SIMILARITYSOLUTION FOR DOUBLE NATURAL CONVECTION
ProceedingsNZ GeothermalWorkshopSIMILARITY SOLUTION FOR DOUBLE NATURAL CONVECTION WITHDISPERSION FROM SOURCES IN AN INFINITE POROUS MEDIUMA.V.and A. F.Geothermal Institute, The University of Auckland, New ZealandSchool of Engineering, University of Ballarat, Ballarat,AustraliaSUMMARY - Scale analysis of coupled heat and mass transfer from a point or a horizontal line source in an infinitesaturated porous medium is reported is this paper. Conservation equations are shown to have solutions of similarityform for generalized variables in the case of flow when dispersion is predominant over molecular diffusion. Closedform solutions are presented for Darcy and non-Darcy natural convection for both point and line sources. Theestimations of distances from sources where the solutions obtained will be valid are given.1. INTRODUCTIONNatural convection phenomena in porous mediacan be associated with simultaneous heat andmass transfer. Buoyancy forces that drive suchdensity differences dueflows arise not onlyto variation in temperature but also from thosedue to variations in solute (chemical species)concentration. Examples are found in manynatural and technological applications such , the dispersion of chemicalcontaminants through water-saturated soil, andunderground disposal of nuclear wastes. Thisproblem also finds applications in chemicalindustry.Transfer processes around concentrated sourceshave a significant place in research of thenaturalmechanism of coupled doubleconvection. Relative to the research activity onthe flow around concentrated sources induced bythermal buoyancy alone, the problem of the nearsource convection driven by two buoyancy effectshas received quite limited attention. Even in thereview by Trevisan and Bejan (1990) devotedsolely to the combined transfer processes bynatural convection, and in the book by Nield andBejan (1999) containing an exhaustivebibliography on convection in porous media, onlya few papers are cited on this problem. Thetransient and steady state flow near a point sourceof heat and mass in the low Rayleigh numberregime was the subject of investigation byPoulikakos (1985).The corresponding problem for the vicinity of ahorizontal line source was analyzed by Larson andPoulikakos (1986). The solutions were obtainedby means of perturbation analysis in the thermalRayleigh number. The effect of species diffusionon the buoyancy induced by temperature and flowfields near the concentrated heat and mass sourcesin porous medium were discussed. The highRayleigh number regime of coupled double201diffusive natural convection from a line source inporous medium for Darcy flow has been(1990). It was shown that theconsidered byboundary layer equations can be written in termsof the similarity variables for power-law variationof centre-line temperature and concentration andhave closed-form solutions for the special case ofLewis number 1.In the comparatively recent paper by Telles andTrevisan (1993) the problem of dispersion innatural convection heat and mass transfer for thecase of vertical surfaces embedded in a porousmedium was analyzed. The authors focused on theboundary layer regime for Darcy flow through aporous medium and studied the effect ofhydrodynamic dispersion in porous media on bothheat and mass transfer in natural convectionflows. It was shown that a few different classes ofproblem exist depending on the dispersioncoefficients. They presented several numericalsolutions of the systems of similarity equations,including the case when the thermal and the massdispersions supersede the molecular diffusion.However the problems of dispersion heat andmass transfer near the source in porous mediahave escaped scrutiny.Before proceeding to an analysis of the combinedeffect of the molecular and the dispersionmechanisms we have to have some asymptoticsolutions for each mechanism individually thatpermit reasonably simple solutions and properphysical interpretation, and thus serve thepurposes of verification and qualitative analysis.2. MATHEMATICAL FORMULATIONNatural convection heat and mass transfer isconsidered in this paper in the steady-state regimefrom a point or a line sources embedded in aporous medium of permeability saturated with aliquid of viscosity, density and the heatcapacityThe sources generate heat at a rateand, at the same time, a substanceat a rate m.
