The Iteratioal Joural Of Egieerig d Sciece (IJES) Volume 4 Issue 10 Pages PP -35-41 2015 ISSN (e): 2319 1813 ISSN (p): 2319 1805 umerical Techique Fiite Volume Method for Solvig Diffusio 2D Problem 1 Mohammed Hasat, 2 Nourddie Kaid, 3 Mohammed Besafi, 4 bdellah Belacem 1,2,3,4 Laboratory of eergy i arid areas (ENERGRID), Faculty of Sciece ad Techology, Uiversity of BEHR, BP 417, 08000 BEHR --------------------------------------------------------BSTRT----------------------------------------------------------- I this paper, a umerical fiite volume techique was used to solve trasiet partial differetial equatios for heat trasfer i two dimesios with the boudary coditio of mixed Dirichlet (costat, ot costat) i a rectagular field. We explaied the procedures step by step, for the digital solutio we used our Fortra code ad a lie by lie TDM solver for algebraic equatios. Fially, the umerical results are compared with the exact solutio. Keywords oductio, Dirichlet boudary coditio, fiite volume method, heat trasfer, TDM. ------------------------------------------------------------------------------------------------------------------------------------------- Date of Submissio: 10 September 2015 Date of ccepted: 30 October 2015 ------------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUTION The partial derivative equatios are very ofte used i sciece ad egieerig. May partial differetial equatios ca t be solved aalytically i a closed form solutio. There are several ad differet techiques for the costructio of umerical methods of the PDE solutio [1]. s the mathematical modelig has become a itegral part of the aalysis of egieerig problems, a variety of techiques for digital etwor have bee developed [2]. The most popular umerical techiques are the fiite differece (FD), fiite elemets (FE) ad the fiite volume method (FVM) which was origially developed as a special formulatio of fiite differece. Each of these methods has its ow advatages ad disadvatages depedig o the problem to solve. The techique of fiite volume is oe of the most versatile ad flexible techique for solvig problems i fluid dyamics [2]. The preset paper deals with the descriptio of the method of fiite volume (FVM) to solve differetial equatios. The compariso is made betwee the aalytical solutio (S), ad the solutio obtaied by implemetig the fiite volume method. This paper is orgaized as follows: Sectio 2: cotais the descriptio of the fiite volume method (FVM), with the help of TDM solver (Tri-diagoal matrix algorithm). Sectio 3: cotais diffusio problem, the aalytical solutio (S) ad the fiite volume method (FVM). Sectio 4: i this sectio the umerical solutios obtaied by this techique are compared with the exact solutio. Sectio 5: cotais the coclusio of this paper. II. FINITE VOLUME METHOD The fiite volume method represets ad evaluates the partial differetial equatios i the form of algebraic equatios [3]. Values are calculated at a discrete odes of the cotrol volume for geometry. "Fiite volume" refers to the small volume surroudig each poit o a mesh odes. I the fiite volume method, volume itegrals of a partial differetial equatio that cotais the divergece term are coverted to the full surface, usig the divergece theorem. These terms are the evaluated as a flow at the surfaces of each fiite volume. Because the iflow ito a give volume is idetical to the outflow at the adacet volume, these methods are coservative. The method is used i may fluid dyamics calculatio software. It is always preferred to use the goverig equatio uder its coservative form i the fiite volume approach for solvig ay problems that esures the preservatio of all properties i each cell / volume cotrol. Now we will discuss the steps ivolvig FVM to solve the differetial equatio [3], ad for more details we ca cosult [4]. www.theies.com The IJES Page 36
