Models of Pipelines in Transient Mode

Mathematical and Computer Modelling of Dynamical Systems

ISSN: 1387-3954 (Print) 1744-5051 (Online) Journal homepage: http://www.tandfonline.com/loi/nmcm20

Models of Pipelines in Transient Mode Drago Matko & Gerhard Geiger To cite this article: Drago Matko & Gerhard Geiger (2002) Models of Pipelines in Transient Mode, Mathematical and Computer Modelling of Dynamical Systems, 8:1, 117-136 To link to this article: http://dx.doi.org/10.1076/mcmd.8.1.117.8341

Published online: 09 Aug 2010.

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Date: 09 June 2017, At: 03:16

Mathematical and Computer Modelling of Dynamical Systems 2002, Vol. 8, No. 1, pp. 117±136

1387-3954/02/0801-117$16.00 # Swets & Zeitlinger

Models of Pipelines in Transient Mode DRAGO MATKO1 AND GERHARD GEIGER2 ABSTRACT The paper deals with the models for pipelines in transient mode. The nonlinear distributed parameter model is ®rst linearized and a two input ± two output system is obtained. The models represented by basic canonical equations are then rearranged into four feed-back models (two hybrid, impedance and admittance) which are capable to describe the oscillations phenomena. Transfer functions in the feed-back models are transcendent with a time delay. Transcendent transfer functions are then approximated with rational transfer functions. The derived models are validated on a real pipeline data during start-up and shutdown phase of the operation.

Keywords: dynamic behavior, oscillation, pipelines.

1. INTRODUCTION Models of pipelines are widely used in designing controllers, leakage monitoring, etc. [1, 2, 3]. The essential model of the pipeline is a set of nonlinear partial differential equations representing the pipeline as a nonlinear distributed parameter system. Numerical solution procedures are available from the ®eld of computational ¯uid dynamics (CFD) [4]. However, sometimes simple models of the pipeline in the form of a lumped parameter system are useful: the classical system theory for Multi-Input Multi-Output (MIMO) systems can be used, e.g. for controller design and system identi®cation. The resulting algorithms are less time-consuming and hence better suited for critical real time applications. Additionally, the analysis of ¯uid transients caused by leaks is much easier. Linearization around an operating point yields a set of linear partial equations, which can be solved analytically. The resulting transfer functions are transcendent [5]. Further simpli®cations enable representation of pipelines as lumped parameter systems. In [6] and [7] simple models are derived which are valid for low frequencies and well damped pipelines, respectively. In this paper models of pipelines in transient mode, i.e. models for oscillatory pipelines are derived. Oscillations on pipelines occur mainly during start-up and shutdown phases of operation. The start-up phase can be used for identi®cation of critical 1 2

Faculty of Electrical Engineering, Univ. of Ljubljana, TrzÏasÏka 25, 1000 Ljubljana Slovenia. Univ. of applied sciences Gelsenkirchen, Neidenburger Straûe 10, 45877 Gelsenkirchen, Germany.

