Simultaneous investigations the effects of non-Newtonian nanofluid flow in different volume fractions of solid nanoparticles with slip and no-slip boundary conditions

In this study, the laminar and forced flow of non-Newtonian nanofluid in a two-dimensional microtube has been numerically simulated. The non-Newtonian, pseudo-plastic fluid is included of a solution with 0.5% wt fraction of CMC in Water as the base
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  Contents lists available at ScienceDirect Thermal Science and Engineering Progress  journal homepage: Simultaneous investigations the e ff  ects of non-Newtonian nano fl uid  fl ow indi ff  erent volume fractions of solid nanoparticles with slip and no-slipboundary conditions Ahmad Reza Rahmati a , Omid Ali Akbari b , Ali Marzban c , Davood Toghraie d, ⁎ , Reza Karimi c ,Farzad Pourfattah e a  Department of Mechanical Engineering, University of Kashan, Kashan, Iran b Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran c  Department of Mechanical Engineering, Aligoudarz Branch, Islamic Azad University, Aligoudarz, Iran d  Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran e  Malek-Ashtar University of Technology, Iran A R T I C L E I N F O  Keywords: Heat transferSlip velocity coe ffi cientnon-Newtonian nano fl uidCMCMicrotube A B S T R A C T In this study, the laminar and forced  fl ow of non-Newtonian nano fl uid in a two-dimensional microtube has beennumerically simulated. The non-Newtonian, pseudo-plastic  fl uid is included of a solution with 0.5% wt fractionof CMC in Water as the base fl uid. In this research, in order to increase the heat transfer rate, the mentioned non-Newtonian  fl uid has been combined with volume fractions of 1 and 1.5% of CuO nanoparticle and has beencreated the non-Newtonian cooling nano fl uid. In this investigation, the e ff  ect of slip velocity boundary conditionon the wall of microtube has been considered. In order to have an accurate estimation of dynamic viscosity of non-Newtonian nano fl uid, the power-law model, for numerical simulation has been used. This research has beeninvestigated in Reynolds numbers of 100, 500, 1500 and 2000. The results indicate that, the increase of volumefraction of solid nanoparticles and slip velocity coe ffi cient, cause the increase of heat transfer. By enhancing theslip velocity coe ffi cient, better mixing accomplishes which causes the reduction of temperature gradients amongthe  fl uid layers close to the surface. In Reynolds numbers of 1500 and 2000, comparing to Reynolds numbers of 100 and 500, Nusselt number, on the microtube wall increases signi fi cantly. 1. Introduction The improvement of heat transfer methods in all of the industrialequipment, especially in heat exchangers and miniature and enormousindustries, is one of the important scienti fi c debates in recent decades.The enhancement of heat transfer and the e ffi ciency of heat exchangersare useful for saving million dollars in the industrious expenditures. Byincreasing the environmental pollutions, because of the globalwarming, the green house e ff  ects and control of heat transfer in in-dustries, the energy consumption is an important debate in the in-dustrial countries. The papers and researches in the universities andinstitutes and the printing of various scienti fi c investigations on heattransfer  fi eld, indicate the importance of this science in modern in-dustries. Using novel methods and techniques in heat transfer can leadto lower consumption of the nonrenewal fuels and control of environ-mental and industrial pollutions. Using e ffi cient methods in high heat fl ux transfer, by saving the time of heat transfer and using heat transfersurfaces with lower consuming materials, is one of the debatable issuesin heat transfer  fi eld. Using nano fl uids with amended rheologicalproperties and micro channel with di ff  erent sections for accomplishingthe surface ratio to higher volume are modern methods in novel in-vestigations. However, using these methods, in addition to their higheradvantages, have some disadvantages such as the limitation of manu-facturing process and high costs of producing these equipment. In re-cent years, various investigations, numerically and experimentally, hasbeen done by researchers in heat transfer of   fl uids and nano fl uids.Therefore, researchers