Forced Convection Heat Transfer over a Horizontal Plate Embedded in a Porous Medium Saturated with a Nanofluid in the Presence of Heat Sources

Transcription

1 .seiub.org/aee Advances in Energ Engineering AEE Volue Issue Aril 3 Forced Convection Heat ranser over a Horizontal Plate Ebedded in a Porous Mediu Saturated ith a Nanoluid in the Presence o Heat Sources Masoud Rabeti * Ainreza Noghrehabadi Ashin Ghanbarzadeh 3 * Deartent o Mechanical Engineering Islaic Azad niversit Dezul Branch Dezul Iran 3 Deartent o Mechanical Engineering Shahid Charan niversit o Ahvaz Iran * asoud.rabeti@gail.co; noghrehabadi@scu.ac.ir; 3 ghanbarzadeh.a@scu.ac.ir Abstract In this aer orced convection heat transer o nanoluids over a horizontal lat late ebedded in a orous ediu saturated ith a nanoluid is nuericall analzed. In the boundar laer heat can be generated or absorbed. It is assued that the nanoarticles are uniorl disersed in the base luid. A siilarit aroach is used to reduce the governing artial dierential equation to an ordinar dierential equation. he resulting ordinar dierential equation is nuericall solved or a te o orous ediu sand and three tes o nanoarticles nael aluina AlO3 coer Cu and titaniu dioide io. he eect o heat generation/absortion as ell as volue raction o nanoarticles on the heat transer enhanceent o nanoluids is theoreticall investigated. Keords Heat Generation/Absortion; Forced Convection; Porous Media; Nanoluid; Darc Model Introduction Noadas nanotechnolog rovides a ne oortunit to roduce nanoarticles ith diensions belo 5 n Wang and Mujudar 8. Disersing nano-size solid articles in a conventional heat transer luid ould increase the theral conductivit o the luid. Nanoluid is a luid hich contains nanoeter-sized solid articles. Successul eloent o nanoluids reduces the size o heat echanger sstes and also enhances the erorance o these sstes Wang and Mujudar 8. Nanoluids have an industrial alications such as nuclear reactors transortation electronics as ell as bioedicine and ood Kaac and Praaujaroenij 9. Forced convection heat transer in orous edia also has an alications such as geotheral energ engineering groundater ollution transort cheical reactors engineering nuclear aste disosal insulation o buildings and ies Zhao and Song. Forced convection heat transer over a horizontal late ebedded in a orous ediu has been studied in an literatures. Becerann and Visanta 986 studied orced convection boundar laer lo and heat transer along a lat late ebedded in a orous ediu. Postelnicu et al. discussed the eect o variable viscosit on the orced convection lo and heat transer ast a horizontal surace laced in a saturated orous ediu in the resence o internal heat generation. Noghrehabadi et al. eained the lo and heat transer o nanoluids over an isotheral lat late ith artial sli boundar condition Noghrehabadi et al. c suction or injection Noghrehabadi et al. a and agnetic ield eects Noghrehabadi et al. b. Maghrebi et al. analzed orced convection heat transer o nanoluids in a orous channel. Rosca et al. investigated the non-darc ied convection o nanoluid over a horizontal surace laced in a orous ediu. Forced convection heat transer o nanoluids over a horizontal lat late ebedded in a saturated orous ediu in the resence o heat generation/absortion has not been analzed et. he resent stud ais to analze the eect o heat generation or absortion on the orced convection heat transer over a horizontal late ebedded in a orous ediu saturated ith a nanoluid. he governing artial dierential equation 46

