Blade Shape Optimisation for Rotor-Stator Interaction in Kaplan Turbine

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1 2 nd International Conference on Engineering Optimization September 6-9, 21, Lisbon, Portugal Blade Shape Optimisation for Rotor-Stator Interaction in Kaplan Turbine Marzena Banaszek, Krzysztof Tesch Gdańsk University of Technology Fluid Mechanics Department ul. G. Narutowicza 11/ -952 Gdańsk Abstract This paper presents an algorithm for simultaneous stator and rotor blade shapes optimisation. The algorithm takes into account the interaction between stator and rotor of a model Kaplan turbine, see figure Evolutionary algorithms (EA) are used as an optimisation method together with artificial neural networks (ANN) for objective fitness function approximation. This is due to high computation demand of CFD calculation. ANN allows for significant calculation time reduction. The advantage of EA over other methods is that it seeks for global rather than local optimum. Reynolds averaged Navier-Stokes (RANS) equations are solved together with two additional transport equations. For this case the standard k ε turbulence model is chosen due to its robustness and popularity. An interesting blade shape description is given which proves useful for optimisation. The optimisation criterion (objective fitness function) is the politropic loss coefficient for the whole stage (two stators plus rotor). Figure 1: A stage of Kaplan turbine Keywords: Fluid Mechanics, Turbomachinery, Optimisation, Genetic Algorithms. Politropic loss coefficient The definition of politropic loss coefficient derives from enthalpy equation which can be written in differential form as ρ dh dt = φ µ + (λ T)+ dp dt. (1) Assuming no heat transfer and averaging through surfaces we have the following definition of average loss coefficient ζ = ρ φ µ dv V (2) ṁ p 1 p 2 where V is a flow domain. In the above equation p describes surface average pressure p := S 1 S pds. The numerator of equation (2) is the dissipated power of mechanical energy N d. It can be calculated from velocity field. In case of incompressible fluid we have N d := φ µ dv = 2µ D 2 dv. (3) V 1 V

2 Equation (2) is valid for both laminar and turbulent flows. The latter case, however, is applicable only if we know the full velocity field. Numerically this can only be obtained with Direct Numerical Simulation. If we know only the average values of velocity, equation (2) cannot be used directly. One must average the enthalpy equation (1). Assuming no heat transfer, no diffusion due to pressure field fluctuation we can write ρ d h dt = 2µ D 2 +ρε+ d p dt. () Comparing equations (1) and () one can notice that there is a new term. It represents turbulent dissipation. Following the same logic as previously the polytropic loss coefficient is written as ζ = ρ φ µ dv V (5) ṁ p 1 p 2 where averaged dissipation function is calculated as φ µ = 2µ D 2 +ρε. 2. Blade profile description The blade profile is defined in terms of centerline (center surface) f and thickness function distributed along the centerline. The centerline is obtained as the arithmetic mean of the pressure- and suction-side profiles of the blade. The thickness can be distributed along the centerline in many ways depending on the distribution coefficient δ. This coefficient can either be constant or vary along the centerline. Generally speaking, it is a function which takes values within the range [;1]. Formally, we have δ : R [;1] on a plane or δ : R 2 [;1] in space. The pressure side of the profile is calculated as f (1 δ), whereas the suction side as f + δ, see figure 2. In this paper we consider a constant value of δ. The original blade profile (before optimisation) is obtained for δ = 1 2. If δ = then the centerline is coincident with the suction side. For δ = 1 the centerline is equivalent to the pressure side. It is also possible to consider δ values out of the range [;1]. One has to bear in mind in that case the centerline will actually be external to the blade. f+δ f (1 δ) f δ 1 Figure 2: Blade profile description 3. CFD 3. Flow domain The flow domain is shown in figure 3 and includes the pre-stator between surface and Its presence is necessary. This is because it mechanically supports and stabilises the whole stage within the pipe. The second part is the stator proper which is located between surfaces 1 and 2. The last part between surfaces 2 and 3 is the rotor Mesh The flow volume was structurally discretised by means of hexahedral elements. The total number of elements for start and optimal case was about 7 million elements. Details are listed in table Figure shows rotor mesh and figure 5 stator mesh for start case δ r = 5, δ s = 5. 2