by using nano fl uids and di ff  erent geometrics,such as micro channels and microtubes, try to deploy these equipmentin various industries like power electronics and automobile industries[1]. The study of convection heat transfer by using of nano fl uid in-dicates that, using nanoparticles in the base  fl uid can change the heattransfer properties of the  fl uid [2 – 4]. The importance of heat transferenhancement in industries and optimization of heat transfer compo-nents have caused numerous investigations on the heat transfer issue, 29 July 2017; Received in revised form 29 August 2017; Accepted 17 December 2017 ⁎ Corresponding author at: Department of Mechanical Engineering, Islamic Azad University, Khomeinishahr Branch, Khomeinishahr 84175-119, Iran.  E-mail address: (D. Toghraie). Thermal Science and Engineering Progress 5 (2018) 263–2772451-9049/ © 2017 Published by Elsevier Ltd.    by using di ff  erent nano fl uids in micron dimensions [5 – 10]. Chhabraand Richardson et al. [11] have investigated the industrial usage of non-Newtonian nano fl uid in order to signi fi cant enhancement of heattransfer in chemical industries, petrochemical, polymer and pharmacy.In recent decades, using the typical  fl uid and non-Newtonian nano fl uidsas cooling  fl uids has been investigated [12 – 26]. Nikkhah et al. [27]numerically investigated the  fl ow of nano fl uid Water/multi-walledCarbon nanotubes (FMWCNT) in a two-dimensional micro channel withslip and no-slip boundary conditions and concluded that the increase of mass fraction of nanoparticles and slip velocity coe ffi cient causes theenhancement of Nusselt number which in higher Reynolds numbers ismore signi fi cant. Lelea et al. [28] numerically studied the heat transferof Water and dielectric  fl uids in a steel microtube. Their results in-dicated that, in lower Reynolds numbers, the thermal conductivity has agreat in fl uence on the enhancement of local Nusselt number. Akbariet al. [29] numerically studied the  fl uid  fl ow and heat transfer of na-no fl uid in a three-dimensional rectangular micro channel. They  fi guredout that, adding nanoparticle to the working  fl uid causes great en-hancement in heat transfer and friction factor. Niu et al. [30] in-vestigated the non-Newtonian slip  fl ow of Water/Al 2 O 3  nano fl uid in atwo-dimensional microtube. They  fi gured out that, the increase of vo-lume fraction of nanoparticles and the slip length have great in fl uenceon the behavior of the non-Newtonian nano fl uid. Minea et al. [31]numerically studied the turbulent  fl ow of Water/Al 2 O 3  nano fl uid involume fractions of 1 – 4% in a microtube. Their results showed that, theincrease of convection heat transfer coe ffi cient of nano fl uid, comparingto the base  fl uid, is more signi fi cant and by increasing volume fractionof nanoparticles and Reynolds number, the convection heat transfercoe ffi cient enhances. Moraveji and Esmaeili [32] numerically studiedthe forced convection heat transfer of Water/Al 2 O 3  nano fl uid in acirclular tube and  fi gured out that, by increasing the volume fractionand Reynolds number, the heat transfer rate enhances. Santra et al.[33,34] investigated the  fl uid behavior and heat transfer of the non-Newtonian Water/CuO nano fl uid in the enclosure geometrics by usingof non-Newtonian power-law model. Chen et al. [35] studied the heattransfer of non-Newtonian  fl uid  fl ow in a microchannel by using of power-law model. They calculated di ff  erent  fl uid parameters for dif-ferent coe ffi cients (K, n). Xi-Wen et al. [36] investigated the  fl ow of Water  fl uid in a microtube. In their study, they  fi gured out that, tran-sition from laminar to the turbulent  fl ow in the microtube accomplishesat the range of Reynolds numbers of 1700 to 1900. El- Genk and Yang[37] and Celata et al. [38] studied the dynamic behavior of Water  fl uid fl ow in a microtube with slip boundary condition. His results showedthat, for Water  fl uid in the microtube, the length of slip is approxi-mately 0.7 to 1µm. Among the investigation of researchers, simulta-neous use of non-Newtonian Water-CMC/CuO in the horizontal mi-crotube and using slip and no-slip boundary conditions has not beenstudied yet. This issue causes the above subjects been studied si-multaneously in this presentation.In this