EFFECTS OF THERMAL STRATIFICATION ON FLOW AND HEAT TRANSFER DUE TO A STRETCHING CYLINDER WITH UNIFORM SUCTION/INJECTION

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1 Iteratioal Joural of Eergy & Techology ISSN X Iteratioal Joural of Eergy & Techology (4) (010) 1 7 EFFECTS OF THERMAL STRATIFICATION ON FLO AND HEAT TRANSFER DUE TO A STRETCHING CYLINDER ITH UNIFORM SUCTION/INJECTION Ali J. Chamkha*, M.M. Abd El-Aziz, Sameh E. Ahmed** *Maufacturig Egieerig Departmet, The Public Authority for Applied Educatio ad Traiig, Shuweikh 70654, Kuwait, achamkha@yahoo.com Departmet of Mathematics, South Valley Uiversity, Qea, Egypt **Departmet of Mathematics, South Valley Uiversity, Qea, Egypt, sameh_sci_math@yahoo.com ABSTRACT I this paper, a umerical solutio of flow ad heat trasfer outside a stretchig permeable cylider with thermal stratificatio ad suctio/ijectio effects is ivestigated. The goverig system of partial differetial equatios is coverted to ordiary differetial equatios by usig suitable similarity trasformatios which are the solved usig a umerical techique. A parametric study is performed to study the effects of the goverig parameters, amely the suctio/ijectio parameter, thermal stratificatio parameter, Pradtl umber ad Reyolds umber o the velocity profiles, pressure distributio ad temperature profiles as well as the ski-frictio coefficiet ad the Nusselt umber. The velocity, pressure ad temperature results are show graphically whereas the values of the ski-frictio coefficiet ad the Nusselt umber are preseted i tables. It is foud that water is a better coolig aget compared to air provided that there is o strog ijectio. Keywords: Heat trasfer, suctio/ijectio, stretchig cylider, thermal stratificatio. 1. INTRODUCTION I may mixed flows of practical importace i ature as well as i may egieerig devices, the eviromet is thermally stratified. The discharge of hot fluid ito eclosed regios ofte results i a stable thermal stratificatio with a lighter fluid overlyig a deser fluid. The thermal stratificatio effects of heat trasfer over a stretchig surface is of iterest i polymer extrusio processes where the object, after passig through a die, eters the fluid for coolig below a certai temperature. The rate at which such objects are cooled has a importat bearig o the properties of the fial product. I the process of coolig the fluids, the mometum boudary layer for liear stretchig of sheet was first studied by Crae [1]. I additio, the study of heat trasfer a roud cyliders have received cosiderable attetio from the researchers. Ahmed et al. [] preseted a umerical study of heat trasfer i a coical aular cylider fixed with saturated porous medium. The problem of lamiar atural covectio boudary layer flow o a isothermal vertical thi cylider embedded i a thermally stratified high porosity medium is studied by Takhar et al. [3]. They foud that, the ski frictio coefficiet vaishes for certai values of the stratificatio parameter or the curvature parameter. Nguye et al. [4] ivestigated heat trasfer from a rotatig circular cylider immersed i a spatially uiform, time-depedet covective eviromet. Hassaie et al. [5] developed a umerical procedure, based o Chebyshev polyomials for the steady micropolar fluid i the viciity of a axisymmetric stagatio poit flow o a cylider. Saeid [6] used the thermally o-equilibrium model to study the free covectio from a horizotal cylider immersed i porous media. The effect of temperaturedepedet viscosity o the atural covectio heat trasfer from a horizotal isothermal cylider of elliptic cross sectio was studied by Cheg [7]. Ishak et al. [8] discussed the effect of uiform suctio/blowig o flow ad heat trasfer due to a stretchig cylider. The study of steady flow i a viscous ad icompressible fluid outside of a stretchig hollow cylider i a ambiet fluid at rest was doe by ag [9]. The mai objective of this paper is to study thermal stratificatio ad uiform suctio/blowig effects o flow ad heat trasfer due to a stretchig cylider. Similarity trasformatios are used to trasform the partial differetial equatios to ordiary differetial equatios which the are solved umerically. The preset study represets a extesio of the work of ag [9] by icludig thermal stratificatio ad suctio/ijectio effects.. PROBLEM FORMULATION Cosider steady lamiar, icompressible flow caused by a stretchig circular tube of radius a i the axial directio i a fluid at rest as show i Fig. 1. The z -axis is measured alog the axis of the tube ad the r -axis is measured i the radial directio. It is assumed that the temperature variatio of the tube is take ito accout ad is also give by the power-law

