Free Convection Along a Vertical Wavy Surface in a Nanofluid

Transcription

1 Cleveland State Universit ETD Archive 01 Free Convection Along a Vertical Wav Srface in a Nanoflid Deepak Ravipati Cleveland State Universit Follow this and additional works at: Part of the Mechanical Engineering Commons How does access to this work benefit o? Let s know! Recommended Citation Ravipati, Deepak, "Free Convection Along a Vertical Wav Srface in a Nanoflid" (01). ETD Archive This Thesis is broght to o for free and open access b EngagedScholarship@CSU. It has been accepted for inclsion in ETD Archive b an athorized administrator of EngagedScholarship@CSU. For more information, please contact librar.es@csohio.ed.

2 FREE CONVECTION ALONG A VERTICAL WAVY SURFACE IN A NANOFLUID DEEPAK RAVIPATI Bachelor of engineering in Mechanical Engineering Achara Nagarjna Universit, India Jne, 008 Sbmitted in partial flfillment of reqirements for the degree MASTER OF SCIENCE IN MECHANICAL ENGINEERING at the CLEVELAND STATE UNIVERSITY December, 011

3 This thesis has been approved for the department of MECHANICAL ENGINEERING and the college of Gradate Stdies b Dr. Rama Sbba Redd Gorla (Thesis Chairperson) Department & Date Dr. Majid Rashidi Department & Date Dr. Asqo Ebiana Department & Date

4 ACKNOWLEDGEMENTS I wold like to thank m academic advisor and thesis committee chairperson Dr. Rama Sbba Redd Gorla for his invalable gidance, encoragement and time dring the period of this work. I appreciate his knowledge in the field and his willingness to share it with me b engaging in new research projects. His appetite for research and his kind and helpfl natre alwas inspire me to work hard. I wold like to thank m thesis committee members Dr. Asqo B. Ebiana and Dr. Majid Rashidi for their encoragement and valable time to review m thesis. I sincerel appreciate love and help of all m friends speciall Kastbh, Lokamana, Sowma Redd, Rama Sree, Prab and Uda who encoraged and helped me a lot in downtimes dring this phase of m life. I wold like to thank m mother, father and brother for allowing me to realize m own potential. I am thankfl to almight God for his grace, which is more than enogh for each new da!

5 FREE CONVECTION ALONG A VERTICAL WAVY SURFACE IN A NANOFLUID DEEPAK RAVIPATI ABSTRACT The std of this paper is to introdce a bondar laer analsis for the flid flow and heat transfer characteristics of an incompressible nanoflid along a vertical wav srface in a nanoflid. The Reslting transformed governing eqations are solved nmericall b an implicit finite-difference scheme (Keller-Bo method). The reslts are presented for the major parameters inclding the wave amplitde, boanc ratio parameter, Brownian motion parameter, Thermophoresis parameter and Lewis nmber. A sstematic std on the effects of the varios parameters of the local frication factor, srface heat transfer rate (Nsselt nmber) and mass transfer rate (Sherwood nmber) characteristics is carried ot. The Obtained reslts are presented graphicall. Kewords: Nanoflid, Free Convection and Wav Srface. iv

6 TABLE OF CONTENTS ABSTRACT.. iv LIST OF FIGURES...vi NOMENCLATURE. viii CHAPTER I. INTRODUCTION...1 II. III. LITERATURE REVIEW...5 NUMERICAL STUDY OF TWO DIMENSIONAL FREE CONVECTION ALONG A VERTICAL WAVY SURFACE IN A NANOFLUID 3.1 Mathematical formlation and analsis Nmerical soltions Reslts and discssions.. IV. CONCLUDING REMARKS. 40 REFERENCES...41 v

7 LIST OF FIGURES Figre Description Page 1. Phsical model and coordinate sstem..16. Effect of on velocit profiles Effect of on temperatre profiles.5 4. Effect of on concentration profiles Effect of on velocit profiles Effect of on temperatre profiles.7 7. Effect of on concentration profiles Effect of on velocit profiles Effect of on temperatre profiles Effect of on concentration profiles Effect of on velocit profiles Effect of on temperatre profiles Effect of on concentration profiles Effect of α on velocit profiles Effect of α on temperatre profiles Effect of α on concentration profiles Effect of on skin- frication coefficient Effect of on heat transfer rate Effect of on mass transfer rate vi

