Ab and cav nu wa ph use usi (SU equ str tra Nu par vol por app oce eng con tra foo sum Inc of and hor gra has con con kn me sub con Engineerin www.etasr Num M La Univer ab bstract—In this d mass transfe vity filled w merically. The alls while the ysical model f e of the Darcy ing a finite vol UR) method i uation. The r reamlines, isoth ansfer rate in usselt and She rameters. Keywords-nat lumes; inclined Double-diffu rous medium plications in eanography, c gineering app ntained in fib ansport in pac od processing, mmary of the cluding differe solutions, mo d mass grad rizontally alo adients impose s been given t nvection wh ncentration ar owledge, no edium where t bject of this w I Consider s nvection in an ng, Technology r.com merica Mass T A. La aboratory of E Faculty of E rsity Frères M Constantin bdelkrim.latre s study, two dim fer generated i with Newtonia e cavity is he solutal gradie for the momen model, and the lume approach is used in the results are pr herms and iso- the cavity is m erwood numb tural convectio d cavity; porous I. IN usive natural m is of signifi n different chemical proce plications such brous insulati cked-bed reac , underground e work done i ent kinds of b st studies are dients: the fi ong the enclo ed vertically [ to the phenom here cross g re imposed [ work is cited this kind of g work. II. MATHEM steady two- n inclined recta y & Applied Sci l Stud Transfe atreche Energetic Phys xact Sciences entouri Consta ne, Algeria che@gmail.co mensional natu in an inclined an fluid has ated and cool ent is impose ntum conservat e set of coupled h. The successi e solution of t resented graph concentrations measured in te bers for vario on; heat and m s media TRODUCTION convection in icant interest fields such esses etc.. It i h as the mig ion, drying p ctors, grain st d disposal of nu in the past is boundary cond concerned wit irst kind is osure [9–14] [15–18]. Relat menon of doub gradients of 19–21] and, d in an incline gradient is imp MATICAL MODE -dimensional angular porou ience Research Latreche & dy of N fer in a sics antine 1 om ural convection rectangular p been invest led along hori d horizontally tion equation m d equations is ive-under-rela the stream fu hically in ter s. The heat and erms of the av ous non-dimen mass transfer; n a fluid-satu due to its v as astroph is also met in gration of mo processes, che torage installa uclear waste e presented in ditions and me th two kinds o gradients im and the seco tively less atte ble-diffusive n temperature to the best o ed saturated p posed, which ELING laminar n us cavity of len h V Djezzar: Nume Natura an Incl n heat porous tigated izontal y. The makes solved xation nction ms of d mass verage nsional finite urated arious hysics, many oisture emical ations, etc.. A [1-8]. ethods of heat mposed ond is ention natural and of our porous is the natural ngth L and imp tem (Th> subj Sh a dep lam is a phy con neg bec den con for rect Vol. 8, No. 4, 20 erical Study of N al Conv lined P L Univ d height H. permeable to th mperatures Th >Tl). The vert bjected to fixed and low conc picted in Fig minar flow are assumed isotro ysical propert ncentration gr glected. The f cause of temp nsity is taken ncentration lev 0 1 T    Fig Under the abo conservation tangular coord 2 2 2 2 x y        018, 3223-3227 Natural Convec vectiv Porou M. D Laboratory of Faculty of ersity Frères M Constan mdjezzar@ The horizon he transport of and Tl at bot tical walls are d concentratio centration Sl a gure 1. Hypo considered, a opic and hom ties. Interacti radients, (So flow is driven perature and c n as a funct vels through th  T l ST T   g. 1. Physical ove assumptio of momentum dinate systems * (cos sin ( Ra T y       7 ctive Heat and ve Hea us Med Djezzar f Energetic Phy Exact Science Mentouri Cons ntine, Algeria @umc.edu.dz ntal walls o f solute and th ttom and top thermally insu ons, high conce at the