CHAPTER 3 ANALYSIS OF NACA 4 SERIES AIRFOILS

Transcription

1 54 CHAPTER 3 ANALYSIS OF NACA 4 SERIES AIRFOILS The baseline characteristics and analysis of NACA 4 series airfoils are presented in this chapter in detail. The correlations for coefficient of lift and drag have been developed and compared with the existing results. The effects of Reynolds number on coefficient of lift and drag for various airfoils are predicted and presented. The basics of NACA airfoils, the developed correlations and the effect of Reynolds number are discussed in the next subsequent sections. 3.1 NACA AIRFOILS The selection of blade profiles for wind rotors was based on NACA (National Advisory Committee for Aeronautics). The standard practices of lettering and numbering adopted by NACA for profiles of airfoils are followed. In general, airfoils are specified by maximum camber height in terms of percentage of the chord in terms of percentage of chord. For example, NACA 4412, the first number used on profiles refers to height (gradient height) on y-axis as percentage ratio based on the fact that the profile is situated at the center of a coordinate system. Whereas, the second digit refers to its location on x-axis as percentage ratio and last two digits indicate the thickness of blade profile as percentage ratio (Dreese 2000). In the above example, the first digit 4 indicates the maximum height of the camber is expressed as a percentage of the airfoil length (i.e.) 4 percentage, the next digit 4 represents the horizontal location of the maximum camber in terms

2 55 of a chord length (40%), the last two digits 12 expresses the maximum thickness of the airfoil expressed as a percentage of the airfoil chord length. The airfoil with camber, camber line, leading edge, trailing edge and chord length are shown in Figure 3.1. Figure 3.1 Airfoil geometry The aim of this study is to present the aerodynamic characteristics of the blade sections which are the most crucial parameter of a wind turbine blade. Some of the NACA profiles are selected for analysis and Reynolds number, Angle of attack, Chord length, Sliding rate, Coefficient of lift and drag have been taken into concern. 3.2 CORRELATIONS FOR COEFFICIENT OF LIFT AND DRAG The coefficient of lift (C l ) and co efficient of drag (C d ) for any airfoil is based on the following parameters. Geometry of the airfoil like maximum camber height, maximum camber position and thickness. Angle of attack (AOA) Wind velocity ( )

3 56 Research studies have been conducted to evaluate different airfoils and found out the coefficient of lift and drag for various airfoils and published their findings. Jacobs et al. (1935) conducted wide range of experiments by varying the airfoil geometrical parameters and angle of attack and wind velocity for various NACA airfoils and published the results of coefficient of lift and drag for 68 airfoils. They conducted experiments in a variable density wind tunnel by keeping a constant velocity of 68.4 ft/s. They have plotted the graph for various angles of attack with the values of ratio for a large group of related airfoils. They have confined their experimental work with the angle of attack ranging from -8 o to 10 o and due consideration has been given for the viscous effects and the phenomenon of stall at higher angle of attack. They have varied the horizontal camber position from 0% to 60% in steps of 20%, vertical camber position from 0% to 7% with a step of 1%, the thickness 0% to 21% in steps of 3%. The predictions for coefficient of lift and drag for the airfoils with parameters other than the above are difficult and need an experimental set-up. Hence, it involves higher cost and time and for study of numerical and simulation with various parameters, the above practical results finds limited applications. Hence there is a need for developing the correlations capable of giving the coefficient of lift and drag for any NACA 4 series by specifying the angle of attack and wind velocity. The correlations are developed with above objective using the experimental results published by Jacobs et al. (1935) and checked for various NACA profiles and further modified suitably for giving better results for any airfoils. The coefficient of lift and drag for various NACA profiles have been taken from the published work of Jacobs et al. (1935) for various angle of attack and velocity by interpolation and extrapolation. The sample of results thus obtained for the NACA 4410 is given in Table A1.1 in Appendix 1.

