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Mathematic Modelling

CFD ANALYSIS OF WIND-POWERED WATER PUMP SYSTEM

Created by:

NOVIAN RISKIANA DEWI (090210101018)

MATH EDUCATION PROGRAME

FACULTY TEACHER TRAINING AND EDUCATION

UNIVERSITAS JEMBER

2012


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I. PRELIMINARY

Access to clean water is a problem for 884 million people around the world, and some of

these problems comes from the inability to pump water from wells and difficulty in moving

water from one region to another. Improved blade design in wind-powered water pumps has the

potential to alleviate these difficulties. This analysis investigated improved wind turbine blade

designs in which blades pivoted out of the wind when not in use. Using locally sourced materials

and labour, the design cost was found to be under two days' wage for the Southern Asian regions

in which it would be built. The comparative computational fluid dynamics analysis found that

torque could be increased by a factor of 2,35 through two factors resulting from blade pivots: the

first was decreased drag on unused blades, and the second was increased torque power-producing

blades due to increased wind availability. A proof of concept was also conducted in order to

confirm computational models.

The purpose of this page is to provide analytical techniques for pre-existing wind-

powered water pumping systems in developing countries. These devices, known as windpumps,

have already been investigated but Remains much work to be done in improvement and

optimization. This device, known as windpumps, have been investigated but is still much work

to be done in the improvement and optimization.

The Following the analysis was therefore conducted using only generic dimensions and

wind speed can be used correlations That in any geography, Provided That members of the

community Deem the use and effects of the Appropriate analysis. The following analysis

because it is done using the correlation dimension and generic wind speed that can be used in

geography, as long as members of the public considers the use and impact of proper analysis.

The designs found in this analysis are also subject to material and labor time constraints: the

design is left to the builder to modify as They see fit. Design found in this analysis are also

subject to the constraints of time material and labor: the design is left to the builder to modify as

they see fit. Such as, the scope does not include local instructions for construction of the blades.

Thus, the scope does not include instructions for the local construction of the blade.


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The conducted analysis consists of two parts: The analysis consists of two parts:

1) The design of a new method to improve a standard

vertical axis wind rotor for use in areas with low wind speeds and variable.

2) Computational Fluid Dynamics (CFD) analysis of new rotor in order to look for possible

improvements.

The overall objective of this project is to improve the community wind turbine was built

for use in pumping air.So most effectively through engineering analysis of the focus, the project

concentrated on the design and modeling of a new rotor to the wind-pump.A proof of concept

and computational fluid analysis performed for the rotor to verify that the fundamental

innovation of the design will function as required. However, this project follows the principles of

sustainable design wherever possible, and as such local materials and expertise should be taken

into account when scaling up the design of a proof of concept to full-scale system.


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II. FORMULATION OF THE EQUATION

2.1 Momentum Equation

P = Note:

P : Power available : Air velocity The air density : The area covered wind turbine

F =

Note:

F : The force created : Pressure on the back of the blade

: Pressure on the front of the blade A : Area

+

-

+

-

= F

+

+

+

+


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+

Change u with u cos and v with v sin

+

Integral both of side to x and y

(1)


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2.2 Energy Equation

+ = k +

+ = k

+ (

)

+

= k

+ (

)

+

= k

+

+ = + +

.. (2)


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Substitute the 2thequation to the 1thequation, so we have :

( [

])

( )

+ ( ) + ( ) + ( ) = -


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= = * +

=

= ( )* + =

= ( )* + =

= ( )* +

*

+ *

+

+

* + * + + * + + * + + *

+ + * + =


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-

[A]

[B]

= [C] [D] [E] F] [G] [H]A = * + B = * +

C = * + * +

D = * +


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E = * +

F = * +

G = * +

H= -

= -


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Discretization Scheme

(i,j+4) (i+1,j+4) (i+2,j+4) (i+3,j+4) (i+4,j+4)

(i,j+3) (i+1,j+3) (i+2,j+3) (i+3,j+3) (i+4,j+3)

(i,j+2) (i+1,j+2) (i+2,j+2) (i+3,j+2) (i+4,j+2)

(i,j+1) (i+1,j+1) (i+2,j+1) (i+3,j+1) (i+4,j+1)

(i,j)

(i+1,j)

(i+2,j)

(i+3,j)

(i+4)

x

y


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Based on discretization scheme, we get the discretization matrix :

C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 H

B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 H

A B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 H0 A B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 H