2 Advances in Energ Engineering AEE Volue Issue Aril 3.seiub.org/aee is transored into an ordinar dierential equation b aid o siilarit variables. he governing equation in its non-diensional or is a unction o heat generation araeter and the ratio o theral diusivit o nanoluid and the base luid. Matheatical Forulation Consider the stead to diensional orced convection heat transer o a nanoluid over an isotheral horizontal surace hich is ebedded in a saturated orous ediu. he scheatic vie o hsical and coordinate sste is illustrated in FIG.. here is the ressure u is the Darc velocit and is the teerature o the nanoluid. he eective theral diusivit o the nanoluid and orous ediu is denoted b n. Here K is the ereabilit o the orous ediu μn is the dnaic viscosit o the nanoluid. Q is the heat generation/absortion rate inside the boundar laer. he subscrit denotes the large values outside the boundar laer. he thero hsical roerties o nanoluid can be evaluated as ollo: n n 7-a C n C C ϕ C ϕ C 7-b n FIG. PHYSICAL MODEL AND COORDINAE SYSEM he nanoluid los ith a constant velocit o through the orous ediu because o an eternal ressure gradient. he teerature o the late is held at constant value o. he nanoluid is heogenius and the nanoarticles are uniorl disersed in the base luid; thereore the concentration gradient in the boundar laer is neglected. B assuing an isotroic Darc orous ediu and neglecting the all eects the governing boundar laer equations in the Cartesian coordinate sste can be ritten as Nield and Bejan 6: u v K P u µ n K P v µ n 3 Q u v n c 4 the boundar conditions are as ollos: : v 5 : u 6 n e n l / ε l / [ l / ε l / ϕ ϕ ] 7-c 7-d Here ε is the orosit o the orous ediu n is the eective theral conductivit o nanoluid and orous ediu is the eective theral conductivit o the base luid and orous ediu Cn is the heat caacit o the nanoluid φ is the nanoarticle volue raction and are the densities o the base luid and the nanoarticles resectivel. is the theral conductivit o orous ediu l is the theral conductivit o the luid hich los through orous nanoluid or ure luid and n is the theral conductivit o the nanoluid. and are theral conductivities o the luid and the nanoarticles resectivel. Eq. 7-c is based on the or o Alizad et al. and Eq. 7-d is based on the or o Bacho et al. and Khanaer and Vaai. In order to evaluate n irst o all n should be evaluated ro Eq. 7-d and then the value o n should be used as l in 7-c. e is hen l is the theral conductivit o ure luid. Eqs. and 3 are siliied using crossdierentiation and the continuit equation is also satisied b introducing a strea unction ψ: u ψ ψ v Solution o Eqs. and 3 leads to: u v

3 .seiub.org/aee Advances in Energ Engineering AEE Volue Issue Aril 3 P µ n here K and is the ree strea velocit. In order to reduce the governing equation Eq 4 subject to the boundar conditions Eqs. 5 and 6 to an ordinar dierential equation the siilarit variables o and are introduced as Pe local Péclet nuber is deined as: Pe Eloing the siilarit variables Eq. and on the Eq. 4 reduces the governing artial dierential equation to the olloing ordinar dierential equation See Aendi A: D λ 3 subject to 4 5 here rie denotes dierentiation ith resect to. he non-diensional araeters o D and λ are deined as ollos: n D 6 Q λ c n 7 Q λ λ λ 8 c n λ is the heat generation/absortion constant and λ is the heat generation/absortion araeter hose ositive values reresent heat generation and negative values reresent heat absortion. In this stud the quantit o ractical interest hich shos the rate o heat transer ro the late is the local Nusselt nuber Nu: Nu here q 9 n q the reduced Nusselt nuber is ritten as ollo: n Nur NuPe Results and Discussion In order to analze the eect o nanoarticles on the heat transer three nanoarticles tes nael Cu AlO3 and io are chosen. he orous ediu is sand and the ater is the base luid. he required therohsical and echanical roerties o the nanoarticles orous ediu and base luid are shon in ABLES and. ABLE PROPERIES OF FLID AND HREE NANOPARICLES BACHOK E AL. Phsical roerties Fluid hase Water Cu AlO3 io C J/g g / W/ ABLE PROPERIES OF SAND POROS MEDIA NIELD AND BEJAN 6 Phsical roerties Sand ε.45 3 he governing ordinar dierential equation Eq. 3 subject to the boundar conditions Eqs. 4 and 5 is solved using Range-Kutta ethod ith shooting technique. he astotic value o ininit is assued as 8. Assuing λ no heat generation/absortion and φ no nanoarticle disersed in the base luid the resent or reduces to the or o Bejan 984. hereore in order to test the accurac o the resent solution the value o ' obtained b Bejan 984 is coared ith ' obtained in the resent stud in the ABLE 3. Moreover in FIG. the teerature roile obtained b Bejan 984 is coared ith in the resent or hen λ and φ. hese coarisons sho eellent agreeent beteen the resent stud and the or o Bejan 984. he results are obtained or dierent values o nanoarticle concentrations as ell as heat generation araeter λ. FIG. 3 shos the eect o heat generation/absortion araeter on the teerature roiles. his igure shos that increase o ould decrease the local 48