8 0. Effect of on skin- frication coefficient Effect of on heat transfer rate.34. Effect of on mass transfer rate Effect of on skin- frication coefficient Effect of on heat transfer rate Effect of on mass transfer rate Effect of on skin- frication coefficient Effect of on heat transfer rate Effect of on mass transfer rate Effect of on skin- frication coefficient Effect of on heat transfer rate Effect of on mass transfer rate.39 vii

9 Brownian diffsion coefficient NOMENCLATURE Thermophoretic diffsion coefficient Lewis nmber Boanc Ratio Brownian motion parameter Thermophoresis parameter U C p Reference velocit Specific heat at constant pressre C f Local skin-friction coefficient f g Dimensionless stream fnction Acceleration de to gravit G r Grashof nmber k (T) Thermal condctivit Local Nsselt Nmber P r Prandtl nmber q w Heat fl at the srface T Temperatre of the flid in the bondar laer T Temperatre of the ambient flid T w Temperatre at the srface viii

10 , v The dimensionless and - component of the velocit û, v The dimensional and ŷ component of the velocit, In the direction along and normal to the srface Greek Smbols: Volmetric coefficient of thermal epansion Stream fnction Amplitde of the wav srface Non-dimensional similarit variable Shear stress Densit of the flid Reference kinematic viscosit Viscosit of the flid Dnamic viscosit of the ambient flid Dimensionless temperatre fnction () Srface profile fnction defined in (1) Nano-particle volme fraction Nano-particle mass densit Flid densit Heat capacit of flid i

11 Sbscript: W X Wall conditions Ambient temperatre Differentiation with respect to Sperscript: Differentiation with respect to

12 CHAPTER I INTRODUCTION The std of convective heat transfer in nanoflids is gaining a lot of attention. The nanoflids have man engineering applications in the indstr since materials of nanometer size have niqe phsical and chemical properties. The term nanoflids refers to a liqid containing a dispersion of sbmicron solid-liqid composite materials consisting of solid nanoparticles or nanofibers with sizes tpicall nm sspended in liqid. Modern technolog makes it possible to prodce ltra fine metallic or nonmetallic particles of nanometer dimensions, which makes revoltions in heat transfer enhancement methods. Considering a ver small particle size and their small volme fraction (<1% volme fraction) of C nanoparticles with ethlene glcol or carbon nanotbes dispersed in oil is reported to increase the inherentl poor thermal condctivit of the liqid b 40% and 150%, respectivel [1,]. Also the relativel large srface area of nanoparticles increases the stabilit and redces the sedimentation of nanoparticles. A more dramatic improvement in that transfer efficienc is epected a reslts decreasing the particle size in a sspension becase heat transfer takes place at the srface of the 1

13 particles. High concentrations (>10%) of particles is reqired to achieve sch enhancement in case of conventional particle-liqid sspensions. In general problems of rheolog and stabilit are amplified at high concentration, preclding the widespread se of conventional slrries as heat transfer flids. In some cases, the observed enhancement in thermal condctivit of nanoflids is the order of magnitde larger than predicted b well-established theories. Other perpleing reslts is that rapidl evolving field inclde a srprising strong temperatre dependence of thermal condctivit [3] and a three-fold higher critical heat fl compared with the base flids [4]. The feasibilit of nanoflids in nclear applications is made b improving the performance of an water-cooled nclear sstem with heat removal limited has been stdies b kim et al. [5] at the Massachsetts Institte of technolog(mit) is eploring the nclear applications of nanoflids, specificall in Main reactor coolant for pressrized water reactors (PWRs), Coolant for the emergenc core cooling sstem(eccs) of both PWRs and boiling water reactors and coolant for in-vessel retention of the molten core dring severe accidents in high powerdensit light water reactors.[6]. Nanoflids can be tilized where straight heat transfer enhancement is ver important as in most Heat transfer applications sch as indstrial cooling applications, smart flids, nclear reactors, etraction of geothermal power and other energ sorces. Other applications, sch as nanoflid coolant, nanoflid in fel, Brakes and other vehiclar nanoflids. Electronic applications sch as cooling of microchips, microscale flidic applications. Biomedical applications sch as nanodrg deliver, cancer therapetics, cropreservation, nanocrosrger, sensing and imaging. Review of eperimental stdies indicates that nanoflids have the potential to conserve 1 trillion Bt