3 Figure 3: A fragment of flow stage together with characteristic cross-sections Table 1: Mesh statistics δ r = 5 δ s = 5 δ r = 5 δ s = 62 Nodes Elements Equations The closed system of equations for turbulent and incompressible flow [9] is composed of the averaged mass conservation equation (6a), and the averaged Navier-Stokes equation (Reynolds equation) (6b). Additional equations depend on the assumed turbulence model. For the two-equation model k ε we utilise Boussinesq hypothesis. The first transport equation is of turbulence kinetic energy k in the form of (6c). The second transport equation is that of turbulent dissipation ε in the form of (6d). Both transport equation are modelled. The last necessary equation is equation for eddy viscosity µ t in the form of (6e). The closed system of equations is [9, 7] U =, (6a) ρ d U = ρ g p e + (2µ e D ), (6b) dt ρ dk (( ) ) dt = 2µ t D 2 µt + +µ k ρε, (6c) σ k ρ dε (( ) dt = C ε ε1 k 2µ t D 2 µt + +µ ε ) C ε2 ρ ε2 σ ε k, (6d) µ t = C µ ρk 2 ε 1. (6e) In the above system p e stands for effective pressure p e := p ρk and µ e for effective viscosity µ e := µ + µ t. There are seven equations and seven unknown functions U x, U y, U z, p, k, ε, µ t. This makes the system closed if we know the values of C µ, σ k, σ ε, C ε1, C ε Boundary conditions The basic boundary conditions are: Inlet. The specified mass flow rate ṁ = 7kgs 1 was directed perpendicularly to surface, see figure 3. Because of two-equation k ε turbulence models it was necessary to specify two additional variable values. Medium turbulence intensity case was considered. This corresponds to turbulence intensity τ t = 5% and viscosity ratio µ t /µ = 1 Knowing τ t and µ t /µ it is possible to calculate k and ε by means of k = 3 U 2 τ t 2 1, ε = c µ ρk 2 µ 1 t. Outlet. Because of strong velocity variations in the outlet of the considered flow domain (crosssection 3 in fig. 3) one should not set the outlet to constant pressure. This is because the constant 3

4 Figure : Example rotor mesh Figure 5: Example stator mesh pressure distribution would force additional changes of velocity field. Mass flow rate was specified here, same as in the inlet, to avoid this problem. Walls. The impermeability and adhesion condition were specified. These demand U = This is true for all the walls except for rotor blades and shaft where the velocity is calculated as in rotating frame of reference U = ω r. Periodicity. The periodic boundary conditions over 6 rotation were set on boundaries resulting from taking a ( 1 6 ) section of the flow volume. Interfaces. Two interfaces are considered. The first fluid-fluid between prestator and stator. This interface is necessary because the cross-section 1 at the prestator side is not exactly the same as cross-section 1 at stator. The second interface stator-frozen rotor is defined on the cross-section 2 to take under consideration interaction between static geometry of stator and rotating geometry of the rotor.. Optimisation The optimisation criterion or objective fitness function for Genetic Algorithms was averaged politropic loss coefficient (5). GA were used to find the global optimum and ANN to approximate the objective fitness function.. ANN approximation An Artificial Neural Network was used for objective fitness function approximation which is loss coefficient 1 δ s δ r ζ Figure 6: Neural network structure ζ. This coefficient depends among other on rotor δ r and stator δ s distribution coefficients. The structure of the network is shown in figure 6. Is is an unidirectional, two-layer network. The total number of

5 weights is 9. The first layer is composed of neurons. There are also two inputs δ r, δ s and one output ζ. The learning method was error back propagation. The error of learning was about CFD ANN CFD ANN Ζ Ζ s Figure 7: ζ distribution as a function of δ r for δ s = 5 Figure : ζ distribution as a function δ s for δ r = Comparison between example teaching data and approximations given by ANN are shown in figures 7 and. Figure 7 presents data for constant value of δ s = 5 whereas figure for constant value of δ r = The accuracy of ANN prediction is acceptable..2. GA optimisation The trained ANN was able to calculate the objective fitness function during optimisation process. This process was carried out by means of GA. The floating point representation of chromosome [6] was used. Table 2 presents basic parameters and statistics of GA. The population size was set as 3 The total (maximal) number of generation as 2 GA converge quickly just after several generations. Table 2: GA parameters and statistics Value Chromosome length 2 Population size 3 Tournament size 3 Crossover prob. 7 Mutation prob. 15 Variable range [; 1] [3; ] Generations 2 Crossovers 193 Mutations 153 The convergence of GA is shown in figure 9. The symbol ζ avg describes averaged loss coefficient for the whole population. ζ min represents global minimum. The optimal value from GA optimisation was δ r = 5 and δ s = 62 with should be compared with existing design δ r = 5 and δ s = 5. As for the rotor distribution coefficient δ r it differs significantly from its original value. For the stator the difference is smaller. Figure 1 presents mutation (mut) and crossover (cross) number variations during optimisation process. This was performed for probabilities taken from table 2. Figure 11 allows to monitor the population irregularity. It shows discrepancy []. A value of discrepancy close to represents a random population and is typical of first generations. Regular populations possess values close to The first generation is random with a corresponding discrepancy of about 35. When the algorithm starts to converge (iteration -6 in figure 9) the value of discrepancy reaches the value of representing a uniform population. Later a slight decrease of discrepancy can be seen, representing the influence of a mutation which brings non-uniformity to a regular population. 5. Results Figure shows loss coefficient distribution ζ for the whole stage as a function of δ r and δ s. The optimal 5