2 Chamkha et al. / Iteratioal Joural of Eergy & Techology (4) (010) 1 7 temperature, Tw T Az where A ad are costats. The viscous dissipatio is eglected as it is assumed to be small. Uder these assumptios, the goverig equatios are. Fig. 1. Physical model ad coordiate system. w u u 0, z r r w w w 1 w w u, z r r r r u u 1 p u 1 u u w u, z r r r r r r T T T 1 T w u, z r r r r subject to the boudary coditios r a : u U, w w, T T r : (z) T (z) Az w 0, T T (z) (1 )T T (z) T0 Az 1 where U ca, w cz, c is a positive costat ad is a costat such that 0 ad 0 correspodig to mass suctio ad mass ijectio, respectively. I additio, is a costat which is the thermal stratificatio parameter such that 0 1. The costat is defied as the thermal stratificatio parameter ad is equal to m1 /(1 m1) of Nakayama ad Koyama [10] where m 1 is a costat. T 0 is a costat referece temperature say, T (0). The suffices ad deote surface ad ambiet coditios. Followig ag [9] we take the similarity trasformatio:, (r / a), u ca F( ) / w czf'( ), ( ) (T T ) /(T T ), (6) where a prime deotes ordiary differetiatio with respect to. Substitutig Eqs. (6) ito Eqs. () ad (4), we will get the followig ordiary differetial equatios: F F ReFF F 0, (7) 1 PrReF PrReF 0, (8) 1 where Re ca / is the Reyolds umber, 0 (1) () (3) (4) (5) Pr C / K is the Pradtl umber where K is thermal coductivity. The dimesioless boudary coditios (5) become F(1), F (1) 1, (1) 1, F 0, (9) 0 as. The pressure p ca ow be determied from Eq. (3) i the form p p p Re F ( ) F ( ), (10) c Physical quatities of iterest are the ski-frictio coefficiet Cf ad the Nusselt umber Nu which are defied as aq Cf, Nu. (11) w / K(T T ) Further, ad q are the shear stress ad the heat trasfer from the surface of the tube, respectively, ad they are give by w T, q K. r (1) ra r ra Substitutig Eqs. (6) ad (1) ito Eq. (11) yields Rez Cf F (1), Nu (1), (13) a 3. NUMERICAL METHOD The umerical procedure used here solves the two-poit boudary value problems for a system of N ordiary differetial equatios i the rage ( x, x1 ). The system is writte as ad the derivatives dyi fi(x, y1, y,..., yn) i 1,,..., N dx are evaluated by a procedure that evaluates the derivatives of y, y,..., at geeral poit x. Iitially, N boudary 1 yn values of the variable y i must be specified, some of which will be specified at x ad some at x 1. The remaiig N boudary values are guessed ad the procedure corrects them by a form of Newtoia iteratio. Startig from the kow ad guessed values of yi at x, the procedure itegrates the equatios forward to a matchig poit R, usig Merso s method. Similarly, startig from x 1 it itegrates backwards to R. The differece betwee the forward ad backward values of yi at R should be zero for a true solutio. The procedure uses a geeralized Newto method to reduce these differeces to zero, by calculatig correctios to the estimated boudary values. This process is repeated iteratively util covergece is obtaied to as specified accuracy. The tests for covergece ad the perturbatio of the boudary coditios are carried out i a mixed form, e.g., if the error estimate for yi iserror i, we test whether ABS(ERRORi ) ERR OR i ( 1 ABS(y i)). This umerical method employed is foud to be accurate ad suitable ad gives results very close to the results obtaid by ag [9]. Table 1 shows a compariso betwee the preset results ad the results obtaied by ag [9]. It ca be see that a very good agreemet betwee the results exists. 4. RESULTS AND DISCUSSION I order to get a clear isight of the physical problem, umerical results are displayed with the help of tables ad graphical illustratios. Represetative umerical results are