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D. MATKO AND G. GEIGER

parameters of the pipeline, such as velocity of sound and of the ¯uid density, which depend on the applied ¯uid and vary signi®cantly with respect to the used ¯uid and its temperature. Main contribution of the paper is the derivation of simple equations which are capable to describe oscillations in pipelines. Transcendent transfer functions are simpli®ed into a closed loop con®guration of rational transfer functions with time delay. The resulting closed loop structure represents the shock waves travelling in the pipeline back and forth. The parameters of the simple transfer functions are obtained by the Pade approximation. The paper is organized as follows: First, nonlinear and linear distributed parameter mathematical models of the pipeline will be given in Section 2. Linear models will be represented in four canonical forms (two hybrid, impedance and admittance). In Section 3 models for pipelines in transient mode are derived. Subsection 3.1 is devoted to the derivation of basic equations which include transcendent transfer functions. These are simpli®ed in the Subsection 3.2 where rational approximations of the transcendent transfer functions are given. In Section 4 the veri®cation on a real pipeline data during start-up and shut-down phase of the operation is presented. 2. MATHEMATICAL MODELS OF THE PIPELINE The analytical solution for unsteady ¯ow problems is obtained by using the equations for continuity, momentum and energy [8]. These equations correspond to the physical principles of mass conservation, Newton's second law F ˆ ma and energy conservation. Applying these equations leads to a coupled nonlinear set of partial differential equations. Further problems arise in the case of turbulent ¯ow, which introduces stochastic ¯ow behavior [9]. Therefore, the mathematical derivation for the ¯ow through a pipeline is a mixture of both theoretical and empirical approaches. The following assumptions for the derivation of a mathematical model of the ¯ow through pipelines are made: Fluid is compressible, viscous, isentropic and homogenous. The resulting nonlinear distributed parameters model is described by the following two partial differential equations [10]: A @p @q ˆÿ …1† 2 a @t @x 1 @q …q† 2 @p ‡ g sin  ‡ q ˆÿ A @t 2DA2  @x

…2†

where A, D and  are the pro®le, the diameter and the inclination of the pipeline respectively, p the pressure, q the mass ¯ow rate,  the friction coef®cient,  the density of the ¯uid, g the gravity constant and a the velocity of sound. Extension to the general non-isentropic case requires the additional use of the equation of energy, enthalpy or entropy [11].

PIPELINES IN TRANSIENT MODE

119

Nonlinear Equations (1, 2) are linearized and written in a form using notations common in the analysis of electrical transmission lines. Also, the gravity effect is included into the working point. The corresponding system of linear partial differential equations is L

@q @p ‡ Rq ˆ ÿ @t @x @p @q ˆÿ C @t @x

…3† …4†

q† q where L ˆ A1 , R ˆ … q is the ¯ow at the working point) and C ˆ aA2 are the inertance A2 D ( (inductivity), resistance and capacitance per unit length, respectively. The next step in the derivation of a simple model of the pipeline is the analytical solution of linear Equations (3, 4). The Laplace transformation of them yields the linear non-causal model

PL …s† ˆ ZK Q0 …s† sinh…nLp † ‡ P0 …s† cosh…nLp † 1 P0 …s† sinh…nLp †: QL …s† ˆ Q0 …s† cosh…nLp † ÿ ZK where Lp is the length of the pipeline and p n ˆ …Ls ‡ R†
Cs r Ls ‡ R Ls ‡ R ˆ ZK ˆ n Cs

…5† …6†

…7† …8†

Finally, the linearized model of the pipeline can be written in one of the following four forms which differ from each other with respect to the model inputs (independent quantities) and outputs (dependent quantities): 1. Hybrid representation: Inputs Q0 , PL , outputs QL , P0 : 2 3 1   ZK tanh…nLp †   6 cosh…nLp † 7 PL P0 7 ˆ6 4 5 Q0 1 1 QL ÿ tanh…nLp † ZK cosh…nLp † 2. Hybrid representation: Inputs QL , P0 , outputs Q0 , PL : 2 3 1   ÿZK tanh…nLp †   6 cosh…nLp † 7 P0 PL 7 ˆ6 4 5 QL 1 1 Q0 tanh…nLp † ZK cosh…nLp †

…9†

…10†

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D. MATKO AND G. GEIGER

3. Impedance representation: Inputs Q0 , QL , outputs P0 , PL : 

P0 PL



2 6 ˆ6 4

ZK coth…nLp † ÿZK

1 sinh…nLp †

3 1   sinh…nLp † 7 7 Q0 5 QL ÿZK coth…nLp † ZK

4. Admittance representation: Inputs P0 , PL , outputs Q0 , QL : 2 3 1 1 1     coth…nLp † ÿ 6 ZK ZK sinh…nLp † 7 Q0 7 P0 ˆ6 4 1 5 PL 1 1 QL ÿ coth…nLp † ZK sinh…nLp † ZK

…11†

…12†

It should be noted that the forms 1 to 4 represent causal models, whilst Equations (5, 6) represent a noncausal model with inputs P0 , Q0 and outputs PL , QL . This completes the derivation of the transfer functions of the linearized pipeline. The resulting transfer functions are transcendent. In the next section, the models for pipelines in transient mode will be given.

