Aerodynamic analysis of a windmill water pump using blade element momentum theory

. A windmill water pump has been designed based on simulation data using the Blade Element Momentum Theory (BEMT) method. According to the simulation data, a 10-blade configuration with an incidence angle of 7 degrees is predicted to produce an output torque of 40 Nm. To simplify the turbine manufacturing process, a turbine cross-sectional profile with a bent flat plate-based airfoil was selected. The simulation results indicate that providing an incidence angle of 7 degrees compensates for the resulting decrease in aerodynamic performance compared to using a cambered airfoil. Furthermore, a dynamic analysis was conducted to predict the turbine's rotational speed. With a 10-blade configuration and a blade material density of 2900 kg/m³ at a wind speed of 5 m/s, it is predicted to rotate at a steady speed of 167 rpm. When the material density is increased to 3500 kg/m³, the rotor's predicted rotational speed is 160 RPM. While the difference in rotational speed due to the increase in material density is not very significant, the time to reach steady-state conditions varies considerably. Specifically, a turbine with a material density of 2900 kg/m³ requires a settling time of 168 seconds, while a turbine with a density of 3500 kg/m³ requires a settling time of 310 seconds. This notable difference suggests that mass inertia primarily influences the dynamic response of the turbine in achieving a steady rotational speed without significantly affecting the turbine's aerodynamic performance.


Introduction
A wind power plant (PLTB) is a renewable energy generator with a high level of intermittency.It happens because of the uncertain wind dynamics that change randomly every time.This phenomenon affects the quality of generated electrical power and has the potential to shorten the life of electrical components, for example in household appliances.It is very detrimental and must be avoided.The most crucial component of a wind power plant is the rotor [1].This component is responsible for the amount of wind energy that can be absorbed and converted into electrical or mechanical energy.Wind turbine rotors have unique and sensitive characteristics to changes in wind conditions.Errors in designing the rotor configuration will impact the degradation of aerodynamic performance, hence it harms the potential of the power generated.Indonesia has quite a large wind energy potential, especially along the north coast of Java (Pantura).In this area, many salt ponds were found.The main process carried out by the salt farmers is to distribute seawater to small ponds to enter the next stage of the crystallization process.The main objective of this research is to explore the design of a windmill water pump using Blade Element Momentum Theory (BEMT) which will be applied as the main driver in distributing seawater to salt farmers' ponds.Even though this method has lower accuracy compared to the finite volume method, it offers the advantage of relatively fast computing times with moderate fidelity, making it easier for us to conduct the initial design exploration of the windmill water pump [2].This research aims to investigate the effects of blade geometry, inertia, and the number of blades on the aerodynamic power and torque generated by the system.Blade geometry will be analysed to optimize aerodynamic efficiency by altering parameters such as blade configuration, blade incidence angle, the number of blades, and blade inertia.This research provides a more comprehensive analysis, not only examining the aerodynamic performance of the windmill but also investigating fluid-structure interactions.The primary objective is to assess the impact of alterations in rotor inertia on the dynamic response and aerodynamic performance of the turbine.Through this analysis, it is expected to deliver a performance overview of the designed turbine, thereby minimizing the potential for errors in 3D model design.The effects of inertia on the system's performance will be investigated through numerical simulations by simplifying the rotor configurations by using multi-body beam formulation based on the Finite Element Analysis (FEA) module of the open-source multi-physics engine Project Chrono [3].The outcomes of this research will contribute to the development of more efficient and effective windmill water pumps for agricultural applications.This research will advance renewable energy utilization in the agricultural sector, promoting sustainable practices and economic growth, especially for Indonesians.

2. Governing equation
BEMT was developed to predict the behaviour of a rotating blade or propeller.This method is also used to determine the load on the blade due to the aerodynamic forces i.e., lift and drag.In the analysis process, the turbine blade is divided into several sections distributed radially.The assumption used in calculating the aerodynamic loads is to consider the fluid flows in the 2D plane of the airfoil [4].The more blade elements are discretized, the more accurate the calculation results using this method will be, but of course, the computing time required will also be longer.The thing to remember about using this method is its inability to capture air flow phenomena in 3D cases, such as tip loss due to induced vortex [5].
In order to improve the accuracy of using the BEMT method, several corrections were added to the aerodynamic performance analysis in this research, namely by implementing the Prandtl Tip Loss Factor [6] and 3D Correction [7].The computational process will be carried out in the Qblade environment developed by David Marten from TU Berlin [8] which is generally used as a reference in research work on wind turbine aerodynamic analysis.A cross-sectional image of a rectangular, untwist and untaper type wind turbine blade is given as shown in Figure 1 above.By referring to [9], [10], and [11] the magnitude of the lift and drag forces along with their projections on the  and  axes can be written as follows: Based on the figure above, for N blades configuration, we can determine the amount of thrust, torque and power output of the turbine by assuming the angle of attack is small sequentially as follows: =   Ω( − ) (7) In the framework of formulating the equation by considering the local tip speed ratio variable (  ) and rotor rigidity (), the magnitude of the thrust, torque, and power coefficient can be written as follows: =   3 (  −   ) 3  (10)

