Natural Convection in a Rectangular Enclosure with Colliding Boundary Layers

Transcription

1 Journal of Appled Mathematcs & Bonformatcs, vol.4, no.2, 2014, ISSN: (prnt), (onlne) Scenpress Ltd, 2014 Natural Convecton n a Rectangular Enclosure wth Colldng Boundary Layers J.W. Mutuguta 1, F.K. Gather 2 and M.M. Kragu 3 Abstract The movement of the flud n natural convecton results from the buoyancy forces mposed on the flud when ts densty n the proxmty of the heat transfer surface s decreased as a result of the heatng process. The obectve of ths paper s the computatonal study of the flow ntated by natural convecton. The problem beng nvestgated s the colldng boundary layer n a rectangular enclosure. The dmensonal governng equatons are frst transformed to non-dmensonal equatons. The purpose of ths transformaton s to reduce the effort requred to make a study over a range of varables. A non-dmensonal scheme was chosen whch led to mproved teratve convergence for faster transents. The three dmensonal analogue of the stream functon-vortcty formulaton was used where the scalar vortcty was replaced by a vector and the scalar stream functon by a 1 School of Pure and Appled Scences, Murang a Unversty College, P.O. Box Murang a. E-mal: mutuguta@gmal.com 2 School of Mathematcs and Statstcs, Techncal Unversty of Kenya, P.O. Box Narob. E-mal: kgather@yahoo.com 3 P.O. Box 474 Karatna. E-mal: mkragu@ymal.com Artcle Info: Receved : February 15, Revsed : March 29, Publshed onlne : June 15, 2014.

2 86 Natural Convecton n a Rectangular Enclosure wth Colldng Boundary Layers vector potental. The equatons were then solved usng the method of varable false transents. The results ndcated that the flow regon s stratfed nto three regons; a cold upper regon, a hot regon n the area of the confluence of hot and cold streams and a warm lower regon. Mathematcs Subect Classfcaton: Appled Mathematcs and Modelng Keywords: Colldng boundary layer; Turbulent Natural Convecton; Heat transfer 1 Introducton When a vscous flud flows past a statonary sold boundary, a layer of flud whch comes nto contact wth the boundary surface adheres to t and the condton of no-slp mples that the velocty of the flud at a sold boundary must be the same as that of the boundary. Thus the layer of flud that cannot slp away from the boundary undergoes retardaton. Ths retarded layer further causes retardaton on the adacent layers thus developng a thn layer n the vcnty of the boundary surface at whch the velocty of the flud ncrease rapdly from zero at the boundary surface to the man stream velocty. Such a layer s known as the boundary layer. Lamnar boundary layers come n varous forms and can be loosely classfed accordng to ther structure and the crcumstances under whch they are created. The thn shear layer that develops on an oscllatng body s an example of the stokes boundary layer, whle Blasous boundary layer (Lao and Campo [2]) refers to the smlarty soluton of the steady boundary layer attached to a flat plate held n an oncomng undrectonal flow. When flud rotates, vscous forces may be balanced by corols 4 effect rather than convectve nerta leadng to the formaton 4 Apparent deflecton of movng obects when vewed from rotatng frame of reference.

3 J.W. Mutuguta, F.K. Gather and M.M. Kragu 87 of an Ekma layer 5. Thermal boundary layers also exst n heat transfer. Multple types of boundary layers can coexst near the surface. The thckness of the velocty boundary layer s normally defned as the dstance from the sold boundary at whch the flow velocty s 99% of the free stream velocty.e. the velocty that s calculated at the surface boundary n an nvscd flow soluton. The boundary layer represents a defct n mass flow compared to an nvscd case wth slp at the wall. It s the dstance by whch the wall would have to be dsplaced at the nvscd case to gve the total mass flow as the vscous case. The no-slp condton requres that the velocty at the sold boundary be zero and the flud temperature be equal to that of the boundary. The flow velocty wll then ncrease rapdly wthn the boundary layer as governed by the boundary layer equatons. Thermal boundary layer thckness s the dstance from the sold boundary at whch the temperature s 99% of the temperature found from an nvscd soluton. The rato of the velocty and thermal boundary layers s governed by the Prandtl number. If the Prandtl number s 1, the two boundary layers are of the same thckness. If the Prandtl number s greater than 1, the thermal boundary layer s thnner than the velocty boundary layer. If Prandtl number s less than 1, the thermal boundary layer s thcker than the velocty boundary layer. For ar of Prandtl Number Pr = 0.71, the thermal boundary layer s 141% of the velocty boundary layer. The present study s desgned to nvestgate the room ar dstrbuton. The mportance of the present study les not only n the room ar dstrbuton but also n room ventlaton. Ventlatng s the process of replacng ar n the room to control temperature or to remove mosture, odor, dust, smoke arborne bactera, carbon doxde and also to replensh oxygen. It ncludes both the exchange of ar to the outsde as well as crculaton of ar wthn the buldng. In the absence of the 5 A layer n whch there exst a balance of forces between pressure gradent, corols forces and turbulent drag.