3. MODELS FOR PIPELINES IN TRANSIENT MODE This section is organized as follows: First four equations will be derived, which express the ¯ow and pressure respectively at the inlet and outlet of the pipeline in dependence of all three remaining quantities. Two equations out of the four will be combined to represent the four causal forms (two hybrid, impedance and admittance) in the Subsection 3.1. The resulting models have a feed-back structure which is with corresponding time delays capable to describe the oscillations phenomena. In Section 3.2 lumped parameter models (rational transfer functions) will be derived from transcendent transfer functions. 3.1. Basic Equation Derivation The ¯ow at the outlet of the pipeline can be expressed from (12) as follows: 2 P0 enLp ‡ eÿnLp PL ÿ ÿnL p ÿe ZK enLp ÿ eÿnLp ZK 2eÿnLp P0 1 ‡ eÿ2nLp PL ˆ ÿ 1 ÿ eÿ2nLp ZK 1 ÿ eÿ2nLp ZK

QL ˆ

enLp

…13†

PIPELINES IN TRANSIENT MODE

121

or equivalently QL ÿ eÿ2nLp QL ˆ 2eÿnLp

P0 PL PL ÿ ÿ eÿ2nLp ZK ZK ZK

…14†

The same ¯ow can be expressed also from (9): 2 enLp ÿ eÿnLp PL Q ÿ 0 enLp ‡ eÿnLp ZK enLp ‡ eÿnLp ÿnLp 2e 1 ÿ eÿ2nLp PL ˆ Q ÿ 0 1 ‡ eÿ2nLp 1 ‡ eÿ2nLp ZK

QL ˆ

…15†

or equivalently QL ‡ eÿ2nLp QL ˆ 2eÿnLp Q0 ÿ

PL PL ‡ eÿ2nLp ZK ZK

…16†

Adding Equations (14) and (16) and dividing by 2 the equation which describes the outlet ¯ow in dependence of the inlet ¯ow and both (inlet and outlet) pressures is obtained: QL ˆ eÿnLp Q0 ‡ eÿnLp

P 0 PL ÿ ZK ZK

…17†

In the same way the inlet ¯ow can be rewritten from Equation (12): 2 PL enLp ‡ eÿnLp P0 ‡ enLp ÿ eÿnLp ZK enLp ÿ eÿnLp ZK 2eÿnLp PL 1 ‡ eÿ2nLp P0 ˆÿ ‡ 1 ÿ eÿ2nLp ZK 1 ÿ eÿ2nLp ZK

Q0 ˆ ÿ

…18†

or equivalently Q0 ÿ eÿ2nLp Q0 ˆ ÿ2eÿnLp

PL P0 P0 ‡ ‡ eÿ2nLp ZK ZK ZK

…19†

and also from (10): 2 enLp ÿ eÿnLp P0 QL ‡ nLp ÿnL p ‡e e ‡ eÿnLp ZK ÿnLp 2e 1 ÿ eÿ2nLp P0 ˆ Q ‡ L 1 ‡ eÿ2nLp 1 ‡ eÿ2nLp ZK

Q0 ˆ

enLp

…20†

or equivalently Q0 ‡ eÿ2nLp Q0 ˆ 2eÿnLp QL ‡

P0 P0 ÿ eÿ2nLp ZK ZK

…21†

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D. MATKO AND G. GEIGER

Adding Equations (22) and (25) and dividing by 2 the equation which describes the inlet ¯ow in dependence of the outlet ¯ow and both (inlet and outlet) pressures is obtained: Q0 ˆ eÿnLp QL ÿ eÿnLp

