NATURAL CONVECTION HEAT TRANSFER WITHIN VERTICALLY ECCENTRIC DOMED SKYLIGHTS CAVITIES

Transcription

1 Poceedings: Building Simulation 007 NATURAL CONVECTION HEAT TRANSFER WITHIN VERTICALLY ECCENTRIC DOMED SKYLIGHTS CAVITIES A. Satipi, A. Laouadi, D. Naylo 3, R. Dhib 4 Depatment of Mechanical and Industial Engineeing, Ryeson Univesity, asatipi@ryeson.ca National Reseach Council of Canada, Abdelaziz.Laouadi@nc-cnc.gc.ca 3 Depatment of Mechanical and Industial Engineeing, Ryeson Univesity, dnaylo@ryeson.ca 4 Depatment of Chemical Engineeing, Ryeson Univesity, Dhib@Ryeson.ca ABSTRACT Domed skylights ae impotant achitectual design elements to delive daylight and sola heat into buildings and connect building s occupants to outdoo. Howeve, most of building enegy simulation pogams do not cove such skylights to quantify thei enegy pefomance when installed in buildings. This pape pesents a numeical study on natual lamina convection within vetically eccentic domed cavities when heated fom the exteio suface. Both inteio and exteio sufaces ae held at unifom tempeatues. A commecial CFD package employing the contol volume appoach is used to solve the lamina convective heat tansfe within the cavity. The obtained esults showed that the flow is mono-cellula fo small and modeate heating intensities as depicted by the Gashof numbe. Fo high Gashof numbes, small votex cells appea within a lage votex cell at the lowe pat of the cavity. The multi-cellula flow inceases heat tansfe. The citical gap spacing that yields the imum heat tansfe was quantified fo seveal dome pofiles. KEYWORDS Natual lamina convection; domed cavity; domed skylight, eccentic domed cavity NOMENCLATURE k Fluid themal conductivity L Maximum gap thickness L min Minimum gap thickness L Local gap thickness P Pessue P Dimensionless pessue q cond Pue conduction heat tansfe q i Convection heat tansfe fom inteio suface q o Convection heat tansfe fom exteio suface R o Oute adius R i Inne adius Position adius Dimensionless position adius T Tempeatue T Dimensionless tempeatue u Velocity component in -diection Velocity component in -diection u Geek Symbols α Fluid themal diffusivity β Fluid themal expansion coefficient δ Maximum dimensionless gap thickness(l /R o ) δ min Minimum dimensionless gap thickness (L min /R o ) δ Dimensionless local gap thickness (L/R o ) ν Kinematic viscosity Position angle ο Dome tuncation angle ρ Fluid density τ Dimensionless time Dimensionless Numbes G Gashof numbe, [gβ(t o -T i )L 3 /ν ] Nu Nusselt numbe, [h L /k] P Pandtl numbe, [ν/α] Ra Rayleigh numbe (Ra G.P) Ra Modified Rayleigh numbe, Eq. (5) INTRODUCTION Domed skylights ae commonly used in buildings such as esidential, industial and commecial buildings. They delive daylight and sola heat into the indoo envionment and connect building s occupants to the outdoo envionment. If popely designed, domed skylights may educe the building enegy usage fo lighting, cooling and heating. Futhemoe, cuent eseach has shown that daylighting and connection to outdoo h positive effects on occupants mood and wellbeing. Despite thei wide spead use, the themal pefomance of domed skylights has not been well undestood. Cuent design tools such as fenestation poduct ating tools and building enegy simulation softwae do not cove this type of skylights. This pape addesses the natual lamina convective heat tansfe in vetically eccentic domed skylight cavities heated fom outside. The inside and outside sufaces ae held at unifom diffeent tempeatues, and the edge sufaces ae sealed and adiabatic. The main objectives ae to investigate the flow patten in the cavity, quantify the citical gap spacing that yields the imum heat tansfe, and develop coelations fo the convective heat tansfe coefficient as a function of the govening paametes

