

σ σ σ σ

−+

−+

−+

− ∗∗∗

subject to hydraulic conditions in tubes. Other constraints like compressor

characteristics are not taken into account.

Here

i p , piσ Input pressure, accuracy of input pressure measurements

o p , poσ Output pressure, accuracy of output pressure measurement

m , mσ Flow rate, accuracy of flow rate measurement

sm , sσ Flow rate driven by set-point requirement, accuracy of flow rate set-point

requirement

∗ Marks the measured values

Accuracy of the flow rate given by set-point requirements is by experience set

to sm07.0 . If any measurement is not available, the corresponding term in the objective

is omitted.





52

2. In the second step, the distances between measured and estimated pressures are

minimized respecting the uncertainty of pressure drops on tube elements computed in

the first step. The minimized objective is

( ) ( ) ( ) ( ) ( ) ( ) _

1

_ _ _

1

_

*1

*

* P P R P P P P R P P P P R P P T

P

T

P

T −−+∆−∆∆−∆+−− −−

∆−

Here

P Vector of estimated pressures

P _ Vector of pressures in the first step

* P Vector of measured pressures (zero in the diagonal matrix ∗ P R corresponds to

non-measured pressure)

∆ P Vector of estimated pressure drops on pipe elements

∆ P _ Vector of pressure drops on pipe elements in the first step

∗ P R Diagonal matrix with non-zero elements in the nodes with pressuremeasurements only; the values of these elements are variances of pressure

measurements

R _

−1 Diagonal matrix with positive elements close to zero; it is used only to

overcome numerical problems arising when in some part of the network the

pressure measurements are missing

R P ∆ Diagonal covariance matrix of pressure drops derived from resistance of tubes

and uncertainty of flow rates in the first step

10.2. Outlying Measured Data

The SIMONE concept is based on suppressing of outliers instead of omitting. The reason is to

avoid the possible loss of observability in case that all pressure measurements are omitted in a

sub-network. In SIMONE, the following method is used:

• If the difference between the results of the first reconstruction step and the pressure

measurement is greater than p prel σ × , where prel is coefficient and pσ is accuracy

of pressure measurement, then the difference value is locally used as modified

accuracy of the measurement.

• If the difference between the first reconstruction step and flow rate measurement is

greater than mmrel σ × , where mrel is coefficient and mσ is accuracy of flow rate

measurement, then this number is locally used as modified accuracy of the

measurement.

The results of this approach evidently depend on parameters prel and mrel .

10.3. State Reconstruction Parameters

The process of state reconstruction requires several parameters defined in program

environment. Different setting of parameters is recommended for the “starting reconstruction

run” where initial conditions may be far from reality and for “normal reconstruction run”

where initial conditions are already rather realistic. Nevertheless, modification of these parameters is recommended only to experienced user.





53

QP rel Real number modifying artificial off-takes (these are defined in all supply/off-

take nodes where pressure measurement is available) as

( )**

p prel

QP p

QP Q −=σ

σ

Here

Qσ is accuracy of supply/off-take

pσ is accuracy of pressure measurement.

Recommended value is 0 2 1. ; .

sigma m _ Coefficient reflecting uncertainty of pipe line pressure drops in the first step of

reconstruction (see matrix R P ∆ ).

Recommended value is 0 03 010. ; .

prel Coefficient for managing outlying pressure measurement data.

Recommended value is 3 10;

mrel Coefficient for managing outlying flow rate measurement data.

Recommended value is 5 20;

For “starting reconstruction run” use values of parameters near to upper bound and for

“normal reconstruction run” to lower bound.

