International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-1, Issue-2, May 2014
Dimensional And Sensitivity Analysis Of Flat Back Model Mrs. Smita Mattalwar, Dr. C.N. Sakhale Abstract: Dimensional analysis and sensitivity analysis conducted on blunt trailing edge airfoil with splitter plates at various angle of attack and velocities provides minimal conclusions on performance of flat back airfoils .The paper contributes ot the study and analysis of the same tested on an educational Low subsonic Wind Tunnel under ambient conditions. Performance through a sweep of angles was needed to determine the behavior of the airfoils at low angle of attack. Dimensional analysis is a practice of checking relations among physical quantities by identifying their dimensions. The sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system can be apportioned to different sources of uncertainty in its output. It also helps to understand the relationship between input and output variables in the system or model. Index Terms— Blunt Trailing Edge , Drag reduction, airfoil, experiment.
Fig. 1 Blunt Trailing Edge with splitter plate
The pressure readings are obtained by connecting the pressure tappings to multi tube manometer . The angle of attack are varied from 0, 5, 10, 13 and 15 degrees and the corresponding velocity changes are varied from 10, 15, 20, 22.5 and 25 m/s. The co-efficient of pressure is calculated from the formula
……….. (1)
Where , pi is the static pressure at any pressure tap on the airfoil surface, p is the free stream pressure (measured on the Pitot static port), is air density, and U is the free-stream velocity, given by
I. INTRODUCTION
…………… (2)
A. Blunt Trailing edge airfoil In case of two dimensional blunt trailing edge airfoil the section selected is 200mm*200 mm sectional area with a chord length of 163 mm.A blunt trailing edge modifications is done to mitigate the drag . The addition of splitter plate to the trailing edge of the airfoil increases the pressure at the trailing edge of the airfoil thereby increasing the pressure .The airfoil is designed according to the specifications occupying the test section area of Low subsonic widn tunnel of 300mm*300 mm. The model is shown in fig.1 is having density of 590 grams and is fabricated in wooden material. The model consists of 13 pressure tapings which will simultaneously read the pressure on upper and lower surface of the airfoil respectively.
Manuscript received April 20, 2014. Mrs.Smita mattalwar, Persuing M.tech in Mechanical Engineering at Priyadarshinin Colllge of Engineering, Nagpur, India. Dr. C.N. Sakhale , Associate Professor, Department of Mechanical Engineering at Priyadarshini College of Engineering, Nagpur, India
1
The pressure readings , are evaluated for the center of pressure and the plot of co efficient of pressure versus no of pressure tappings with respect to the chord are plotted on the graph. The table 1.1 represents the readings and the plot is shown in fig 2 Velocity = 15 m/s Angle of Attack x /c
0
5
10
15
20
0 .6 7
1
1 .4
1 .3
1 .1
0 .9
0 .5 0 5
0 .9
1 .5
1 .4
1 .2
1
0 .3 4
0 .9
1 .6
1 .5
1 .3
1 .1
0 .1 7 5
0 .4
1 .7
1 .6
1 .5
1 .2
0 .1 2 5
-0 . 3
1 .6
1 .4
1
0 .8
0 .0 7 5
-0 . 2
1 .6
1 .4
0 .9
0 .7
0
2 .4
2 .2
2 .2
2 .2
2 .2
0 .0 7 5
-0 . 1
0 .4
0 .8
0 .7
0 .9
0 .1 2 5
0 .2
0 .4
0 .9
1
1 .2
0 .1 7 5
0 .8
0 .9
1 .1
1 .2
1 .3
0 .3 4
0 .7
1
1
1
1 .1
0 .5 0 5
0 .8
1
1
1
1
0 .6 7
0 .8
0 .9
0 .9
0 .9
0 .8
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Dimensional And Sensitivity Analysis Of Flat Back Model Table 1.1. Coefficient of pressure at various angle of attack with respect to the position of pressure tappings with respect to the chord.
