The improvement of pipeline mathematical model for the purposes of leak detection

Transcription

1 The improvement of pipeline mathematical model for the purposes of leak detection A. Bratek (a), M. Słowikowski (a), M. Turkowski (b) (a) Industrial Research Institute for Automation and Measurements, Al. Jerozolimskie 0, Warszawa, Poland (b) Warsaw University of Technology, Institute of Metrology and Measurement Systems, ul. Boboli 8, 0 55 Warszawa, Poland Abstract The paper presents the improvements of the mathematical model of liquid flow dynamics in long distance pipelines. The model was formulated on the basis of the real pipeline system as a result of research concerning the leak detection and localization algorithms. The model takes into account the liquid pressure and velocity in the pipeline, but also the impact of the others important system elements such as pumps, valves and a receiving tank. This allows more precise detection and localization of the pipe leaks. 1. Introduction The aim of the research was to improve the pipeline system model developed for the purposes of leakage detection and localization. The modelling of liquid flow dynamics for long distance transfer pipelines is essential for the analysing of physical phenomena that occur in the pipeline, which are extremely difficult to be detected and examined without being supported with mathematical and computational methods. The mathematical model presented in the paper enables the simulation of various events related to liquid transportation, such as start up or shut down the pumping, switching inlet or outlet of the pipeline from one tank to another, liquid transport in steady state conditions, stand-by conditions and the situations when two or more media of diverse physical characteristics are being transported subsequently through the same pipeline. The accurate mathematical model describing the real pipeline in all technological situations is of greatest importance for the analytical methods of

2 56 A. Bratek, M. Słowikowski, M. Turkowski the leak detection and localization. Existing models, however, rarely take into account other than the pipe elements of the system.. Short description of a pipeline transport system Figure 1 presents the simplified scheme of the pipeline transport system [1]. A liquid is drawn from the supply tank TA by the main pump and the auxiliary pump 1, and pumped through a pipeline to the receiving tank TB. The tanks are filled to different levels, with products of various physical characteristics. Remotely controlled gate valve stations are installed along a pipeline every several kilometres. T A T B Fig. 1. Simplified scheme of a pipeline transport system Each tank can be switched from one tank to another in the both supply A or receiving B tank set, which may result in a sudden disturbance of pressure at the pipe input or output. 3. The model of the pipeline For the modeling purposes the pipeline was arbitrarily divided into sections at points where measuring transmitters were installed and each section between these points was divided into shorter, equal segments. Every segment fulfils a set of partial differential equations as a result of the law of mass and momentum conservation. In the case of a leak-proof pipeline these equations, according to [4] can be written as ( x t) ( x, t) 1 p( x, t) w x (, t) + E t = 0 ( x) ( x) p, w x λ ρ + ρ( x) = ρ( x) gsinα x t d w t ( ) w( t) (1) ()

3 The improvement of pipeline mathematical model for the purposes of leak detection 563 where E is the elasticity coefficient of a liquid-pipeline system, λ(x) is the pipe friction factor and α - the angle of inclination of a pipeline segment. The mathematical model should however also reconstruct as accurately as possible the static and dynamic characteristics of all other elements of a pipeline system as well as interactions between these elements. 4. The model of primary pump The static characteristics of the pump for petrol of the density 755 kg/m 3, was approximated with reference to the pump catalogue data as ( q) 6 3 H =, q + 0, q , q + 340, 95 3 (3) where H(q) = HT HS is the increment of a liquid column between pump suction and force sides (in m) and q is the value of the volumetric flow rate (in m 3 /h). In the pump nominal working conditions (00 m 3 /h < q < 300 m 3 /h) the relative error of the approximation (3) is less than 0.%. The pressure directly behind the pump is given by ( t) + g H ( q) p( 0, t) = ps ρ (4) where ps(t) is the pressure at the auxiliary pump outlet, ρ is the density of the pumped medium and g is the acceleration due to the gravity. Considering the liquid delivery, the pump can be treated as a non inertial object. The pump reaches its nominal speed in 5-7 s while the volumetric flow rate q(0,t) achieves more than 80 % of its steady state value. Later variation of q(0,t) is implied by the pipeline dynamics and back coupling delay in flow influence on the pressure behind the pump. Disregarding the delay reason, the transmittance obtained experimentally in form G(s)=(Ts+1) - (5) was included into the model to relate q(0,t) with H(q) in the equation (3). 5. The model of receiving tank The real pipe installation has been equipped with the outlet tank of diameter DT = 56 m, height of 13 m with a floating roof causing small constant