0 0 A B C 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 H

F 0 0 0 0 C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 H

0 F 0 0 0 B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 7 H

0 0 F 0 0 A B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 8 H

0 0 0 F 0 0 A B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 9 H

0 0 0 0 F 0 0 A B C 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 10 H

E 0 0 0 0 F 0 0 0 0 C D 0 0 0 G 0 0 0 0 0 0 0 0 0 11 H

0 E 0 0 0 0 F 0 0 0 B C D 0 0 0 G 0 0 0 0 0 0 0 0 12 H

0 0 E 0 0 0 0 F 0 0 A B C D 0 0 0 G 0 0 0 0 0 0 0 13 = H

0 0 0 E 0 0 0 0 F 0 0 A B C D 0 0 0 G 0 0 0 0 0 0 14 H

0 0 0 0 E 0 0 0 0 F 0 0 A B C 0 0 0 0 G 0 0 0 0 0 15 H

0 0 0 0 0 E 0 0 0 0 F 0 0 0 0 C D 0 0 0 G 0 0 0 0 16 H

0 0 0 0 0 0 E 0 0 0 0 F 0 0 0 B C D 0 0 0 G 0 0 0 17 H

0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D 0 0 0 G 0 0 18 H

0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D 0 0 0 G 0 19 H

0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C 0 0 0 0 G 20 H

0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 0 0 C D 0 0 0 21 H

0 0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 0 B C D 0 0 22 H

0 0 0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D 0 23 H

0 0 0 0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D 24 H

0 0 0 0 0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C 25 H


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MATLAB

disp('==========================================================');

disp('CFD analysis of wind-powered water pumping system');

disp('Created by:');

disp('NOVIAN RISKIANA DEWI');

disp('090210101018');disp('==========================================================');

g=9.8;

rho=0.8;

deltax=input('Enter the discretization length of x=');

deltay=input(Enter the discretization length of y=');

tetha=input('Enter the angle of the turbine blades=');

u=input('Enter the speed of the wind on the x axis=');

v=input('Enter the speed of the wind on the y axis=');

miu=input('Enter the thickness of=');

Cv=input('Insert the power coefficient of rotor=');

Pf=input('Enter the pressure on the front of the blade=');

Pb=input('Enter the pressure on the back of the blade=');

A2=input('Enter the area covered by wind turbines =);

k=input('Enter a value of k=');

A=(1/8*rho*u*cos(tetha))-(1/8*rho^2*u*cos(tetha))*deltay;


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B=((((1/8)*rho^2*u*cos(tetha))-((1/8)*rho*u*cos(tetha))+((3/4)*rho^2*u*cos(tetha))-((3/4)*rho*u*cos(tetha)))*deltay);

C=((((3/4)*rho*u*cos(tetha))-((3/4)*rho^2*cos(tetha))+((3/8)*rho^2*u*cos(tetha))-

((3/8)*rho*u*cos(tetha)))*deltay)+(((3/4)*rho*v*sin(tetha))-((3/4)*rho 2*v*sin(tetha))+((3/8)*rho^2*v*sin(tetha))-

((3/8)*rho*v*sin(tetha))*deltax);

D=((((3/8)*rho*u*cos(tetha))-((3/8)*rho^2*u*cos(tetha)))*deltay);

E=((((1/8)*rho*v*sin(tetha))-((1/8)*rho^2*v*sin(tetha)))*deltax);

F=((((1/8)*rho^2*v*sin(tetha))-((1/8)*rho*v*sin(tetha))+((3/4)*rho^2*v*sin(tetha))-((3/4)*rho*v*sin(tetha)))*deltax);

G=((((3/8)*rho*v*sin(tetha))-((3/8)*rho^2*v*sin(tetha)))*deltax);

H=((rho*g*deltay)+(rho*g*deltax)-((1/2)*rho*u^3*A2*deltay)-

((1/2)*rho*v^3*A2*deltax)+((miu*u*cos(tetha))*(deltay/deltax))+((miu*v*sin(tetha))*(deltay/deltax))+((miu*u*cos(tetha))*(deltax/d

eltay))+((miu*v*sin(tetha))*(deltax/deltay))-((miu*v*sin(tetha))*(deltax/deltay))-(((k*u*cos(tetha))/Cv)*(deltay/deltax))-

(((k*v*sin(tetha))/Cv)*(deltax/deltay))-((((2*miu*u 2)/Cv)*(cos(tetha))^2)*(deltay/deltax))-