Equation 1.5 is a regression equation of Z on A, B, C, and D. in an n dimensional co-ordinate system. This represents a regression hyper plane .To determine the regression hyper plane, determines a1, b1, c1, and d1 in equation 1.5 so that ΣZ1 = nK1 + a1*ΣA + b1*ΣB + c1*ΣC + d1* ΣD ΣZ1*A = K1*ΣA +a1*ΣA*A + b1*ΣB*A + c1*ΣC*A + d1* ΣD*A ΣZ1*B = K1*ΣB +a1*ΣA*B + b1*ΣB*B + c1*ΣC*B + d1* ΣD*B ΣZ1*C = K1*ΣC +a1*ΣA*C+ b1*ΣB*C + c1*ΣC*C + d1*
Figure. 2 co-efficient of pressure versus no of pressure tappings
ΣD*C
B : Dimensional Analysis The data of the independent and dependent parameters of the system has been gathered during experimentation. In this case there are two dependent and independent pi terms for each processing operations. It is necessary to correlate quantitatively various independent and dependent pi terms involved in this airfoil model. This correlation is nothing but a mathematical model as design tool for such workstation. Lift (L) = k1 x (π1)a1 x (π2)b1 x (π3)c1 x (π4)d1 ------ ( 1.1)
ΣZ1*D = K1*ΣD +a1*ΣA*D + b1*ΣB*D + c1*ΣC*D + d1* ΣD*D -------------- (1.6) In the above set of equations the values of the multipliers K1, a1, b1, c1, and d1 are substituted to compute the values of the unknowns (viz. K1, a1, b1, c1, and d1 ). The values of the terms on L H S and the multipliers of . K1, a1, b1, c1, and d1 in the set of equations are calculated and tabulated in the Table 1.1 to Table 1.4. After substituting these values in the equations 1.6 one will get a set of 5 equations, which are to be solved simultaneously to get the values of K1, a1, b1, c1, and d1. The above equations can be verified in the matrix form and further values of K1, a1, b1, c1, and d1 can be obtained by using matrix analysis.
The values of exponential a1, b1, c1, d1 are established, considering exponential relationship between dependent pi term lift and Independent π terms π1, π2, π3, π4, independently taken one at a time, on the basic of data collected through classical experimentation. There are four unknown terms in the equation curve fitting constant K1 and indices a1, b1, c1, d1, to get the values of these unknown we need minimum four sets of values of (π1, π2, π3, π4). Solution to establish the relationship for π11is as follows: Model of dependent pi term π11 for lift obtained for this model is (FL / ds*ρ )= k1 x (π1)a1 x (π2)b1 x (π3)c1 x (π4)d1 ------ (1.2)
‘MATLAB’ is given below.
π11 = k1 x (π1)a1 x (π2)b1 x (π3)c1 x (π4)d1
( 1.3)
a nd d 1
Taking log on the both sides of equation for π11, , Log π11 = log k1+ a1log π1+ b1log π2+ c1log π3 + d1log π4
Then,
----- (
------
X1 = inv (W) x P1
…………….(1.7)
The matrix method of solving these equations using W = 5 x 5 matrix of the multipliers of K1, a1, b1, c1, and d1 P1 = 5 x 1 matrix of the terms on L H S and X1 = 5 x 1 matrix of solutions of values of K1, a1, b1, c1,
The matrix obtained is given by,
1.4)
Let, Z1= log π11, K1 = log k1, A = log π1, B = log π2, C = log π3, D = log π4, Putting the values in equations 1.4, the same can be written as
Zx
P1 = W1 x X1
Z1 = K1+ a1 A + b1 B + c1 C + d1 D -----(1.5)
2
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International Journal of Engineering Research And Management (IJERM) ISSN : 2349- 2058, Volume-1, Issue-2, May 2014 impact of the geometric variable of airfoils , angle of attack and weight of the airfoil model are the governing parameters which effect on the aerodynamic forces i.e lift and drag force .i.e the term π1 = ( l*c/ds2 ) which represents the geometric variable of the airfoil model, π2 = (θ /α) , which represents the dependency on angle of attack of the airfoil, π3 = (μ / ds*v*ρ) which is a relation of viscosity of fluid medium, and π4 = (w/ds*ρ) which competes with the weight or density of the airfoil section. The research and calculations clear the effect and impact of the various components on designing of the airfoil model. The dimensional analysis helps to get the relation of dependent and independent parameters to be considered on checking the performance of the section or model.
