Natural Convection with Localized Heating and Cooling on Opposite Vertical Walls in an Enclosure

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1 The SIJ Transactons on Computer Networks & Communcaton Engneerng (CNCE), Vol. 1, No. 4, September-October 013 Natural Convecton wth Localzed Heatng and Coolng on Opposte Vertcal Walls n an Enclosure Marura O. Edward*, Johana K. Sgey**, Jeconah A. Okello*** & James M. Okwoyo**** *Department of Pure and Appled Mathematcs, Jomo Kenyatta Unversty of Agrculture and Technology, Narob, KENYA. E-Mal: edoma03@gmal.com **Department of Pure and Appled Mathematcs, Jomo Kenyatta Unversty of Agrculture and Technology, Narob, KENYA. E-Mal: ksgey@kuat.ac.ke ***Department of Pure and Appled Mathematcs, Jomo Kenyatta Unversty of Agrculture and Technology, Narob, KENYA. E-Mal: okelo@kuat.ac.ke ****School of Mathematcs, Unversty of Narob, Narob, KENYA. E-Mal: mkwoyo@uonb.ac.ke Abstract Ths s a numercal study of turbulent natural convecton flow n a rectangular enclosure. The flow of heat s one form of Newtonan moton. We consder natural convecton n a three dmensonal rectangular enclosure n the form of a room wth heaters placed on opposte walls and two wndows each on the adacent opposte walls. The study of free convecton n the past fve decades focused manly on two dfferent smple geometres, frst the sngle sothermal or constant flux vertcal plate n sothermal stagnant surroundng, secondly, the enclosed rectangular cavty wth heated floor and cooled walls. There has also been much emphass on Reylegh number as opposed to the Reynolds number used n ths study. To analyze the flow and heat transfer rates, a complete set of non-dmensonalzed equatons governng Newtonan flud and boundary condtons are recast nto vector potental to elmnate the need for solvng the contnuty equatons. A Boussnesq flud moton n a three dmensonal cavty s consdered. The governng equatons wth the boundary condtons are descrtzed usng three pont central dfference approxmatons for a non-unform mesh. The resultng fnte dfference equatons are solved usng Matlab smulaton software. The solutons are presented at the Reynolds number 5500, wth Prandtl number The results are presented on graphs to show velocty profles and temperature dstrbuton n the room. The room s dvded nto a number of regons wth those near the heaters havng hgh temperatures as those near the wndows have low temperatures. Convectve currents caused by buoyancy forces play a maor role n determnng the velocty profles n the room. Keywords Heat Convecton n an Enclosure; Heat Transfer; Natural Convecton n a Rectangular Enclosure. I. INTRODUCTION Aflud s a substance whose consttuent partcles contnuously change ther postons relatve to one another when subected to energy change or force. In ths concept a flud s perceved to flow. Moton n fluds could be caused by dfferent forces, one such force s heat. Natural convecton s caused by local buoyancy dfferences brought about by the presence of hot and cold body surfaces. Natural convecton occurs when the ar n a space s changed wth outdoor ar wthout the use of mechancal systems, such as a fan. Most often natural convecton s assured through operable wndows but t can also be acheved through temperature and pressure dfferences between spaces. The flow pattern n fluds could also be turbulent or lamnar. Lamnar flow s where we have a steady flow whle turbulent flow s rregular where varous quanttes show random varaton wth tme and space co-ordnates. In an enclosure, convecton could be shown by movement of ar due to buoyancy up and around the room subect to dfferental heatng encountered n many practcally mportant engneerng problems. They nclude thermal nsulaton of buldngs, heat transfer through open wndows, refrgeraton, etc. The study of free convecton n the past fve decades focused manly on two dfferent smple geometres, frst the sngle sothermal or constant flux vertcal plate n sothermal stagnant surroundng, secondly, the enclosed rectangular cavty wth heated floor and cooled walls [Mynet & Duxbury, 1974; Olson & Glckman, 1991]. The motvaton of ths study s to nvestgate heat dstrbuton and velocty profles brought about by heatng two opposte vertcal walls and coolng the other two vertcal walls n an enclosure. A turbulent flow s consdered and ISSN: Publshed by The Standard Internatonal Journals (The SIJ) 7