Aerodynamics simulation
Initially, the airfoil aerodynamics analysis was carried out using Xfoil software by varying the Reynolds number range from 40,000 to 1.4 million.In order to consider an easy and cheap manufacturing process, a flat plate-based airfoil type was chosen.Of course, the aerodynamic performance is not as good as with a camber airfoil, therefore modifications were made to the shape of the flat base plate by providing curves with the following details as seen in Table 1 below:  Based on the airfoil simulation results above, it can be seen that the Aerodynamic efficiency value increases as the Reynold number increases.Apart from that, it can also be observed that at high Reynolds numbers, aerodynamic efficiency values are achieved at relatively low angle of attack (AoA).The results above give us the idea that in order to get high aerodynamic efficiency we need to add an incidence angle in the range of 6 to 7 degrees to our blade.Of course, this result is a preliminary analysis and needs to be evaluated in more detail for the 3D case.Therefore, the next scenario is to analyse a 3D blade with a rectangularuntapered configuration with a span of 0.6 m and a chord length of 20 cm.In this scenario, we will see the effect of varying the incidence angle on the resulting aerodynamic efficiency.Based on this blade configuration, an aerodynamic analysis was then carried out by varying the incidence angle from 0 to 15 degrees.Figure 3 and Figure 4 below summarizes the simulation results as follows: The power coefficient generally increases as the incidence angle rises, but this trend is only valid for  ≤ 11 o .Remarkably, there is no significant difference in aerodynamic performance between  = 7 o and  = 11 o , it means that the saturated condition is achieved.However, at  = 15 o the aerodynamic performance is dropped for TSR > 3. It can be said that in this phase a stall condition has occurred where an increase in the incidence angle value that is too large has an impact on reducing the aerodynamic efficiency of the turbine.On the other hand, the torque coefficient reaches its maximum value at TSR 2.5, while at TSR 3.5 the power coefficient reaches its highest value.It can be observed that the highest torque coefficient value tends to shift to the left (lower TSR value) when the incidence angle value increases.Because the windmill application in this research is used for distributing the seawater to salt ponds, the torque coefficient value is treated as the main consideration.Therefore, a windmill water pump configuration will be designed with an operating condition around TSR 2.5 at incidence angle 7 0 .The next scenario aims to determine the effect of wind speed variations on resulting aerodynamic quantities (aerodynamic torque and aerodynamic power).The turbine configuration remains the same as before, with a constant blade incident angle of 7 degrees.The analysis is limited to cases with laminar and steady wind profiles coming uniformly from upstream.Wind speed varies from 3 m/s to 15 m/s using a three-blade configuration with structural flexibility is neglected.Figures 5, 6, and 7 below summarize the simulation results.It can be observed that variations in wind speed have almost no effect on the power coefficient value's profile.However, even this small change in value can lead to a significant impact when converted into an output power variable.Notably, a 5 m/s difference in wind speed results in an almost 1 kW change in the output power, as seen in the graph between wind speeds of 10 m/s and 15 m/s.Conversely, at a wind speed of 15 m/s, the designed windmill water pump is predicted to generate a torque of 40 Nm.The next scenario that is carried out is to analyse the effect of the number of blades on the power and torque coefficient values.The simulation scenario is set by varying the number of blades from 3 to 10 at constant wind speed values.Figures 8 and 9 below summarize the simulation results.Based on the picture above, it can be seen that the number of blades has almost no significant effect on the value of the power coefficient.However, it's quite remarkable to note that an increased number of blades results in the maximum power coefficient occurring at a low Tip Speed Ratio (TSR).This phenomenon is a result of the heightened inertia, which prevents the wind's kinetic energy from effectively rotating the turbine.Consequently, the turbine's rotational speed decreases significantly.In addition, it can be seen that the maximum value of the power coefficient in this scenario remains relatively constant, standing at 0.325 for several blades ranging from 5 to 10 pieces.On the other hand, the number of blades has a significant effect on the resulting torque coefficient.According to the graphical data, the number of blades plays a major role in increasing the resulting torque value.Notably, when the turbine's blade count falls within the range of 5 to 10 blades, the increase in torque value can reach 50% compared to the data from blade counts below 5.
Based on simulation data from various scenarios, we have decided to design a windmill water pump with untapered, untwisted blade configuration.The blades will have a chord length of 20 cm and a span of 0.6 m, with an incidence angle of 7 degrees and a total of 10 blades.However, it's essential to note that this conclusion is based on initial simulation data, assuming rigid blades without accounting for the effects of inertia and structural flexibility.The actual values will differ when considering the impact of fluid interaction and blade structure.This interaction is referred to as the aeroelasticity phenomenon, which becomes dominant in flexible structures with high aspect ratios [12] and [13].

