A PRAGMATIC APPROACH ON RATIONALIZATION AND AUTOMATIC SELECTION OF WATER PUMP IMPELLERS: A CRUCIAL ASPECT OF WATER RESOURCES AND ENVIRONMENTAL ENGINEERING

Discussed in this work is the outlined mathematical modeling procedure that can help to automate the often-difficult design and selection of the optimum water pump impeller for a given task. Also discussed are, ways and how to automate the selection process; how to approximate the surface by a mathematical model-through the group of How rate — speed pressure points modeled by a function P = f (Q,n). The methodology was based on the Multilevel B Spline Approximation Algorithm (MBA) which made it very easy to select automatically, the most appropriate water pump impeller. However, the difficulties involved in the process of designing a water pump impeller suggests the re-use of knowledge acquired during the past projects, meaning that, for selection criteria, it is necessary (o use the characteristic curves of previously designed impellers 'by way of mathematical modeling. • Introduction The impeller is the heart of a water pump and the main origin of its behavioural efficiency. An impeller that is badly designed in relation to a specific work point produces losses at the inlet, along the blade aricTalso separation losses. Many mono-dimensional models allow estimation of these losses; the best known models are those of Japikse (1984); Neumann (1991); Tuzson (1993); Stepanoff (1993); Japikse, Marsher and Furst (1997) and Tuzson (2000);) and There are also many other studies, building more or less on these previous models, that suggest some procedure lo design water pumps (Veres, 1994; Moore & Atkinson 1997; Oshima & child 1999; and Zoz, Thclen, Alcenius and Wiseman 2001). The main limitation of these models is their approximated behaviour: an optimization process cannot start from these models. The answer to this problem is given by the Computational Fluid Dynamics (CFD) simulation technique. This technique allows calculation of the flux distribution by the solution of the Navier stokes partial differential equations. Optimization still remains a critical process and it is difficult to automate it. The designer usually develops a first attempt and tests it. According to the result, the geometry is modified to improve the flux distribution, and re-tried. This iterative process is repeated until the designer thinks that the i-th step gives a good result ( in relation to his own experience). Some studies according to Deguchi, Fujita and Nomato (2000), also propose analysis of the contribution of each geometrical parameter based on a . multivariate analysis (Principal Component Analysis). Here the authors repeat different Computational Fluid Dynamics (CFD) simulations with different geometrical parameters and study the influence of each factor. These studies help the designer to analyse only a few geometrical parameters, reducing the size of the manual optimization, and therefore, the time for the optimization the number of iterations. This process has been much improved in recent years, kudos to the development of multi-objective optimization software. Software based on the use of genetic algorithm parallel h i l l Climbing, parallel architectures and Pareto frontiers has according to Bze, Obiegbu and Jude-eze (2005); Spicer, (1998); Igbe (1995), and Igbe (1999), provided the possibility to automatically run generic computational Fluid Dynamics (CFD) software. -:but in general, a generic design tool. The optimizer, interacting with the CFD software, progressively modifies the geometry of the impeller until the optimum geometry is reached (Spicer, 1998; Spicer, Cook, Poloni and Sen, 1998). In recent years, some new optimization processes have been developed based on a different philosophy; the main exponent of this development are those of Zangeneh (1997); Yiu and Zangeneh (1998); and Zangeneh, Goto and Harada (1999). The approach is different, or, put in a better way, is, "inverse": a specific blade loading is assigned and the geometry that produces that blade loading is calculated using 3D inverse design methods. An optimizer can modify some parameters that define blade loading, so that the optimization is an easier task than previously. Manual or automatic optimization processes involve many difficulties. This observation justifies the necessity to record new knowledge resulting from the optimization process. This problem becomes particularly important if this optimization process is a daily task, as for water pump builders. The majority of well-designed impellers suggest the re-usability of the impellers. So the problem is; "how to proceed in this operation? " The main idea is to use the experimental curves of the water pumps. These curves represent the pressure-flow rate relationship at a specific speed-figure 2b shows the flow rate relationship. The group of such curves is a surface space. This surface has to be compared with the specific flow rate, head and speed required-by the customer. Mathematical Approximation of the Surface If we want to automate the selection process, it is necessary to approximate the surface by a mathematical model. The group of flow rate speed pressure points has to be modeled by a function P=f(Q,n). There are many mathematical ways to approximate the surface-scattered data points; neural networks according to Haykin (1998), nurbs (non-uniform rational B -splines) and Bspline a p p r oximations according to Piegi and Tiller (1996), are only some examples. The modification of [• r last-mentioned method has been chosen because, the algorithm is very fast. Methodology The methodology is based on the Multilevel B Spline Approximation (MBA) Algorithm, pnposed by Lee, Wolberg and Shin (1997); it consists of the surface approximation by a uniform cubic BSpline function defined by its control lattice O (See figure I below). Also, the mathematical details are given below. Figure I: Lattice Overlaid to Scattered Data Points Methodology of the Approximation The MBA algorithm methodology proposed here consists of the surface approximation by a uniform cubic B-spline function defined by its control lattice O overlaid on its definition domain Control points <Dy have to be chosen in such a way in the assigned points xc> yc that the next condition (where zc is the pressure value) is satisfied: A possible choice consists in the minimization of the quadratic mean error. This method brings us to the solution: The second behaviour of the algorithm is that it is multilevel. Given the lattice, it is possible to proceed with the calculation of the approximation function. This approximation function in the assigned points Xc, yc (flow rate and speed) returns an error. So we use a second, finer lattice (second level) that has the scope to approximate the previous error, the difference between the desired output and the estimated value obtained by the function defined previously. The function is modeled by two approximations. With the increment of the number of levels, the marginal error is reduced, but in the siimc way the complexity of the approximation function is improved. The software implementation of the previous method gives the result as shown in figure 2 below. Characteristic Head Flow rate (Fig. 2.a) PA6.8109. 0 Curves Speed Surface of a Water Pump (Fig. 2.b)