[P1] = [W1] [X1] Using Mat lab, X1= W1\ P1 , after solving X1 matrix with K1 and indices a1, b1, c1, d1 are as follows -1.6644 -0.3426 -0.4117 -1.7968 -0.3234
K a1 b1 c1 d1
APPENDIX Table 1.1 Sample calculations for calculating lift of airfoil section
But K1 is log value so convert into normal value take antilog of K1 Antilog (-1.6644) = 0.021657084 Hence the model for dependent term π11 a1
b1
c1
π11 = k1 x (π1) x (π2) x (π3) x (π4)
d1
(FL / ds*ρ )= 0.021657084 (π1)-.3426 (π2)-0.4117 (π3)-1.7968 (π4) -0.3234 Hence , Fl =0.021657084 ( (
-1.7968
(
)
) -0.3426 (
-0.4117
-0.3234
Sr. no
Π1=L s*C/d s ^2
Π2=θ/α
Π3=µ/ds*v*ρ
Π4=W/ds* ρ
Π01=FL/ ds*ρ
1
40000
1 1 .9 9 2 9 0 9
0 .0 0 0 1 5 4 2 2 9
4 0 6 .3 0 1 8 2 4
1 6 2 .6 8 6 6
2
40000
5 .9 9 6 4 5 4 5
0 .0 0 0 1 5 4 2 2 9
4 0 6 .3 0 1 8 2 4
3 2 5 .3 7 3 1
3
40000
3 .9 9 7 6 3 6 4
0 .0 0 0 1 5 4 2 2 9
4 0 6 .3 0 1 8 2 4
3 6 6 .0 4 4 8
4
40000
2 .9 9 8 2 2 7 3
0 .0 0 0 1 5 4 2 2 9
4 0 6 .3 0 1 8 2 4
3 1 7 .2 3 8 8
5
40000
1 1 .9 9 2 9 0 9
0 .0 0 0 1 0 2 8 1 9
4 0 6 .3 0 1 8 2 4
4 0 6 .7 1 6 4
Table 1.2 sample calculations for log of pi terms for lift of a i r f o i l s e c t i on Sr.no
SENSITIVITY ANALYSIS The influence of the various independent π terms has been studied by analyzing the indices of the various π terms in the models. Through the technique of sensitivity analysis, the change in the value of a dependent π term caused due to an introduced change in the value of individual π term is evaluated. In this case, of change of ± 10 % is introduced in the individual independent π term independently (one at a time).Thus, total range of the introduced change is 20 %. The effect of this introduced change on the change in the value of the dependent π term is evaluated .The average values of the change in the dependent π term due to the introduced change of 20 % in each independent π term. This is defined as sensitivity. The total % change in output for ±10% change in input are shown in Table .Nature of variation in response variables due to increase in the values of independent pi terms. sequence of influence of independent pi terms on dependent pi terms for all models . II. CONCLUSION Dimensional analysis helps to find the relation of the various dependent and independent variables for the particular airfoil model. The results clear that the
3
pi terms for
logZ
Logπ1
Logπ2
Logπ3
Logπ4
2 .2 1 1 3 5 2
4 .6 0 2 0 6
1 .0 7 8 9 2 5
-3.81183
2 .6 0 8 8 4 9
2
2 .5 1 2 3 8 2
4 .6 0 2 0 6
0 .7 7 7 8 9 5
-3.81183
2 .6 0 8 8 4 9
3
2 .5 6 3 5 3 4
4 .6 0 2 0 6
0 .6 0 1 8 0 3
-3.81183
2 .6 0 8 8 4 9
4
2 .5 0 1 3 8 6
4 .6 0 2 0 6
0 .4 7 6 8 6 5
-3.81183
2 .6 0 8 8 4 9
5
2 .6 0 9 2 9 2
4 .6 0 2 0 6
1 .0 7 8 9 2 5
-3.98793
2 .6 0 8 8 4 9
Table 1.3 Sample Calculations of Multipliers of the R.H. S. terms of equation 1.6 for formulation of model for calculating lift of airfoil model A 4 .6 0 4 .6 0 4 .6 0 4 .6 0 4 .6 0
A 4 .6 0 4 .6 0 4 .6 0 4 .6 0 4 .6 0
A A 21. 18 21. 18 21. 18 21. 18 21. 18 A A 21. 18 21. 18 21. 18 21. 18 21. 18
A
AA
AB
AC
AD
B
4. 60 4. 60 4. 60 4. 60 4. 60
21. 18 21. 18 21. 18 21. 18 21. 18
4 .9 7 3 .5 8 2 .7 7 2 .1 9 4 .9 7
-1 7 . 5 4 -1 7 . 5 4 -1 7 . 5 4 -1 7 . 5 4 -1 8 . 3 5
12. 01 12. 01 12. 01 12. 01 12. 01
1. 08 0. 78 0. 60 0. 48 1. 08