4 57 The correlation for coefficient of lift and drag depends upon the airfoil geometry, angle of attack and wind velocity. The linest function method () (y=m 1 x 1 +m 2 x 2 +m 3 x 3 +b) is used to develop the correlation. shown below. The correlation to find coefficient of lift and drag can be written as = [( 4 ) + ( ) + ( ) + ] (3.1) Where 4XXX - Airfoil geometry in usual notation AOA - Angle of attack in degrees - Wind velocity in m/s C 1, C 2, C 3 & C 4 - Constants to be evaluated Using a best fit straight line regression method, the constant C 1, C 2, C 3 & C 4 are obtained for coefficient of lift and drag separately using the function available in M.S. Excel software as Linest, substituted in the equation (3.1) and presented below in equations (3.2) and (3.3). C = {( XXX) + ( AOA) + ( ) } (3.2) C = {( XXX) + ( AOA) + ( ) } (3.3)

5 58 The statistical measures for the correlation (3.2) is given below R 2 (Coefficient of determination) is Standard error for C 1 is , C 2 is , C 3 is C 4 is 2.65and C l is The statistical measures for the correlation (3.3) is given below R 2 (Coefficient of determination) is Standard error for C 1 is 1.36e -5, C 2 is 1.29e -5, C 3 is 1.52e -5 C 4 is 0.067and C d is In order to improve the R 2 value, the correlations are modified by refining the constants C 1, C 2, C 3 & C 4 for coefficient of lift and drag separately. The correlations are applied to various airfoils that are grouped as shown below. I Group - NACA4401 to NACA4410 II Group - NACA4411 to NACA4420 III Group - IV Group - NACA4421 to NACA4430 NACA4431 to NACA4440 The procedure followed in the modifications of constants (C 1, C 2, C 3 & C 4 ) for predicting coefficient of lift and drag are separately discussed in the following subsections and Modified Correlation for Coefficient of Lift The correlation used to evaluate the coefficient of lift in equation (3.2) is considered and modified to yield results closer to the experimental

6 59 results by identifying the terms C 1 and C 2 as these two parameters are directly related to the geometry of the airfoil. The other two terms C 3 & C 4 are assumed to be constant. The term C 3 is associated with wind velocity and not directly related to the geometry of the profile. Further, the term C 4 is an independent constant. In the equation (3.2) the values of C 3 and C 4 and are taken as and respectively in the modified correlation. However any profile may be considered for evaluating the constants, NACA 4415 is chosen to evaluate the modified values of C 1 and C 2. The modified values are substituted in equation (3.2) for the coefficient of lift of NACA 4415 with an angle of attack as 0 o and -8 o. By substituting 0º angle of attack in the equation (3.2) the term C 2 becomes zero and the term C 1 can be arrived by equating the coefficient of lift as 0.3 which is the experimental result derived from Jacobs et al. (1935) for the wind velocity m/s. {( 4415) + ( 0) + ( ) } = The modified value of C 2 is evaluated by substituting the angle of attack as -8 o and experimental coefficient of lift as -0.3, the modified value of C 1 in the equation (3.2) as shown below. {( ) + [ ( 8)] + ( ) } = 0.3 = By substituting the modified constants and in equation (3.2), the revised correlation for coefficient of lift is obtained as in equation (3.4). = {( ) + ( ) + ( ) } (3.4)

7 60 The revised correlation (3.4) is applied to NACA 4412 airfoil which belongs to the Group II as mentioned above and the coefficient of lift is verified. The variation of the coefficient of lift with respect to angle of attack for NACA 4412 is shown in Figure NACA 4412 Wind velocity m/s Cl (wind tunnel) Cl (Correlation) Angle of attack (Degree) Figure 3.2 Validation of coefficient of lift of NACA 4412 It is evident from the Table 3.1 and Figure 3.2 that the revised correlation with modified constants yields much closer results. The correlation can be applied to any profile coming under the Group II (NACA 4411 to NACA 4420). The above procedure was adopted for other Groups and the revised correlation with modified constants. In Group I, the values corresponding to NACA 4406 is used to obtain the modified constants and the revised correlation is shown in the equation (3.5). The modified constants are given below. C 1 = , C 2 = The above constants are substituted in equation (3.2) with the values corresponding to NACA 4409 series airfoil and the results are compared with the experimental values and presented Figure 3.3.

