T. Raja Rani. Military Technological College, Muscat, Oman. Abstract

Transcription

1 ISSN: 78-8 Vol. Issue, Octobe - Fee Convection ove a Vaying all Vetical Cylinde embedded in a Poous medium ith effect of Radiation, Vaiable Fluid Popeties and Statification. T. Raja Rani Militay Technological College, Muscat, Oman Abstact The aim of the pesent pape is to study fee convection bounday laye flos ove a vetical cylinde hich is embedded in a poous medium ith effects of Radiation, vaiable viscosity, vaiable themal conductivity and statification ae studied numeically. Tempeatue of the lateal suface of the cylinde as ell as the ambient tempeatue is assumed to be linea functions of the axial coodinate. A similaity tansfomation is used to educe patial diffeential equations govening the poblem into odinay diffeential equations and equations ae solved numeically subject to bounday conditions by the use of Runge-Kutta-Gill method ith shooting technique. Unde the assumption of stable statification similaity solutions ae pesented fo the poblem. Solutions fo an unstatified medium ae obtained as a special case by assigning zeo value to the statification paamete S. Qualitatively inteesting gaphs ae discussed as functions of paametes of the poblem.. Intoduction The study of fee convection fom the oute suface of a heated vetical cylinde embedded in a fluid satuated poous medium has impotant geophysical and engineeing applications (efe Nield and Bejan [7]). As quoted in the efeence, such studies ill aid in the assessment and evaluation of geothemal esouces duing geophysical exploation. Futhe, the heat tansfe coefficients of such a study ill be useful in estimating the cooling ate of intusive bodies and consequently the life span of a geothemal esevoi. It ill also be useful to calculate the heat loss fom undegound casing and piping systems fo the optimum design of geothemal poe plants. In many pactical situations the egion of flo and the hole configuation ill be confined to a finite egion of space and so the heated bounday laye eventually hits the ceiling. This heated laye emains at the top of gadually colde layes and this leads to themal statification (efe Bejan[ ]). Minoycz and Cheng [6] discussed fee convection at a vetical cylinde embedded in a poous medium. Cheng and his co-oes, Cheng and Minoycz [4], Cheng and Chang [] have obtained similaity solutions fo fee convection in a poous medium adjacent to vetical and hoizontal plates ith all tempeatues being a poe function of distance. Hossain and Naayamma [5] discussed non- Dacy fee convective flo along a vetical cylinde embedded in a poous medium ith suface mass flux. Hoeve, in the above studies the effect of stable ambient statification on heat and mass tansfe as not discussed. It is non that stable ambient statification phenomenon usually occus in cooling ponds, laes, sola ponds and atmosphee. In vie of this, seveal eseaches have discussed the effect of statification on diffeent heat tansfe studies. Taha.et.al.,[] discussed natual convection at vetical cylinde in a themally statified high poosity medium. Saha and Hossain [9] discussed the natual convection flo ith combined buoyancy effects due to themal and mass diffusion in themally statified media. Angiasa and Sinivasan [] studied the natual convection flos due to the combined buoyancy of heat and mass diffusion in a themally statified medium. Raja Rani and CNB. Rao [8] studied the effects of vaiable fluid popeties on fee Convection Flos and heat tansfe at an isothemal vetical plate embedded in poous medium in the pesence of magnetic field and adiation. In the pesent pape the fee convection flo about a vetical cylinde embedded in a satuated poous medium is discussed. Hee the tempeatues on the suface of the cylinde as ell as the ambient tempeatue ae assumed to vay linealy ith x, hee x is the distance fom the leading edge of the cylinde. It is also assumed that ambient tempeatue is less than the tempeatue of the cylinde and, as a esult, stable statification occus. Hoeve in an unstatified medium, the ambient tempeatue can be loe o highe than the tempeatue of the cylinde. In the pesent poblem possible cases of hot cylinde and cold cylinde ae consideed in the case of an unstatified medium. The effects of adiation, statification, IJERTVIS55 745