PL P0 ‡ ZK ZK

…22†

Similarly the outlet pressure can be expressed from Equation (9): 2 enLp ÿ eÿnLp PL ‡ nLp ZK Q0 ÿnL p ‡e e ‡ eÿnLp 2eÿnLp 1 ÿ eÿ2nLp ˆ P ‡ ZK Q 0 L 1 ‡ eÿ2nLp 1 ‡ eÿ2nLp

P0 ˆ

enLp

…23†

or equivalently P0 ‡ eÿ2nLp P0 ˆ 2eÿnLp PL ‡ ZK Q0 ÿ eÿ2nLp ZK Q0

…24†

and also from Equation (11): 2 enLp ‡ eÿnLp Z Q ‡ ZK Q0 K L enLp ÿ eÿnLp enLp ÿ eÿnLp 2eÿnLp 1 ‡ eÿ2nLp ˆÿ ZK QL ‡ ZK Q0 ÿ2nL p 1ÿe 1 ÿ eÿ2nLp

P0 ˆ ÿ

…25†

or equivalently P0 ÿ eÿ2nLp P0 ˆ ÿ2eÿnLp ZK QL ‡ ZK Q0 ‡ eÿ2nLp ZK Q0

…26†

Adding Equations (24) and (26) and dividing by 2 the equation which describes the inlet pressure in dependence of the outlet pressure and both (inlet and outlet) ¯ows is obtained: P0 ˆ eÿnLp PL ÿ eÿnLp ZK QL ‡ ZK Q0

…27†

Finally from Equation (10) the outlet pressure can be expressed as: 2 enLp ÿ eÿnLp P ÿ ZK QL 0 enLp ‡ eÿnLp enLp ‡ eÿnLp 2eÿnLp 1 ÿ eÿ2nLp ˆ P ÿ ZK QL 0 1 ‡ eÿ2nLp 1 ‡ eÿ2nLp

PL ˆ

…28†

or equivalently PL ‡ eÿ2nLp PL ˆ 2eÿnLp P0 ÿ ZK QL ‡ eÿ2nLp ZK QL

…29†

PIPELINES IN TRANSIENT MODE

123

Fig. 1. Hybrid representation: inputs Q0 , PL , outputs QL , P0 .

and also from Equation (11): 2 enLp ‡ eÿnLp ZK Q0 ÿ nLp ZK Q L ÿnL p ÿe e ÿ eÿnLp 2eÿnLp 1 ‡ eÿ2nLp ˆÿ Z Q ÿ ZK Q L K 0 1 ÿ eÿ2nLp 1 ÿ eÿ2nLp

PL ˆ ÿ

enLp

…30†

or equivalently PL ÿ eÿ2nLp PL ˆ 2eÿnLp ZK Q0 ÿ ZK QL ÿ eÿ2nLp ZK QL

…31†

Adding Equations (29) and (31) and dividing by 2 we get the equation which describes the outlet pressure in dependence of the inlet pressure and both (inlet and outlet) ¯ows: PL ˆ eÿnLp P0 ‡ eÿnLp ZK Q0 ÿ ZK QL

…32†

The four causal representations for low damped pipelines according to the enumeration 1 to 4 in Section 2 are shown in Figures 1, 2, 3 and 4 respectively. 3.2. Lumped Parameter Models for Low Damped Pipelines Simple models for pipelines in transient pmode involve three different transcendent ÿ …Ls‡R†CsLp 1 . They will be approximated by second transfer functions, ZK , ZK and e order rational transfer functions (as a second order lumped parameter system) G…s† ˆ

b2 s2 ‡ b1 s ‡ b0 ÿsTd e a 2 s 2 ‡ a1 s ‡ a0

…33†

where Td is the dead time. The transcendent transfer functions will be approximated by a rational transfer function with dead time. The procedure of the approximation is

124

D. MATKO AND G. GEIGER

Fig. 2. Inputs QL , P0 , outputs Q0 , PL .

Fig. 3. Impedance representation: inputs Q0 , QL , outputs P0 , PL .