8 NACA 4409 Wind velocity m/s Angle of attack (Degree) Cl (wind tunnel) Cl (Correlation) Figure 3.3 Validation of coefficient of lift of NACA 4409 = {( ) + ( ) + ( ) } (3.5) The above correlation is suitable of predicting coefficient of lift for any profile in Group I (NACA NACA 4410). From the equations (3.5) and (3.4) corresponding to Group I and II, the modified constants C 1 and C 2 are given in Table 3.1. Table 3.1 Modified constants C 1 and C 2 Group C 1 C 2 Group I Group II The difference between the constants C 1 of Group I and II is 2.9 x Similarly, the difference between the constant C 2 of Group I and Group II is 0.55 x The constants corresponding to Group III can be obtained by adding the difference with the value of constant corresponding to

9 62 Group II and the corresponding revised correlation for Group III is given in equation (3.6) = = = = {( ) + ( ) + ( ) } (3.6) The above constants are used in finding the coefficient of lift for the airfoil NACA 4421 (Group III) and it is compared with the experimental data for validation and the results are given in Figure 3.4. In the same way, the constants corresponding to Group IV can also be evaluated. 1.5 NACA 4421 Wind velocity m/s Cl (wind tunnel) Cl (Correlation) Angle of attack (Degree) Figure 3.4 Validation of coefficient of lift of NACA 4421 Thus the general correlation with modified constants of C 1 and C 2 can be written as shown in the equation (3.7). = {( ( ) + ( ) + ( ( )) + ( ) + ( ) } (3.7)

10 63 shown in Table 3.2. Values of M in equation (3.7) for various Groups of airfoils are Table 3.2 Values of M for NACA airfoils M NACA Airfoil By substituting the values of M in equation (3.7) the value of for various airfoils is obtained. The R 2 value for the correlation (3.7) is identified as Hence, the above correlation proved the goodness of the fit with experimental results. The correlation developed is also useful in predicting power coefficient of wind turbine systems and in reducing the time and cost. For the airfoil shapes not having experimental results, the above correlation will be useful in determining the coefficient of lift and other useful parameters Modified Correlation for Coefficient of Drag Similar to the correlation for coefficient of lift, the correlation for finding coefficient of drag is also developed and modified further to yield closer results to the experimental values. In this section, the step by step procedure adopted for developing correlation for finding coefficient of drag is illustrated. Let us consider the basic correlation in equation (3.3). The terms C 1 and C 2 are modified as detailed in the section by keeping other two terms C 3 & C 4 constant and their values are and respectively. As discussed in the previous section, in the modification of correlation for finding coefficient of drag, NACA 4415 is considered. The

11 64 modified terms C 1 and C 2 are calculated by substituting the AOA and wind velocity. The AOA varied from -8 o to 10 0 and the wind velocity is fixed as m/s. While substituting AOA as 0º in the equation (3.3) the term C 2 becomes zero and the term C 1 can be arrived by equating the coefficient of drag as which is the experimental result for the above wind velocity. (( 4415) + ( 0) + ( ) + ( ) = The modified term 2 is evaluated by substituting the value of term 1 in the equation (3.3). As the AOA varied from -8 o to 10 0 in the experimental result, 2 is separately calculated for negative angle of attack (-8 o to 0 o ) and positive angle of attack (0 o to 10 0 ). The value of 2 is determined by selecting angle of attack as -5 o with the corresponding experimental coefficient of drag. (( ) + ( 5) + ( ) + ( ) = The value of above is substituted in equation (3.3) and the correlation for coefficient of drag for negative angle of attack (( ( )) is derived and is shown in equation (3.8). The term C 2 is the same for all selected group of airfoils as the coefficient of drag ( negative values of angle of attack (-8º to 0º). ) remains the same for

12 65 ( ) = {( ) + ( ) + ( ) } (3.8) In a similar way, the modified value of term C 2 is determined for the positive angle of attack (0º to 10º) by considering angle of attack as 6º with corresponding experimental coefficient of drag ( ) as ( ) + ( 6) + ( ) + ( ) = = By substituting the term C 1 and C 2, the correlation of coefficient of drag for the positive angle of attack ( ( ))is developed and shown in equation (3.9). ( ) = {( ) + ( ) + ( ) } (3.9) The revised correlations (3.8) and (3.9) are applied to NACA 4412 airfoil of Group II, the coefficient of drag is found out. The graph showing the variation of the coefficient of drag with respect to angle of attack is shown in Figure 3.5.