it For the density variations, the assumption of aBoussinesq fluid has been made, which meansthat the density is assumed constant everywhereexcept in the body force term in the momentum)and theequations via the thermalconcentration () volumetric expansioncoefficients. Dwhere is the stagnant thermal conductivity ofa fluid and a saturated porous medium; D is themolecular mass diffusivity; is the characteristiclength (an analog of the mixing length inturbulence); andrepresent the dispersioncoefficients.Under the boundary layer approximation, thegoverning equations that describe the flow in theplume above a point and a line source are given asfollows K (C -bIn the next section we consider the boundary-layerheat and mass transfer problem with scaleanalysis. Then the similarity analysis of theproblem when the dispersion mechanismsupersedes the molecular dispersion will also bepresented.(1)3. SCALE ANALYSISThe algorithm of the scale analysis of boundarylayer-type transfer processes was detailed in byGhukhman (1967). For the problem of the sametype for porous media this approach wasextensively used by A. Bejan (see, for instance,Nield Bejan (1999) for details and references).with the boundary conditions given by0,- o,y oThe total energy and mass diffusion conservationconditions across any horizontal plane in theplume are-0 -Hereand 1 for a line and a point source,whererespectively; inertial coefficientform-drag constant is of the order of magnitude(Ward, 1964;Nield Bejan, 1999).We will focus on two limiting flow regimes. Thefirst one is Darcy flow-3.1. MolecularmechanismHeat transfer drivenflowsConsider the case of heat transfer driven flowswhen the molecular diffusion is predominantin the momentum equation.Using the scale analysis one can obtain the nextestimations for a line source and Darcy flow onthe basis of the momentum, the energy and themass conservationequationsor(CVIn the other case the role of the linear resistance isnegligible, so that one can use the next version ofthe momentum equation (an inertial flow)bThe simplest case of the dominant diffusionmechanism will be our initial concern.(10)The coefficients andrepresent overall thethermal and the mass diffusivities, respectively.They embody both molecular diffusion anddispersion.One can say, at the present time the commonlyaccepted typical representation of overalldiffusivity for boundary-layer-type problemsthe problem of interest) is (Aerov Umnik, 1951;Ranz, 1952;Wakao Kaguei, 1982)202where is the Rayleighnumber for a line source and Darcy flow; and Hare the heat plume thickness and height.can beThe concentration plume thicknessestimated following Bejan(1985). Weobtain from the concentrationequation (3)ACHFor the casethe Lewis numberusingas a scale length andaccountingfor the estimations (13) we haveaand for the case(Le
The Table 1 summarizes all for this class of flow.It is worthy of note that these results for the heatplume are the same obtained for pure heat transfernatural convection.Here,, -molecular diffusion. Whereas for the case ofDarcy flow regime it seems quite reasonable formany kinds of liquids, Darcy flow needsadditional estimations to find the field ofapplication.The Darcy flow regime is realized if Re Taking into account the velocity and theplume thickness scales (13) we haveare the modified Rayleigh numbers.RaPOINT SOURCELINE SOURCEBut from the relations (10) it follows thata, a , Da The dispersion is predominant ifThese estimations are compatible ifwhich can be realised for oils and chemicalsolutions.With the procedure given in Sec.3.1 one canobtain the next scales of flow (for the safe ofsimplicity we suggest the mixing lengths for theheat and the mass dispersion are the same. But itis not difficultto carry out the full analysis):(inertial regime)POINT SOURCELINE SOURCEDarcyflowTable 1 Transport scales for molecular diffusion only(inertial regime)Mass transfer drivenflowFor these flows the buoyancy effect due tovariations in solute concentration is dominantin the momentum equation).New scales we obtain from the momentum andthe constituent conservation equations, keeping inmind that the scale length of the flow isaHaH(1Table 2 Scales for dispersion only4.SIMILARITY SOLUTIONScale analysis allows an understanding of the roleof the main parameters controlling thephenomenon, and is very useful for experimentaldata processing and finding possible self-similartransformationsin the governing equations.linesourceThe similarity solutions for the particular' casewhere dispersiondominates are now presented.The energy and the constituent conservationequationstake the formHere N -is the 'buoyancy' ratio.3.2. HydrodynamicdispersionIn this section we will consider that thehydrodynamic dispersion supersedes the203
F'inertialflowThese equations coupled with the momentum(9) (ormass conservationequationequation (4) and the boundary conditions (5) - (8)form the complete set of equations.be the generalizedFurthermore let'temperature'(F'Q')3where primes denote differentiation with respecttoThese systems are subject to the boundaryconditionsThen the problem of interest will be formulated asand the generalizedconservation constraintW 00These equations have the following analyticalsolutionsfor a line sourceDarcyflow1inertialflowThe foregoing boundary layer scales suggest thefollowing similarity transformations0for Darcy flowfor a point source00Darcy flowfor inertial flowwith the independent variables x / l and stream functionwhere A 3Using these variables andEqs(14) - (20) reduce to the following sets of theordinary differentialequations:Darcyflown l4204 0.7036 ;andarethe Bessel functions of the first kind, respectively,is the firstof zeroth and first-order;root of the functionThe distinctive feature of the solution of thestatement under discussion is that the temperature(concentration) perturbation is localized with