Fig. 4. Admittance representation: inputs P0 , PL , outputs Q0 , QL .

as follows: the static gain and the high frequency gain will be determined from transcendentpfunction  and known dead time will be applied. In the case of the transfer function eÿ …Ls‡R†CsLp also the closed loop static gain will be used to serve one additional equation. In this way, three and four coef®cients respectively of the transfer function (33) will be determined. The remaining three will be obtained using a Pade approximation of the transcendent transfer function. Since the Taylor extension will be made on variable sÿ1 , the derived models are only valid for high frequencies and so for low-damped pipelines. The derived models are as follows:

PIPELINES IN TRANSIENT MODE

3.2.1. Characteristic Impedance ZK ˆ

r Ls ‡ R Cs

125

…34†

connects quantities on the same side of the pipeline, so the time delay is zero Td ˆ 0 The static and high frequency gains are obtained by: r Ls ‡ R ˆ1 lim ZK ˆ lim s!0 s!0 Cs r r Ls ‡ R L ˆ lim ZK ˆ lim s!1 s!1 Cs C

…35†

…36† …37†

From Equation (36) it follows that the transfer function has an integral character, so the coef®cient a0 ˆ 0. In this case a2 can be chosen to be 1 without loss of generality. The high frequency gain b2 can be determined by Equation (37) r L …38† b2 ˆ C The remaining coef®cients b1 , b0 and a1 are determined using a Pade approximation of (34) i.e. by comparing the corresponding terms of the following two series expansions: r r r r r r L R L L R ÿ1 L R2 ÿ2 L R3 ÿ3 1‡ ˆ ‡ s ÿ s ‡ s ÿ


…39† C Ls C C 2L C 8L2 C 16L3 and r L 2  s ‡ b1 s ‡ b0 r L C ˆ ‡ s 2 ‡ a1 s C ‡ a1

r ! r ! L L 2 ÿ2 ÿ1 b1 ÿ a1 s ‡ b2 ÿ b1 a1 ‡ a s C C 1 r ! L 2 ÿ3 a s ‡


ÿb2 ‡ a1 b1 ÿ C 1

The resulting transfer function is r r r L 2 L R L R2 s ‡ s‡ CL C 8L2 ZK  C R s s2 ‡ 2L

…40†

…41†

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D. MATKO AND G. GEIGER

It must be noted again that this approximation q is a good approximation for high frequencies only. For low frequencies ZK  CR p1s which has no lumped parameter equivalence unless R ˆ 0. 3.2.2. Characteristic Admittance 1 ˆ ZK

r Cs Ls ‡ R

…42†

is the inverse of the characteristic impedance. It has the derivative characteristic with the transfer function r r C 2 C R s s ‡ 1 L L 2L  ZK R R2 s2 ‡ s ‡ 2 8L L

…43†

p ˆ 1:41, Its damping is 28 q p so the response is an overdamped response. Again for low 1 C frequencies ZK  R s which has no lumped parameter equivalence unless R ˆ 0.

3.2.3. Transfer Function G…s† ˆ eÿ

p

…Ls‡R†CsLp

…44†

connects quantities on the different sides of the pipeline, so the time delay is Td ˆ

p LC Lp

…45†

The static and high frequency gains are obtained by: lim G…s† ˆ lim eÿ s!0

p

…Ls‡R†CsLp

s!0

lim G…s† ˆ G1 ˆ lim eÿ

s!|1

s!|1

ˆ1 p

…Ls‡R†CsLp

ˆ eÿ

pC RLp L
2

…46† …47†

The static gain (46) is 1 so the coef®cient b0 ˆ a0 . a2 is chosen to be 1 and the high frequency gain determines that b2 ˆ G1 . There are two possibilities to determine the remaining coef®cients a0 , a1 and b1 . With the ®rst one all three are determined by the Pade approximation while with the second one the static gain of the hybrid closed loop models is used as one additional equation and only two equations of the PadeÂ