14 of energ for U.S. indstr b replacing cooling and heating water with nanoflid. For the U.S. electric power indstr, sing nanoflids in closed-loop cooling ccles cold save abot trillion Bt per ear (eqivalent to the annal energ consmption of abot 50, ,000 hoseholds). The related emissions redctions wold be approimatel 5.6 million metric tons of carbon dioide; 8,600 metric tons of nitrogen oides; and 1,000 metric tons of slfr dioide [7]. There is a growth in the se of colloids which are nanoflids in the biomedical indstr for sensing and imaging prposes. This is directl related to the abilit to design novel materials at the nanoscale level, alongside recent innovations in analtical and imaging technologies for measring and maniplating nonmaterials. This has led to the fast development of commercial applications which ses a wide variet of manfactred nanoparticles. The prodction, se and disposal of manfactred nanoparticles will lead to discharges in air, soils and water sstems. Negative effects are likel, so qantification and minimization of these effects on environmental health is necessar. Tre knowledge of concentration and phsicochemical properties of manfactred nanoparticles nder realistic conditions is important to predicting their fate, behavior and toicit in the natral aqatic environment. The aqatic colloid and atmospheric ltrafine particle literatre both offer evidence as to the likel behavior and impacts of manfactred nanoparticles [8]. In geothermal power, energ etraction from the earth s crst involves high temperatres arond 5000C to 10000C, nanoflids can be emploed to cool the pipes eposed to sch high temperatres. When drilling, nanoflids can serve in cooling the machiner and eqipment working in environments with high temperatres. Nanoflids cold be sed as a working flid to etract energ from the earths core [9], However 3

15 previos research has concentrated on the problem of natral convection for the case of flat plates, on the other hand, few stdies have been carried ot to eamine the effect of geometric compleit, sch as irreglar srfaces, the convection heat transfer. That is becase complicated bondar conditions or eternal flow fields are difficlt to deal with. However, the prediction of heat transfer form an irreglar srface is of fndamental importance, and is encontered in several heat transfer devices, sch as flat-plate solar collectors and flat-plate condensers in refrigerators. For eample, in cavit wall inslating sstems, grain storage installations geothermal and indstrial applications, sch as the dispersion of chemical contaminants throgh water-satrated soil, the migration of moistre throgh air contained in fibros inslations heat radiator in indstr. Irreglarities freqentl occr in the process of manfactre. Moreover, srfaces are sometimes intentionall roghened to enhance heat transfer becase the presence of rogh srfaces distrbs the flow and alters the heat transfer rate. Despite of man investigations done in irreglar srfaces, these investigations have onl concentrated on the problems of free convection along a vertical wav srface in nanoflids. 4

16 CHAPTER II LITERATURE REVIEW Heat transfer problems have several engineering applications sch as thermal energ storage, crde oil etraction, geothermal energ recover, grond water polltion and flow throgh filtering media. Described below are a few stdies which show work done in the area of the geometric compleit, sch as irreglar srfaces, on the convection heat transfer in nanoflids. That is becase complicated bondar conditions or eternal flow fields are difficlt to work with. However, the prediction of heat transfer from an irreglar srface is of fndamental importance, and is encontered in several heat transfer devices, sch as flat-plate solar collectors and flat-plate condensers in refrigerators. Moreover, srfaces are sometimes intentionall roghened to enhance heat transfer for the presence of rogh srfaces distrbs the flow and alters the heat transfer rate. However, all of the previos stdies considered onl the case of flat plate or simple two-dimensional bodies, and few have been on wav srfaces. 5

17 Ghosh and Yao [10] have stdied laminar free convection along a semi-infinite vertical wav srface. This is a model problem for the investigation of heat transfer from roghened srfaces in order to nderstand heat transfer enhancement. In man applications of practical importance, however, the srface temperatre is non-niform. In this note, the case of niform srface heat fl rate, which is often approimated in practical applications and is easier to measre in a laborator, has been investigated. Nmerical reslts have been obtained for a sinsoidal wav srface. The reslts show that the Nsselt nmber varies periodicall along the wav srface. The wavelength of the Nsselt nmber variation is half of that of the wav srface, while the amplitde gradall decreases downstream where the bondar laer grows thick. It is hoped that eperimental reslts will become available in the near ftre to verif the reslts of this investigation. Molla, Hossain and Yao [11] investigated the effect of internal heat generation /absorption on a stead two-dimensional natral convection flow of viscos incompressible flid along a niforml heated vertical wav srface. The eqations are mapped into the domain of flat vertical plate, and then solved nmericall emploing the implicit finite difference method, known as Keller-bo scheme. Effects of the pertinent parameters, sch as the heat generation/absorption parameter (Q), the amplitde of the waviness (α) of the srface and Prandtl nmber Pr on the rate of heat transfer in terms of the local Nsselt nmber (N ), isotherms and the streamlines were discssed. Hossain and Rees [1] stdied the combined heat and mass transfer in natral convection flow from a vertical wav srface. In this paper, effects of combined boanc forces from mass and thermal diffsion b natral convection flow from a 6