2 Poceedings: Building Simulation 007 MATHEMATICAL FORMULATION A double-laye domed skylight with the eccenticity of (ε) is the subject of this study. The dome cavity is chaacteized by its inteio and exteio adii (R i, R o ), minimum and imum gap thicknesses (L min, L ) and tuncation angle ( 0 ). The inteio and exteio domed sufaces ae maintained at unifom tempeatues T i and T o, espectively. The edges of the domed cavity ae sealed and adiabatic. The cavity is filled with a gaseous fluid. The tuncation angle may vay fom 0 o to 90 o, coveing almost flat hoizontal cavities o fully hemispheic cavities. Since the dome is symmetic with espect to the Y-axis (evolution axis), the flow is consideed two-dimensional. Figue shows a schematic epesentation of a double-laye domed cavity. Substituting those dimensionless vaiables into the govening equations, the esulting tansient dimensionless govening equations educe to: Continuity: ( U ) ( U sin) 0 -momentum: -momentum sin () U U U U U P U G T cos (3) τ ( ) U U sin sin U U U U U U U τ U sin P G T ( U sin) sin U (4) Enegy: U P U τ sin sin (5) Figue Schematic desciption of a double-laye domed cavity Assumptions The govening equations fo mass, momentum and enegy tansfe balances ae witten in the spheical coodinate system. The following assumptions ae made to simplify the govening equations; The fluid is incompessible, Newtonian and lamina; The physical popeties of the fluid ae constant, except the density in the body foce tems; Boussinesq appoximation is used fo the buoyancy tems; and The viscous dissipation is neglected in the enegy equation. Govening Equations The simplified govening equations ae cast in a dimensionless fom using the following dimensionless vaiables; P L ρ i, τ p t, U, U ( L / υ) ( υ/ L) ( υ/ L) T Ti T ( υ/ L ) To Ti, u u, () Dimensionless Bounday Conditions The dimensionless govening equations () to (5) ae subject to the following bounday conditions: the dome ends ae sealed and adiabatic; the exteio and inteio sufaces ae maintained at unifom and constant tempeatues and no-slip condition. The dimensionless bounday conditions ae as follows: Edge suface, δmin U U 0, 0 at 0, < < (6) Symmety suface, U 0, 0 at 0, < < (7) Exteio suface, U U 0, T at (8) Inteio suface, δ (9) U U 0, T 0 at with: ; ; ; (0) δ L δ L δ L δ L min min Whee L (L L min )/ and L is the local gap thickness at position angle. At the initial conditions (τ 0), the fluid is assumed quiescent and, theefoe, the heat tansfe is by pue

