NUMERICAL SIMULATION OF HYDRODYNAMIC FIELD FROM PUMP-TURBINE RUNNER

Transcription

1 UMERICAL SIMULATIO OF YDRODYAMIC FIELD FROM PUMP-TURBIE RUER A. Iosif and I. Sârbu Department of Building Serices, Politehnica Uniersity of Timisoara, Romania STRACT One of the fundamental hypotheses of turbomachine design is to assume that the stream surfaces in the machine bladed zones are of reolution type. This can be obtained by accepting the flow as axisymmetric in the hypothesis of incompressible and ideal fluid and by determining the hydrodynamic field constituted by the stream lines and equipotential ones, respectiely, in the meridian plane. In this paper is deeloped a computational model of the hydrodynamic field in the meridian plane by the boundary element technique. The results are obtained by soling a boundary-limit conditions problem for Stokes equation for the elocity potential. From the connection between the functions and ψ is determined the stream function ψ and the elocity field along stream lines. The proposed computational model is applied to a pump-turbine runner and the numerical results are compared to those obtained by the finite element method. Keywords: model, pump-turbine runner, numerical simulation, axisymmetric motion, boundary element method, elocity distributions.. ITRODUCTIO Many times, the mathematical formulation of some physical phenomena will lead to partial differential equations that together with the corresponding to the limit conditions gie the so called limit problems. In most limit problems there is the impossibility of constructing analytical solutions which has led to the elaboration of numeric methods for the purpose of obtaining some approximated solutions. From this point of iew the Boundary Element Method (BEM) was deeloped. In this paper the dimensionless formulation in the potential functions will be used to proide a generality of the results. Therefore in Figure- in the meridian plane are presented: the analysis domains and the boundary conditions in pump case, respectiely water turbine.. BOUDARY ELEMET METOD FORMULATIO Within the BEM, taking into account the formulated hypothesis we well adopt a cylindrical coordinate system (r,θ z). The dimensionless way of dealing with the problem in the potential functions will impose the next change of ariable and of function: z zl ax. ax. r rl () π Q L ax. where Q, L ax. represent the fluid flow rate and the axial extension of analysis domain. Gien that the axisymmetric domain is notated with Ω and its boundary with, d r dθ d, the following expression of integral equation on the boundary will result [5]: c ( ζ ) ( ζ ) + q ( ζ, x) r in which n is the ( ζ ; x) d d () Ω domain boundary in the axial ζ, x and semiplane in conformity with Figure- and ( ) q ( ζ, x) are the following integrals: q ( ζ, x) ( ζ, x) dθ ( ζ, x) π π n ( ζ, x) dθ The fundamental ( ζ, x) (3) for Laplace s equation in three-dimensional case, where ζ is the source point and x is field point: ( ζ ) + r r ( ζ ) r [ θ( ζ ) θ ] + z ( ζ ) z r ( ζ, x) (4) cos [ ] allows calculating the integrals (3) knowing that the normal deriatie expression of the fundamental solution is determined with no difficulty. The first integral from (3) has the form [4], [6]: ( m) 4K (5) ( a + b) 6

2 in which the notation below hae been used: m b a r b r ( a + b) ( ζ ) + r + [ z ( ζ ) z ( ζ ) r ] (6) and the complet elliptic integral of first kind K ( m) is replaced with polinom of approximation [], [3]: K 4 ( m) [ a m b m ln( m ) ε( m) + ] (7) where m m, and ε ( m) is the error terms. The second integral from (3) in conformity with [5] has the expression: a) Pump Figure-. Analysis domains and the boundary conditions. q 4 ( a + b) r { [ r E( m) K( m)] n r ( ζ ) r z + ( ζ ) z a b + [ z a b ( ζ ) z E( m) n z ] } where E ( m) represents the complete elliptic integral of second kind. This is replaced with the next polinom of approximation: E 4 [ ] ε( m) ( m) + c m d m ln( m ) The discretization of the + boundary elements that hae the (8) (9) boundary in constant boundary makes the following discretized form of the (3) equation obtainable: π i + i G i () n The coefficients G i i also q and i and i ( ζ, x) r d ( ζ, x) r d G are gien by: () n are notations for the alues on the element of the function and its normal deriatie. The integral equation () after implementation of the boundary conditions will lead to a linear system of equations with ariables. If on the boundary of the constant element are considered for the z, following parametric equations: r ariables z Aξ + B; r Cξ + D; ξ [, ] () 7

