APPENDIX A: Laplace transformation of the RCP side TH system.

APPENDIX A: Laplace transformation of the RCP side TH

system.

It is possible to linearize and then Laplace transform the TH system of §6.2.2.

Assumptions of §6.2.1 are applied also in this section and equations of §6.2.2 are recalled for completeness.

The describing equations are:

Volume A          = − + − − − − = − − = ) ( ) ( ) ( )) ( ( ) ) ( )( ( ) )( ( ) ( ) ( ) ( ) ( ) ( , 2 1 1 , 2 1 t RT t m V t P t T T A h c T t T t m c T T t m dt t dT c t m t m t m dt t dm A A A A A W conv p ref tot s p ref A v A tot s A

τ

τ

& & & & Set of Equations 1 TR pipe section

[

( ) ( )

]

) ( 2 ) ( 1 ) ( ) ( , 2 2 2 3 2 2 4 2 1 , tot l A TR line A TR tot tot TR tot tot s A TR t t t A t m K K A A K t P t P

τ

ρ

ρ

τ

ρ

τ

− = −         + +       + − − = & Equation 1

where

τ

l,totis the time required to a plug of gas with density

ρ

A to travel from volume a to TR

in nominal flow rate conditions.

These equations require “a little” effort to be linearized around the nominal working condition.

             + + = + + − + − + − + + − − + = + + − − − + = + )) ( ))( ( ( )) ( ( ))) ( ( ( ) ) ( ( )) ( ( ) ))( ( ( )) ( ))( ( ( ) ( ) ( )) ( ( 0 , 0 , 0 , 0 , 0 , , 2 0 , 2 1 1 0 , 1 0 , 0 , , 2 0 , 2 1 0 , 1 0 , t T T t m m R V t P P t T T T A h T t T T c t m m c T T t m m c t m m t T T dt d t m m t m m t m m dt d A A A A A A A A W c ref A A p tot s p ref v A A A A tot s A δ δ δ δ δ τ δ δ δ δ τ δ δ δ & & & & & & & & Set of Equations 2

Assuming for state 0 the nominal condition, the steady state value of the “states” of the system (m_A(t) and T_A(t)) can be determined.



















=

=

−

+

−

−

−

=

0 , 0 , 0 , 0 , 0 , 0 , 2 1 0 , 1 2 0 , 1

0

)

(

)

(

)

(

A A A A W c ref A p ref p

RT

V

P

m

T

T

A

h

T

T

c

m

T

T

c

m

m

m

&

&

&

&

Resolving for m_A,0 and T_A,0 it yields:

) ( ) ( 0 , 2 0 , 2 1 0 , 1 0 , 0 , 0 , 0 , p c W c ref p ref p A A A A c m A h AT h T c m T T c m T RT V P m & & & + + + − = =

Neglecting second order terms and substituting for

T

0 (only where necessary to simplify some

             + = − − − − − + − + + − − − + − + − = − − = )) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 ; 0 , 0 , 2 1 0 , 1 , 2 0 , , 2 0 , 2 0 , 2 1 1 1 0 , 1 0 , , 2 1 0 , t m T t T m R V t P AT h T c m T T c m t T A h AT h T c t m T c t m t T c m T c m T T c t m T T c m c m t T dt d t m t m t m dt d A A A A A A W c ref p ref p A c W c ref p tot s A p tot s A p ref p ref p ref p v A A tot s A δ δ δ δ τ δ τ δ δ δ δ τ δ δ δ & & & & & & & & & & Set of Equations 3

Grouping and simplifying the second equation yields:

            + = + − − + − − − − = − − = )) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 , 0 , , 2 0 , , 2 0 , 2 1 1 0 , , 2 1 t m T t T m R V t P t T A h T c t m T c t m t T c m T T c t m c m t T dt d t m t m t m dt d A A A A A A A c ref p tot s A p tot s A p ref p v A A tot s A δ δ δ δ τ δ τ δ δ δ δ τ δ δ δ & & & & & & Set of Equations 4

A further grouping in the second equation for

δ

T_A(t),

δ

m&₁(t) and

δ

m&₂ yields

(

)

(

)

         + = + − − + − + + − = − − = )) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 , 0 , 0 , , 2 1 1 0 , 2 0 , , 2 1 t m RT t T m R V t P T c T c t m T T c t m c m A h t T c m t T dt d t m t m t m dt d A A A A A A ref p A p tot s ref p p c A v A A tot s A δ δ δ τ δ δ δ δ τ δ δ δ & & & & & Set of Equations 5

(

)

(

)

                 −           +                     =                 −         + − − − +               + − =       ) ( ) ( 0 0 0 0 0 0 ) ( ) ( 1 0 0 1 ) ( ) ( ) ( ) ( ) ( ) ( 1 1 ) ( ) ( 0 0 0 ) ( ) ( , 2 1 0 , 0 , , 2 1 0 , 0 0 , 1 0 , 0 , 2 tot s A A A A A A A A A tot s v A ref p p v A ref p A A v A p c A A t m t m t T t m V Rm V RT t P t T t m t m t m c m T c T c c m T T c t T t m c m c m A h t T t m dt d

