Blade Polymeric Material Study of a Cross-Flow Water Turbine Runner

Although Romania has a consistent hydro energetic potential, till now is valuated just approximate 30 percent of it. On the big rivers there are already installed high power hydro plants, but a lot of small and medium rivers are not valuated energetically. Installing a new high power hydro plant tends to affect the zone, being necessary a lot of changes in the environment. The Cross-Flow hydraulic turbines don’t need very complex hydro settlements, being very suitable for small and medium power hydro plants. Also a quite big potential in the use this type of hydraulic machines is the energy recovery in the water treatment and sewage plants. The turbine’s blades surfaces enters in contact with the pressurized water jet. The water jet creates a hydrodynamic force that tends to stress the blade. Mainly the Cross-flow turbine blades are made from steel. This article presents the hydrodynamic design and the possibility of using new polymeric material Delrin ®AF for the Cross-Flow turbine runner blades, together with the stress analysis.

The Cross-Flow water turbine [1][2][3][4][5] is an action type turbine, having a cylindrical runner mainly, composed from two discs and curved blades flushed in a cylindrical ring. The water enters the runner through a wicket gate, which drives it such as at entrance the attack angle is constant. The water which gets out from the nozzle, passes through the runner blades two times, giving 2/3 of the hydraulic energy at the first passing and 1/3 on the second one. In figure1 is presented a vertical section through a Cross-Flow runner and the water flow path inside it. * email: adriana.manea@upt.ro Generally the Cross-Flow turbine blades are circle arcs with constant width. The variable thickness blade has the advantage of a smoothly modification of the inter blade channel. The blade shape was designed using O. Popa method [6], the geometry being generated using a trigonometrical 6 th grade polynomial, (fig. 2). The sharp blade edge ensures with minimal losses the stream distribution at runner inlet. Although with classic machining was quite difficult and expensive to make the shape of the blade, the new methods and materials make this possible in quite an easy manor. Usually, the Cross-Flow turbine blades are made from steel. The development of the new polymeric materials, and of the new technologies, make possible the implementation of these two in the Cross-Flow runners and blades manufacturing.
This paper presents the hydro-dynamic blade design and a study from strength of materials point of view of some suitable polymeric materials class for the turbine blades.

The Cross-Flow turbine hydrodynamic runner and blade sizing
The analysed blade is intended to be used for a Cross-Flow turbine with a head of H=40 m and a volumetric flow rate of Q=1.8 m 3 /s. The stereo-mechanical power of the turbine is: (1) in which it is assumed an efficiency η T =0.8, water density ρ=1000 kg/m 3 and gravitational acceleration g=9.8065 m/s 2 . The speed of rotation of the hydraulic turbine's runner is selected between the synchronization speeds with a.c. frequency 50 Hz namely: (2) in which pp=1,N are the pole pairs of the electric generator. The specific speeds for the Cross-Flow hydraulic turbines are n s =50...150.
Runner diameter is: (3) Absolute flow velocity at the entrance in the runner: (4) where k v1 =0.98 depends on the nozzle hydrodynamics. Runner tangential velocity is: Fig.2. Cross-Flow turbine blade with variable thickness Absolute velocity angle is: (6) Runner angles are: Inner runner diameter was obtained through an iterative procedure (9) The relative velocities are: The literature gives different formulas for the turbine hydraulic efficiency. So it is accepted: So we consider acceptable the estimated overall efficiency of the cross-flow hydraulic turbine ηT=0.8. The blades number after an empirical formula: z=4 x INT (10 . D 1 -1)=24 . The transport velocity by the inner diameter of the runner: (12) The flow angle here is: Runner width ( 15) The Cross-Flow turbine runner has the main geometric sizes D 1 =0.76 m, D 2 =0.44 m and the width L=0.884 m. The turbine blade geometr y has been generated transposing the circles network in an airfoils network, (fig.  3).
The network characteristics results from the singularization of the network, obtaining the coefficients of approximation trigonometric polynomial. The blade geometr y was approximated through a 6 order trigonometric polynomial. The blade coordinates are: . In order to obtain the trigonometrical polynomial coefficients are considered 12 values for Y p , p = 0, 1, 2 …. (2N-1). With these values are calculated the coefficients of the trigonometrical polynomial: (17) For the studied blade, the trigonometrical polynomial coefficients are presented in table 1.