2 The SIJ Transactons on Computer Networks & Communcaton Engneerng (CNCE), Vol. 1, No. 4, September-October 013 hence the use of hgh Reynolds number. Due to the ellptc nature of nternal buoyant flows, natural convecton flow requres a complete soluton of the Naver-Stokes equaton. Most of the flud dynamcs problems nvolvng natural convecton could be solved by expermental, theoretcal and numercal methods. Hgh Reynolds number natural convecton has been dentfed as one of the man subects for future research, extenson to three dmensonal calculatons for practcal and real lfe stuatons s consdered of prme mportance. The obectve of ths numercal study s to nvestgate; The temperature dstrbuton n the enclosure caused by non-unformty n the boundary temperature as a result of heatng the vertcal opposte walls and coolng the two opposte adacent vertcal walls of the rectangular enclosure. Velocty profles of ar n the enclosure caused by the resultng buoyancy effects and energy ganed from heater. II. LITERATURE REVIEW A number of researchers modeled a number of turbulent flows near vertcal sothermal walls n ther nvestgatons. These nclude studes by Cebec & Smth (1974), Leornard (1984), de Vahl Daves (1986), Plumb & Kennedy (1984), Lankhorst (1991), Sgey (1999), Kpng eno Joel (006), among others. Ths led to a better understandng of turbulence n natural convecton boundary layers. Recent studes were undertaken by Sgey et al., (004) who nvestgated n detal turbulent flow n a three dmensonal enclosure n the form of a room wth a convectonal heater bult nto one of the walls and havng a wndow n the same wall. Kpng eno (006), also studed turbulent natural convecton wth localzed heatng at the bottom wall (the floor) and two wndows each on the vertcal opposte walls of a rectangular enclosure. Sgey et al., (011) also studed buoyancy drven free convecton turbulent heat transfer n an enclosure. They nvestgated a three-dmensonal enclosure n form of a rectangular enclosure contanng a convectonal heater bult nto one wall and havng a wndow n the same wall. The heater s located below the wndow and the other remanng walls are nsulated. The results were that the enclosure s stratfed nto three regons: a cold upper regon, a hot regon n the area between the heater and the wndow and a warm lower regon. Sheremet (011) also studed mathematcal smulaton of conugate turbulent natural convecton n an enclosure wth local heat source. Much work has been done but for even dstrbuton of heat, a dfferent approach that would ncrease effcency especally n buldngs s to be consdered. In ths study, the postonng of the heaters and smulaton of hgh Reynolds heat transfer by Matlab smulaton software creates a smplfyng dfference from the prevous use of reylegh number. III. MATHEMATICAL FORMULATION Turbulent natural convecton n an enclosure arsng from localzed heatng and coolng s encountered n a number of practcal occurrences, such as the use of convectonal heaters n rooms. The temperature and velocty felds n a room depend manly on temperature of any heat source and wndow as well as ar ventlaton flow rate. Ths s a numercal study of turbulent flow n a rectangular enclosure. The flow of heat s one form of Newtonan moton. We consder natural convecton n a three dmensonal rectangular enclosure n the form of a room wth heaters placed on two opposte vertcal walls and wndows on the other two vertcal walls. Fgure 1: Model of a Room Fgure 1 above s a model of a room wth heaters placed on two opposte walls (on the X-Z planes) and wndows on the other two walls (Y-Z planes). The floor and the top planes are taken to be nsulated (X-Y planes). IV. GOVERNING EQUATIONS The fundamental equatons of flud dynamcs are based on the followng unversal laws of conservaton; conservaton of mass (contnuty), momentum and energy. Consder a flud n whch the densty ρ s a functon of poston x (=1,, 3), let u (=1,, 3) denote the components of the velocty. Hence n wrtng the varous equatons, use of the notaton of Cartesan tensors wth the usual summaton convecton s appled Curre (1974). ( u ) 0 (1) t x u u u ( u k ) ( u u ) g s () x t x x x x x x k t X T p u p c T p T c p u T x W Z H x W Wndow; H - Heater (3) x t x The equatons (1), () and (3) for the contnuty, mean momentum and mean energy respectvely are however n the most general form, hence non-dmensonalzaton, Reynolds H W Y ISSN: Publshed by The Standard Internatonal Journals (The SIJ) 73