PIPELINES IN TRANSIENT MODE

127

approximation are used. In both cases the transcendent transfer function is expanded into Taylor series as follows: r p p L LpR2 ÿ1 s e LCLp s eÿ …Ls‡R†CsLp ˆ G1 ‡ G1 C 8L2 p LpR3 …CLpR ÿ 8 LC † ÿ2 ‡ G1 s 3 128L p  p LpR4 …120 LCL ÿ 24RLCLp ‡ LC CR2 Lp2 † ÿ3 ‡ G1 s ‡


…48† 3072L5 3.2.3.1. All Three Equations by the Pade Approximation: The rational transfer function approximate is expanded as follows: G 1 s 2 ‡ b1 s ‡ a0 ˆ G1 ‡ …b1 ÿ G1 a1 †sÿ1 s 2 ‡ a1 s ‡ a0 ‡ …a0 ÿ G1 a0 ‡ …ÿb1 ‡ G1 a1 †a1 †sÿ2 ÿ
‡ …ÿb1 ‡ G1 a1 †a0 ‡ …ÿa0 ‡ G1 a0 ‡ a1 b1 ÿ G1 a21 †a1 sÿ3 ‡




…49†

Comparing the second to the fourth term of Equations (49) and (50) and solving the system of equations the following coef®cients of the resulting transfer function are obtained:   1 2 2 1 L R C ÿ L Lp R3 CG1 384 p 8 p p
…50† a0 ˆ ÿ 3 L 8G1 LC ‡ G1 Lp RC ÿ 8 LC ‡ CLp R  p 1  R ÿ2L2p R2 G1 C ÿ 120L ‡ 120LG1 ‡ 24Lp R LC ÿ L2p R2 C p p
ÿ …51† a1 ˆ 24 L 8G1 L ‡ G1 Lp R LC ÿ 8L ‡ Lp R LC p 1 G1 R…L2p R2 G1 C ÿ 120L ‡ 120LG1 ‡ 2L2p R2 C ‡ 24 LC Lp RG1 † p p
ÿ b1 ˆ 24 L 8G1 L ‡ G1 Lp R LC ÿ 8L ‡ Lp R LC

…52†

3.2.3.2. Two Equations from Pade and the Static Gain of the Hybrid Closed Loop: From the static gain of the hybrid models (Figs. 1 and 2) and from the static gain of the hybrid representation lim ZK tanh…nLp † s!0 r p  Ls ‡ R tanh …Ls ‡ R†CsLp ˆ Lp R ˆ lim s!0 Cs

…53†

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D. MATKO AND G. GEIGER

it follows that lim ZK s!0

1 ÿ G2 …s† ˆ Lp R 1 ‡ G2 …s†

…54†

Inserting (41) and (33) we get

and consequently

ÿ p
1 ÿ G2 …s† R a0 LC Lp ‡ a1 ÿ b1 p ˆ lim ZK s!0 1 ‡ G2 …s† 8a0 CL

…55†

p a1 ˆ b1 ‡ 7a0 LC Lp

…56†

The coef®cients b1 and a0 are determined by comparison if the second and third term of (49) and the following Taylor series expansion   p G 1 s 2 ‡ b1 s ‡ a0 p ˆ G1 ‡ b1 ÿ G1 …7Lp a0 LC ‡ b1 † sÿ1 s2 ‡ …b1 ‡ 7a0 LC Lp †s ‡ a0     p p ‡ …1 ÿ G1 †a0 ‡ ÿb1 ‡ 7G1 Lp a0 LC ‡ G1 b1 7Lp a0 LC ‡ b1 sÿ2 ‡




…57†

The result is: a0 ˆ

p p R3 Lp G1 ……Lp G1 C ‡ CLp †R ÿ 8 LC ‡ 8G1 LC †   16L2 ÿ7L2p CG1 R2 ‡ 8L ÿ 16G1 L ‡ 8LG21

 p p Lp R2 G1 LC 56Lp RG1 LC ‡ 7L2p CG1 R2 ‡ 16G1 L ÿ 16L   b1 ˆ 16L2 7L2p CG1 R2 ÿ 8L ‡ 16G1 L ÿ 8LG21