13 NACA 4412 Wind velocity m/s Angle of attack (Degree) Cd (Wind tunnel) Cd (Correlation) Figure 3.5 Validation of coefficient of drag of NACA 4412 As it is seen in the case of coefficient of lift, the revised correlation for coefficient of drag also yields closer values with experimental values for various AOA that is illustrated using the Figure 3.5. The above correlation can be applied to any profile of Group II (NACA 4411 to NACA 4420). In order to validate the consistency of the correlation for different airfoils of other groups, the above correlation is applied for the airfoil of group I with the same procedure. The NACA 4406 is used to obtain the modified constants and revised correlation and it is shown in the equation (3.10). The term C 1 and the terms C 2 for both negative and positive angle of attack is also found out and given below. The term C 1 is same for both negative and positive angle of attack for an airfoil group =

14 67 The modified terms C 1 and C 2 are substituted in equation (3.3) and revised correlation for Coefficient of drag for negative and positive angle of attack are shown in equation (3.10) and (3.11) respectively. ) = {( ) + ( ) + ( ) } (3.10) (+ ) = {( ) + ( ) + ( ) } (3.11) The above equations (3.10) and (3.11) are applied to NACA 4409 airfoil and analyzed. The results are illustrated graphically in Figure NACA 4409 Wind velocity m/s Cd (Wind tunnel) Cd (Correlation) Angle of attack (Degree) Figure 3.6 Validation of coefficient of drag of NACA 4409 The above revised correlations (3.10) and (3.11) are suitable for predicting coefficient of drag of any profile in Group I. The values of terms C 1 and C 2 for Group I and II are given in the Table 3.3.

15 68 Table 3.3 Values of C 1 and C 2 of Group I & II Group C 1 for AOA (-8 o to 10 o ) C 2 for AOA (-8 o to 0 o ) C 2 for AOA (0 o to 10 o ) Group I Group II As described in the previous section, the difference between the constants is added to predict the constants of other preceding groups. The difference between the term C 1 of Group I and II is 1.69 x Similarly, the difference between the term C 2 for negative angle of attack of Group I and II is zero. Further, the difference between the constant C 2 for positive angle of attack of Group I and II is 1.3 x The terms C 1 and C 2 corresponding to Group III is obtained by adding the difference between the terms which belongs to Group I and II with the term corresponding to Group II as shown below. = (AOA -8 o to 10 o ) = (AOA -8 o to 0 o ) = = (AOA 0 o to 10 o ) The revised correlation for negative and positive angle of attack is given in equations (3.12) and (3.13) respectively. ( ) = {( ) + ( ) + ( ) } (3.12) ( ) = {( ) + ( ) + ( ) } (3.13)

16 69 The above constants are used in finding the coefficient of drag for the airfoil NACA 4421 (Group III) and it is compared with the experimental values for validation. The values are graphical illustrated in Figure NACA 4421 Wind velocity m/s Angle of attack (Degree) Cd (Wind tunnel) Cd (Correlation) Figure 3.7 Validation of coefficient of drag of NACA 4421 In the same way, the constants corresponding to Group IV are also be evaluated. The generalized correlation for predicting the coefficient of drag for any profile belonging to any group is formulated and given in the equations (3.14) and (3.15). The value of M is given in the Table 3.7 for various Groups. ( ) = {( ( ) + ( ) + ( ) + ( ) } (3.14) ( ) = {( ( ) + ( ) + ( ( )) + ( ) + ( ) } (3.15)