Force on blade calculation
In this case we analyzed the hardest condition, this was when the runner is completely blocked and on it acts the whole force of the water jet.
As it is known from the fluid mechanics theory the force of the jet is: (18) where: ρ=1000 [kg/m 3 ] is the water density, Q=1.8 [m 3 / s] is the volumetric flow rate, ∆ν=28 [m/s] is the absolute velocity variation (we considered the most hard condition, when the velocity variation is equal with the velocity from the pipe). As we know from the geometry the jet attacks three blade channels, it means that for one blade force calculation we must consider the formula: (19) After calculus results that jb F =16800 N.

Material study
In many mechanical solutions, the thermoplastic resins are a very good alternative solution for the conventional materials [7][8][9][10][11][12]. From this material class, the acetal homo polymer present a combination of physical and mechanical properties which permits them to compete easy with metals. The acetals are formed from a group of aldehyde or ketone which reacts nucleophile with alcohol in the presence of an acid catalyst [13]. The acetal ensures a high resistance and rigidity associated with a good dimensional stability and machinability. Also it is characterized through a low friction coefficient and good wear properties, especially in wet medium. The acetal absorbs minimal humidity quantities, keeping it's properties constants in a big variety of media. The low water absorption confers an excellent dimensional stability for the parts machined with high precision. From acetals homo polymers Delrin®AF gain a large scale recognition because of it's reliability and performance in many domains of Table 1 mechanical engineering [14]. From producer datasheet Delrin®AF presents the following mechanical properties given in table 2.

The finite element simulation
Taking into account the blade geometry and the load resulted from hydro dynamic calculus, (fig. 4), it has been made a numerical analysis of the blade stress and strain. The material is defined elastic linear with properties similar with the acetal Delrin®AF. The blade support conditions have been established according to it's mounting solution in the runner, considering also two intermediary supports, (fig. 5). The figure 6 and 7 presents the Von-Mises equivalent stress and strain distributions for the considered loading case. As a result of blade geometry and deformation modulus can be seen stress concentrations in the blade upper part, nearby the supports. However these stress concentrations give a relatively good safety coefficient accepted in the design standards of these components, (fig. 8).
The use of thermoplastic materials in Cross-Flow hydraulic turbines blades and runners has many advantages as: good mechanical resistance, easy machining because of lower density than metals, the hydraulic machine components having a reduced weight. For the studied case the blade volume is de 0.00127652 m 3 . In the case in which the blade will be machined from steel, for a medium steel density of 7850 kg/m 3 , will give a blade mass of 10 kg. If Delrin material is used, having the density of 1420 kg/m 3 , the blade will have the mass of 1.812 kg, being almost 5.5 times lighter.

Conclusions
The Cross-Flow turbines with polymeric material runners, can be used as energy recovering units in water treatment plants and also in sewage plants, being suitable for relatively small heads and variable flows.
The analytical knowledge of the geometry is useful for the future analysis of the blade behavior. The O. Popa method permits the establishment of any blade design using trigonometrical polynomials.
In order to analyze the blade it is necessary to size the turbine runner, knowing mainly the geometric parameters, the sizing method is given in the presented paper.
The use of new materials in the accomplishment of Cross-Flow hydraulic turbines components, permits the utilization of these machines for energy production also with dirty waters.
From variable thickness blade analysis designed for a Cross-Flow turbine, is concluded that the thermoplastic material Delrin, presents advantages which recommends it to be used for this turbine type runners.