3 The SIJ Transactons on Computer Networks & Communcaton Engneerng (CNCE), Vol. 1, No. 4, September-October 013 (1976) decomposton, Boussnesq (1903) approxmaton and approprate boundary condtons together wth these equatons are necessary for determnng the velocty component u and the flud propertes such as the densty ρ, pressure P and the temperature T whch apply turbulent flows. The fnal set of equatons then become; U U M 1 U U M g M 3 (4) t x x x R U t x T x Equatons (4) for momentum and (5) for energy are descrtzed usng three pont central dfference approxmaton as shown below; (5) V. METHOD OF SOLUTION Equaton (4) wth respectve substtutons of M 1, M and M 3 can be wrtten as; u 1 u u u u Eu Re v t x y x y R Usng Taylor s central dfference approxmaton; y g Fr (6) n1 4k n k vk n k vk n u, 1 u, 1, 1, u u Re Re h Re h h h h k vk n k vk n Eu g u, 1, 1 u h Re h h Re h R Fr Equaton (5) wth respectve substtutons T can also be wrtten as; 1 v t Pr Re x y x y Usng Taylor s central dfference approxmaton; n1 n,, 1 n n n n n n,, k 1, 1,, 1, 1 PrReh v n n 1, 1, n, 1 n, 1 h Equatons (7) and (9) together wth boundary condtons are used n a code wth Matlab software to smulate the expected results. (7) (8) (9) VI. RESULTS AND DISCUSSIONS In an enclosure, each boundary s consdered mpermeable and capable only of moton n ts own place (statc moton). Ths mples that the normal component of velocty at each boundary s zero. ux, y, z,0 snx cos y. In the smulaton of the graphcal results the followng parameters were used; Reynolds number = 5,500 Prandtl number = 0.71, Euler number =.7188 and Frounde number = The Reynolds Number (Re) s an mportant parameter of forced vscous flow. If the Reynolds number of the system s small, the vscous force s predomnant and the effects of vscosty are mportant, whle f t s large, the nerta force s predomnant and the vscous effects are only mportant n the narrow layer near the sold boundary. The Prandtl Number (Pr) represent the rato of momentum dffusvty (v) to thermal dffusvty (k) that gves the measure of effectveness of energy and momentum transport of dffuson n thermal and velocty boundary layer respectvely Temperature Dstrbuton Fgure (a), and fgure 4(a) shows the vertcal temperature felds n the planes x = 0.1 and x = 0.9 respectvely. Beng close to the heaters there are hgher temperatures n the lower regons where the heaters are placed. The temperature decreases as you move up towards the top of the enclosure as a result of the warm flud rsng and mxng up wth the cold flud from the wndow areas. The lowest temperatures are recorded on the walls contanng the wndows. It s warmest at these two planes n the enclosure. Fgure 3(a) shows the sotherms at the vertcal plane x = 0.5. It s the regon concdng wth the wndows and equdstant from the two wndows. At ths plane, the temperatures are low. Ths s because of the dstance from the heaters but there are hgher temperatures at bottom wall of ISSN: Publshed by The Standard Internatonal Journals (The SIJ) 74

4 The SIJ Transactons on Computer Networks & Communcaton Engneerng (CNCE), Vol. 1, No. 4, September-October 013 the enclosure due to the effect of the heaters. The walls here recorded the lowest temperatures due to the coolng from the wndows. The horzontal plane z = 0.5 s shown n fgure 5 concdng wth the wndows and close to the heaters, the plane records mxed temperatures wth hgh temperatures on the regon close to the heaters and low temperatures n the mddle of the room due to the effect of the wndows. The room s dvded nto a number of regons wth those near the heaters havng hgh temperatures as those near the wndows have low temperatures. Mxng of hot and cold flud and turbulence makes the other areas warm. Natural convecton also plays a key role n the varaton of temperature dstrbuton n the room Fgure (a): Isotherms at the Plane x = 0.1 for Re = Fgure 3(a): Isotherms at the Plane x = 0.5 for Re = Fgure 4(a): Isotherms at the Plane x = 0.9 for Re = 5500 ISSN: Publshed by The Standard Internatonal Journals (The SIJ) 75