Effect of inertia to aerodynamic performance
In this simulation scenario, we will implement aerodynamic equations based on the Lifting Line Free Vortex Wake method, coupled with structural equations using the Multi-Body Beam Formulation, which relies on the FEA module of the open-source multi-physics engine Project Chrono [3].In this case, the turbine geometry and configuration will be kept constant but the turbine material will be varied by changing the density value.The effect of inertia on turbine aerodynamic performance will be observed by computing within the Qblade software environment.The aerodynamic performance of the turbine was then compared by varying the density value from 2900 to 3700 kg/m 3 .Figure 10 and Table 2 below describe and summarize the structural definition of the simulated turbine.The figure 10 above is a 3D model of a windmill water pump designed based on the previous aerodynamic performance results.In this design, we opted for a blade configuration with a rectangular profile featuring a cambered plate airfoil.The blades have a span of 0.6 m and a chord length of 20 cm.We use 10 blades, and their orientation angle is set at 7 degrees.Additionally, it is equipped with a gear and chain transmission system with a 4:1 ratio, connected to a rod that functions as a single-acting reciprocating pump.Based on the graph above, it is clear that the increase in mass inertia does not significantly affect the resulting turbine rotational speed.However, the increase in mass inertia has an impact on the length of time required for the turbine to reach a steady rotational speed condition.We can see a striking time difference with a difference of almost two times longer between the both of them in order to achieve a steady state level.
It can be seen that mass inertia does not have a significant effect on the aerodynamic performance produced by the turbine.This can be observed from the graph below which has almost the same similarity both in terms of profile and magnitude.Of course, this conclusion is valid as long as the assumption that the turbine blade is stiff is still acceptable.Meanwhile, when aeroelastic effects are dominant, we hypothesize that there will be quite significant differences in the resulting aerodynamic performance results.The same pattern can also be seen in the graph above, where mass inertia only influences when the maximum torque value is reached.It can be seen that for a turbine with a density of 2900 kg/m 3 , the maximum torque value is achieved in around 110 seconds, while for a turbine with a density of 3700 kg/m 3 , it is achieved in 240 seconds.Figure 13 above depicts a prediction of the magnitude of the aerodynamic force generated by a turbine with a configuration of 10 blades, rectangular and untapered, with an initial angle of incidence of 7 degrees at varying speed values from 5 m/s to 8 m/s.It is evident that the optimal rotational speed value to achieve maximum power for each wind speed is around 100 to 150 rpm.Understanding this pattern will facilitate the development of a speed control system in the future, enabling the turbine to operate within its optimal operating range.

Conclusion
Based on the data obtained from the simulation results, it can be concluded that the greater number of blades has a significant effect on the torque produced but does not have much of an effect on the amount of power produced.Apart from that, the effect of mass inertia only influences the dynamic response of the turbine to reach steady rotational speed conditions without much affecting the aerodynamic performance of the turbine by considering that the aeroelasticity effect is not dominant.
Figures 11 and 12  below illustrate the effect of changes in mass inertia on the prediction of turbine rotational speed and its impact on the value of the torque produced.

Table 1 .
Airfoil parameters.Figure 2 below illustrates the aerodynamic efficiency values of plate camber-type airfoil with various Reynolds number values.

Table 2 .
Structural definition of the simulated turbine for  = 2900   3 ⁄