18 vertical wav srface have been investigated sing the implicit finite difference method. The std was focsed on the evoltion of the srface shear stress, f (0), rate of heat transfer, g (0), and srface concentration gradient, h (0) with effect of different vales of the governing parameters, sch as the Schmidt nmber Sc ranging from 7 to 1500 which are appropriate for different species concentration in water (Pr = 7.0), the amplitde of the waviness of the srface ranging from 0.0 to 0.4 and the boanc parameter, w, ranging from 0.0 to 1. A previosl proposed transformation has been applied b Yao, L. S [13] to the natral-convection bondar laer along a comple vertical srface created from two sinsoidal fnctions, a fndamental wave and its first harmonic. The nmerical reslts demonstrate that the additional harmonic sbstantiall alters the flow field and temperatre distribtion near the srface. The conclsion that the averaged heat-transfer rate per nit-wetted wav srface is less than that of a corresponding flat plate has been confirmed for this more comple srface. On the other hand, the total heat-transfer rates for a comple srface are greater than that of a flat plate. According to the nmerical reslts the enhanced total heat-transfer rate seems to depend on the ratio of amplitde and wavelength of a srface. Neild and Kznestov [14] have stdied the convective heat transfer in a nanoflid past a vertical plate. According to this std a model was sed in which Brownian motion and thermophoresis are acconted with the simplest possible bondar conditions, namel those in which both the temperatre and the nanoparticle fraction are constant along the wall. Using this, soltion was obtained which depends on five dimensionless parameters, namel a Prandtl nmber, a Lewis nmber, a boanc-ratio 7

19 parameter, a Brownian motion parameter, and a thermophoresis parameter. The have eplored the wa in which the wall heat fl, represented b a Nsselt nmber N and then scaled in terms of - local Raleigh nmber defined in the paper to prodce a redced Nsselt nmber, depends on these five parameters. Chamkha and Khaled [15] have solved the problem of copled heat and mass transfer b natral convection from a semi-infinite inclined flat plate in the presence of an eternal magnetic field and internal heat generation or absorption effects. Srface of the plate has a power- law variation of both wall temperatre and concentration and is permeable to allow for possible flid wall sction or blowing. The reslting governing eqations were solved b sing similarit transformation and solved nmericall b implicit iterative and finite-difference scheme methods. The performed parametric std of all involved parameters and condcted a set of nmerical reslts for the velocit and temperatre profiles as well as the skin-friction parameter, average Nsselt nmber, and the average Sherwood nmber is illstrated graphicall to show tpical trends of the soltions. Wang and Chen [16] stdied mied convection bondar laer flows of Non- Newtonian flids over the wav srfaces sing the coordinate transformation and the cbic spline collocation nmerical method. The effects of the wav geometr, the boanc parameter and the generalized Prandtl nmber for psedoplastic flids, Newtonian flids and dilatant flids on the skin-friction coefficient, local and mean Nsselt nmbers have been graphicall stdied. As per the reslts both higher generalized Prandtl nmbers and boanc parameters were seen to enhance the inflence of wav srfaces on the local Nsselt nmber, irrespective of whether the flids 8

20 were Newtonian flids or non-newtonian flids. Moreover, the irreglar srfaces were having higher total heat fl than that of corresponding flats plate for an flid. Yang, Chen and Lin [17] applied Prandtl transformation method to std the natral convection of Non-Newtonian flids along a wav vertical plate in the presence of a magnetic field. A simple transformation was proposed, to transform the governing eqations into the bondar laer eqations, and solved nmericall b the cbic spline approimation. A simple coordinate transformation was emploed to transfer the comple wav srface to a vertical flat plate for a constant wall temperatre b the nmerical method. The effects of the magnetic field parameter, the wav geometr and the non-newtonian natre of the flids on the flow characteristics and heat transfer were discssed in detail. It was fond that the action of the magnetic field decelerates the flow, does decreasing the Nsselt nmber. Hossain and Pop [18] formlated the problem of the bondar laer flow and heat transfer on a continos moving wav srface in a qiescent electricall condcting flid with a constant transverse magnetic field. The reslting parabolic differential eqations were solved nmericall sing the Keller-bo scheme. The presented the detailed reslts for the velocit and temperatre fields, and also the reslts for the skin-friction coefficient and the local Nsselt nmber. Those reslts are given for different vales of the amplitde of the wav srface and magnetic parameter when the Prandtl nmber eqals 0.7. The showed that the flow and heat transfer characteristics are sbstantiall altered b both the magnetic parameter and the amplitde of the wav srface Hossain, Kabir and Rees [19] stdied the effect of a temperatre dependent viscosit on natral convection flow of viscos incompressible flid from a vertical wav 9

