AN APPROXIMATE SOLUTION OF BOUNDARY VALUE PROBLEM IN CONVECTION PROBLEM WITH CHEMICAL REACTION
Bulletin of Mathematics
ISSN Printed: 2087-5126; Online: 2355-8202
Vol. 08, No. 02 (2016), pp.109-115 http://jurnal.bull-math.org
AN APPROXIMATE SOLUTION OF
BOUNDARY VALUE PROBLEM IN
CONVECTION PROBLEM WITH CHEMICAL
REACTION
Yulita Molliq Rangkuti, Wulandari, and Penny Charity
Abstract.Lumban Raja
In this work, we firstly use the application of Finite Difference Method(FDM) and Homotopy Perturbation Method (HPM) to find the approximate so-
lution of Two-Point boundary Value Problem (BVPS) which appear in the fully
developed mixed convection with chemical reaction. HPM is a simple method to
control the convergent solution using He polynomial which is optimally determined.
The result shows the effectiveness and realibility of FDM than HPM for this con-
vection problem1. INTRODUCTION The study of combined and free convection flow has received consider- able attention of geothermal reservoir, cooling of nuclear reactors, conserva- tion, chemical, etc. Tulus et al. [1] studied the fluid dynamic phenomenon in the chemical reaction in stirred tank by using finite element method. In this work, a steady flow of a viscous, incompressible fluid and the heat assumed supplied around the surface by an exothermic surface reaction from [2] are
Received 15-08-2016, Accepted 20-09-2016. 2010 Mathematics Subject Classification: 30E25, 65L12, 70E20
Key words and Phrases: Fully Developed Mixed Convection, Homotopy Perturbation Method,
He Polynomial, Boundary Value Problem, Finite Difference MethodRangkuti et al., An Approximate Solution ...
considered. The reaction is located only on the surface which is constructed by first-order Arrhenius kinetics. The following system are written as
2
d U θ − A,
= G R (1)
2
dt
2
d θ
θ
e = −K F (2)
2
dt subject to U (0) = 0, U (1) = 0 and θ(0) = R , θ (1) = −R . (3)
T T
Here, U (x) is a velocity and θ(y) is a temperature. Saleh and Hashim [3] have solved this equations analytically via dsolved from MAPLE which was reached convergence. The method is very arduous, so another alternative to solve (1) and (2) is semi-analytical method. The selected method in this research are homotopy perturbation method (HPM) [4] and finite difference method (FDM).
Figure 1: Schematic representation of the model
2. FINITE DIFFERENCE METHOD Every operator of finite difference is obtained from Taylor expansions. Firstly we consider two-order differential equation as follow
′
y ” = p(x)y + q(x)y + r(x). (4)
In interval [a, b], Eq (3) subject to the the following boundary conditions y (a) = α, y(b) = β (5)
Rangkuti et al., An Approximate Solution ...
For all of cases, we divided all of range into the point in even sum x =
b−a
a, x
N = b, x i = x + ih, h = . For any function y = x, with x ∈ [a, b], it N i
can be defined the value of point y i = y(x ). If y(x) is adequately smooth,
i
we can also define the approximation into the derivative of y(x ) for every
i ′
point x . Now we put the derivative y (x) and y”(x) with the centered difference approximation of Taylor theorem.
1
′
2
− y h (x) = (y(x i + h) − y(x i )) + O(h )
2h
1
2
= (y(x i ) − y(x )) + O(h ) (6)
- 1 i−1
2h
1
′
2
(x) = (y(x i + h) − 2y(x i ) + y(x i )) + O(h )
− y h
2
2h
1
2
= (y(x i ) − y(x i ) − y(x )) + O(h ) (7)
- 1 i−1
2
2h For i = 1, 2, · · · , N − 1. Now, we have
1
1 (y(x i ) − y(x i ) − y(x )) = p(x) (y(x i ) − y(x )
- 1 i−1 +1 i−1
2
2h 2h
- q(x)y(x i ) + r x (8) For i = 1, 2, · · · , N − 1. Because the value of p(x i ), q(x i ) and r(x i ) are some unknown functions representing linear algebra equations involving y (x i ), y(x i ), y(x ). We reclaim y(a) = y = α and y(b) = y N = &be
- 1 i−1 hp i hp i (x ) (x )
2
− Rewritten (5) into − 1 + y + (2 + h q (x i ))y i 1 − y i =
i−1 +1
2
2
2
h r (x ). The value of y for i = 1, 2, · · · , N − 1 can be found by solving this
i k form Ay = B.
3. HOMOTOPY PERTURBATION METHOD Firstly, a differential equation system which satisfies the following re- lation can be formed as:
Du , u , . . . , n , u , . . . , u
i (t) = L i (t, u ) + N i (t, u n ) + g i (t), (9)
1
2
1
2
where L i is a linear operator, N i is a non-linear operator, and g i is a known analytical function. By using homotopy perturbation technical, we construct the homotopy form [4] :
Du , u , . . . , n , u , . . . , u
i (t) = p[L i (t, u )+N i (t, u n )+g i (t)], 1 ≤ i ≤ n, (10)
Rangkuti et al., An Approximate Solution ...
where p is a embedded parameter which changes from zero to the unity. If p , v , . . . , v
= 0, (10) becomes a linear equation and v n are an initial approx-
1
2
imation which subjected to the condition that given to (10). Obviously, the embedded parameter is p = 0. The equation (10) becomes a linear equation system and we obtain a non-linear equation system when p = 1. Assuming the initial approximation such that: u
(t) = v (t) = u (t ) = c
1,0
1
1
1
u (t) = v (t) = u (t ) = c
2,0
2
2
2 ..