6 Ζ avg Ζ min 6. 1 mut cross 1 Ζ Number Generation Generation Figure 9: GA convergence Figure 1: GA crossovers and mutations D Generation Figure 11: Discrepancy value can be localised. Calculations show that this value is ζ =. for δ r = 5 and δ s = 62. The rotor loss coefficient distribution ζ r as a function of δ r i δ s is shown in figure 13. It is obvious that both δ r and δ s have strong influence on ζ r value. Figure1presentsthelosscoefficientofproperstatorζ s asfunctionofδ r andδ s. Obviously, thiscoefficient depends strongly on δ s and weakly on δ r. Figure 15 shows what the loss coefficient of pre-stator ζ p is like. It depends on δ s and does not depend on δ r. Figure 16 shows the distribution of φ µ over the stage. Graph scale has been truncated on the upper bound for the plot. Regions of high dissipation values are visible (red colour). Figure 17 presents averaged pressure p distribution over the whole stage. Both figure are drawn for the optimal solution. 6. Conclusions The best blade profile shape in terms of loss coefficient ζ is obtained for δ s = 62 and δ r = 5. This means that the curvature of rotor blade is increased compared with its original shape (δ = 5). As for the stator on the contrary, it is decreased. The loss coefficient for the whole stage depends on δ r and δ s. For the proper stator it depends only on δ s whereas for prestator it almost does not depend on δ r and δ s. Changing the coefficients δ r and δ s simultaneously makes it possible to decrease loss coefficient from ζ = 5.332% to the level of ζ =.%. Changing only δ r we have ζ =.9%. It can be seen that there is stronger influence of δ r on the loss coefficient ζ compared with δ s. The highest values of ζ are found in rotor. Figure suggests that increasing the curvature of the rotor one could expect further decrease of loss coefficient. This is because the optimal value is located near the boundary of considered region. However, this conclusion if only of speculative character. Nomenclature 6

7 s s Figure : ζ distribution as a function of δ r and δ s Figure 13: ζ r distribution as a function of δ r and δ s s s Figure 1: ζ s distribution as a function of δ r and δ s Figure 15: ζ p distribution as a function of δ r and δ s D deformation rate tensor f blade profile centerline g gravity acceleration h enthalpy k turbulence kinetic energy ṁ mass flow rate N d dissipated power p, p e hydrodynamic and effective pressure r radius t time T temperature U velocity V volume δ r rotor thickness distribution coefficient δ s stator thickness distribution coefficient 7

8 Figure 16: φ µ distribution Figure 17: p distribution thickness ε turbulent dissipation ζ politropic loss coefficient λ heat conductivity coefficient µ, µ t, µ e molecular, eddy and effective viscosity ρ density τ t turbulence intensity φ µ dissipation function ω angular velocity ( ) surface average time average References [1] M. Banaszek and K. Tesch, Rotor Blade Geometry Optimisation in Kaplan Turbine, XIX National Conference on Fluid Mechanics, Poznan, Poland, 21 [2] M. Banaszek, Theoretical and experimental flow analysis through the model water turbine stage, Cieplne Maszyny Przeplywowe, 1 (), 5 52, 25 [3] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University, 2 [] W. Jones, B. Launder, The prediction of laminarization wit a two-equation model of turbulence, Int. J. Heat Mass Transfer, 15 (2), 31 31, 1972 [5] V. Kecman, Learning and Soft Computing, The MIT Press, 21 [6] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer 1996 [7] S. B. Pope, Turbulent Flows, Cambridge University Press, 2 [] E. Thiémard, An Algorithm to Compute Bounds for the Star Discrepancy, Journal of Complexity, 17 (), 5, 21 [9] D.C. Wilcox, Turbulence Modeling for CFD, DCW Industries, 199