5 The SIJ Transactons on Computer Networks & Communcaton Engneerng (CNCE), Vol. 1, No. 4, September-October Fgure 5: Isotherms at the Plane z = 0.5 for Re = Velocty Flow Felds The structure of the flow can easly be seen n the vector plots whch have been selected from the Y-Z plane. The vector plots are n the planes x = 0.1, x = 0.5 and x = 0.9 (see fgure (b), fgure 3(b) and fgure 4(b)). In fgure 3(b) where the plane s at x = 0.5, the structure of the flow shows two crcular motons n dfferent drectons; one n the clockwse drecton (to the rght) whle the other (to the left) s ant-clockwse. Ths s due to the strong convectve moton developed by the heatng and hgh Reynolds number. The warm flud gans energy becomes less dense and gans n velocty resultng n an upward moton at the centre where the effect of the heaters s felt, whle on the sdes the cold flud descends. Ths s due to buoyancy effects. The velocty of the descendng flud s strongest near the wndows whle the rsng velocty s strongest near the heaters (at the bottom). Fgure (b) and fgure 4(b) shows the flow patterns n the planes x = 0.1 and x = 0.9 respectvely. They have smlar patterns snce they have the same temperature effects wth more strength n upward moton snce they are close to the source of heat. On the sdes there s mxng up of warm and cold flud resultng nto slow movement of the flud partcles. Fgure 3(b): Velocty Vector Plot at the Plane x = 0.5 for Re = 5500 Fgure 4(b): Velocty Vector Plot at the Plane x = 0.9 for Re = 5500 VII. CONCLUSION In ths study we nvestgate temperature dstrbuton and velocty profles n an enclosure brought about by heat transfer. Natural convecton played a maor role n the study. A three dmensonal rectangular enclosure n the form of a room wth heaters placed on opposte walls and two wndows each on the adacent opposte walls was consdered for the purposes of the study. The flow was consdered turbulent Fgure (b): Velocty Vector Plot at the Plane x = 0.1 for Re = 5500 ISSN: Publshed by The Standard Internatonal Journals (The SIJ) 76

6 The SIJ Transactons on Computer Networks & Communcaton Engneerng (CNCE), Vol. 1, No. 4, September-October 013 hence the solutons were obtaned for Reynolds number 5,500 and Prandtl number To analyze the flow and heat transfer rates, a complete set of non-dmensonalzed equatons governng Newtonan flud and boundary condtons are recast nto vector potental to elmnate the need for solvng the contnuty equatons. A Boussnesq flud moton n a three dmensonal cavty s consdered. The governng equatons wth the boundary condtons are descrtzed usng three pont central dfference approxmatons for a non-unform mesh. The resultng fnte dfference equatons are then solved usng Matlab smulaton software. The results show that turbulent natural convecton plays a maor role n temperature dstrbuton n an enclosure. The room s dvded nto a number of regons wth those near the heaters havng hgh temperatures as those near the wndows have low temperatures. Ths helps n keepng some of the tems at stated temperatures. Convectve currents caused by buoyancy forces due to temperature dfferences between room ar and ar n contact wth hot and cold surfaces also play a maor role n determnng the velocty profles. The study has a wde range of applcatons n engneerng. A good example s n an oven where drect heatng may not be requred. VIII. RECOMMENDATIONS / FUTURE WORK There s need to nvestgate forced convecton and ts effect on temperature dstrbuton and velocty profles n a room. A more practcal approach n engneerng would reduce theoretcal assumptons made n ths work. Investgaton of buoyancy drven natural convecton n non-rectangular enclosures. REFERENCES [1] J. Boussnesq (1903), Theore Analytque Deha Chanteur, Gauther-vllars,. [] T. Cebec & A.M.O. Smth (1974), Analyss of Boundary Layers, Appled Mathematcs and Mechancs, Vol. 15, Academc Press; New York. [3] T.A. Mynet & D. Duxbury (1974), Temperature Dstrbuton wthn Enclosed Plane Ar Cells assocated wth Heat Transfer by Natural Convecton, Proceedngs of 5th Internatonal Heat Transfer Conference. Tokyo, Vol. 38, Pp [4] I.G. Curre (1974), Fundamentals of Fluds, McGraw Hll. [5] W.C. Reynolds (1976), Computaton of Turbulent Flows, Annual Revew of Flud Mechancs, Vol. 8, Pp [6] E. Leornard (1984), A Numercal Study of Effects of Flud Propertes on Natural Convecton, Ph.D. Thess. Unversty of New South Wales; Australa. [7] G. de Vahl Daves (1986), Fnte Dfference Methods for Natural and Mxed Convecton n Enclosures, Proceedngs 8th Internatonal Heat Transfer Conference, San Francsco, Vol. 1, Pp [8] A. Lankhorst (1991), Lamnar and Turbulent Natural Convecton n Cavtes, Ph.D. Thess, Dept Unversty of Technology; Netherlands. [9] D.A. Olson & L.R. Glckman (1991), Transent Natural Convecton n Enclosure at Hgh Raylegh Number, Transton of the ASME Journal of Heat Transfer, Vol. 113, Pp [10] O.A. Plumb & L.A. Kennedy (1997), Applcaton of K-E Model to Natural Convecton from a Vertcal Isothermal Surface, ASME Journal of Heat Transfer, Vol. 99, Pp [11] J.K. Sgey (1999), Buoyancy Drven Turbulent Natural Convecton n