3 for coefficients G i are obtained expressions which contain integrals that can be numerically ealuated using a Gauss quadrature. These are [3], [7]: G G i ii l ( ) K( m) dξ; i l ξ (3) l K ( m) ( ξ ) ξ dξ ( ξ ) ln dξ; i In the relation aboe K ( m) proposed for K ( m) : ( m) K ( m) G( ξ ) ln( ξ ) (4) comes from the decomposing K + (5) The coefficients i are calculated regarding the following expressions that can be numerically ealuated using a Gauss quadrature: i ii + K E ( m) ( ξ ) dξ ( m) ( ξ ) dξ; i [ ] ( ξ ) E( m) K ( m) ξ ( ξ ) ln dξ; i dξ + (6) (7) The i alues in ζ i Ω are determinated with the help of following integral representation written under discretized form: ( ) i 4π Gi i (8) n The coefficients G and from (8) are i i determined with the relations (3) and (6) remembering that ζ is replaced with Ω. On the basis of the BEM ζ i there were elaborated in FORTRA programming language the computer programs FIELFR and FICTAXS for IBM-PC compatible systems. The first soles the equation () and the second, on the basis of the integral representation (8) determines the alues i in ζ i Ω. The it calculates, through numerical deriatie the alues of components, belonging to the elocity z r in ζ i Ω, determines streamlines ct. ψ and equipo- tential ct. also the elocity and pressure field along the streamlines taking into account the relations [8]: ; p (9) 3. UMERICAL RESULTS In Figures- is presented the hydrodynamic field in the pump and turbine cases. otable is the fact the besides the results obtained with BEM in the figures below are also shown those calculated with the Finite Element Method (FEM) [9], [], []. In Figures-3 and 4 are presented the elocity and pressure distributions along the streamlines in the pump and turbine cases. These are obtained taking into account eq. (9) with the obseration that / ies. where ies. is the elocity corresponding to the last point on the streamlines found on the boundary. From Figures 3 and 4 of interest are the alues max., p and length l of the streamline ψ also for the pump case as for the turbine case. These alues are reported in Table-. From this table is noticeable that alues p.45 (pump) p.9 (turbine); obtained with the BEM and respectiely p.8; p.7 computed with FEM shows that the operating in pumping regime is less conenient caitationally. In Table- there were centralized the alues of the elocity and of the p pressure from exit in the pump case, respectiely, p operating as a water turbine, alues obtained with BEM and FEM for ψ.;.;.6;.8;.. Case Table-. Maximal elocities and minimal pressures. ψ BEM max. p l FEM max. p l Pump Turbine

4 a) Pump b) Turbine Figure-. ydrodynamic field. ψ Table-. Velocities and pressures on the boundary and. Pump Turbine BEM FEM BEM FEM p p p p

5 Figure-3. Velocity and pressure distributions along streamlines in the pump case. 4. COCLUSIOS Figures 3 and 4 suggest the way in which kinetic energy becomes potential energy, along the streamline established for the runner with no blades, meaning the field defined by the solid boundariesψ, ψ and also the input and output boundaries. The dimensionless alues of the minimum presure p.45 (pump); p.9 (turbine) obtained with BEM and p.8 ; p.7 obtained with FEM show that the operation in pumping regime is the most unfaourable caitationally. It has been obsered that the alues p.75 (BEM); p.73 (FEM) obtained in the pumping regime are the same for all case p (BEM); ψ alues and in the turbine p (FEM) ariates according to ψ. The method presented offers the possibility of determining the position of the point pertaining to the streamline ψ *, from which the pressure p is obtained. This result can be the basis for accomplishing a geometrical optimization of the solid boundary ψ *, thus resulting a conenient alue for p which would diminish as much as possible the unsuccessful operating of the runner from a caitational point of iew.

6 Figure-4. Velocity and pressure distributions along streamlines in the turbine case. REFERECES [] M. Abramowitz and I. A. Stegun andbook of Mathematical Functions, Doer, ew York. [] I. Anton Turbine hidraulice. Editura Facla, Timisoara, Romania. [3] P. K. Baneree and R. Butterfield. 98. Boundary Element Methods in Engineering Science. Mc. Graw- ill, London, ew York. [4] A. A. Beker. 99. The Boundary Element Method in Engineering. Mc Graw-ill, London. [5] C. A. Brebbia, J. C. F. Telles and L. C. Wrobel Boundary Element Techniques. Springer-Verlag Berlin, eidelberg, ew York. [6] C. A. Brebbia and J. Dominquez. 99. Boundary Elements. Computational Mechanics Publications, Southampton and Mc. Graw-ill, ew York. [7] G. Chen and J. Zhou. 99. Boundary Element Methods. Academic Press, ew York. [8] T. J. Chung Finite Element Analysis in Fluid Dynamics. Mc. Graw-ill, ew York. [9] A. J. Daies. 98. The Finite Element Method. Clarendon Press, Oxford. [] J.. Reddy An Introduction to the Finite Element Method. Mc. Graw-ill, ew York. [] O. C. Zienkiewicz. 97. The Finite Element Method in Engineering Science. Mc. Graw-ill, London.