sources when one can usedistancesanalytical solutionsspace. This is common with non-linear problemsof heat conduction (Landau1987).From Eqs (21)-(28) it follows that the temperature(concentration) difference falls on the plume ofthe terminal thickness:Darcyflow line source 0 3.401(29)It is well to bear in mind that these solutions canbreak down in the neighbourhood of the plumeborders for the conductivity component can be ofconsiderable importance in transfer processesbecause of small convective velocities.- point source (30)inertial flow- line source (31) 4.1655. CONCLUSIONS- point source (32)In this paper only the analysis of limitingsituations of the dominant effects of eithermolecular or dispersion diffusion mechanism ofcoupled heat and mass transfer from point or linesources embedded in a porous medium ispresented. The results of scale analysis havedemonstrated a key part of these mechanisms onthe heat and mass transfer laws and have enabledus to obtain new simple analytical solutions forthe case of predominant part of dispersion on theassumption of identical mechanism for heat andmass transfer.The plume has a parabolic form.When equations (29) and (30) are compared withequations (31) and (32) it is apparent that thequadratic resistance causes an increase in thetemperature (concentration)boundary layer.The temperature on the centre-linehas the form:point sourceThe limited space of the paper has not allowedpresentation of an analysis with coupled effectsboth of these mechanisms.6. REFERENCESline sourceAerov, M. E.Umnik, N. N. (1951). Heattransfer from the tubes with packed beds.Technicheskoi Phiziki , Vol.No. 11, pp.1364- 137 1 (inRussian).point sourceBejan, A. Khair, K. R. (1985). Heat and masstransfer by natural convection in a porousHeat Mass Transfer, Vol. 28,medium. Int.908-918.linesourceGhukhman, A.A. (1967). Application of similaritytransfer. Vischayatheory to heat andshkola, Moscow, 1967 (in Russian).jlowThe temperature and the concentration profilescan be obtained by reference to the definition ofthe generalized 'temperature' (16)F. C. (1990). Coupled heat and mass transferby natural convection from a horizontal linesource in saturated porous medium. Int.Heat Mass Transfer, Vol. 17, pp.Landau, L. D.E. M. (1987). Fluidmechanics. 2-nd ed. Pergamon Press. N -Y.The solutions obtained would hold ifThis condition gives the next estimations for the205Larson, S. E.Poulikakos, D. (1986). Doublediffusion from a horizontal line source in aninfinite porous medium.Heat MassTransfer, Vol. 29, No. 3, pp.492-495.Nield, D. A.Bejan. A. (1999). Convection inporous media. 2-nd ed., Springer-Verlag. N- Y.
Poulikakos, D. (1985). On buoyancy induced heatmass transfer from a concentrated source in aninfinite porous medium.J. Heat MassTransfer, Vol. 28, No.3, pp.621-629.Telles, R. S. & Trevisan, 0. V. (1993). Dispersionin heat and mass transfer natural convection alongvertical boundaries in porous media. Znt. J. HeatMass Transfer,Vol. 36, No. 5, pp. 1357-1365.Trevisan, 0. V.Bejan, A. (1990). Combinedheat and mass transfer by natural convection in aporous medium. Advances in Heat Transfer, Vol.315-352.20,Ranz, W. E. (1952). Friction and transfer coefficients for single particles and packed beds. Chem.Engng Progress, Vol. 48, No.5,247-253.N.Kaguei, S . (1982) Heat and masstransfer in packed beds. Gordon and Breach, N-Y.Ward, J. C. (1964). Turbulent flow in porousDiv., Vol. 90,media.206
Bejan, A. Khair, K. R. (1985).Heat and mass transfer by natural convection in a porous medium. Int. Heat Mass Transfer, Vol. 28, 908-918. Ghukhman,A.A. (1967).Application of similarity theory to heat and transfer. Vischaya shkola,Moscow, 1967 (in Russian). F. C. (1990).Coupled heatand mass transfer by n
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
Boiling water CONDUCTION CONVECTION RADIATION 43. Frying a pancake CONDUCTION CONVECTION RADIATION 44. Heat you feel from a hot stove CONDUCTION CONVECTION RADIATION 45. Moves as a wave CONDUCTION CONVECTION RADIATION 46. Occurs within fluids CONDUCTION CONVECTION RADIATION 47. Sun’s rays reaching Earth CONDUCTION CONVECTION RADIATION 48.
10 tips och tricks för att lyckas med ert sap-projekt 20 SAPSANYTT 2/2015 De flesta projektledare känner säkert till Cobb’s paradox. Martin Cobb verkade som CIO för sekretariatet för Treasury Board of Canada 1995 då han ställde frågan
service i Norge och Finland drivs inom ramen för ett enskilt företag (NRK. 1 och Yleisradio), fin ns det i Sverige tre: Ett för tv (Sveriges Television , SVT ), ett för radio (Sveriges Radio , SR ) och ett för utbildnings program (Sveriges Utbildningsradio, UR, vilket till följd av sin begränsade storlek inte återfinns bland de 25 största
Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid
LÄS NOGGRANT FÖLJANDE VILLKOR FÖR APPLE DEVELOPER PROGRAM LICENCE . Apple Developer Program License Agreement Syfte Du vill använda Apple-mjukvara (enligt definitionen nedan) för att utveckla en eller flera Applikationer (enligt definitionen nedan) för Apple-märkta produkter. . Applikationer som utvecklas för iOS-produkter, Apple .
convection, pure solutal convection, heat transfer driven flows, and mass transfer driven flows were taken into account. Mobedi et al.,8,9 analyzed double diffusive convection in partially heated cavities. Nikbakhti and Rahimi10 studied numerically the flow, heat and mass transfer in
xeach gene is a section of DNA with a specific sequence of bases that acts as the 'instructions' or code for the production of a specific protein. xthe human genome has 20 000-25 000 genes xthe average gene has about 3000 bases xthe genes make up only 2%* of the human genome; the rest of the DNA is