4 564 A. Bratek, M. Słowikowski, M. Turkowski overpressure above the liquid surface. The liquid flows to the tank or from the tank through manifolds installed in the bottom part of the tank. The change of the liquid level in any tank connected to a pipeline results in a change of the pressure at either the input or the output of a pipeline, introducing in this way a disturbance to the transportation process. The change of the liquid level rate is given by dh T D = 3600 w( xt, t) (6) D T and the pressure variation rate at the pipeline output is given by dp T ( t) dht = ρ( xt, t) g (7) where D is the pipe inner diameter (m), xt is the pipeline length (m). The liquid velocity in a pipeline, w(t), is less than 1,1 m/s, then the maximum change rate of the liquid level in a tank is smaller than 0.1 m/h, so the maximum change rate of the pressure at the output is about 1000 Pa/h. However, switching the fully filled tank to the empty one causes strong disturbances, and it generates a sudden pressure jump at the output side of the system. This change can achieve even pt = ρg HT 0.11 MPa. 6. The model of gate valve The pins of the real pipeline valves are moved by electrical drives with constant linear movement speed. The average time period of switching between one to another extreme position was about 150 s. The valve pressure loss coefficient z is defined as the ratio of the pressure drop across the valve and the total kinetic energy of a flowing medium: p z = (8) ρw For numerical simulation the following relation was used p p w = = K(x) (9) z ρ ρ

5 The improvement of pipeline mathematical model for the purposes of leak detection 565 where z is the pressure loss coefficient, p is the pressure drop across the valve (Pa), w is the flow velocity of the liquid (m/s). K(x) is the coefficient related to the valve s aperture x=h/d, H is the linear shift of the valve pin and D is the pipeline inner diameter. It was also assumed that K(x) = KZ ϕ(x), where ϕ(x) is the ratio of a valve s clearance surface to the cross section of the pipe given by the formula ϕ 1 ( x) = arccos(1 x) ( 1 x) x x π π (11) The value KZ = 0.45 was calculated from pressure balance along a pipe. 7. Conclusions The inclusion of the pump, valves and tanks to the mathematical model enabled the increase of the performance of the leak detection system. It allows to detect the position of the leak with accuracy ±(100 00) m for steady flow and during transient process (pumping start up, switching the outlet tank, changing the medium being transported) with accuracy ±(00 400) m. It must be underlined that for unsteady flow the existing systems [, 3, 4] cannot detect the leak or they generate false alarms. Unfortunately some parameters were not measured: the density of the liquid ρ, the pressure at the primary pump inlet ps and the pressure just before the recieving tank ph. There is no doubt that these measurements would increase the quality of the model even more. The paper is a result of research financed as a research project within the scope of the Multi-Year Programme PW-004 Development of innovativeness systems of manufacturing and maintenance References [1] Michałowski Witold S., Trzop S.: Rurociągi dalekiego zasięgu, wydanie IV, Fundacja Odysseum, Warszawa, 005 [] Bilman, L. Isermann R.: Leak detection methods for pipelines, Automatica, vol.3, no. 3, 1987, Pages [3] H. Siebert: "Dynamische Leckuberwachnng bei Pipelines", Erdol Ergas Kohle, 116 Jahrgang, Heft 11, November 000 [4] Sobczak R: Detekcja wycieków z rurociągów magistralnych cieczy, Nafta, Gaz, nr /01, pp