τ

δ

δ

δ

δ

δ

δ

δ

τ

δ

δ

δ

δ

δ

δ

& & & & & By Laplace transforming:

[

]

[

]













+

=

−

+

−

−

+

−

=

−

=

− − 0 , 0 , 1 2 0 , 2 1 1 0 2 1

)

(

)

(

(

)

(

)

(

)

(

)

(

)

(

,

)

(

)

(

)

(

)

(

, , A A A A A A ref s p c p A ref p v A s A

T

s

m

s

T

m

R

V

s

P

T

T

e

s

m

c

A

h

c

m

s

T

T

T

c

s

m

mA

c

s

sT

e

s

m

s

m

s

sm

tot s tot s τ τ

&

&

&

&

&

Grouping the second equation for

T(s)

yields:

[

,0 2,0

]

1 1 2

[

,0 1

]

, ) ( ) ( ) ( ) (s c m s m c h A m s c T T c m s e T T T_{A v A} + _p + _c = _p − _ref − _p −τstots _A − & & &

that can be substituted in the state equation to find the required transfer function. It can be seen that the pressure, that is the output signal, is influenced by the inlet flow (i.e. the controlled variable) and by the outlet flow, that is, the disturbance.

A clarifying scheme follows (the dependence of inputs and outputs upon

s

has been suppressed to light the notation).

A h c m s m c T T c m T c p A v ref p A + + − = 0 , 2 0 , 1 1 ) ( & & s c p A v ref p A stot e A h c m s m c T T c m T , 0 , 2 0 , 1 2 ) ( −τ + + − − = & & s m mA 1 1 = & A A A A V Rm T P ,0 = A A A A V RT m P ₌ ,0

The same approach is used for TR (Equation 1) and it yields:

(

)

(

( )

)

2 ) ( ) ( 2 1 )) ( ( ) ( , 0 , 2 2 2 2 0 , 2 2 0 , 2 3 2 2 4 2 1 , 0 , 0 , tot l A A TR tot tot TR tot tot s A A TR TR t A t m t m m m K K A A K t P P t P P τ δρ ρ δ δ τ δ δ − + + +         + +       + − − + =

+ & & & &

Equation 2

Multiplying by the density term which appears at the denominator of the last RHS term and neglecting 2nd order terms Equation 2 becomes:

(

)

2 2 0 , 2 2 0 , 2 , 0 , , 0 , 0 , 0 , 0 , 0 , , 0 , 0 , ) ( ) ( ) ( ) ( ) ( TR h tot s A A tot l A A A A TR A TR tot l A TR A A m t m m R t P t P P t P P t P & & & + + − − + − + = + − + δ τ δ ρ τ δρ ρ δ ρ τ δρ ρ Equation 3 where:         + +       +

= tot TR tot tot

h K K A A K R _{2 3} 2 4 2 1 1

The steady state conditions can now be found from Equation 4:

2 2 0 , 2 0 , 0 , 0 , 0 , TR h A A TR A

A

m

R

P

P

=

ρ

−

&

ρ

Equation 4

(

ρ

_A,0is expressed by the following relation

A A A V m _,0 0 , =

ρ

)

Substituting Equation 4 into Equation 3t and dividing by the steady state density yields:

(

)

₂ 0 , 2 0 , 2 , 0 , , 0 , 0 , ) ( ) ( ) ( ) ( TR A h tot s A A tot l A TR A TR A t m m R t P t P P t P

ρ

δ

τ

δ

ρ

τ

δρ

δ

= − − + − − & & Equation 5

(

)

(

)

          − −                   − − = ) ( ) ( ) ( 1 ) ( , 2 , 2 0 , 0 , 2 0 , 0 , 0 , tot s A tot l A TR A h A TR A TR t P t m t A m R P P t P

τ

δ

δ

τ

δρ

ρ

ρ

δ

& & By Laplace transforming:

(

)

       = − + − = − − A A A TR A h s A s A A TR A TR V s m s A s m m R e s P e s P P s P ltot stot ) ( ) ( ) ( ) ( ) ( ) ( ₂ 0 , 2 0 , 2 0 , 0 , 0 , , ,

δρ

ρ

δ

δ

ρ

δρ

δ

τ τ & &

It is possible to notice that the pressure at PT1 is influenced by the density of the gas flux, the mass flow rate flowing into the pipe and the back pressure in volume A. All dependences apart that one upon Volume A pressure are disturbances according to the needs of the process.

(

)

s A TR A A TR stot

e

P

P

P

, 0 , 0 , 0 , τ

ρ

ρ

−

−

=

2 0 , 0 , 2 2 A TR h TR

A

m

R

m

P

ρ

&

&

=

−

s A TR stot

e

P

P

=

−τ , TR

P