10.4. Tools for Scanning State Reconstruction

For scanning and checking the state reconstruction results, the following values can be

displayed by SIMONE graphic support:

nodename.phat Change of pressure in node „nodename“ in the second step of

reconstruction

node1^node2.qrec Mean value of „extra flow rate“ corresponding to the change of pressures

in the element „node1^node2“since the last writing of results to the

database

subsystem.aqrec Mean value of „extra flow rate “ corresponding to the change of

pressures in the subsystem „subsystem“ since the last writing of results to

the database. This gives information about imbalance of in-takes and off-

takes in the subsystem

subsystem.aqp Mean value of flow rate corresponding to the sum of artificial off-takes in

the subsystem „subsystem“ since the last writing of results to database

subsystem.asupp Mean value of flow rate corresponding to the sum of supplies in the

subsystem „subsystem“ in the reconstruction since last writing of results

to database.

subsystem.aoff Mean value of flow rate corresponding to the sum of off-takes in thesubsystem „subsystem“ in the reconstruction since last writing of results





54

to database

The reason for using the “mean value of flow rate since the last writing’’ is the calculation of

mass balance under the circumstances of different computation time step and writing time

step.

There are two integrals in the graphic arithmetic: INTG for integrating by trapezoidal rule and

INTG2 for rectangular rule. To compute balances use INTG only for supplies and off-takes

where continuous linear interpolation between samples is assumed. For all other flow rates

use INTG2.

There is possible to use in topology visualization the statement tv qrec to have quick

look over the situation in network at one time instant.

Using subsystem TOTAL, the above data for the whole network are displayed.

Because gasses of different quality may stream within the network, the above subsystemvariables are expressed in volumetric units using the default gas defined in topoeditor.





55

11. Steady-State Optimization

The steady-state optimization for fixed compressor configurations is based on gradient

method of feasible directions. The attention is focused on the design of the control strategy

with respect to commercial policy taking into account the network possibilities, the storage

capacities, the structure of contracts, etc. Also the on-line use of the steady-state optimizationis described.

11.1. Objective Function

The objective function is represented as a sum of different penalty functions (for fuel

consumption, for price of purchased gas, for pressure condition, etc.). The components of

penalty functions can be classified into two groups:

• Terms expressing the basic aims of optimization (fuel consumption, prices of gas etc.).

• Terms substituting some constraints by penalty functions (e.g., for pressure and flowrate conditions) to reach the feasibility.

If the components of the above two classes are merged together in one objective function, the

„valley problem” may emerge what decreases the speed of convergence. Therefore, we

recommend to use only the terms from the second group for finding a feasible solution, and

then apply the first group (completed by all constraints) for the optimization itself.

11.2. Constraints

The constraints can be classified into three basic groups:

1. Constraints related to pipe elements. These constraints are equalities.

2. Constraints related to non-pipe elements and representing the fixed operation limits of

built-in technological de-vices (compressor stations, control valves). These constraints

are inequalities.

3. Constraints determined by contractual conditions. These constraints are inequalities.

The constraints 1 and 2 can be derived from the simulation model; the constraints 3 are

defined (and changed) by the user.

11.3. SIMONE Method of Steady-State Optimization

The method is based on the generalized reduced gradient method and generalized projection

gradient method with the use of a steady-state simulation model.

The set of constraints can be divided into two basic groups: the equality constraints (group 1)

and the inequality ones (groups 2 and 3).

The constraints read after the local linearization at the working point 0 x , for the first group:

00 x x a x A =





56

and for the second group

00 x x b x B ≤

The number of equality constraints can be great (it is determined by the network extent), but

the following transformation can simply be computed (using the knowledge of the structure of

0 x A ):

y D x x0=

where y is a choice of linearly independent variables from x and matrix 0 x D has a special

structure given by the configuration of the network. From this follows, that it is advantageous

to respect these constraints in the computation of a new step of optimization by gradient

reduction.

The inequality constraints will then be transformed to:

000

00

x x x

x x

D B M

b y M

=

≤

It should be noted that the vector y represents the set of control variables. The set points of

non-pipe elements need not necessarily correspond to real control variables used for the

optimization – it is better to interpret them as a way how to describe the final optimum

solution (may be completed by some additional information about the pressure and flow rate

distribution).

The number of active constraints belonging to the second group is in comparison to the

number of constraints of the first group much smaller, but it changes within iterations. The

integration of these constraints in the first group would significantly complicate the

computation as well as the structure of the matrix 0 x D .Moreover, the procedure of

determining the 0 x D matrix would have to be repeated any time the number of active

constraints changes! For these reasons, the method of gradient projection was chosen to

respect the constraints. It enables to change the number of active constraints very simply. This

method also allows solving the problem of linearly dependent constraints (this problem does

not occur in the first group of constraints).