…58†

…59†

4. VERIFICATION OF THE PIPELINE MODELS IN TRANSIENT MODE The models were applied to a real pipeline with the following data: Length of the pipeline Lp ˆ 9854 m, diameter D ˆ 0:2065 m, relative roughness kc ˆ 0:0561 mm, inclination  ˆ ÿ0:1948 and the ¯uid data: Density  ˆ 654 kg=m3, kinematic viscosity v ˆ 7:0  10ÿ7 m2 =s and velocity of sound a ˆ 951 m=s. First the comparison of step responses of the transcendent transfer functions Z1K and p  ÿ …Ls‡R†CsLp e with their rational approximations will be given. Step responses are obtained by the integral of the inverse Fourier transform of the frequency response.

PIPELINES IN TRANSIENT MODE

129

Frequency response is obtained by replacing s with |! in transcendent transfer functions. Then the treated models will be veri®ed on start-up and shut down phase of the real pipeline operation. The working point of the pipeline is chosen at ¯ow velocity v ˆ 0:6 m=s, which is about 30% of the steady state ¯ow velocity. In this working point the damping factor [7] r 1 C
R
Lp ˆ 0:37 …60†  2 2L which indicates a low damped pipeline. 4.1. Veri®cation of the Step Responses First the step responses for the transfer functions Z1K and enLp and their approximations are evaluated. In Figure 5 the step responses of the transfer function Z1K (solid line) and of its approximation with the rational transfer function 43 (dotted line) are shown. It can be seen that both responses coincide for t ! 0. This is due to the high frequency Taylor series expansion of the transcendent transfer function. In Figure 6 the step responses of the transfer function (44) (solid line) and of its two approximations with the rational transfer function (33) are shown. The dotted line

Fig. 5. Step response of the Z1K : transcendent (solid) and rational (dashed).

130

D. MATKO AND G. GEIGER

Fig. 6. Step response of the enLp : transcendent (solid) and rational (dashed and dotted).

denotes the ®rst approximation, i.e. the evaluation of all remaining coef®cients by the Pade approximation, while the dashed line denotes the approximation where the static gain of the hybrid models is taken into account. It can be seen that all responses coincide for t ! 0 and that the dashed line better approximates the solid line, i.e. that the approximation using static gain of the hybrid models is better. So in the sequel this approximation will be used. 4.2. Veri®cation of the Hybrid Models In Figures 7 and 8 the start-up phase of the pipeline operation is shown. In Figure 7 the measured (solid line) and simulated (dashed line) velocities at the pipeline inlet are depicted, while Figure 8 shows the measured (solid line) and simulated (dashed line) pressure at the outlet of the pipeline. Good accordance of the measured and simulated responses can be stated. This means that the derived models are capable to describe all phenomena of low damped pipelines. In Figures 9 and 10 the shut-down phase of the pipeline operation is shown. At this phase both valves at the inlet and outlet of the pipeline are shut, forcing both ¯ows to zero. The inlet and outlet pressures are oscillating as shown in the Figure 10 for the outlet pressure. The simulated inlet ¯ow should be 0. This is due to the subtracting of

PIPELINES IN TRANSIENT MODE

131

Fig. 7. Veri®cation of the hybrid models ± start-up: the measured (solid line) and simulated (dashed line) inlet ¯ow.

Fig. 8. Veri®cation of the hybrid models ± start-up: the measured (solid line) and simulated (dashed line) outlet pressure.

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D. MATKO AND G. GEIGER

Fig. 9. Veri®cation of the hybrid models ± shut-down: the measured (solid line) and simulated (dashed line) inlet ¯ow.

Fig. 10. Veri®cation of the hybrid models ± shut-down: the measured (solid line) and simulated (dashed line) outlet pressure.