21 srface has been investigated sing an implicit finite difference method. The focsed on the evalation of the local skin-friction and the local Nsselt nmber. The governing parameters were, the Prandtl nmber, Pr, ranging from 1 to 100, the amplitde of the waviness of the srface, alpha, ranging from 0.0 to 0.4 and the viscosit variation parameter, epsilon, ranging from 0.0 to 6. Kmari, Pop and Takhar [0] considered a theoretical analsis of laminar freeconvection flow over a vertical isothermal wav srface in a Non-Newtonian power-law flid. First the casted the governing eqations into a non dimensional form b sing sitable bondar-laer variables that sbtract ot the effect of the wav srface from the bondar conditions. Then the solved the bondar-laer eqations nmericall b a ver efficient implicit finite-difference method known as the Keller-Bo method. A sinsoidal srface was sed to elcidate the effects of the power-law inde, the amplitde wavelength, and the Prandtl nmber on the velocit and temperatre fields, as well as on the local Nsselt nmber. The reslts obtained showed that the local Nsselt nmber varies periodicall along the wav srface. The wave-length of the local Nsselt nmber variation is half that of the wav srface, irrespective of whether the flid is a Newtonian flid or a Non-Newtonian flid. Lakshmi and Sibanda [1] In their article, stdied free convection of heat and mass transfer along a vertical wav srface in a Newtonian flid satrated Darc poros medim b considering cross diffsion (namel the Soret and the Dfor effects) in the medim. The vertical wav wall and the flow governing eqations are transformed to a plane geometr case b sing a sitable transformation. The then presented a similarit soltion to the problem nder the large Darc Raleigh nmber assmption. The 10

22 governing partial differential eqations were redced to a set of ordinar differential eqations that are integrated sing nmerical methods to std the natre of the nondimensional heat and mass transfer coefficients in the medim. The reslts are presented for a range of the flow governing parameters sch as the diffsivit ratio parameter, the boanc ratio parameter, the Soret parameter, the Dfor parameter and the amplitde of the wav srface. Hossain, Kabir and Rees [] in their paper presented the effect of a temperatre dependent viscosit on natral convection flow of viscos incompressible flid from a vertical wav srface has been investigated sing an implicit finite difference method. The have focsed their attention on the evalation of the local skin-friction and the local Nsselt nmber. The governing parameters are the Prandtl nmber, Pr, ranging from 1 to 100, the amplitde of the waviness of the srface, ranging from 0.0 to 0.4 and the viscosit variation parameter,, ranging from 0.0 to 6. In this std, Chen, Yang and Chang [3] sed Prandtl's transposition theorem to stretch the ordinar coordinate-sstem in certain direction. The small wav srface can be transferred into a calclable plane coordinate-sstem. The new governing eqations of trblent forced convection along wav srface were derived from complete Navier Stokes eqations. A simple transformation was proposed to transform the governing eqations into bondar laer eqations for soltion b the cbic spline collocation method. The effects sch as the wav geometr, the local skin-friction and Nsselt nmber were stdied. The reslts of the simlation show that it is more significant to increase heat transfer with small wav srface than plat srface 11

23 Kmar and Shalini [4] analzed the natral convection heat transfer from a vertical wav srface in a thermall stratified flid satrated poros medim nder Forchheimer based Non-Darcian assmptions. The redced the governing eqations to bondar laer eqations based on non-similarit transformation dedced b scale analsis. Those simplified partial differential eqations are solved nmericall b a finite difference scheme following the Keller Bo approach. Etensive nmerical simlations are carried ot for varios vales of wavelength-to-amplitde ratio of wav vertical srface at different thermal stratification levels of a poros medim both nder Darcian and Non-Darcian assmptions. Reslts of their std are compared with those available in literatre. In the Darcian case, local heat fles along the wav vertical srface are periodic with an oscillator pattern of period, which is eactl half of the period of a vertical wav srface. In the non-darcian case, the local heat fles contine to be periodic bt with a comple oscillator pattern of period eactl the same as that of the vertical wav srface. Increasing S or or leads to a fall in local Nsselt nmber. Polidori, Fohanno and Ngen [5] have stdied the problem of natral convection flow and heat transfer of Newtonian almina water nanoflids over a vertical semi-infinite plate from a theoretical viewpoint, for a range of nanoparticle volme fractions p to 4%. The analsis is based on a macroscopic modeling and nder the assmption of constant thermophsical nanoflid properties. The proposed semianaltical formlas of heat transfer parameters for both the niform wall temperatre (UWT) and niform heat fl (UHF) srface thermal conditions and fond that natral convection heat transfer is not solel characterized b the nanoflid effective thermal 1