57

12. Hydrate formation risk analysis12.1. Equilibrium condition

The Hydrate Formation Risk analysis is based on detection of three-phase equilibrium

condition gas – free water – hydrate within the pipeline system. This equilibrium is expressed

by relation of

• Local gas temperature T

• Water dew point temperature d T

• Hydrate equilibrium temperature hT .

This can be seen on the following picture showing the longitudinal profiles of pressure and

temperatures along a pipeline.

p

T <T d Free water

present

T h( p)

T

T d ( p,WC )

Gas

inlet

Gas

outlet

T <T d

T <T h

Hydrate formation

risk area

The water content of gas WC [g/m3 std. dry] is being tracked over the network (as a gas

quality parameter). For known water content and given pressure, the water dew point

temperature can be calculated anywhere in the network.

The hydrate equilibrium temperature calculation can take into account the effect of hydrate

inhibitor (methanol, glycols). For this purpose, the inhibitor content of gas IC [g/m3 std. dry]

is being tracked over the network.





58

12.2. Water dew point

The water dew point temperature d T is calculated from the water content of gas

( ) xWC pT T d d

,,= .

Here

p Pressure [Pa]

WC Water content of gas [g/m3 std. dry]

x Gas composition using molar fractions (dry gas) [1]

12.2.1. Bukacek aprroximation

The equilibrium water content of gas is expressed as a function of pressure and temperature:

B p

AWC +=0

Here

0WC Equilibrium water content [g/m3 at 101.325 kPa and 0ºC, dry] p Gas pressure [bar]

A , B Temperature-dependent coefficients:

The coefficients A and B are given by the following equations:

( )( )2

2

1000198660400535703270exp04487390

20003067391007374330exp92655734

:C40,C40

t .t .. B

t .t .. A

t

−=

−=

°+°−∈

( )

( )2

2

7930.0001655390.05504279exp0.0405058

3450.000130960.0620686exp6.1569611

:C90,C40

t t B

t t A

t

−=

−=

°+°+∈

( )( )t .. B

t .t .. A

t

0326378940exp07007370

50001309634006206860exp15696116

:C130,C90

2

=

−=

°+°+∈

Bukacek approximation is suitable for sweet natural gases ( 8.0≤r , %22<CO x , %5.1

2<S H x ).





59

12.3. Hydrate equilibrium

The hydrate equilibrium temperature is given by a function

( ) x pT T hh ,=

Here

p Pressure [Pa] x Gas composition using molar fractions (dry gas) [1]

12.3.1. Motiee approximation

Simple relation involving gas relative density only (published in form of charts)

( )r pT T hh ,=

has been approximated by Motie using the equation

pr br br b pb pbbt logloglog 6

2

54

2

321 +++++=

Here (note the imperial units!)

t Temperature [ºF] p Pressure [psia]

r Gas relative density [1]61 bb K Coefficients

The values of coefficients are

i ib

1 -238.24469

2 78.996674

3 -5.352544

4 349.473877

5 -150.8546756 -27.604065

The original charts exhibit relatively high uncertainty, especially for gas relative density

around 0.6. However Motiee approximation seems to perform well for sweet, methane-rich

gas.





60

12.3.2. Carson-Katz method

Carson-Katz vs K -model is based on vapor-solid distribution coefficients. The hydrate

equilibrium temperature is calculated from the condition

( )∑ =i hivs

i

T p K

x1

,,

Here

ivs K , Vapor-solid distribution coefficient for gas component i [1]

i x Molar fraction of component i in dry gas [1]

Subscript i Hydrate-forming components of natural gas: CH4, C2H3, C3H8, i-C4H10, n-

C4H10, N2, CO2, H2S

The ivs K , coefficients are calculated using the approximation published by Sloan:

( )4233321212

112211

, lnln

t Rt pQt P tpO pt N pt M tp L p K

pt J pt I p H t G pt F p E t D pC t B A K

iiiiiiii

iiiiiiiiiiivs

+++++++

++++++++++=−−−−−−

−−−−

Here

t Temperature [ºF] p Pressure [psia]

ii R A K Coefficients

12.3.3. Ponomarev approximation

Ponomarev approximation of hydrate equilibrium

( )( )