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133

the outlet pressure (PL ) ®ltered by enLp and the inlet pressure, as shown in Figure 2. In the beginning of the transient the simulated ¯ow in Figure 9 is not 0 and the simulated pressure PL in Figure 10 deviates from the measured one. This is due to different steady state conditions prior to the shut-down of the pipeline our model is not valid for this operation point. After vanishing of the in¯uence of this transient, both simulated responses are satisfactory. 4.3. Veri®cation of the Admittance Models In Figures 11 and 12 the start-up phase of the pipeline operation is shown. Again good accordance of the measured (solid line) and simulated (dotted line) responses can be stated. In Figures 13 and 14 the shut-down phase of the pipeline operation is shown. As described in Subsection 4.2 at this phase both valves at the inlet and outlet of the pipeline are shut, forcing both ¯ows to zero. The inlet and outlet pressures are oscillating. As both pressures oscillate in the opposite phase, after ®ltering of one pressure by enLp (which reverses the phase) and subtracting form the other pressure, the resulting simulated ¯ow should be zero. Again due to the phenomenon described in Subsection 4.2 this is valid only after vanishing of the transition effects.

Fig. 11. Veri®cation of the admittance models ± start-up: the measured (solid line) and simulated (dashed line) inlet ¯ow.

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Fig. 12. Veri®cation of the admittance models ± start-up: the measured (solid line) and simulated (dashed line) outlet pressure.

Fig. 13. Veri®cation of the admittance models ± shut-down: the measured (solid line) and simulated (dashed line) inlet ¯ow.

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Fig. 14. Veri®cation of the admittance models ± shut-down: the measured (solid line) and simulated (dashed line) outlet ¯ow.

5. CONCLUSIONS The models for pipelines in transient mode were derived from linear models. Four representations are given and successfully tested on real pipeline data. Investigations into adaptation of the linear models to the current working point are under way as well as the knowledge±based gray box modelling [12]. REFERENCES 1. Geiger, G., Gregoritza, W. and Matko, D.: Leak Detection and Localisation in Pipes and Pipelines. European Symposium on Computer Aided Process Engineering-10. ESCAPE-10, 7±10 May, Florence, Italy, 2000, pp. 781±786. 2. Siebert, H.: Untersuchung verschiedener Methoden zur LeckuÈberwachung bei Pipelines. PhD thesis, TH Darmstadt, 1981. 3. Matko, D. and Geiger, G.: Pipeline Supervision: Comparative Evaluation of Different Models for Leak Detection and Localization. ECC Porto, September 2001. 4. Anderson, J.D.: Basic Philosophy of CFD. In: J.F. Wendt (ed.): Computational Fluid Dynamics. Springer-Verlag, Berlin Heidelberg, 2nd edition, 1996. 5. Matko, D., Geiger, G. and Gregoritza, W.: Pipeline Simulation Techniques. Mathematics and Computer Simulation 52 (2000), pp. 211±230.

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6. Matko, D., Geiger, G. and Werner, T.: Modelling of the Pipeline as a Lumped Parameter System. 9th Mediterranean Conference on Control and Automation, Dubrovnik, June 2001. 7. Matko, D., Geiger, G. and Gregoritza, W.: Veri®cation of Various Pipeline Models. 3rd MATHMODSymposium. Vienna, February 2± 4, 2000. 8. Streeter, V.L. and Wylie, E.B.: Fluid Mechanics. McGraw-Hill, New York, 8th edition, 1985. 9. Anderson, J.D. jr.: Modern Compressible Flow. McGraw-Hill, New York, 2nd edition, 1992. 10. Wylie, E.B. and Streeter, V.L.: Fluid Transients in Systems. Prentice-Hall, Englewood cliff, New Jersy, 1993. 11. Rao, C.V. and Eswaran, K.: On the Analysis of Pressure Transients in Pipelines Carrying Compressible Fluids. Int. J. Pres. Ves. and Piping 56 (1993), pp. 107±129. 12. Geiger, G., Werner, T. and Matko, D.: Knowledge Based Leak Monitoring for Pipelines. 4th IFAC Workshop on On-Line Fault Detection & Supervision in the Chemical Process Industries (CHEMFAS-4). Jejudo, Korea, 2001, pp. 289±294.