right wa otheses of in and the saturat mogeneous wit ion between oret and Du n by combined concentration tion of both he Boussinesq  S lS S  l model and geom ons, the dimen m, energy and s are given by ( s ( )) T S N x x S N x          3223 Mass Transfer at and dia ysics es stantine 1 z of the cavity hey are subject walls, respect ulated and the entration at th all of the cavi ncompressible ted porous me th constant th the thermal ufour effects) d buoyancy f variations. So h temperature approximation (1) metry nsionless equa species transpo (2)-(4): ) S x (2) r in … y are ted to tively ey are he left ity as and edium hermo and are forces o, the and ns: ations ort in Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3223-3227 3224 www.etasr.com Latreche & Djezzar: Numerical Study of Natural Convective Heat and Mass Transfer in … 2 2 . . 2 2 T T T T y x x y x y                          (3) 2 21 . . 2 2 S S S S y x x y Le x y                          (4) where the dimensionless variables are: T Ty ψx , , , T T S S S S lx y T H H a h l lS h l           (5) The dimensionless boundary conditions are: 0 : 1, 0 0 S x and y T and y         (6) 1: 0, 0 0 S x and y T and y         (7) 0 : 1, 0 0 T y and x S and x         (8) : 0, 0 0 T y and x A S and x         (9) The present problem is governed by 5 parameters: the buoyancy ratio N, the Lewis number Le, Darcy-modified Rayleigh number Ra*, the inclination angle and the aspect ratio A. The average values of Nusselt number evaluated on the bottom wall and Sherwood number evaluated on the left side wall are given by: 11 , 0 00 0 A T S Nu dx Sh dy A y xy x          (10) III. NUMERICAL SOLUTION The volume finite method [22] is employed to solve numerically the governing equations together with the boundary conditions. The computation domain is divided into rectangular control volumes with one grid located at the center of the control volume that forms a basic cell. The set of conservation equations are integrated over the control volumes, leading to a balance equation for the fluxes at the interface. The iterative process, employed to find the stream function, temperature and concentration fields, was repeated until the following convergence criterion was satisfied:  , , 6 , 10 new old i j i j i j new i j i j        (11) where Φ stands for Ψ, T and S. The subscripts i and j denote grid locations in the (x, y) plane. A further decrease of the convergence criteria, 10−6, does not cause any significant change in the final results. Numerical tests, using various mesh sizes, were done for the same conditions in order to determine the best compromise between accuracy of the results and computer time. A mesh size of 121×61 was adopted. The accuracy of the code was checked, modifying the thermal and solutal boundary conditions, to reproduce the results reported in 20. Good agreement can be seen from Table I with a maximum deviation of about 3.4%. TABLE I. VALIDATION OF THE NUMERICAL CODE Le max Nu Sh Present work [20] Present work [20] Present work [20] 0.1 11.625 11.706 4.484 4.633 1.209 1.221 1 9.505 9.609 4.130 4.276 4.840 5.086 10 9.104 9.171 3.983 4.078 15.870 17.02 =0, Ra*=200, n=0.3 and various le, in terms of  Max, nu and sh IV. RESULTS AND DISCUSSION A. Considered Situations In present work, the results are displayed in the form of stream, iso-thermal and iso-concentration lines to investigate the effect of inclination angle (0°≤α≤90°), buoyancy ratio (−5≤N≤5) and Lewis number (0.1≤Le≤10) while Darcy- modified Rayleigh number, aspect ratio and Prandtl number are taken as 200, 2 and 0.71 respectively. The rate of heat and mass transfer at different conditions in the cavity is measured in terms of the average Nusselt and Sherwood numbers. B. Flow Structure, Temperature and Concentration Fields, and Heat and Mass Transfer Visualization. To show the effects of the inclination angle α, isotherms, stream