4 Advances in Energ Engineering AEE Volue Issue Aril 3.seiub.org/aee teerature. he teerature on the surace is one and decreases to reach its astotic value o zero ar ro the late. It is clear that heat generation gives an increase in the teerature o the nanoluid hereas heat absortion decreases the teerature o the nanoluid. Variation o reduced Nusselt nuber ith volue raction o nanoarticles or three tes o nanoarticles is shon in FIG. 4. In this igure the orous ediu is sand and the λ4. FIG. 4 shos that the reduced Nusselt nuber ould increase ith an increase o nanoarticles volue raction. It should be noted that or a constant value o nanoarticle volue raction the coer nanoarticles rovide the highest value o reduced Nusselt nuber and ater that the aluiniu oide nanoarticles rovide high values o reduced Nusselt nuber. It is orth noticing that the theral conductivit and heat caacit o coer nanoarticles also are higher than that o titaniu dioide and aluinu oide nanoarticles. FIG. 3 EFFECS OF HEA GENERAION PARAMEER ON EMPERARE PROFILES FOR SAND POROS MEDIA AND ALO3 NANOPARICLES WHEN Φ. ABLE 3 COMPARISON BEWEEN RESLS OF BEJAN BEJAN 984 AND PRESEN WORK FOR λ AND Φ Bejanbejan 984 Present or FIG. 4 VARIAION OF REDCED NSSEL NMBER WIH Φ FOR SAND AND HREE YPES OF NANOPARICLES WHEN λ4 Conclusion Eect o heat generation or absortion as ell as tes and concentration o nanoluids on the orced convection heat transer over a horizontal lat late ebedded in a orous ediu is investigated nuericall. he results can be suarized as ollos: FIG. COMPARISON BEWEEN EMPERARE PROFILE OF PRESEN SDY AND HA OF BEJAN 984. Heat generation increases teerature o nanoluid hereas heat absortion decreases the teerature. 49

5 .seiub.org/aee Advances in Energ Engineering AEE Volue Issue Aril 3. For all studied cases an increase o nanoarticles volue raction ould signiicantl increase the reduced Nusselt nuber. REFERENCES Alizad K. Vaai K. and Shaahi M. ''"heral erorance and oerational attributes o the startu characteristics o lat-shaed heat ies using nanoluids." International Journal o Heat and Mass ranser Bacho N. Isha A. and Po I. ''"Flo and heat transer characteristics on a oving late in a nanoluid." International Journal o Heat and Mass ranser. Becerann C. and Visanta R. ''"Forced convection boundar laer lo and heat transer along a lat late ebedded in a orous ediu." ht. J. Hear Mass ranser Bejan A. "Convection Heat ranser." Wile 984. Kaac S. and Praaujaroenij A. ''"Revie o convective heat transer enhanceent ith nanoluids." Int. J. Heat Mass ranser Khanaer K. and Vaai K. ''"A critical snthesis o therohsical characteristics o nanoluids." International Journal o Heat and Mass ranser Maghrebi M. J. Nazari M. and Araghani. ''"Forced convection heat transer o nanoluids in a Porous channel." rans Porous Med Nield D. A. and Bejan A. "Convection in Porous Media." Sringer _64 6. Noghrehabadi A. Ghalabaz M. Ghalabaz M. and Ghanbarzadeh A. ''"Coaring theral enhanceent o Ag-ater and SiO-ater nanoluids over an isotheral stretching sheet ith suction or injection." Journal o Coutational and Alied Research in Mechanical Engineering a. Noghrehabadi A. Ghalabaz M. and Ghanbarzadeh A. ''" Heat transer o agnetohdrodnaic viscous nanoluids over an isotheral stretching sheet." Journal o herohsics and Heat ranser b. Noghrehabadi A. R. Pourrajab R. and Ghalabaz M. ''"Eect o artial sli boundar condition on the lo and heat transer o nanoluids ast stretching sheet rescribed constant all teerature." International Journal o heral Sciences 53-6 c. Postelnicu A. Grogan. and Po I. ''"he eect o variable vscosit on orced convection lo ast a horizontal lat late in a orous ediu ith internal heat generation." Mechanics Research Coucations 33_337. Rosca A. V. Rosca N. C. Grosan. and Po I. ''"Non-Darc ied convection ro a horizontal late ebedded in a nanoluid saturated orous edia." International Counications in Heat and Mass ranser Wang X. Q. and Mujudar A. S. "''A revie on nanoluids - art i: theoretical and nuerical investigations. " Brazilian Journal o Cheical Engineering Zhao. S. and Song Y. J. ''"Forced convection in a orous ediu heated b a ereable all erendicular to lo direction: analses and easureents." International Journal o Heat and Mass ranser Noenclature seciic heat at constant ressure ereabilit theral conductivit local Nusselt nuber local Péclet nuber luid teerature abient teerature velocit coonents Cartesian coordinates Gree sbols theral diusivit orosit siilarit variable diensionless teerature heat generation/absortion araeter dnaic viscosit densit nanoarticle volue raction Subscrits abient condition luid orous edia eective value or orous ediu and base luid eective value or nanoluid and orous ediu nanoluid article Suerscrit dierentiation ith resect to C K Nu Pe uv ε λ μ φ n n 5

6 Advances in Energ Engineering AEE Volue Issue Aril 3.seiub.org/aee 5 Aendi A: Details o transoring the PDE Energ equation to ODE: Dierentiation ith resect to and : 3 / Pe A and dierentiation ith resect to and : 3 / A Pe A3 A4 Energ equation ollos as: c Q v u n n A5 Substituting dierentiations into energ equation: n 3 c Q c Q n n n λ D A6