an Enclosure, DAAD Scholars Workshop, Egerton Unversty, Noro, Kenya. [1] J.K. Sgey (004), Three-Dmensonal Buoyancy Drven Natural Convecton n an Enclosure, PhD Thess, JKUAT, Kenya. [13] Kpng eno Joel (006), Turbulent Natural Convecton wth Localzed Heatng and Coolng on Opposte Vertcal Walls of an Enclosure, MSc Thess, Kenyatta Unversty, Kenya. [14] M.A. Sheremet (011), Mathematcal Smulaton of Conugate Turbulent Natural Convecton n an Enclosure wth Local Heat Source, Thermophyscs and Aeromechancs, Vol. 18, No. 1, Pp [15] J.K Sgey, F. Gather & M. Knyanu (011), Buoyancy Drven Free Convecton Turbulent Heat Transfer n an Enclosure, Journal of Agrculture, Scence and Technology, Vol. 1, No. 1. Edward Okemwa Marura, was born on 15 th August, 1985 n Ks town, Nyanza provnce, Kenya. He holds a Bachelor of Educaton degree n Mathematcs & Gudance and Counselng from Mo Unversty, Kenya and s currently pursung a Master of Scence degree n Appled Mathematcs from Jomo Kenyatta Unversty of Agrculture and Technology, Kenya. Afflaton: Jomo Kenyatta Unversty of Agrculture and Technology, (JKUAT), Kenya. Teachng Experence: He s currently an assstant lecturer at JKUAT Ks Cbd Campus, n appled mathematcs (May 01 to date) besdes beng a full tme teacher at Bshop Mugend Nyakegog secondary school (January 010 to date) near Ks, Kenya. He has much nterest n the study of heat transfer n dfferent medums and geometres and ther respectve applcatons to engneerng. Dr. Johana Kbet Sgey, holds a Bachelor of Scence degree n mathematcs and computer scence frst class honors from Jomo Kenyatta Unversty of Agrculture and Technology, Kenya, Master of Scence degree n Appled Mathematcs from Kenyatta Unversty and a PhD n appled mathematcs from Jomo Kenyatta Unversty of Agrculture and Technology, Kenya. Afflaton: Jomo Kenyatta Unversty of Agrculture and Technology, (JKUAT), Kenya. Teachng Experence: He s currently the actng drector, kuat, Ks cbd where he s also the deputy drector. He has been the substantve charman - department of pure and appled mathematcs kuat (January 007 to July- 01). He holds the rank of senor lecturer, n appled mathematcs pure and appled Mathematcs department kuat snce November 009 to date. He has publshed 9 papers on heat transfer n respected ournals. ISSN: Publshed by The Standard Internatonal Journals (The SIJ) 77

7 The SIJ Transactons on Computer Networks & Communcaton Engneerng (CNCE), Vol. 1, No. 4, September-October 013 Dr. Okelo Jecona Abonyo, holds a PhD n Appled Mathematcs from Jomo Kenyatta Unversty of Agrculture and Technology as well as a Master of scence degree n Mathematcs and frst class honors n Bachelor of Educaton, Scence; specalzed n Mathematcs wth opton n Physcs, both from Kenyatta Unversty. I have dependable background n Appled Mathematcs n partcular flud dynamcs, analyzng the nteracton between velocty feld, electrc feld and magnetc feld. Has a hand on experence n mplementaton of currculum at secondary and unversty level. I have demonstrated sound leadershp sklls and have ablty to work on new ntatves as well as facltatng teams to acheve set obectves. I have good analytcal, desgn and problem solvng sklls. Afflaton: Jomo Kenyatta Unversty of Agrculture and Technology, (JKUAT), Kenya. 011-To date Deputy Drector, School of Open learnng and Dstance e Learnng SODeL Examnaton, Admsson &Records (JKUAT), Senor lecturer Department of Pure and Appled Mathematc and Assstant Supervsor at Jomo Kenyatta Unversty of Agrculture and Technology. Work nvolves teachng research methods and assstng n supervson of undergraduate and postgraduate students n the area of appled mathematcs. He has publshed 10 papers on heat transfer n respected ournals. Dr. Okwoyo James Marta, holds a Bachelor of Educaton degree n Mathematcs and Physcs from Mo Unversty, Kenya, Master Scence degree n Appled Mathematcs from the Unversty of Narob and PhD n appled mathematcs from Jomo Kenyatta Unversty of Agrculture and Technology, Kenya. Afflaton: Unversty of Narob, Chromo Campus School of Mathematcs P.O Narob, Kenya. He s currently a lecturer at the Unversty of Narob (November 011 Present) responsble for carryng out teachng and research dutes. He plays a key role n the mplementaton of Unversty research proects and nvolved n ts publcaton. He was an assstant lecturer at the Unversty of Narob (January 009 November 011). He has publshed 7 papers on heat transfer n respected ournals. ISSN: Publshed by The Standard Internatonal Journals (The SIJ) 78