°<+

°>+=

C0for 0171.0

C0for 0541.0log

1

0

t t P

t t P p

Here

p Pressure [at]

10 , P P Coefficients dependent on reduced relative density r ′ [1]

∑∑

=′

i

i

i

ii

x

xr

r Reduced relative density [1]

ir Relative density of component i [1]

i x Molar fraction of component i in dry gas [1]

Subscript i Stands for CH4, C2H6, C3H8, C4H10 (sum of i-C 4H10 and n- C 4H10), CO2,H2S





61

The values of 10 , P P are given by the following table:

r ′ 0 P 1 P r ′ 0 P

1 P r ′ 0 P 1 P

0,56 24,25 77,4 0,71 13,85 43,9 0,86 12,07 37,6

0,57 21,80 70,2 0,72 13,72 43,4 0,87 11,97 37,20,58 20,00 64,2 0,73 13,57 42,9 0,88 11,87 36,8

0,59 18,53 59,5 0,74 13,44 42,4 0,89 11,77 36,5

0,60 17,67 56,1 0,75 13,32 42,0 0,90 11,66 36,2

0,61 17,00 53,6 0,76 13,20 41,6 0,91 11,57 35,8

0,62 16,45 51,6 0,77 13,08 41,2 0,92 11,47 35,4

0,63 15,93 50,0 0,78 12,97 40,7 0,93 11,37 35,1

0,64 15,47 48,6 0,79 12,85 40,3 0,94 11,27 34,8

0,65 15,07 47,6 0,80 12,74 39,9 0,95 11,17 34,5

0,66 14,76 46,9 0,81 12,62 39,5 0,96 11,10 34,2

0,67 14,51 46,2 0,82 12,50 39,1 0,97 11,00 33,9

0,68 14,34 45,6 0,83 12,40 38,7 0,98 10,92 33,60,69 14,16 45,0 0,84 12,28 38,3 0,99 10,85 33,3

0,70 14,00 44,5 0,85 12,18 37,9 1,00 10,77 33,1

12.3.4. Remarks on accuracy and choice of models

These models for hydrate equilibrium temperature calculation have been verified against the

codes CSMHYD (detailed thermodynamic model of hydrate equilibrium) and HYDK

(implementation of vs K -model) of Colorado School of Mines for the pressure range up to

about 100 bar (1500 psia) for three typical composition of natural gas:

Component [%mol.] L-gas “Groningen”CH4-rich “Russian”H-gas “North Sea”

CH4 81,29 96,52 85,91

N2 14,32 0,26 1,00

CO2 0,89 0,60 1,49

C2H6 2,87 1,82 8,49

C3H8 0,38 0,46 2,31

H2O 0,00 0,00 0,00

H2S 0,00 0,00 0,00

H2 0,00 0,00 0,00

CO 0,00 0,00 0,00O2 0,01 0,00 0,00

i-C4H10 0,00 0,10 0,35

n-C4H10 0,15 0,10 0,35

i-C5H12 0,00 0,05 0,05

n-C5H12 0,04 0,03 0,05

n-C6H14 0,05 0,07 0,00

Relative density r 0,645 0,581 0,650

Reduced relative density r ′ 0,589 0,578 0,646





62

The results of CSMHYD, HYDK, Motiee, Ponomarev and our implementation of vs K -model

(referred below as HETKVS) are shown on the following pictures:

Hydrate equilibrium curve for "Groningen gas"

1

10

100

1000

-5 0 5 10 15 20 25 30

t [ºC]

p [ b a r ]

CSMHYD p->T

CSMHYD T->p

CSM K-code

Motie

Ponomarev

HETKVS

Hydrate equilibrium curve for "Russian gas"

1

10

100

1000

-5 0 5 10 15 20 25 30

t [ºC]

p [ b a r ]

CSMHYD

CSMHYD T->p

CSM K-code

Motie

Ponomarev

HETKVS

Hydrate equilibrium curve for "North Sea gas"

1

10

100

1000

-5 0 5 10 15 20 25 30

t [ºC]

p [ b a r ]

CSMHYD

CSMHYD T->p

CSM K-code

Motie

Ponomarev

HETKVS





63

The results of hydrate equilibrium temperature calculation for given pressure may be

summarized in the following points:

• The computationally simplest approximation by Motiee performs with precision

about 0.5 ºC for CH4-rich gas (“Russian gas”). For other natural gases (C2H6-rich H-

gas “North Sea” and N2-rich L-gas “Groningen”) the error may significantly increase(up to 5 ºC).