24 condctivit and that the sensitivit to the viscosit model sed seemed ndeniable and plas a ke role in the heat transfer behavior. Mamn, Hossain and Gorla [6] investigated the natral convection laminar flow of a viscos incompressible flid along a niforml heated vertical wav srface with temperatre dependent viscosit and thermal condctivit. The considered the bondar laer regime when the Grashof nmber is large. The appropriate transformation variables were sed. The basic governing eqations are first transformed to non-similar bondar laer form and then solved nmericall emploing the implicit finite difference method together with Keller-Bo scheme. The presented the nmerical reslts in terms of the streamlines and isothermals as well as flid flow and heat transfer characteristics, namel, the local skin-frication coefficient and the local Nsselt nmber for a wide range of vales of the viscosit and thermal condctivit variation parameter, srface waviness and the Prandtl nmber Bachok, Ishak and Pop [7] have stdied the stead bondar-laer flow of a nanoflid past a moving semi-infinite flat plate in a niform free stream. Assmptions were made that the plate is moving in the same or opposite directions to the free stream to define the reslting sstem of nonlinear ordinar differential eqations. Governing eqations were solved sing the Keller-Bo method. The reslts are obtained for the skinfriction coefficient, the local Nsselt nmber and the local Sherwood nmber as well as the velocit, temperatre and the nanoparticle volme fraction profiles for the governing parameters, namel, the plate velocit parameter, Prandtl nmber, Lewis nmber, the Brownian motion parameter and the thermophoresis parameter. 13

25 Xan and Li [8] have eperimentall investigated flow and convective heat transfer characteristics for C water based nanoflids throgh a straight tbe with a constant heat fl at wall. Reslts showed that the nanoflids give sbstantial enhancement of heat transfer rate compared to pre water. Khan and Pop [9] have stdied the problem of laminar flid flow which reslts from the stretching of a flat srface in a nanoflid and investigated it nmericall. The model the sed for the nanoflid incorporates the effects of Brownian motion and thermophoresis and fond soltion which depends on the Prandtl nmber, Lewis nmber, Brownian motion nmber and thermophoresis nmber. The showed variation of the redced Nsselt and redced Sherwood nmbers with and for varios vales of and in tablar and graphical forms and conclde that a redced Nsselt nmber is a decreasing fnction while the redced Sherwood nmber is increasing fnction of each vales of the parameters,, and considered for std. Koo and Kleinstreer [30] have stdied stead laminar liqid nanoflid flow in microchannels for condction-convection heat transfer for two different base flids water and ethlene glcol having copper oide nanospheres at low volme concentrations. The conjgate the heat transfer problem for microheat-sinks solved nmericall. The emploed new models for the effective thermal condctivit and dnamic viscosit of nanoflids in light of aspect ratio, viscos dissipation and enhanced temperatre effects for comptation of the impact of nanoparticle concentrations in these two mitre flows on the microchannel pressre gradients, temperatre profiles and Nsselt nmbers. 14

26 CHAPTER III NUMERICAL STUDY OF FREE CONVECTION ALONG A VERTICAL WAVY SURFACE IN A NANOFLUID 3.1 Mathematical formlation and analsis We consider the stead free convection bondar laer flow past a vertical wav plate placed in a nano-flid. The co-ordinate sstem is chosen sch that -measres the distance along the wav srface and ŷ measres the distance normal otward, see figre (1). Here we consider a wav srface that is described nder the foregoing assmptions with introdcing Overbeck-Bossinesq approimations, the eqations governing the stead state flow can be written in two dimensional Cartesian coordinates(, ŷ ), in sal notation as 15

27 16 ŷ g L L w sin v û T w T Figre 1: Phsical model and coordinate sstem L w sin (1) where L is half of the wavelength of the wav srface. The geometr of the wav srface and the two-dimensional Cartesian coordinate sstem are as shown in Figre1. 0 v () Momentm eqation f f p f f g T T g p C - C 1 1 v (3) p f 1 v v v f v (4)