3 Poceedings: Building Simulation 007 conduction. The dimensionless govening equations and bounday conditions show that the cavity flow is govened by the Gashof numbe (G), Pandtl numbe (P), dimensionless gap spacing (δ ), the atio of the minimum to imum gap spacing (L min /L ) and tuncation angle ( 0 ). NUMERICAL METHOD The contol volume appoach is used to discetize the dimensionless govening equations. A commecial CFD package is used to solve the discetized equations. The heat tansfe poblem was investigated fo a wide ange of geomety and bounday conditions: the Gashof numbe (G) vaies fom 0 3 to 0 7, non-dimensional gap spacing (δ ) vaies between and 0.5, and thee diffeent tuncation angles ( o ) of 30 o, 45 o and 90 o. The atio of the imum to minimum gap thickness is fixed to L /L min. Fo all geometies, diffeent mesh sizes wee used to ensue mesh-independent esults. A non-unifom gid in the adial diection with fine mesh sizes nea the walls was used to account fo the tempeatue and velocity gadients nea the inteio and exteio sufaces. The govening equations wee solved numeically using the implicit and segegated scheme [4]. A second ode upwind scheme is used to discetize the momentum and enegy equations (3) to (5). The SIMPLEC algoithm [4] was used fo the pessue-velocity coupling and PRESTO algoithm fo the pessue coection, which is ecommended fo buoyancy-diven poblems [5]. Validation of the Numeical Method Fo the model validation puposes, the esults fom the unifom gap thickness model (L min /L ) ae compaed with those obtained by Laouadi and Atif [], Raithby and Holand [] and Gag [3] fo concentic sphees when heated fom inside. Table shows this compaison. The imum diffeence between the pesent model and the othe models is less than %. Table. Nusselt numbe fo concentic sphees heated fom inside fo δ δ min 0.5 RA Pesent model Laouadi and Atif [] Gag [3] Raithby and Hollands [] EVALUATION OF HEAT TRANSFER The heat tansfe fom the inteio and exteio sufaces of the domed cavities ae evaluated as follow: q πr k sind () i i o 0 Ri q πr k sind () o o o 0 Ro Fo cavities of non-unifom gap thicknesses, the conduction heat tansfe is two-dimensional, which has a complicated equation in the spheical co-odinate system. In this pape fo convenience, the Nusselt numbe is defined fo the inteio and exteio sufaces as follows: qi Nui q' cond qo Nuo q' cond ( R / R ) i o δ δ cos δ δ cos o 0 δ o 0 sin d sind (3) (4) Whee q cond is the conduction heat flux of a concentic cavity with an age gap thickness equal to L (L L min )/. Unde the steady state conditions, the heat tansfe and Nusselt numbe at the inteio suface ae equal to the exteio suface ( q q q & Nu Nu Nu ). i o i o RESULTS AND DISCUSSION The numeical esults ae pesented when the exteio suface is hotte than the inteio one and ai (P 0.7) is used in the cavity gap. Small, modeate and lage gap thicknesses ae the subject of this study. The flow eaches the steady state conditions fo small and modeate Gashof numbes, independently of the tuncation angles and gap thickness. Fo the Gashof numbe aound.5x0 6, small unstable votex cells fom inside a big cell, which causes small oscillations in the Nusselt numbe. Figue shows typical oscillations fo a tuncation angle of 30 o, δ and G 7.5x0 6. It is found that in most of the times these oscillations ae peiodic. Unde such peiodic tansient conditions, the Nusselt numbe is aged ove two o thee oscillation peiods. Figue 3 shows the steamlines and isothems fo small, modeate and lage gap cavities fo two diffeent tuncation angles of 90 o and 45 o. The Gashof numbe is fixed at 5x0 4. Fo all cases, the cavity flow eaches the steady-state conditions with only one votex cell. As the Gashof numbe inceases, one o moe small cells fom inside a big cell. Figue 4 shows the compaison of steamlines and isothems fo thee diffeent gap thicknesses and two tuncation angles of 90 o and 45 o. Fo lage cavity gap thicknesses (δ > 0.), the flow eaches the steady-state conditions with one small cell