A
AA
AB
AC
AD
B
4. 60 4. 60 4. 60 4. 60 4. 60
21. 18 21. 18 21. 18 21. 18 21. 18
4 .9 7 3 .5 8 2 .7 7 2 .1 9 4 .9 7
-1 7 . 5 4 -1 7 . 5 4 -1 7 . 5 4 -1 7 . 5 4 -1 8 . 3 5
12. 01 12. 01 12. 01 12. 01 12. 01
1. 08 0. 78 0. 60 0. 48 1. 08
A B 4. 97 3. 58 2. 77 2. 19 4. 97 A B 4. 97 3. 58 2. 77 2. 19 4. 97
BB 1 .1 6 0 .6 1 0 .3 6 0 .2 3 1 .1 6
BB 1 .1 6 0 .6 1 0 .3 6 0 .2 3 1 .1 6
B C -4 . 11 -2 . 97 -2 . 29 -1 . 82 -4 . 30 B C -4 . 11 -2 . 97 -2 . 29 -1 . 82 -4 . 30
BD 2 .8 1 2 .0 3 1 .5 7 1 .2 4 2 .8 1
BD 2 .8 1 2 .0 3 1 .5 7 1 .2 4 2 .8 1
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Dimensional And Sensitivity Analysis Of Flat Back Model Table 1.4 Sample Calculations of Multipliers of the L.H. S. terms of equations 1.6 for formulation of model for lift L.H.S Multipliers for Lift ZA
ZB
ZC
ZD
-144.49438
1 .2 5 5 9 5 2 5
-8.4293064
5 .7 6 9 0 8 2 2
-144.49438
0 .4 7 0 7 1 9 5
-9.5767829
6 .5 5 4 4 2 3 9
-144.49438
0 .2 1 7 9 5 3 4
-9.7717678
6 .6 8 7 8 7 3 1
-144.49438
0 .1 0 8 4 3 8 9
-9.5348703
6 .5 2 5 7 3 8 6
Table 1.5 Sample calculations for pi terms for lift
Π1=Ls * C/ds^2
Π3=µ/ds* v*ρ
Π2=θ/α
Π4=W/ds* ρ
Π01=FL/ ds*ρ
40000
1 1 .9 9 2 9 0 9
0 .0 0 0 1 5 4 2
4 0 6 .3 0 1 8
1 6 2 .6 8 7
40000
5 .9 9 6 4 5 4 5
0 .0 0 0 1 5 4 2
4 0 6 .3 0 1 8
3 2 5 .3 7 3
40000
3 .9 9 7 6 3 6 4
0 .0 0 0 1 5 4 2
4 0 6 .3 0 1 8
3 6 6 .0 4 5
40000
2 .9 9 8 2 2 7 3
0 .0 0 0 1 5 4 2
4 0 6 .3 0 1 8
3 1 7 .2 3 9
40000
1 1 .9 9 2 9 0 9
0 .0 0 0 1 0 2 8
4 0 6 .3 0 1 8
4 0 6 .7 1 6
Acknowledgment I take this opportunity to express a deep sense of gratitude towards Dr. Chandrashekhar N Sakhale, Associate Professor, Department of Mechanical Engineering, Priyadarshini College of Engineering , Nagpur, India, who has functioned not only as a supervisor for this research work but has performed a role of a guide and philosopher. I feel short of words to express thanks to him REFERENCES [ 1] [ 2] [ 3] [ 4]
[ 5] [ 6] [ 7]
[ 8]
[ 9]
Numerical simulations of Flat back airfoil for wind Turbine applications by Mathew FULLER , B.S. Eng,, University of Louisivle, 2009. Computational Design and Analysis of Flatback Airfoil Wind Tunnel Experiment by C.P. “Case” van Dam, Edward A. Mayda, and David D. Chao ; Flatback Airfoil Wind Tunnel Experiment by Jonathon P. Baker, C.P. “Case” van Dam, and Benson L. Gilbert PIV measurements of the flow around airfoil models equipped with the plasma actuator Artur Berendt1, Janusz Podlinski1*, Annie Leroy3, Pierre Audier4, Dunpin Hong4 and Jerzy Mizeraczyk Chapter 4, Hawt Aerodynamics and models From wind tunnel Measurements Laird, Daniel, and Thomas Ashwill. WIND ENERGY RESEARCH AT SANDIA NATIONAL LABORATORIES. Rep. 2004. Print. McNally, Amy, Darrin Magee, and Aaron T. Wolf. "Hydro Power and Sustainability: Resilience and Vulnerability in Chinas Power sheds." Journal of Environmental Management 90 (2009): S286-293. Print Aerodynamcic and Aeroacoustic properties of Flatback airfoil by Dale E. Berg* and Matthew Barone† Sandia National Laboratories‡Albuquerque, NM 87185-1124 USA Airfoil Performance Analysis and Wind Tunnel Engineering by Linn-Benton Community College Chapter of the Society of Physics Students Albany, Oregon
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