3 F' F' Chamkha et al. / Iteratioal Joural of Eergy & Techology (4) (010) preseted i Figs. -10 ad Tables 1-4 to illustrate the ifluece of several o-dimesioal parameters, amely Reyolds umber Re, Pradtl umber Pr, thermal stratificatio parameter ad suctio/ijectio parameter. For the referece case for the results Pr 0.7, , 0., ad Re 3.0. Table 1. Compariso of F (1) with ag [9] for impermeable surface case Re=0.5,,3.0 solid () dash dot() Re ag [9] Preset work Fig. 3. Velocity profiles for various values of Re 4.1 Velocity profiles It is clear from the mometum equatio (7) that the thermal stratificatio parameter ad the Pratdl umber Pr have o effect o the velocity profile as it is ucoupled from the eergy equatio (8). Fig. shows the velocity profiles for various values of the suctio/ijectio parameter. It is observed from this figure ad is true i all cases, that the velocity vaishes at some large distace from the surface of the tube ad it decreases with icreasig values of. I additio, Fig. 3 displays the effect of Reyolds umber Re o the velocity distributios. The Reyolds umber represets the relative sigificace of iertia effects compared to the viscous effects. Thus, as Re icreases, the velocity profiles i the boudary layer decrease. It is also observed from Figs. ad 3 that icreasig the suctio/ijectio parameter ad the Reyolds umber Re results i decreases i the hydrodyamic boudary-layer thickess. 4. Pressure distributios After the velocity distributio F ( ) is obtaied, the pressure distributio P i terms of ( P P )/ c ca be foud from Eq. (10). The umerical results for the pressure distributio are show i Figs. 4 ad 5 for various values of the suctio/ijectio parameter ad the Reyolds umber Re, respectively. All curves show that P P far away from the surface ( ). It is observed from Fig. 4 that icreasig the suctio or ijectio parameter produces reductios i the pressure distributio whereas Fig. 5 shows that icreasig the Reyolds umber results i higher pressure distributio. - - P P ) -1. c ( Fig.. Velocity profiles for various values of Fig. 4. Pressure distributio ( P P )/ c obtaied from Eq. (10) for various values of

4 F Chamkha et al. / Iteratioal Joural of Eergy & Techology (4) (010) Temperature distributios Re=0.5,1.5, P P ) c ( The physical model of the problem (Fig. 1) eglects the effect of gravitatio force or thermal buoyacy force i favor of the stretchig effect. This meas that the flow problem is ucoupled from the thermal oe. Represetative dimesioless temperature profiles are preseted for various values of the thermal stratificatio parameter, suctio/ijectio parameter ad the Pradtl umber Pr i Figs. 7-10, respectively. These figures clearly show that the fluid temperature decreases by icreasig either of, or Pr. I all cases, the temperature vaishes at some large distace from the surface ad therefore, gives some iformatio o the thermal boudary layer thickess. It is see from these figures that the thermal boudary layer thickess decreases sigificatly by icreasig either of, or Pr ad that this effect is compouded if they are icreased together. Fig. 5. Pressure distributio ( P P )/ c obtaied from Eq. (10) for various values of Re ad The similarity profiles F( ) surface() The similarity profiles F( ) for various values of Re ad are show i Fig. 6. It is clear from the boudary coditio (9) that F( ) startig from the value of ad icreases util takes the fial predicted value F( ) far away from the cylider surface (see Table ). Therefore, as icreases, the similarity profiles F( ) icrease. Moreover, it is predicted that icreasig the value of Re leads to decreases i the similarity profiles F( ) =0.,, Fig. 7. Temperature distributios for various values of Re=0.5,,3.0 solid dash dot Fig. 6. Similarity profiles F( ) for various values of Re ad 0. permeable surface solid dash () =0.,, Fig. 8. Temperature distributios for various values of ad

5 Chamkha et al. / Iteratioal Joural of Eergy & Techology (4) (010) Table. Values of the ski-frictio coefficiet F (1) for various values of Re ad 0. Permeable Re Fig. 9. Temperature distributios for various values of Table 3. Values of F( ) for various values of Re ad Re Permeable solid() dash () dot() Air (Pr=0.7) ater (Pr=7.0) Fig. 10. Temperature distributios for various values of Pr 4.5 Ski-frictio coefficiet Table presets some values of F (1) (which represets the ski-frictio coefficiet) for various parametric coditios [see Eq. (13)]. It is observed that all these values are egative. Physically, a egative sig of F (1) implies that the stretchig tube or cylider exerts a draggig force o the fluid ad a positive sig implies the opposite. I additio, Table shows that icreasig the value of the traspiratio parameter results i a icrease i the absolute values of the skifrictio coefficiet F (1). Moreover, as the Reyolds umber Re icreases, the absolute value of F (1) or the ski-frictio coefficiet icreases. Table 3 shows values of F( ) for various values of Re ad. It is see that F( ) decreases with Re ad icreases with Nusselt umber Tables 4-6 illustrate the ifluece of the Reyolds umber Re, Pradtl umber Pr, suctio/ijectio parameter ad the thermal stratificatio parameter o the rate of heat trasfer or Nusselt umber. It is predicted that icreasig the values of either of the Reyolds umber Re, suctio/ijectio parameter or the thermal stratificatio parameter produces icreases i the values of the Nusselt umber. Also, for the case of impermeable surface ad the case suctio coditio, icreasig the value of the Pradtl umber Pr causes the Nusselt umber to icrease but, for the ijectio case, as the Pradtl umber Pr icreases from 0. to 7.0, the Nusselt umber decreases from to 3593 (see Tables 4-6) Table 4. Values of Nusselt umber (1 ) for various values of Re ad Re Permeable