4 Poceedings: Building Simulation 007 inside a big cell fo both 90 o and 45 o tuncation angles. Howeve, fo modeate and small gap thicknesses, small multi-cells appea inside the big votex cell. Figue Typical oscillations in the Nu o fo G7.5x0 6, o 30 o and δ The small votex cells in Figue 4 ae unstable. They fom inside the big cell and move inside the cavity and then mege with the big cell. This behavio causes small oscillations in the Nusselt numbe, paticulaly fo vey high Gashof numbes. In some of the cases, depending on the gap thickness and tuncation angle, it is found that the Gashof numbe of 7.5x0 6 indicates the tansition to tubulent flow. Figues 5 and 6 show the vaiation of the Nusselt numbe (Nu o ) as a function of the modified Rayleigh numbe (Ra ) fo tuncation angles of 90 o and 45 o, espectively. The modified Rayleigh numbe, which was fist intoduced by Raithby and Hollands [], is expessed as follows: Ra ( /) Ra (5) 3/5 4/5 ( δ ) ( δ ) [ ] 5 The figues show that the changes in the Nusselt numbe as a function of Ra /4 depend on the non-dimensional gap spacing δ fo a given tuncation angle. This dependence is moe significant fo the tuncation angle of 45 o. Figue 7 shows the effect of the gap thickness on the Nusselt numbe evaluated at G 5x0 4, 5x0 5 and 5x0 6 fo thee tuncation angles of 90 o, 45 o and 30 o. the figue shows that the Nusselt numbe inceases with the dome tuncation angle o. The citical gap thickness that imizes the heat tansfe vaies with the tuncation angle o, dimensionless gap thickness δ and Gashof numbe. Fo a tuncation angle of o 90 o the Nusselt numbe inceases with the gap thickness and then conveges to an asymptotic value. Fo tuncation angles of o 45 o and 30 o the citical gap thickness inceases with the Gashof numbe. CONCLUSION Natual lamina convection in vetically eccentic domed skylight cavities was investigated when the exteio suface is hotte than the inteio one. The contol volume appoach was used with a commecial CFD package to solve the tansient govening equations. The esults wee obtained fo small, modeate and lage gap cavities and high and low pofile domes ( o 90 o, 45 o and 30 o ) with a wide ange of Gashof numbe (0 3 < G < 0 7 ). Fo small Gashof numbes, the flow is monocellula and eaches the steady state conditions, independently of the gap thickness and tuncation angle. Fo lage and modeate Gashof numbes depending on the gap thicknesses and tuncation angle, the flow is steady state and may be multi-cellula with one big cell and some small cells inside. The Nusselt numbe is highe fo tuncation angle of 90 o, and it inceases with the dimensionless gap thickness to an asymptotic value. Fo tuncation angels of 45 o and 30 o the Nusselt numbe inceases with the dimensionless gap thickness and eaches to the imum value and then deceases. The citical gap thickness that imizes the heat tansfe vaies with the tuncation angle and Gashof numbe. The citical gap thickness inceases with the Gashof numbe and gap thickness. Fo instance at Gashof numbe of 5x0 6, the citical gap thickness fo o 45 o and 30 o is aound δ 0. and 0.05 espectively. With inceasing the Gashof numbe, small cells appea inside a big cell. These votex cells become stonge and bigge upon inceasing the Gashof numbe. Depending on the tuncation angle and gap thickness, at the Gashof numbe aound 7.5x0 6 the flow might be unstable and gets into the tansition condition to the tubulent egime. ACKNOWLEDGEMENTS This eseach was funded by NSERC of Canada, the National Reseach Council of Canada, the Panel fo Enegy Reseach and Development (PERD), BC Hydo, and Natual Resouces Canada. REFERENCES A. Laouadi and M.R. Atif (00), Natual Convection Heat Tansfe Within Multi-laye Domes, Intenational Jounal of Heat and Mass Tansfe, Vol. 44, pp G.D. Raithby and K.G.T. Hollands (998), Chapte 4, Natual Convection, in Hand book of heat tansfe, Editos: W.M. Rohsenow, J.P. Hatnett and Y.I. Cho, McGaw-Hill, New Yok V.G. Gag (99), Natual Convection Between Concentic Sphees Intenational Jounal of Heat and Mass Tansfe, Vol. 35(8), pp H.K. Vesteeg and W. Malalasekea (995), An Intoduction to Computational Fluid Dynamics the Finite Volume Method, Pentice Hall FLUENT 6. Documentation (005), Fluent Incopoated, Evanston, Illinois

5 Poceedings: Building Simulation 007 Figue 3 Steamlines and isothems fo o 90 o & 45 o and G 5x0 4 Figue 4 Steamlines and isothems fo o 90 o & 45 o and G 5x

6 Poceedings: Building Simulation 007 Figue 5 Pofiles of the Nusselt numbe as a function of Ra /4 fo a tuncation angle of 90 o Figue 6 Pofiles of the Nusselt numbe as a function of Ra /4 fo a tuncation angle of 45 o

7 Poceedings: Building Simulation o 90 o 0.0 Nu / Ra / o 45 o G 5e4 G 5e G 5e o 30 o δ Figue 7 Pofiles of the Nusselt numbe as a function of Ra /4 and gap thickness δ