• For Ponomarev approximation, the similar results as those for Motiee formula hold

(about 1ºC for “Russian gas” and significant decrease of accuracy for other two types

of natural gas). The higher computational complexity (gas composition tracking)

makes however this method inferior to Motiee formula.

• The vs K -model performs much better even for very different gases as it takes the

composition of gas into account (observed accuracy to 1ºC for all three above

mentioned basic gas types, to 0.5 ºC for “Russian gas”). Therefore this model seems

to be the most recommendable general solution, well compromising accuracy and

speed.

12.4. Effect of inhibitors

The lowering of hydrate equilibrium temperature due to presence of inhibitor is calculated by

Hammerschmidt equation

X

X

M T h −

=∆100

1297

Here

hT ∆ Lowering of hydrate equilibrium temperature [K]

Molar weight of inhibitor [g.mol-1]

X Weight fraction of inhibitor in solution with free water [% wt.]

Currently, methanol, ethylenglycol and diethylenglycol are supported as hydrate inhibitor.

The weight fraction X of inhibitor in solution with free water is calculated from the

equations:

( ) l

WC xT pWC WC += ,,0

( ) X

X WC X T p MC IC

v

−×+=

100, l

for methanol

100×+

= IC WC

IC X

l for glycols

Here

WC Total (tracked) water content of gas [g/m3 std. dry]

0WC Equilibrium water content of gas [g/m3 std. dry]lWC Free water content [g/m3 std. dry]





64

IC Total (tracked) inhibitor content of gas [g/m3 std. dry]

( )T p MC v , Methanol losses to the hydrocarbon phase –

amount of methanol vaporized into gas per 1%

wt. of methanol in solution with free water

water freeinmethanolwt.%

std.g/m3

An approximation built upon the classic chart of ( )T p MC v , published by Campbell iscurrently used:

( )

=

p

p

bbb

bbb

bbb

t t MC v

2

963

852

741

2

log

log

1

,,1

Here

( )T p MC

v

, Methanol losses to thehydrocarbon phase ( )

°

water freeinmethanolwt.%C15kPa,101.325g/m

3

t Temperature [ºC] p Pressure [bar]

Hammerschmidt equation is reported to be accurate to certain upper limit of the inhibitor

fraction (30% for methanol and 50% for glycols). Therefore the value of hT ∆ is limited to the

corresponding maximum.

12.5. Risk indicators

The results of the equilibrium analysis over pipe element are derived from the detailed

internal temperature profile used by the SIMONE Heat Dynamics module (see chapter 5). The

following parameters show, how far the local gas condition are from the equilibrium ones. For

each pipe element the worst case, i.e. minimum of these sub cooling values over the element

length, is stored:

• Gas sub cooling with respect to water dew point temperature:

[ ]( )d

LT T DTDP −=

∈ ;0minξ

• Gas sub cooling with respect to hydrate equilibrium temperature:

[ ]( )h

LT T DTHE −=

∈ ;0minξ

• Gas sub cooling with respect to the three-phase equilibrium boundary temperature:

[ ]( )( )hd

LT T T DTHF ,minmin

;0−=

∈ξ

Here



SIMONE Equations and Methods 65

L Pipe length [m]

ξ Longitudinal coordinate [m]

Note: If the profiles of all temperatures along the pipe element are monotone (what the most

frequent case), these three indicators show only two different values:

( ) DTHE DTDP DTHF ,max=

However the monotonicity of all profiles along pipe element is generally not guaranteed

(assume for example the possibility of a “batch” of gas of different water content moving

inside the pipe element), so all three indicators have to be stored.

simone (equations and methods)

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