4 88 Natural Convecton n a Rectangular Enclosure wth Colldng Boundary Layers heater, cold ar flows nto the room and down to the floor. Warm ar n the room rses and flows out through the wndow. The room s therefore flled wth cold ar. Mohammed O. and Galvans [3] studed natural turbulent convecton n a dfferentally heated 2D cavty whch they smulated usng the Shear Stress Transtonal (SST) k ω turbulence model. Comparsons wth expermental benchmark values show that, n ths case, conducton n the horzontal walls has a sgnfcant effect on the calculated results. They also show that agreement between calculated and measured values at md-heght of the cavty does not suffce to establsh the valdty of the model and numercal procedure. Vsser et al [4] nvestgated buoyancy drven vscous flow through saturated packed pebble beds usng a set of homogeneous volume averaged conservaton equatons. Comparson of the smulated results and expermental data of Nessen and Stocker [7] ndcated an acceptable correlaton. Kandaswamy et al [1] studed natural convecton heat transfer n a square cavty nduced by heated plate. The top and bottom of the cavty were adabatc. The study was performed for dfferent values of Grashof number rangng from 10 3 to 10 5 for dfferent aspect ratos. They found that the rate of heat transfer ncreases wth ncreasng Grashof number. Rundle and Lghtstone [5] nvestgated turbulent natural convecton n a square cavty heated and cooled on opposte walls and obtaned the soluton usng three turbulent models.e. models (SST). They found that the SST models due to ts hgh accuracy. Nomenclature τ Vscous stress tensor μ Frst coeffcent of vscosty µ s Second coeffcent of vscosty g Force due to gravty Dfferental operator k ε, wlcox k ω and Shear Stress Transtonal wlcox k ω model was superor to k ε and

5 J.W. Mutuguta, F.K. Gather and M.M. Kragu 89 ρ Densty Φ The dsspaton functon or scalar transport unknown λ Thermal conductvty β Volumetrc coeffcent of expanson C P η k T Specfc heat at constant pressure Coordnate normal to the boundary Knetc energy of turbulence Thermodynamc temperature x, y, z Coordnate drecton n the ˆ, ˆ, kˆ drecton Θ P Ra Pr ε Non-dmensonal mean temperature Thermodynamc pressure. 2 ρrc Raylegh Number Ra = µ Prandtl number Pr = κ R C PR R g β TL μ κ Dsspaton rate of turbulent knetc energy PR R R 3 r 2 Mathematcal formulaton The problem beng consdered s shown schematcally n Fgure 1 below. The dmensons of the model are 3.0m long, 3.0m wde and 3.0m hgh. The heater s mounted below the wndow and the remanng walls are adabatc. Some of the assumptons used n ths analyss are that flud s Newtonan and that there are no heat sources wthn the enclosure. The change n the flud densty s ncorporated through the Boussnesq approxmaton. Turbulent flow of ar s descrbed mathematcally by the Reynolds Averaged Naver Stokes equatons (RANS) ncludng the tme averaged energy equaton for

6 90 Natural Convecton n a Rectangular Enclosure wth Colldng Boundary Layers the mean temperature feld that drves the flow by buoyancy force. These equatons are; Fgure 1: The schematc dagram for the case of colldng boundary layers wth buoyancy effects ) Equaton of Contnuty. ρ + t ( ρu + ρu) = 0 (1) ) Equaton of Momentum t ( ρu + ρu) + ( ρuv + U ρv) = p + ρg + ( τ U ρu ρuv ρuv) (2) ) Equaton of Energy t ( cpρt + cpρt ) + ( cpρut ) ( λ p ρt pρ t) p = + U P + u p + T cu c u +Φ t (3)