((((2*miu*v^2)/Cv)*(sin(tetha))^2)*(deltax/deltay))-((((miu*v^2)/Cv)*(sin(tetha)) 2)*(deltay/deltax))-

(((miu*u*v)/Cv)*sin(2*tetha))-((((miu*u^2)/Cv)*(cos(tetha))^2)*(deltax/deltay))+(((Pf-Pb)/A)*deltax*deltay));

M=[C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;A B C D 0 0 0 G 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0;0 A B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 A B C 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;F 0 0 0 0 C D 0 0 0G 0 0 0 0 0 0 0 0 0 0 0 0 0 0;0 F 0 0 0 B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0;0 0 F 0 0 A B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0;0 0 0 F

0 0 A B C D 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0;0 0 0 0 F 0 0 A B C 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0;E 0 0 0 0 F 0 0 0 0 C D 0 0 0 G 0 0 0 0 0 0

0 0 0;0 E 0 0 0 0 F 0 0 0 B C D 0 0 0 G 0 0 0 0 0 0 0 0;0 0 E 0 0 0 0 F 0 0 A B C D 0 0 0 G 0 0 0 0 0 0 0;0 0 0 E 0 0 0 0 F 0 0 A B C D

0 0 0 G 0 0 0 0 0 0;0 0 0 0 E 0 0 0 0 F 0 0 A B C 0 0 0 0 G 0 0 0 0 0;0 0 0 0 0 E 0 0 0 0 F 0 0 0 0 C D 0 0 0 G 0 0 0 0;0 0 0 0 0 0 E 0 0

0 0 F 0 0 0 B C D 0 0 0 G 0 0 0;0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D 0 0 0 G 0 0;0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D 0 0 0 G 0;0

0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C 0 0 0 0 G;0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 0 0 C D 0 0 0;0 0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 0


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B C D 0 0;0 0 0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D 0;0 0 0 0 0 0 0 0 0 0 0 0 0 E 0 0 0 0 F 0 0 A B C D;0 0 0 0 0 0 0 0 0 0 0 0 0

0 E 0 0 0 0 F 0 0 A B C]

L=[H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H;H]

V=inv(M)*L

plot(V)

Matlab results:

Suppose we enter the following numbers:

pemodelan1

==========================================================

CFD analysis of wind-powered water pumping system

Created by:

NOVIAN RISKIANA DEWI

090210101018

==========================================================


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Enter the discretization length of x=0.6

Enter the discretization length of y=0.6

Enter the angle of the turbine blades=45

Enter the speed of the wind on the x axis=20

Enter the speed of the wind on the y axis=15

Enter the thickness of=0.7

Insert the power coefficient of rotor=2

Enter the pressure on the front of the blade=2

Enter the pressure on the back of the blade=1

Enter the area covered by wind turbines =24

Enter a value of k=0.8


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FLUENT

Grid Visualization


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Iteration Graphics 1 Iteration Graphics 2

Iteration Graphics 3


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Filled Contours of Velocity Magnitude 1 Filled Contours of Velocity Magnitude 2

Filled Contours of Velocity Magnitude 3


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Filled Contours of Static Pressure 1 Filled Contours of Static Pressure 2

Filled Contours of Static Pressure 3


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The Velocity Visualization by Vektor Appearance 1 The Velocity Visualization by Vektor Appearance 2

The Velocity Visualization by Vektor Appearance 3


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Analysis:

In the picture I present turbine in front, I distinguish the point of curvature on the turbine blades.

Figure 1, the turbine blade angle greater than the figure 2 and figure 3. Whereas Figure 3 has a

blade angle of curvature smaller.

Curvature of the blade angle will result in greater wind speeds are small, while the pressure of

the wind turbines around. The smaller the angle of curvature of turbine blades, the greater the

wind speed and the smaller the pressure around the turbine. On the turbine which has a small

angle of curvature, will move faster, and be able to pump more water.

Whereas in Figure 1, the wind pressure is greater when compared with figure 2 and figure 3.

Greater curvature of the blade, the greater the pressure that goes around the turbine.

Conclusion:

Of the three images, we can conclude that if the turbine blade angle of curvature is smaller then

the wind speed will be greater, and vice versa.

So, if we're going to make a water pump with the use of wind systems, we have to make a

smaller angle of turbine blades for wind speeds greater that the speed of water pumped will be

even greater.

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