functions and iso-concentration lines are presented in Figure 2 for Le=10 and N=0.1. For no inclination angle (α=0°) four regions of important concentration gradients can be observed. Two of these regions are located in the vicinity of the upper part of the right wall and the lower part of the left wall where the cells exchange important quantity of solute with these boundaries. The two other regions are located in the vicinity of the upper and the lower central parts of the enclosure, in the interface between the cells where the solute exchange is by diffusion. For 0°<α<15° the previous regions exchange just their positions (the upper becomes lower and vice versa). The vortex strength of the fluid in the porous medium is lower when α15° and it increases when the angle of inclination decreases. When α>15° the secondary circulations disappear and the vortex strength of the fluid in the porous medium increases with the increase of the angle of inclination. The temperature field obtained shows the presence of thermal boundary layers near the upper and lower walls and parallel to the two other walls stratification of the temperature in the core region. The two regions of important concentration gradients located in the vicinity of the upper and the lower central parts of the enclosure disappeared. Fig Ra* str and the bif eng in the lay inc Engineerin www.etasr g. 2. Isotherm *=200, Pr=0.71, A Figure 3 pres ream functions d for the oppo e strength of furcates into tw genders a decr favor of the ere is a tende yers, this beha creases. ng, Technology r.com s, iso-concentrat A=2, Le=10 and N sents the isoth s with differen osed flow (N= the flow circ wo weak analo rease of the si upper one. T ency to the de avior decrease y & Applied Sci (a) (b) (c) (d) tionand and stre N=0.1 with differe herms, iso-con nt angles of inc =−2). It is seen culation decre ogical circulati ize and intensi The temperatu evelopment of es when the a ience Research Latreche & eam function lin ent inclination an ncentration line clination α for n that as α incr eases and the ions. Exceedin ity of the lowe ure field show f thermal bou angle of inclin h V Djezzar: Nume nes for ngles. es and r Le=5 reases e flow ng 45° er cell ws that undary nation Fig. Ra* Vol. 8, No. 4, 20 erical Study of N 3. Isotherms =200, Pr=0.71, A 018, 3223-3227 Natural Convec ( ( ( ( ( , iso-concentrat A=2, Le=5 and N= 7 ctive Heat and (a) (b) (c) (d) (e) ion and stream =-2 with different 3225 Mass Transfer m function line inclination angle r in … es for s. (α= ver the gra 15 upp wh the wh enc exc C. Nu Ra nu Nu Fu wh sig Sh wh tha Sh Fig A=2 Nu α= sig N Sh sam and N wh Engineerin www.etasr The concent =0°), shows th rtical walls an e core region adients can be °. Two of the per part of the here the cells ese boundarie hich links the closure (the in change is by d Heat and Ma To show th usselt and She a*=200, Pr=0. mber. In gen usselt number urthermore, wh hen the angle gnificant chang herwood numb hen the angle at the maximum happens when g. 4. Average 2, N=2 with diffe Figure 5 sho usselt and She =45° for differe gnificant chang increases un herwood numb me behavior f d 1, where the increases. Th hen N increas ng, Technology r.com tration field, o he presence of nd vertical stra n. Three regi e observed wh ese regions ar e left wall and exchange im s. The third r two previous nterface betwe diffusion. ass Transfer P he effects of erwood numbe .71, A=2, N=2 neral, when th decreases and hen (α<30°) of inclination ge on the Nu ber decreases a of inclination m Nu happens n (α20°). Nusselt and Sh rent Lewis numbe ows the effect erwood Numb ent Lewis num ge on the Nuss ntil the value bers increase w for (Le=1 and e Nusselt and S he Nusselt an ses only when y & Applied Sci obtained for n f solutal bound atification of t ions of impo hen the inclina re located in d the lower pa mportant quan region is loca regions