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Abstract

Flow and pressure transients in closed pipes are described by non-linear hyperbolic partial differential equations. In many control applications a discrete-space lumped-parameter ordinary differential equation model is required. With such representations, care is needed to ensure that they retain essential physical characteristics of the phenomena described. Intentionally or otherwise, most of these fluid-system models are related to the numerical discretisation technique known as the Method of Lines. Aspects of implementing this are investigated, including the form of the basic equations and variables, representation of pipe friction, choice of interpolating polynomials in space and method of integration in time. These are compared with the standard Method of Characteristics solution for the original partial differential equations to eliminate any potential numerical instabilities, parasitic transient solutions or inconsistencies between steady and transient states which might influence adversely system control or stability studies. The Method of Lines is most appropriate for parabolic problems, and many existing applications to fluid transients are found to be satisfactory only where changes are gradual and continuous. The preferred formulation is free from this restriction, and applied to a stability investigation (prediction of compressor surging) is shown to perform reliably.

Keywords

Unsteady flow hyperbolic partial differential equations continuous systems system stability compressor surging.

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References

Ames, W.F. 1977. Numerical methods for partial differential equations , (2nd edn), Academic Press, New York, USA.

Anderson, A. 1983. 'Interaction of surge shafts and penstocks', Proc. 4th Int Conf on Pressure Surges, Canterbury , BHRA, Cranfield, Paper G2, 297-312.

Anderson, D.A. et al. 1984, Computational fluid mechanics and heat transfer, Chap 3 Hemisphere, New York, USA.

Bharath, S. et al. 1990. 'A distributed mathematical model for pressure transient analysis in a railway brake pneumatic system', Intl J Mech Sci, 32, (2), 133-145.

Brunelle, P.E. 1977. 'La simulation des transitoires dans les systèmes hydroélectriques avec chambre d'equilibre et régulateur de vitesse lors de variations de la demande', Proc IAHR Hydraulic Symposium, Bucharest, Romania Sept. 1976, Studii si cercetari de hidraulica, 27, Institutual de cercetari hidrotehnice, Paper 12, 151-162.

Collins, J.I. and Fersht, S.N. 1968. 'Mixed technique for computing surges in channels ', J Hydraulics Div, ASCE Proc, 94, (HY2), 349-362.

Digernes, T. 1980. 'Real-time failure detection and identification applied to supervision of oil transport in pipelines', Modelling, Identification and Control, 1, (1), 39-49.

Dondoe, S. and Halanay, A. 1977. 'Stability of hyperbolic systems with controlled boundary conditions. Applications to governors for hydraulic devices', Proc IAHR Hydraulic Symposium, Bucharest, Romania Sept. 1976, Studii si cercetari de hidraulica , 27, Institutul de cercetari hidrotehnice, Paper 25, 313-318.

Galler, B.A. and Westervelt, F.H. 1961. 'The digital computer for fluid-flow calculations' in Handbook of Fluid Dynamics, ( Streeter, V. L. , ed.), Section 25, 19-22, McGraw-Hill , New York, USA.

10.

Greitzer, E.M. 1981. 'The stability of pumping systems - the 1980 Freeman Scholar lecture', J Fluids Engng, ASME Trans, 103, 193-242.

11.

Holloway, M.B. and Chaudhry, M.H. 1985. 'Stability and accuracy of waterhammer analysis', Advances in Water Resources, 8, 121-128.

12.

Holt, M. 1984. Numerical methods in fluid dynamics, 2nd edn, Springer Verlag, Berlin , Germany.

13.

IEEE Committee Report. 1973. ' Dynamic models for steam and hydro turbines in power system studies', in Stability of Large Electric Power Systems, ( Byerly & Kimbark , eds), IEEE, New York, USA. 128-139.

14.

Jeffrey, A. 1976. Quasilinear hyperbolic systems and waves, Research Notes in Mathematics, 5, Pitman, London.

15.

Jones, D.J. et al. 1972, 'On the numerical solution of elliptic partial differential equations by the method of lines', J Computational Physics, 9, (3), 496-527.

16.

Lakshminarayanan, P.A. et al. 1979. 'A finite difference scheme for unsteady pipe-flows', Int J Mech Sci, 21 (9), 557-566.

17.

Maudsley, D. 1984. 'Errors in the simulation of pressure transients in a hydraulic system', Trans Inst MC, 6 (1), 7-12.

18.

NAG Library. 1975. Fortran Mark 7, Routine F02AGF, NAGFLIB, Numerical Algorithms Ltd., Oxford, 3 (MK5), 987/623.

19.

Nougaro, J. et al. 1967. 'Critique des méthodes numériques de calcul des intumescences et examen d'une nouvelle méthode', L'Energia Elettrica, (2), 85-93.

20.

Oosterveld, M. and Adamowski, K. 1976. 'Hybrid computer model of St Venant equations' , J Hydraulics Div, ASCE Proc, 102 (HY10), 1491-1501.

21.

Ortega, J.M. and Poole, W.G. 1981. An introduction to numerical methods for differential equations, Pitman, Massachusetts , USA. Chap 7.

22.

Osiadacz, A.J. 1987. Simulation and analysis of gas networks, Spon, London.

23.

Rektorys, K. 1982. The method of discretization in time and partial differential equations, Reidel, Dordrecht, Netherlands.

24.

Rothe, P.H. and Runstadler, P.W. 1978. 'First-order pump surge behaviour ', J Fluids Engng, ASME Trans, 100, (4), 459-466.

25.

Smith, J.R. et al. 1983. 'Assessment of hydroturbine models for power-systems studies', Proc IEE, Part C, 130, (1), 1-7.

26.

Sucena Paiva, J.P. and Betamio de Almeida, A. 1977. 'Estabilidade da regulacao carga-velocidade em centrais hydroelectricas', Congresso, Ordem dos Engeneiros, Lisboa, Nov. 1977, Section 8, Paper 11.

27.

Suwan, K. 1989. The method of lines applied to waterhammer computation , PhD Thesis, University of Newcastle upon Tyne.

28.

Thirriot, C. et al. 1969. 'Simulation numérique des écoulements permanents et transitoires dans les systèmes d'alimentation en eau', Proc 13th Congress, IAHR, Kyoto , Japan, 1, 533-542.

29.

Vichnevetsky, R. 1969. 'Hybrid computer integration of hyperbolic partial differential equations by a method of lines', Proc 4th Australian Computer Conference, Adelaide, 457-461.

30.

Wood, D.J. 1967. 'Waterhammer analysis by analog computers', J Hydraulics Div, ASCE Proc, 93, (HY1), 1-11.

31.

Wylie, E.B. 1983. 'The microcomputer and pipeline transients', J Hydraul Engng, ASCE, 109, (12), 1723-1739.

32.

Wylie, E.B. and Streeter, V.L. 1984. Fluid Transients, FEB, Ann Arbor, USA.

33.

Wylie, E.B. et al. 1985. 'Fundamental equations of waterhammer' (Discussion and closure). J Hydraul Engng, ASCE, 111, (8), 1185-1200.

Lumped-parameter representation of fluid-system transients

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