17 70 The R 2 value for the correlation (3.14) is found as 0.93 and for correlation (3.15) is Hence the above correlations are confirmed with the experimental results. The above correlations developed for coefficient of lift and drag are being used to analyze the effect of Reynolds number on lift and drag forces of airfoils and discussed elaborately in the forthcoming subsections of this chapter. The same correlations have also been used in optimization of power coefficient using genetic algorithm and it is presented in chapter EFFECT OF REYNOLDS NUMBER ON LIFT AND DRAG In this section, the effect of Reynolds number (R e ) on coefficient of lift (C l ) and drag (C d ) has been analyzed. This study is useful in determining the characteristics of blade profile and cross section that will be used in optimizing the design and thereby reduces the manufacturing cost. Further, the ratio of lift and drag (sliding rate) which is a function of C l and C d has been evaluated with respect to R e. The R e is varied from to in steps of The various airfoils of NACA series are considered for analysis. The modified correlations developed in the previous section are used to evaluate C l and C d. In general, the rotor diameter varies from 2 to 100m and chord length varies from 0.1 to 5m depending on the power generation. Piggott (2000) predicted that R e varied with respect to chord length (c), wind velocity (v) and the geometry of the airfoil. He suggested the following equation (3.16) relating the above parameters. = (3.16)

18 71 Where, the constant is proposed by him to represent the ratio of air density and viscosity. The chord length is a significant factor for blade profile. It is the length between the leading edge and the trailing edge. Further, he proposed the following equation (3.17) to predict the chord length (c) using rotor diameter (D), tip speed ratio ( ) and number of blades (N). = (3.17) In the above equation the tip speed ratio ( ) is calculated using the following equation (3.18). = (3.18) The velocity triangle as discussed by Lee and Flay (1999) for an airfoil is shown in Figure 3.8 that indicates the lift and drag forces, AOA ( ), pitch angle ( ), air inflow angle ( ) wind velocity (v), velocity of rotor ( r) and relative velocity (W). Figure 3.8 Blade velocity diagram

19 72 From the tip speed ratio, the rotor linear velocity ( r) can be calculated and using the above velocity triangle, wind inflow angle ( ) can be determined using the following equation (3.19). tan = v / r (3.19) The angle of attack ( ) can be found out as and are known. (3.20) and (3.21). The lift and drag forces are calculated by the following equations ( ) = (3.20) ( ) = (3.21) Where - Density of air, kg/m 3 c - Chord length, m r - Radius of the blade element, m W - Relative velocity of air in m/s = v + ( r) The Sliding rate ( ) is defined as the ratio between the Lift (L) and Drag (D). In the above equation (3.20) and (3.21) as other parameters for a particular airfoil are the same, the sliding rate can also be taken as ratio between coefficient of lift and drag and is given as the equation (3.22). = C l /C d (3.22) In this work, the NACA series airfoils of NACA 4410, NACA 4412, NACA 4414, NACA 4416, NACA 4418 and NACA 4420 are considered for analysis. For the airfoil NACA 4410, the sliding rate has been evaluated for the angle of attack from 0 o to 10 o at R e = and it is

20 73 presented in the Table 3.4. The coefficient of lift and drag has also been evaluated using modified correlations and used in predicting sliding rate. Table 3.4 Sliding rate for various AOA NACA 4410, R e S.NO AOA (Degree) C l C d L/D The above values have been shown in Figure 3.9 as a graph to show the variation of sliding rate with respect to AOA. From the Figure 3.9 it is noticed that at 5 o of AOA, the sliding rate attains the maximum value of and hence the power output is optimum at the conditions specified above.

21 74 Figure 3.9 Variation of L/D ratio for various angle of attack The same procedure is adopted for predicting optimum sliding rate for various airfoils by varying angle of attack from 0 o to 10 o, Reynolds number in the range of to in steps of The results are shown as graph in Figures 3.10(a) to 3.10(e) Figure 3.10(a) Variation of sliding rate for various angle of attack at R e =100000