28 17 Concentration Eqation T T D C D C C T B v (6) where, are the dimensional coordinates in the vertical and horizontal directions, v, û are the velocit components parallel to,, g is the acceleration de to gravit, p is the pressre of the flid, (T) is the dnamic viscosit, f, and are the densit, viscosit, and volmetric volme epansion coefficient of the flid while p is the densit of the particles. B D Brownian diffsion coefficient and T D is the thermophoretic diffsion coefficient. The associated bondar conditions are: T T w w at 0, 0, v p p T T as, 0, where T w is the srface temperatre, T is the ambient temperatre of the flid. Following Yao (1983), we now introdce the following non-dimensional variables: p Gr L p Gr L Gr L L,,, 1 3 1/ 4 / 1 Energ Eqation T T D T C D T T T T B v (5)

29 18 C C C C T T T T Gr L W w,, 4 1/ v v (7) On introdcing the above dimensionless dependent variables into the eqations (1) (6) The following dimensionless form of the governing eqations are obtained at leading order in the Grashof nmber Gr 0 v (8) r N p Gr p 4 1/ 1 1 v (9) p Gr 4 1/ 1 1 v (10) B combining eqations (7) and (8) r N v (11) Energ Eqation Pr Pr 1 Pr 1 N N t b v (1) Concentration Eqation 1 T T D D v T B v (13) Now we introdce the following transformations to redce the governing eqation to a convenient form:

30 19,,,, 4 1/ 4 3/ f, (14) Where is the psedo similarit variable and is the stream-fnction that satisfies the eqation and is defined b v, (15) Introdcing the transformations given in eqation (14) into the eqations (11), (1) and (13) we have, ' ' '' ''' ' ' r N f ff f f f f ' ' f (16) ' ' ' r '' ' P 1 1 r t b r P N N P f f ' + ' (17) ' '' '' ' f P L N N f f e r b t ' (18) Where prime denotes differentiation with respect to and parameters are defined b

31 (19) The bondar conditions now take the following form:, 0 f,0 0,,0 1 f (0),, 0 f Soltions of the non-similar partial differential sstem given b 16, 17 and 18 sbject to the bondar conditions (0) are obtained b sing the implicit finite difference method developed b Keller (1978). However, once we know the vales of the fnction f and and their derivatives, it is important to calclate the vales of the local Nsselt nmber (N ), Sherwood nmber (, and skin-friction (C f ) from the following relations and (1) Where and () here n is the nit normal to the srface. Using the transformation (14) N and C f take the following forms: (3) (4) ( = 1,0 (5) 0

32 3.. Nmerical Soltions Keller Bo Method: The reslting nonlinear sstem of the partial differentials eqations is solved nmericall b the Keller-Bo method which is an implicit finite difference method. The method allows for non-niform grid discretion and converts the differential eqations into algebraic ones which are then solved sing Thomas algorithm. Thomas algorithm is essentiall the reslt of appling gass elimination to the tri-diagonal sstem of eqations. The nmber of grid points in both directions affects the nmerical reslts. To obtain accrate reslts, a mesh sensitivit std was performed. 1

33 3.3 Reslts and discssions The nonlinear ordinar differential Eqations (16-18) were solved nmericall to satisf the bondar conditions (0) for parametric vales of (Lewis nmber), (boanc ratio nmber), (Brownian motion parameter) and (Thermophoresis parameter) sing implicit finite difference method (Keller-Bo method). The effect of varios parameters on the free convection with heat and mass transfer along a vertical wav srface is eamined and discssed in this section. The parameters that the soltion affected are the. Figre -4 represents the effect of variations in the velocit, temperatre and concentration for vales of it is clear that as the Boanc ratio increases, the velocit decreases while the temperatre and concentration increases. So, we conclde that Boanc has a minor effect on the particle diffsion. Figre 5-7 represents the effect of variations in the velocit, temperatre and concentration for vales of, it is clear that as the thermophoresis parameter increases, the velocit, temperatre and concentration increases. plas a strong role in determining the diffsion of heat and nanoparticle concentration. Figre 8-10 represents the effect of variations in the velocit, temperatre and concentration for vales of, it is clear that as the Brownian motion increases, the velocit and temperatre increases while concentration decreases. Brownian diffsion promotes heat condction and also redces nanoparticle diffsion.