at the en the two cel Parameters f the inclinati ers are presen 2 and differen he Lewis num d the Sherwood the Sherwood n increases, w usselt number. and the Nussel increases. Pre s when (α=90° erwood number er and inclination of buoyancy ber for Ra*=2 mbers. For (Le selt and Sherw e N=1 where when N increa 5) until N re Sherwood num nd Sherwood n 1 1N   . ience Research Latreche & no inclination dary layers ne the concentrat ortant concent ation angle ex the vicinity o art of the righ ntity of solute ated in the dis central parts lls) where the ion angle, av nted in Figure nt values of L mber increase d number incr d number incr while there is When (α>30 lt number incr evious results °) and the max for Ra*=200, P n angle values . ratio N on av 200, Pr=0.71, e=0.1), there is wood numbers e the Nussel ases. We can s eaches the valu mbers increase numbers dec The maximu h V Djezzar: Nume angle ear the tion in tration xceeds of the ht wall e with stance of the solute verage 4 for Lewis es the reases. reases not a 0°) the reases, show ximum Pr=0.71, verage A=2, s not a when lt and ee the ues -3 when crease um of both (N= an hor gen equ this (0°≤ (−5 rati resp dec num the max resp N=5 Fig. A=2 Vol. 8, No. 4, 20 erical Study of N h Nu and Sh h =5). Numerical ste inclined por rizontal solut neralized mod uations. A num s paper for dif ≤α≤90°), Lew 5≤N≤5), while o and Prandtl pectively. Res creases and the mber increases inclination a ximum Nu a pectively, and 5. 5. Average N 2, α=45° with diffe A as a th D so Da D g gr H ca k th K pe L en Le L N bu Nu N P pr P di Pr Pr Ra R Ra* D S di 018, 3223-3227 Natural Convec happens when V. CO eady of double rous media w tal gradients del has been mber of relevan fferent values o wis number (0. e Darcy-mod l number have sults show th e Sherwood nu s, although, th angle and the and Sh happe d both of them Nusselt and She ferent Lewis numb NOMEN spect ratio, [=L/H hermal diffusivity olutal diffusivity, Darcy number, [=K ravitational accele avity height, m hermal conductivi ermeability of po nclosure length, m ewis number, [=a uoyancy ratio, [= Nusselt number ressure, kg.m-1s-2 imensionless pres randtl number, [= Rayleigh number, Darcy-modified Ra imensional conce 7 ctive Heat and n N reaches th ONCLUSIONS e-diffusive na with vertical has been used to sol nt results have of inclination 1≤Le≤5), and dified Rayleig e been taken a hat the averag umber increas hey change th e buoyancy r en when (α= m reach max erwood number f ber and buoyancy NCLATURE H] y, m2.s-1 m2.s-1 K/H2] eration, m.s-2 ity of porous layer orous layer, m2 m a/D] SSh-Sl)/TTh-Tl ssure, [=ε2H2P/0a =/a] [=gβT(Th-Tl)H 3/ ayleigh number, [ ntration, kg.m-3 3226 Mass Transfer he maximum v atural convecti temperature investigated. lve the gove e been present angle of the c the buoyancy gh number, a as 200, 2 and ge Nusselt nu ses when the L heir behavior ratio. Furtherm =90°) and (α ximum value w for Ra*=200, Pr y ratio values. r, W.m-1.K-1 l)] a2] a] [=Ra.Da] r in … value ion in and The rning ted in cavity ratio aspect d 0.71 umber Lewis with more, 20°) when r=0.71, Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3223-3227 3227 www.etasr.com Latreche & Djezzar: Numerical Study of Natural Convective Heat and Mass Transfer in … S dimensionless concentration Sh average Sherwood number T dimensional temperature, K T dimensionless temperature, [= (T-Tl) /(Th-Tl)] x,y coordinates system, m x, y dimensionless coordinates system, [=x(y) /H]  inclination angle, ° S solutal expansion coefficient, K -1  thermal expansion coefficient, m 3.kg-1  dimensional stream function, m2.s-1  dimensionless stream function  density of fluid, kg.m-3 h higher l lower max maximum value REFERENCES [1] R. W. Schmitt, “Double diffusion in oceanography”, Annual Review of Fluid Mechanics, Vol. 26, pp. 255–285, 1994 [2] J. S. 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