7 J.W. Mutuguta, F.K. Gather and M.M. Kragu 91 where ( u + v) + µ s w τ = µ δ ( u + v) u, and Φ = τ U + µ. v) Conservaton Equaton for the Turbulent Knetc Energy t u u 1 ( ρk) ( ρu k) u μ + = + ( ρu u u ) x x + ρu g u x p x x 2 x ρu u U x (4) v) Equaton of rate of Dsspaton of Turbulent Knetc Energy. ρε u u u p ε u u u + ρu ε = μμ + 2 v μ 2μ t x x x x x x x x x x k k k k k k 2 2 u u ρ u u u u k u xkx x x xk x x x xk 2ρ + 2v g 2μ + u u 2 2μμk 5 xxk x ( ) The boundary condtons descrbng ths moton are; u = v = w = 0 on each boundary On the hot boundary, T = T H = 70 0 c On the cold boundary, T = T c 0 = 20 c The non dmensonal temperature s defned by Θ T T T T * =. H C

8 92 Natural Convecton n a Rectangular Enclosure wth Colldng Boundary Layers where T * Quescent temperature. The choce of Θ ensures that t s bounded and ts value les between 0 and 1. Thus the non-dmensonal temperature boundary condtons are; Θ Θ =1 on the hot boundary, Θ = 0 on the cold boundary and = 0 η boundares. on the other 3 Method of Soluton The system of non-lnear partal dfferental equatons wth approprate boundary condtons was solved usng Alternatng Drecton Implct method (Samask and Andreyev [6]). The convectve terms were dfferenced usng the hybrd scheme whle the dffuson terms were dfferenced usng the central dfference scheme. The Ellptc Partal Dfferental Equatons were converted to Parabolc Partal Dfferental Equatons by nserton of false transent dervatves. The soluton was then obtaned by matchng the parabolc equatons through tme. The procedure nvolves usng dfferent false transent factors n dfferent flow regons. Ths results n small tme steps n the boundary layer and large tme steps n the rest of the enclosure. 4 Results and dscusson 4.1. Flow Felds: Ra=10 8 The man obectve of ths study s to nvestgate the structure of the flow due to the collson of two opposed natural convecton boundary layers drven by temperature dfference along a vertcal wall and the mechansm whch causes the

9 J.W. Mutuguta, F.K. Gather and M.M. Kragu 93 formulaton of vertcal structures n the flow that ets away from the wall. 8 Solutons have been obtaned for Ra = 10 and Pr = To reveal the fne structure of the thermal boundary layers, most of the vector plots n the x z plane have been selected on the symmetry plane and near sde wall. In the case of x y the vector plots near the actve wall and at the rear wall of the enclosure where most of the flow s concentrated have been selected. In the y z plane, vector plots n the borderlne between the wndow and the heater have been selected. Heatng over the lower half and coolng over the upper half creates an unstable stratfcaton of flud at the colldng boundary layer. At suffcently hgh Raylegh number (hgh temperature dfference), a strong convectve moton develops and heat s transferred from the lower half of the enclosure to the upper half. Flud n the upper half moves so that the cold flud descends adacent to the actve wall and colldes wth the rsng flud from the lower half. Ths collson takes place halfway between the vertcal walls. After collson, the boundary layers curve nto the room and then on to form a horzontal et whch moves across the cavty as shown n Fgure 2 (a) and (b). The et mpnges on the rear wall and bfurcates formng two streams. One of the streams moves towards the celng and flows back to the actve wall through the celng whle the other stream flows towards the floor and returns to the actve wall through the floor of the enclosure. A clockwse cell s formed n the lower half of the enclosure between the upward movng stream on the actve wall and the downward movng stream on the rear wall as shown n fgure 2 (c). In the confluence regon of the colldng boundary layers, there s sgnfcant heat transfer whch results from the mxng of the hot and cold streams. Ths s shown by the upward and downward moton of vectors n Fgure 2 (b). As the horzontal et moves towards the rear wall, there s ncreased mxng of flud as seen n Fgure 2 (a). As the stream moves towards the rear wall t becomes

10 94 Natural Convecton n a Rectangular Enclosure wth Colldng Boundary Layers congregated towards the center formng a strong et whch mpnges on the rear wall and s dsplaced n all drectons as may be seen n Fgure 2 (b). (a) (b) (c) Fgure 2: Velocty vectors plots at 8 Ra = 10 : (a) End elevaton at the plane z = 0.1, 0.5 and z = 0.9 respectvely, (b) sometrc vew at x = 0.5 and at z = 0. 1 and (c) sde elevaton at y = Temperature Felds The hot and cold sotherms are congregated n a narrow strp due to fast rsng and fallng streams near the heater and the cold wndow. Fgure 3(b) show temperature dstrbuton along the heated and cooled surfaces. The sotherm 8 separates the contour plots between the upper and the lower regons. The sotherms of Fgure 3 (a) ndcate that the flow regon may be dvded nto three regons. () The lower regon where the temperature of the flud s everywhere above the mean value of Θ = () The unstable zone wthn the confluence regon of the heater and the wndow where temperature ranges from 0.24 to Ths s the regon where mxng of the cold and hot streams of the flud occurs. Ths regon