umerical Techique Fiite Volume Method for 2. 1 Grid geeratio The first step of the fiite volume method is the geeratio of the grid by dividig the area ito small discrete cotrol volumes. The limits of the cotrol volumes are positioed halfway betwee adacet odes. Thus, each ode is surrouded by a cotrol volume or cell [5]. It is commo practice to set up the cotrol volumes close to the edge of the field so that the physical boudaries coicide with boudaries betwee cotrol volumes. geeral odal poit P is defied by its eighbors, i a two-dimesioal geometry, odes o the west ad east, orth ad south are defied as follow; W, E, N ad S, respectively. The four faces of the volume cotrol are defied at their sides by w, e,, ad s, respectively. The distaces of these poits are give i Fig 1. Fig 1- part of the two-dimesioal grid 2. 2 Discretizatio The most importat characteristics of the method of fiite volume are the itegratio of the goverig equatio o a volume cotrol to obtai a discretized equatio to its odal poits P. a T + p p a T S (1) p p u where Σ idicates summatio over all eighborig odes (p), are the coefficiets of the eighbor odes,, is the value of the property T at the eighborig odes; ad is the liearized source term. I all cases, the coefficiets surroudig the poit P satisfy the followig relatioship: a - p a S (2) p p 2. 3 Solutio fter discretizatio o each volume cotrol, we fid a system of algebraic equatio. Which fills a sparse matrix that ca be easily solved TDM algorithm (Tri-Diagoal Matrix lgorithm). The algorithm of tridiagoal matrix (TDM), also ow uder the ame of Thomas algorithm, is a simplified form of Gaussia elimiatio which ca be used to solve the system of equatios with three diagoal. TDM is based o Gaussia elimiatio procedure ad cosist of two parts - a forward elimiatio phase ad a bacward substitutio phase. TDM is i fact a direct method, but ca be applied iteratively i a lie by lie fashio, to solve multidimesioal problems ad is widely used i FD programs as show i [4].osider the equatio system 1 for 1, - - -, ad we use equatio (1) the geeral form of the TDM solver which is give by: a T + a T a T a T + a T + S (3) S S p p N N W W N N u To solve the above TDM system alog North-South lies, the discrete equatio (3) is reorgaized i the form: T + D T T (4) Where 1 1 1 + (5) 1 T T www.theies.com The IJES Page 37
umerical Techique Fiite Volume Method for D 1 D + 1 1 III. PROBLEM FORMULTION We cosider solvig two-dimesioal steady heat coductio problems i rectagular plate made of uiform material thus the thicess of the plate is about D 0.01 m ad the thermal coductivity of material is 1W/m0. The boudary coditios are as follow Fig. 2: East boudary: fixed temperature T0 100. West boudary: fixed temperature T0 100. South boudary: fixed temperature T0 100. North boudary: variable temperature T f(x). Fig 2 - Solutio regio with mixed boudary coditio The mathematical formulatio of this problem is give by T T + 0 X X Y Y The exact temperature field is (6) X Y T ( X, Y ) T ( X, Y ) + s i ( ) s i h ( ) 0 (7) b b Where is give by: X b 1 b /2 X ( ) (, ) si ( ) L b si b f X T 0 X Y d X (8) X 0 www.theies.com The IJES Page 38
For > 1, the reader may cosult [6]; the fial solutio of equatio (6) is therefore umerical Techique Fiite Volume Method for s i h ( y / b ) x T ( x, y ) T s i ( ) T ( x, y ) (9) m o s i h ( L / b ) b The error of temperature field is evaluated by: (10) E rror x, y B S T T N um E xact The grid size, x 0.1 m, y 0.2 m THE FINITE VOLUME DISRETIZTION: Equatios the geeral form of discretized for problems is give by equatio (3). a T a T + a T + a T + a T + S (11) p P W W E E S S N N u d we use the coefficiets surroudig the poit P, equatio (2). a a + a + a + a - S (12) p W E S N p The area D y, D x w Where a W w x t iterior poits 7, 8 ad 9 Where e S 0, S 0 u p a E x t a boudary ode the discretized equatio taes the form For Fixed value T S u e 2 s a S T IV. RESULTS ND DISUSSION ll umerical calculatios were carried out with volume cotrol method ad aalytical solutio usig computig FORTRN code. The coefficiets ad the source term of the discretizatio equatio for all odes are summarized i Table 1. y ad Node a N a S a W a E a P S u 1 0.005 0 0 0.02 0.075 5 2 0.005 0.005 0 0.02 0.07 4 3 0.005 0.005 0 0.02 0.07 4 4 0.005 0.005 0 0.02 0.07 4 5 0.0 0.005 0 0.02 0.075 5.099 6 0.005 0. 