2 ISSN: 78-8 Vol. Issue, Octobe - vaiable viscosity and vaiable themal conductivity on fee convection at the vetical cylinde ae discussed. Because of linea vaiation of tempeatues of the cylinde and that of the ambient ith the axial coodinate, similaity solutions exist fo the poblem.. Fomulation and Solution Let a vetical cylinde of adius a immesed in a satuated poous medium (the physical model is shon in fig.). Oigin is chosen at the cente of cylinde at its leading edge hee the bounday laye thicness is zeo. The axial coodinate ( x ) is taen vetically along the length of the cylinde and the adial coodinate () pependicula to it. Tempeatue of the suface of the cylinde T is assumed to vay linealy ith the distance x fom its leading edge. The tempeatue of the ambient fluid T is also assumed to vay as a linea function of x. Stable statification occus unde the assumption thatt T (hot plate). Tempeatue of the fluid at the leading edge is taen as T hich is the minimum tempeatue of the configuation. Viscosity of the fluid and effective themal conductivity of the fluid satuated poous medium is assumed to vay as linea functions of tempeatue. Othe fluid popeties ae assumed to be constant except fo density hich is taen to be a function of tempeatue only in the body foce tem one of the momentum equations. P Ps x P v K T T c p u v x Bounday Conditions ae: at u K g () a, v, T T s T m T q Ax, x () (4) as, u, T T T Bx, x (5) hee u, v ae fluid velocity components, T is fluid tempeatue, K is Pemeability, is effective themal conductivity of the poous medium and is the dynamic viscosity, q is adiative heat flux and the Rosseland appoximation used in the enegy equation to descibe the themal adiative heat tansfe. by q In fact the adiative heat flux 4 s e T y 4 m q, hee s the is given Stefan- Boltzmann s constant and e is the mean absoption coefficient. Assuming tempeatue diffeences ithin the flo to be sufficiently small, the tem q y 4 equation (4) is simplified by expanding T into the Taylo seies aboutt, and neglecting highe ode tems. It may be noted that because of the use of Rosseland appoximation, the pesent analysis gets limited to optically thic fluids. of The govening equations of the Dacy model bounday laye fee convection flo in cylindical coodinates ae as follos: v u x () Taing ( T T )] in the [ body foce tem, and eliminating fluid pessue fom equations () and (), the govening equations ae obtained as T T T q c p u v m (6) x A steam function is intoduced though the elations IJERTVIS55 746

3 ISSN: 78-8 Vol. Issue, Octobe - u & v (7) x Intoducing non- dimensional vaiables, f, and paametes o though the elations K g A m m f m a K g A T T T T f e F 4 T 4 st Rd K g A f s e a a ax f ( ) The govening equations in non-dimensional fom ae obtained as f f Rd Rd f 4 Bounday conditions in tems of f and ae: at, as, f f, S, (8) (9) f () () becomes a stably statified medium. Hoeve S = coesponds to an unstatified medium in hich case, T need not be geate thant. The case hen the paametes, tae non-zeo values is efeed to as the vaiable fluid popety case, o in shot VFP case. The case hen the paametes, tae zeo values is efeed as the constant fluid popeties case, o in shot CFP case. When, tae non-zeo values, fou cases aise (i), (ii), (iii), (iv), ( Refe [8]). As a esult, depending on the numeical values of and, the esults become applicable to cetain fluids in specific anges of tempeatues.. SOLUTION OF THE PROBLEM.. Paametes of the poblem The paametes of the pesent poblem ae S, Rd,,,. The paamete S is the statification paamete. As pointed out ealie, this paamete assumes values beteen & and hence the medium ill be a stably statified medium. S= coesponds to an unstatified medium. Rd is the adiation paamete. Zeo value fo the paamete Rd coesponds to the case hen the heat enegy though adiation fom the fluid is neglected. Positive numeical values of Rd coespond to intensity of themal adiation fom the fluid. Solutions ae found fo the values of,.5 and of the paamete Rd. Themal adiation can causes thicening of the themal bounday laye and hence inceasing values of the paamete Rd can incease themal bounday laye thicness. In this pape solutions ae found fo the values of -, and of both. Calculations ae done fo the value &,.5 fo S. Hee is the cuvatue paamete. The value infinity coesponds to flat plate case. Calculations ae done fo the values.5,.5 and of. B hee S is the statification paamete. A Because of the assumption thatt T, it follos that S and the fluid satuated poous medium.. Numeical Solution Equations (9) and () ae solved subject to the conditions () by the use of Runge-Kutta Gill method togethe ith a shooting technique. Appopiate esults of the pesent analysis ae IJERTVIS55 747