11 J.W. Mutuguta, F.K. Gather and M.M. Kragu 95 s domnated by hgh temperature gradents thus there s sgnfcant heat transfer between the hot and cold streams. () The upper regon where temperature s below the mean room temperature Θ = 0.5 Fgure 3(c) shows the sotherms at the md-plane of the hot regon of the enclosure. There s steep temperature gradent near the actve wall whch decreases towards the rear. (a) (b) (c) Fgure 3: Isotherms at y = 0.5 and (c) top vew at x = Ra = 10 : (a) end elevaton at z = 0.5, (b) sde elevaton at 5 Concluson In ths study, buoyancy nduced turbulent flow and heat transfer of ar nsde a three dmensonal enclosure heated and cooled on the same vertcal wall s nvestgated numercally. Dfferent temperature condtons resulted n colldng boundary condton. It was observed that two boundary layers are formed whch collde n the regon between the wndow and the heater. After collson the two layers curve to form a horzontal et that moves towards the rear of the enclosure. The et then mpnges on the rear wall of the enclosure and spreads to form two

12 96 Natural Convecton n a Rectangular Enclosure wth Colldng Boundary Layers streams. One stream moves towards the celng whle the other stream flows towards the floor of the enclosure. The flud then flows back to the actve wall through the celng and the floor of the enclosure. In the enclosure two large, equally szed, counter-rotatng cells are formed. The upper cell s drven by the cold wndow whereas the lower cell s drven by the heater. The upper cold cell rotates n a counter-clockwse drecton whereas the hot lower cell rotates n a clockwse drecton. The two streams collde n the regon between the wndow and the heater. Mxng process takes place n ths regon leadng to the exchange of heat between the two flud layers. Ths process transfers heat energy from the hot to the cold regons of the enclosure The present study s desgned to nvestgate the room ar dstrbuton. The results of the study ndcates that placng the heater next to the wndow mnmzes condensaton and offsets the convectve ar current formed n the room as a result of the ar next to the wndow. The mportance of the present study les not only n the room ar dstrbuton but also n room ventlaton. Ventlatng s the process of replacng ar n the room to control temperature or to remove mosture, odor, dust, smoke arborne bactera, carbon doxde and also to replensh oxygen. It ncludes both the exchange of ar to the outsde as well as crculaton of ar wthn the buldng. In the absence of the heater, cold ar flows nto the room and down to the floor. Warm ar n the room rses and flows out through the wndow. The room therefore s flled wth cold ar. When the heater s placed below the wndow, hot ar from the heater rses and colldes wth the cold ar from the wndow, mxng of the two layers takes place and as a result warm ar flows nto the room. The results of ths analyss therefore show that the heater should be placed below the wndow.

13 J.W. Mutuguta, F.K. Gather and M.M. Kragu 97 References [1] P. Kandaswamy, J. Lee and A. K. Abdul Hakeem, Natural convecton n a square cavty n the presence of a heated plate, Non-lnear Analyss, Modelng and Control, ASME Journal of Heat Transfer, 12(2), (2007), [2] S.J. Lao, A. Campo, Analytc solutons of the temperature dstrbuton n blasus vscous flow problems, Journal of Flud Mechancs, 453, (2002), [3] O. Mohamed and N. Galans, Numercal analyss of turbulent buoyancy flows n enclosures: nfluence of grd and boundary condtons, Internatonal Journal of Thermal Scences, 46(8), (2007), [4] C.J. Vsser, A.G. Malan, and J.P. Meyer, An artfcal compressblty method for buoyancy drven flow n a heterogeneous saturated packed beds, Internatonal Journal of Numercal Methods for Heat and Flud Flow, 18(7/8), (2008), [5] C.A. Rundle, and M.F. Lghtstone, Valdaton of turbulent natural convecton n a square cavty for applcaton of computatonal flud dynamcs modelng to heat transfer and flud flow n atra geometres, Second Canadan Solar Buldng Conference, Calgary, June 10-14, [6] A.A. Samask, and V.B. Andreyev, Hgh accuracy dfference scheme for an ellptc equaton wth several space varables, USSR Computatonal Mathematcs, 3, (1963), [7] H.F. Nessen, and B. Stocker, Data Sets of SANA Experment: , JUEL-3409, (1997).