0.02 0.02 0.055 1 7 0.005 0.005 0.02 0.02 0.05 0. 8 0.005 0.005 0.02 0.02 0.05 0. 9 0.005 0.005 0.02 0.02 0.05 0. 10 0.0 0.005 0.02 0.02 0.055 1.2 11 0.005 0. 0.02 0. 0.075 5 12 0.005 0.005 0.02 0. 0.07 4 13 0.005 0.005 0.02 0. 0.07 4 14 0.005 0.005 0.02 0. 0.07 4 15 0. 0.005 0.02 0. 0.075 5.099 Table 1- The coefficiets ad source term for all odes s S p a N y 2 www.theies.com The IJES Page 39
umerical Techique Fiite Volume Method for The umerical solutio of the discretized equatios system is calculated usig TDM as show i Table 2. Node D 1 0 0.075 0.005 5. 0,07 66,67 2 0.005 0.07 0.005 4. 0,07 62,20 3 0.005 0.07 0.005 4. 0,07 61,90 4 0.005 0.07 0.005 4. 0,07 61,88 5 0.005 0.075 0.0 5.099 0,00 72,46 6 0. 0.055 0.005 2.4226 0,09 44,05 7 0.005 0.05 0.005 1.3397 0,10 31,48 8 0.005 0.05 0.005 1.3343 0,10 30,14 9 0.005 0.05 0.005 1.3417 0,10 30,15 10 0.005 0.055 0.0 2.6494 0,00 51,38 11 0. 0.075 0.005 5.944 0,07 79,25 12 0.005 0.07 0.005 4.6977 0,07 73,12 13 0.005 0.07 0.005 4.6742 0,07 72,37 14 0.005 0.07 0.005 4.7068 0,07 72,78 15 0.005 0.075 0. 6.1276 0,00 86,97 Table 2- The Numerical coefficiets TDM after first iteratio Fig 3- The Numerical solutio after first iteratio Fially, the solutios obtaied by umerical fiite-volume techiques, were compared with a exact solutio, to verify the accuracy of the results o calculatig errors, as show i Table 3. Node FVM Exact Error 1 100.002 100.000 0,002 2 100.014 100.006 0,008 3 100.086 100.053 0,033 4 100.502 100.432 0,07 5 102.928 103.509 0,581 6 100.004 100.001 0,003 7 100.029 100.013 0,016 8 100.172 100.106 0,066 9 101.005 100.864 0,141 10 105.857 107.018 1,161 11 100.002 100.000 0,002 12 100.014 100.006 0,008 13 100.086 100.053 0,033 14 100.502 100.432 0,07 15 102.928 103.506 0,578 Table 3- ompariso betwee Numerical ad Exact Solutios www.theies.com The IJES Page 40
umerical Techique Fiite Volume Method for Fig 4- comparisos betwee Fiite Volume Numerical Solutio with Exact Solutio V. ONLUSION I this paper fiite volume umerical grid techique for steady state heat flow problems was studied ad umerical solutio of the two dimesioal heat flow equatio with Dirichlet boudary coditios was obtaied. We have used TDM solver for solvig algebraic equatios ad the results obtaied by this techique are all i good agreemet with the exact solutios uder study. Furthermore, this techique is effective, reliable, precise ad easier to implemet i programmig i FORTRN, compared to other costly techiques. REFERENES [1] P. V Patil ad J. S. V. R. K. Prasad, umerical grid ad grid less ( Mesh less ) techiques for the solutio of 2D Laplace equatio, dvaces i pplied Sciece Research, Pelagia Research Library, 5(1), 2014, 150-155. [2] J.S.V.R. Krisha Prasad ad Patil Parag Viay, Fiite Volume Numerical Grid Techique for Solvig Oe ad Two Dimesioal Heat Flow Problems, Res. J. Mathematical ad Statistical Sci, 2(8), 2014, 4-9. [3] J. D. J. derso, omputatioal Fluid Dyamics: The Basics with pplicatios (6th ed. McGraw-Hill Publicatios, 2008). [4] H. K. V. ad W. Malalaseera, Itroductio to omputatioal Fluid Dyamics(Secod Ed, Bell & Bai Limited, Glasgow, 2007) [5]. Shula,. K. Sigh, ad P. Sigh, omparative Study of Fiite Volume Method ad Fiite Differece Method for ovectio-diffusio Problem, m. J. omput. ppl. Math., 1(2), 2012, 67-73 [6] S. Whitaer, Fudametal priciples of heat trasfer, vol. 28, o. 1, 1985. www.theies.com The IJES Page 41