4 ISSN: 78-8 Vol. Issue, Octobe - compaed ith the esults of Mynoycz & Cheng [6]. Some of the qualitatively and quantitatively inteesting esults ae pesented though figues to and though table... DISCUSSION OF THE RESULTS Vaiations in sin fiction f () ae pesented in figues and ; coesponding vaiations in all heat tansfe ate ( ()) ae pesented in figues 4 and 5 espectively. Sin fiction assumes negative values fo all values of the paametes hen 5. As the adiation paamete taes. inceasing values, absolute values of the sin fiction can be seen to diminish. The diminishing tend is moe ponounced in an unstatified medium athe than in a statified medium. Fo fluids lie Methyl chloide absolute value of sin fiction in the vaiable fluid popety (VFP) case ill be lage than that in the constant fluid popety (CFP) case. Hoeve fo fluids lie Dichloo Difluo methane, opposite is the behaviou of sin fiction. Fom figue, it may be noted that sin fiction doesn t vay ith hen adiation paamete taes a constant value. The values hoeve change ith changing values of Rd. The vaiation in sin fiction ith changing values of the statification paamete and ith changing values of ealie., is as pointed out Fom figues 4 and 5, all heat tansfe ate can be obseved to tae positive values fo all values of the paametes. ' ()' assumes lage numeical values in the absence of statification than in its pesence. ' ()' Diminish ith inceasing values of the adiation paamete Rd. Fo small values of Rd, fo fluids lie Dichloo Difluo methane the heat tansfe coefficient in the VFP case assumes smalle values than in CFP case. Opposite is the tend fo fluids lie Methyl chloide. Fom Figue 5, it may be noted that, lie sin fiction, the heat tansfe coefficient also doesn t change ith changing values of fo a fixed value of Rd. IJERTVIS55 748

5 ISSN: 78-8 Vol. Issue, Octobe - medium than in an unstatified medium. Also Themal Bounday laye thicness is lage in a statified medium than in an unstatified medium. Vaiations in fluid velocity ( f ( ) ) ae pesented in figue.6. Reduced velocities can be obseved in a stably statified medium than in an unstatified medium. Velocities nea the plate ae elatively lage in an unstatified medium than in a stably statified medium. Effect of vaiable fluid popeties on fluid velocity is moe significant in unstatified medium than in a stably statified medium. Vaiations in fluid tempeatue ae pesented in figues 7 and 8. Effect of inceasing values of Rd and inceasing values of on fluid tempeatue is to incease tempeatue in the bounday laye egion nea the cylinde and, effect inceasing values of S is to diminish the fluid tempeatue. Fom figue 8, in a stably statified medium, fo fluids lie Dichloo Difluo methane tempeatue in the VFP case is lage than the one in the CFP case. Hoeve fo fluids lie Methyl chloide tempeatue in the VFP case is smalle than the one in the CFP case. Tempeatue in the vicinity of the cylinde is smalle in a stably statified Vaiations in steam function ae pesented in figues 9 and. As can be expected, steam function stats fom zeo value at the suface of the cylinde and asymptotically appoaches a constant value fa aay fom it. Fom figue 9, both in an unstatified medium and in a stably statified medium, ith inceasing intensity of adiation and ith inceasing values of, steam function assumes lage numeical values. The values of steam function ae smalle in a stably statified medium than in an unstatified medium. IJERTVIS55 749