34 Figre represents the effect of variations in the velocit, temperatre and concentration for vales of. It is clear that as the increases, the velocit increases while the temperatre and concentration within the bondar laer decreases. is the ratio of moleclar thermal diffsivit to mass diffsivit. As increases, thermal diffsivit dominates. Figre indicates the effect of variations in the srface wave amplitde in velocit, temperatre and concentration. It is clear that as the increases, the velocit decreases while the temperatre and concentration increases. So, we conclde that mch roghness of the srface reslts in an increase of viscos effect within the bondar laer and hence, the flow and thermal fles redces. Figre represents the effect of variations in the local skin-frication coefficient,, Nsselt nmber and Sherwood nmber for vales of, it is clear that as the increases the Skin-frication coefficient, Nsselt nmber and Sherwood nmber decreases. So, we conclde that mch vale of the Boanc ratio, the Skin-frication coefficient, heat and mass transfer rate are decreasing. Figre 0- represents the effect of variations in the local skin-frication coefficient,, Nsselt nmber and Sherwood nmber for vales of, it is clear that as the increases, the Skin-frication coefficient, Nsselt nmber and Sherwood nmber decreases. So, we conclde that mch vale of Thermophoresis parameter, the Skin-frication coefficient, heat and mass transfer rate are decreasing. 3

35 Figre 3-5 indicates the effect of Brownian motion parameter on local skinfrication coefficient, Nsselt nmber and Sherwood nmber,, and it is clear that increasing the vale of increases the Skinfrication coefficient and Sherwood nmber, where as Nsselt nmber decreases. So, we conclde that mch vale of Brownian motion parameter, the Skin-frication coefficient, mass transfer rate increasing and heat transfer rate decreasing. Figre 6-8 indicates the inflence of Lewis nmber on local skin-frication coefficient,, Nsselt nmber, and Sherwood nmber )for vales of, and it is clear that as the increases the rate of heat transfer decreases, while the rate and amplitde of mass transfer rate and Skin-frication coefficient increases. Figre 9-31 represents the effect of variations in the srface wave amplitde on local skin-frication coefficient,, Nsselt nmber and Sherwood nmber, for vales of.we observe that, as the vale of increases,the Skin-frication coefficient, Nsselt nmber and Sherwood nmber decreases. So we conclde that mch roghness of the srface increasing, the Skinfriction coefficient, heat and mass transfer rate decreasing. The Brownian motion and thermophoresis of nanoparticles increase the effective thermal condctivit of the nanoflid. Both Brownian motion and thermophoresis give rise to cross diffsion terms similar to soret and dfor cross diffsion terms that arise with a binar flid. 4

36 Figre Effect of on velocit profiles Figre 3 Effect of on temperatre profiles 5

37 Figre 4 Effect of on concentration profiles Figre 5 Effect of on velocit profiles 6

38 Figre 6 Effect of on temperatre profiles Figre 7 Effect of on concentration profiles 7

39 Figre 8 Effect of on velocit profiles Figre 9 Effect of on temperatre profiles 8

40 Figre 10 effect of on concentration profiles Figre 11 Effect of on velocit profiles 9

41 Figre 1 Effect of on temperatre profiles Figre 13 Effect of on concentration profiles 30

42 Figre 14 Effect of α on velocit profiles Figre 15 Effect of α on temperatre profiles 31

43 Figre 16 Effect of α on concentration profiles Figre 17 Effect of on skin- frication coefficient 3

44 Figre 18 Effect of on heat transfer rate Figre 19 Effect of on mass transfer rate 33

45 Figre 0 Effect of on skin- frication coefficient Figre 1 Effect of on heat transfer rate 34

46 Figre Effect of on mass transfer rate Figre 3 Effect of on skin- frication coefficient 35

47 Figre 4 Effect of on heat transfer rate Figre 5 Effect of on mass transfer rate 36

48 Figre 6 Effect of on skin- frication coefficient Figre 7 Effect of on heat transfer rate 37

49 Figre 8 Effect of on mass transfer rate Figre 9 Effect of on skin- frication coefficient 38

50 Figre 30 effect of on heat transfer rate Figre 31 Effect of on mass transfer rate 39

51 CHAPTER IV CONCLUDING REMARKS In this std, we presented a bondar laer analsis for free convection along a vertical wav srface in a nanoflid. Nmerical reslts for friction factor, srface heat transfer rate and mass transfer rate have been presented for parametric variations of boanc ratio parameter, Brownian motion parameter, thermophoresis parameter and Lewis nmber. The reslts indicate that as and increases, the friction factor, the heat transfer (Nsselt nmber) and mass transfer rates (Sherwood nmber) decreases. As increases, the friction factor and srface mass transfer rates increases, where the srface heat transfer rates decreases. As increases, the friction factor increases, where as mass transfer rates and heat transfer rates decreases. As increase, the friction factor, mass transfer and heat transfer rates increases. 40

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