6 Chamkha et al. / Iteratioal Joural of Eergy & Techology (4) (010) Table 5. Values of Nusselt umber (1 ) for various values of Pr ad Pr Permeable Table 6. Values of Nusselt umber (1 ) for various values of ad Permeable CONCLUSION This paper focused o steady two-dimesioal flow of a icompressible fluid due to a stretchig permeable tube with uiform suctio/ijectio ad thermal stratificatio effects. The goverig equatios are trasformed ito self-similar equatios ad are solved umerically by a itegratio techique. A parametric study of the physical parameters such as the thermal stratificatio parameter, Pradtl umber, Reyolds umber ad the suctio/ijectio parameter is coducted to illustrate the ifluece of these parameters o the flow ad heat trasfer characteristics. From this ivestigatio, we ca draw the followig coclusios: 1. The fluid velocity ad the hydrodyamic boudarylayer thickess decreased as either of the suctio/ijectio parameter or the Reyolds umber icreased.. The pressure distributio reduced as the suctio/blowig parameter was icreased. 3. The fluid temperature ad the thermal boudary-layer thickess decreased sigificatly by icreasig either of the suctio/ijectio parameter, thermal stratificatio parameter or the Pradtl umber. The ski-frictio coefficiet icreased with icreasig values of either the Reyolds umber or the suctio/ijectio parameter while it remaied costat with icreasig values of the thermal stratificatio parameter ad Pradtl umber. 4. The rate of heat trasfer icreased by icreasig either of the suctio/ijectio parameter, thermal stratificatio parameter or the Pradtl umber. Nomeclature A a c C C f p costat radius of cylider positive costat ski frictio coefficiet specific heat of the fluid F similarity variable K thermal coductivity thermal stratificatio parameter Nu Nusselt umber P pressure Pr Pradtl umber Re Reyolds umber T dimesioal temperature u,w dimesioal velocity compoets r,z dimesioal coordiates Greek symbols thermal diffusivity dimesioless coordiate dyamic viscosity kiamatic viscosity desity suctio/ijectio parameter dimesioless temperature Subscript Coditios at wall Coditios far away from wall 6. REFERENCES [1] L. J. Crae, Z. Agew, Secod-order fluid flow past a stretchig sheet with heat trasfer, Math. Phys. Vol. 1, pp , [] N. J. S. Ahmed, I. A. Badruddi, Z. A. Zaial, H. M. T. Khaleed, J. Kaesa, Heat trasfer i a coical cylider with porous medium, It. J. Heat Mass Trasfer, vol 5, pp , 009. [3] H. S. Takhar, A. J. Chamkha, G. Nath, Natural covectio o a vertical cylider embedded i a thermally stratified high-porosity medium, It. J. Therm. Sci. vol, 41, pp , 00. [4] H. D. Nguye, S. Paik, R.. Douglass, Usteady mixed covectio about a rotatig circular cylider with small fluctuatios i the free-stream velocity, vol. 39, pp , [5] A. Hassaie, H. M. El-hawary, A. A. Salama, chebyshev solutio of axisymmetric stagatio flow o a cylider, Eergy Covers. Mgmt. vol. 37, pp , 1996.

7 Chamkha et al. / Iteratioal Joural of Eergy & Techology (4) (010) [6] N. H. Saeid, Aalysis of free covectio about a horizotal cylider i a porous media usig a thermal o-equilibrium model, It. Commu. Heat MassTrasfer vol. 33, pp , 006. [7] C-Y. Cheg, The effect of temperature- depedet viscosity o the atural covectio heat trasfer from a horizotal isothermal cylider of elliptic cross sectio, It. Commu. Heat MassTrasfer vol. 33, pp , 006. [8] A. Ishak, R. Nazar, I. Pop, Uiform suctio/blowig effect o flow ad heat trasfer due to a stretchig cylider, Appl. Math. Modell. Vol. 3, pp , 008. [9] C. Y. ag, Fluid flow due to a stretchig cylider, Phys. Fluids vol. 31, , [10] A. Nakayama, H. Koyama, Similarity solutios for buoyacy iduced flows over a o isothermal curved surfacei a thermally stratified porous medium, Appl. Sci. Res. 46 (1989) THE MATERIAL ITHIN THIS PAPER, AT THE AUTHOR S (AUTHORS ) RESPONSIBILITY, HAS NOT BEEN PUBLISHED ELSEHERE IN THIS SUBSTANTIAL FORM NOR SUBMITTED ELSEHERE FOR PUBLICATION. NO COPYRIGHTED MATERIAL NOR ANY MATERIAL DAMAGING THIRD PARTIES INTERESTS HAS BEEN USED IN THIS PAPER, AT THE AUTHOR S (AUTHORS ) RESPONSIBILITY, ITHOUT HAVING OBTAINED A RITTEN PERMISSION.