22 75 For Reynolds number , the optimum angle of attack at the highest sliding rate for various NACA airfoils are depicted in Figure 3.10(a) as 5 o that yielded optimum power output NACA 4410 NACA 4412 NACA 4414 NACA 4416 NACA 4418 NACA Angle of Attack (Degree) Figure 3.10(b) Variation of sliding rate for various angle of attack at R e = The optimum sliding rate for NACA 4410 and NACA 4412 is arrived at 4 o of angle of attack, where as for other airfoils it is at 5 o of angle of attack for the Reynolds number

23 NACA 4410 NACA 4412 NACA 4414 NACA 4416 NACA 4418 NACA Angle of Attack (Degree) Figure 3.10(c) Variation of sliding rate for various angle of attack at R e = Various NACA airfoils yield the optimum sliding rate at 5 o of angle of attack for Reynolds number NACA 4410 NACA 4412 NACA 4414 NACA 4416 NACA 4418 NACA Angle of Attack (Degree) Figure 3.10(d) Variation of sliding rate for various angle of attack at Re = The airfoils NACA 4410, NACA 4412, NACA 4414, and NACA 4416 have optimum sliding rate at 5 o of angle of attack whereas

24 77 NACA 4418 and NACA 4420 have optimum sliding rate at 6 o of angle of attack for Reynolds number NACA 4410 NACA 4412 NACA 4414 NACA 4416 NACA 4418 NACA Angle of Attack (Degree) Figure 3.10(e) Variation of sliding rate for various angle of attack at Re = The optimum sliding rate for NACA 4420 is at 6 o angle of attack, other profiles achieved optimum sliding rate at 5 o of angle of attack for Reynolds number The optimum angle of attack for various NACA airfoils with respect to Reynolds number is presented in Table 3.5.

25 78 Table 3.5 Optimum angle of attack with respect to Reynolds number Reynolds number NACA 4410 Optimum Angle of attack (Degree) NACA 4412 NACA 4414 NACA 4416 NACA 4418 NACA From the above table, it is inferred that the optimum angle of attack varies from 4 o to 6 o for different airfoils corresponding to optimum sliding rate for various Reynolds numbers. The variation of coefficient of lift (C l ) at various angles of attack for different airfoils are found out at Reynolds number of and presented in Table 3.6. The flow around the airfoil is attached to certain limit of angle of attack and beyond which the flow is separated from the surface of the airfoil. Hence, the coefficient of lift increases for all airfoils as the angle of attack increases and then started decreasing after attaining the maximum value. This optimum coefficient of lift is attained at 14 o to 15 o for various airfoils. The results are shown graphically to show the variation of coefficient of lift with respect to angle of attack in Figure 3.11.

26 79 Table 3.6 Variation of coefficient of lift with respect to angle of attack Angle of Coefficient of lift (C l ) at R e = attack NACA NACA NACA NACA NACA NACA (Degree)

27 NACA 4412 NACA 4414 NACA 4416 NACA 4418 NACA 4420 NACA Angle of Attack (Degree) Figure 3.11 Variation of coefficient of lift with angle of attack The airfoil with lower thickness (NACA 4410) have maximum coefficient of lift than others at higher angle of attack. The coefficient of lift for various airfoils is very close to each other for angle of attack from -5 o to 5 o and started deviating to a larger extent as the angle of attack increases. It attains the maximum value for the airfoils of NACA 4410, NACA 4412, and NACA 4414 at 15 o and for airfoils of NACA 4416, NACA 4418 and NACA 4420 is at 14 o of angle of attack. 3.4 SUMMARY The baseline characteristics and analysis of NACA 4 series airfoils are presented in this chapter. The correlations for coefficient of lift and drag of NACA 4 series airfoils have been developed and compared with the experimental results. The effects of Reynolds number on coefficient of lift and drag for various NACA airfoils are predicted using the correlations and

28 81 presented. The NACA airfoils have higher Lift/Drag ratio at the angle of attack from 4 o to 6 o for Reynolds number in the range of to The variation of coefficient of lift (C l ) at various angles of attack for different airfoils is found out at Reynolds number of It is found that the airfoil with lower thickness (NACA 4410) have maximum coefficient of lift than others at higher angle of attack. The coefficient of lift for various airfoils is very close to each other for smaller angle of attack and the variations are more as the angle of attack increases.