6 ISSN: 78-8 Vol. Issue, Octobe - Fom figue it may be noted that, the vaiations in the values of the steam function beteen CFP and VFP cases ae significant and they can be intepeted in an appopiate fashion hen S= and hen S=.5. When S=.5, fo fluids lie Dichloo Difluoo methane steam function assumes lage values in the VFP case than in the CFP case hile fo fluids lie Methyl chloide opposite is the behaviou beteen VFP and CFP case Conclusions:. Sin fiction assumes negative values fo all values of the paametes hen 5. As. the adiation paamete taes inceasing values, absolute values of the sin fiction diminish. The diminishing tend is moe ponounced in an unstatified medium athe than in a statified medium.. Fo fluids lie Methyl chloide absolute value of sin fiction in the vaiable fluid popety (VFP) case ill be lage than that in the constant fluid popety (CFP) case. Hoeve fo fluids lie Dichloo Difluo methane, opposite is the behaviou of sin fiction.. Velocities nea the cylinde ae elatively lage in an unstatified medium than in a stably statified medium. Effect of vaiable fluid popeties on fluid velocity is moe significant in unstatified medium than in a stably statified medium. 4. Effect of inceasing values of Rd and inceasing values of on fluid tempeatue is to incease tempeatue in the bounday laye egion nea the cylinde and, effect inceasing values of S is to diminish the fluid tempeatue. 5. When S=.5, fo fluids lie Dichloo Difluoo methane, the steam function assumes lage values in the VFP case than in the CFP case hile fo fluids lie Methyl chloide opposite is the behaviou beteen VFP and CFP cases. Acnoledgements Raja Rani ishes to than the authoities of Militay College of Technology, Muscat fo thei encouagement and also fo poviding the facility fo eseach. IJERTVIS55 75

7 ISSN: 78-8 Vol. Issue, Octobe - Refeences [] Angiasa, D., Sinivasan, J., Natual Convection flos due to the Combined Buoyancy of Heat and Mass Diffusion in a Themal Statified Medium, Int. J. of Heat Tansfe,,989, [] Bejan, Adian., Convection heat tansfe, A Wiley-Inte Science Publication, John Wiley and Sons, INC, II- Edition,995. [] Cheng, P. Dee Chang, I., Buoyancy induced flos in a poous medium Adjacent to impemeable hoizontal sufaces, Int.J.Heat Mass Tansfe, 9, 976, [4] Cheng, P., and Minoycz, W.J, Fee Convection about a Vetical flat Plate Embedded in a Poous Medium ith application to Heat tansfe fom a Die, J.Geophys. Res, 8, 977, [5] Hossain, M.A. and Naayama, A.C., Non-Dacy Fee Convective flo along a Vetical Cylinde Embedded in a Poous medium ith Suface Mass Flux, Int.J.Heat Fluid Flo, 4, [6] Minoycz, W.J., and Cheng, P., Fee Convection about a Vetical Cylinde Embedded in a Poous Medium, Int.J.Heat Mass Tansfe, 9, 976, [7] Nield, D.A, Bejan, A., Convection in Poous Media, Thid ed., Spinge, Ne Yo, 6. [8] Raja Rani.T, C.N.B.Rao., Effects of Vaiable Fluid popeties on Fee Convection Flos and Heat Tansfe at an Isothemal Vetical plate Embedded in Poous Medium in the pesence of Magnetic field and Radiation, Intenational Jounal of Advanced Reseach in Compute science and Softae Engineeing, (5),, -8. [9] Saha,S.C., Hossian, M.A., Natual Convection flo of a Combined Buoyancy Effects due to Themal and Mass Diffusion in a Themally Statified Media, Nonlinea Analysis: Modelling and Contol., 9(), 4, 89-. [] Taha, H.S., Chamha, A.J., Nath,G., Natual Convection on a Vetical Cylinde Embedded in a Themally Statified High Poosity Media, Int.J.Themal Sci., 4,, IJERTVIS55 75