. u m, (t) = v m (t) = u m (t ) = c m (11) and
2
3 u (t) = u (t) + pu (t) + p u (t) + p u (t) + . . . 1 1,0 1,1 1,2 1,3
2
3
u u u (t) = u (t) + pu (t) + p (t) + p (t) + . . .
2 2,0 2,1 2,2 2,3 ..
.
2
3
u u u
m (t) = u m, (t) + pu m, 1 (t) + p m, 2 (t) + p m, 3 (t) + . . . (12)
where u i,j , (i = 1, 2, . . . , m; j = 1, 2, . . .) is a function that will be deter- mined. Substituting (??) into (10) and the forming of coefficient from rank p , obtained
L , u , . . . , u (u 1,1 ) + L(v
1 ) + N
1 (u 1,0 2,0 m, ) − f
1 = 0, u 1,1 (t ) = 0,L (u ) + L(v ) + N (u , u , . . . , u m, ) − f = 0, u (t ) = 0,
2,1
2 2 1,0 2,0 2 1,1 ..
. L , u , . . . , u
(u m, ) + L(v m ) + N m (u m, ) − f m = 0, u m, (t ) = 0,
1 1,0 2,0
1 L , u , . . . , u
(u 1,1 ) + N
1 (u 1,0 2,0 m, ) − f = 0, u 1,2 (t ) = 0,
L , u , . . . , u (u ) + N (u m, ) − f = 0, u (t ) = 0,
2,1 2 1,0 2,0 2,2 ..
. L , u , . . . , u
(u m, ) + N m (u m, ) − f m = 0, u m, (t ) = 0,(13)
1 1,0 2,0
2 etc.
The form above is solved for u , (i = 1, 2, . . . , m; j = 1, 2, . . .), an
i,j
unknown unsure using inverse operator
Rangkuti et al., An Approximate Solution ...
u
2
(t) =
n =1 X k =0
u
2,k
(t) .. . φ
m 1,n (t) = u m (t) = lim p→1
m (t) = n =1 X k
p→1
=0
u
m,k (t) (14)
4. RESULTS The problem of temperature and the velocity of combined and free convection flow have been solved by using MAPLE. The results are repre- sented in 2 and 3. Figure 2 shows the approximation of temperature by using FDM and the second term of HPM for Frank-Kamenetskii number (KF ), KF = 1.5 and G
R
is a combined convection parameter, while figure 3 shows the velocity from 0 to 1. From 2 and 3, FDM approach the exact solution more than HPM.
Figure 2: Temperature FOR KF = 1.5 and G
R
u
(t) = lim
L
p→1
−1
(•) = Z t (•)dt Hence, according to HPM , the approximation of n-term for the solu- tion (16) can be expressed as
φ
1,n
(t) = u
1
(t) = lim
u
2
1
(t) =
n =1 X k =0
u
1,k
(t) φ
2,n
(t) = u
= 10 Rangkuti et al., An Approximate Solution ...
Figure 3: Velocity for KF = 1.5 and G R = 10
5. CONCLUSION The zones of parameter for the reversal fluid occurrences by fully de- veloped mixed convection in a vertical channel and the chemical reaction are presented. The solution by numerical for this problem is also discussed. The method of FDM and HPM have been compared to reveal the accuracy of method by using MAPLE. We can conclude that FDM method is more accurate than HPM method. This method can be an alternative method to solve this problem specifically and more complicated boundary value prob- lem.
REFERENCES
1. Tulus, O. S. Sitompul, S. Nasution, Mardiningsih, Mathematical Model-
ing of Fluid Dynamic in the Continuous Stirred Tank, Bulletin of Math- ematics, 07 (02), 1-12, 2015.
2. I. Pop, T. Grosan, and R. Cornelia, Effect of heat generated by an exother-
mic reaction on the fully developed mixed convection flow in a vertical channel, Commun nonlinear Sci. Numer. Simulat, 15 (3), 474-474, 2010.
3. H. Saleh, I. Hashim, dan S. Basriati, Flow reversal of fully developed
mixed convection flow in a vertical channel with chemical reaction, Int. J chemical engge. 2013ID 310273
4. J. H. He, Homotopy perturbation Technique, Computers meth. Appl.
Mech. 178, 257-262 1999
Rangkuti et al., An Approximate Solution ...
Second Edition, Springer, 1999
Yulita Molliq Rangkuti : Mathematics Department, Faculty Mathematics and Natural Science, Universitas Negeri Medan, Medan, Sumatera Utara.
E-mail: yulitamolliq@yahoo.com
Wulandari : Mathematics Department, Faculty Mathematics and Natural Science, Universitas Negeri Medan, Medan, Sumatera Utara.
E-mail: wulandari 0910@yahoo.com
Penny Charity Lumban Raja : Mathematics Department, Faculty Mathematics and Natural Science, Universitas Negeri Medan, Medan, Sumatera Utara.
E-mail: pennycharitylumbanraja@gmail.com