study, computational  fl uid dynamics of laminar  fl ow andheat transfer of non-Newtonian solution of Water/CMC nano fl uid0.5%wt fraction of CMC in Water as the base  fl uid and volume fractions of 1 Nomenclature A Area, m 2 C p  Speci fi c Heat, J/kg.KD h  Hydraulic diameter, md p  Nanoparticle diameter, nmh Convection heat transfer coe ffi cient, W/m 2 .Kk Thermal conductivity, W/m.KK Consistency index, N. sec n m − 2 K b  Boltzmann constant, J/KL Length, mn Power-Law indexNu Nusselt numberP Pressure, PaPe Peclet numberPr Prantdl numberq ″  Heat  fl ux, W/m 2 Re Reynolds numberT Temperature, KU Dimensionless velocityu Inlet velocity, m/s Greek symbols α  Thermal di ff  usivity, m 2 /s Δ  Di ff  erent λ  Mean free path, nm π  Phi number 3.14 ρ  Density, kg/m 3 φ  Volume fraction β  Slip velocity coe ffi cient, m β ∗ Dimensionless slip velocity coe ffi cient ν   Kinematics viscosity, m 2 /s θ  Dimensionless temperature  Super- and sub-scripts b Bulkbr Browniane ff   E ff  ectivef Fluidin Inletnf Nano fl uidP Particles Solidw Wall Fig. 1.  The schematic of studied two-dimensionalhorizontal microtube.  A.R. Rahmati et al. Thermal Science and Engineering Progress 5 (2018) 263–277  264  and 1.5% CuO nanoparticles in the horizontal two-dimensional micro-tube with slip and no-slip boundary conditions has been numericallysimulated. The rheological behavior of studied nano fl uid has been in-vestigated by using of non-Newtonian and the power-law model and byconsidering K as consistency index and n as power-law index for eachvolume fraction. The advantage of present study is the simultaneousinvestigations the e ff  ects of non-Newtonian nano fl uid in di ff  erentvolume fractions of solid nanoparticles and slip and no-slip boundaryconditions. 2. Mathematical modeling  2.1. Problem statement  In this study, the forced and laminar  fl ow of nano fl uid, for volumefractions of 1% and 1.5% of CuO solid nanoparticles in the non-Newtonian  fl uid of 0.5% wt of CMC in Water, in a two-dimensionalhorizontal microtube has been investigated. In order to study the heattransfer and laminar non-Newtonian  fl ow, the velocity, temperature,the e ff  ects of friction in di ff  erent volume fractions and Reynolds num-bers and slip and no-slip boundary conditions have been studied. Fig. 1indicates the schematic of two-dimensional microtube. In this study, ahorizontal microtube with the ratio of L/D h =66.67 has been nu-merically investigated and simulated. Among the horizontal micro-tubes, the constant heat  fl ux of q ″ =1000W/m 2 has been applied.The inlet temperature of   fl uid in the horizontal microtube is 301K.The laminar  fl ow has been investigated for Reynolds numbers of 100,500, 1500 and 2000. The base  fl uid is included of non-NewtonianWater/CMC solution with 0.5% wt fraction of CMC and solid nano-particles of CuO with volume fractions of 1 and 1.5%. The diameter of solid nanoparticle is 100nm and has spherical shape. The thermo-physical properties, according to the related equations for the base fl uidand CuO nanoparticles have been presented in Table 1.In this study, the  fl ow has been considered as two-dimensional,steady, incompressible, non-Newtonian, laminar and single phase, andnano fl uid properties have been considered with constant temperature.In the inlet of the horizontal microtube, the  fl uid has uniform velocity.The problem has been solved in the axisymmetric cylindrical co-ordinate. K and n coe ffi cients along the horizontal microtube for eachvolume fraction related to the temperature are constant. Also, thesource terms, the radiation and the Brownian motion e ff  ect are negli-gible. Table 1 The thermophysical properties of base  fl uid and solid nanoparticles.Material Pr  ρ  (kg/m 3 ) Cp (J/kg.K)K (W/m.K)Pure Water 6.2 997.1 4179 0.613CuO [39]  –  6500 535.6 200.5wt% of CMC aqueous solution inWater [40] –  1002 4500 0.6CMC(0.5%)+1%CuO  –  1044.3 3976 0.6251CMC(0.5%)+1.5%CuO  –  1071 3869 0.6311 Fig. 2.  Figures of a and b for determining K and n coe ffi cients in Power-Law method [35]. Table 2 The investigation of independence from grid in present study.Number of mesh(width×length)Average of frictionfactor (f)Average Nusseltnumber (Nu ave )30×300 0.1578 14.005150×500 0.1612 15.064170×700 0.1620 15.198 Fig. 3.  Yhe validation of present numerical procedure with the study of Akbari [64].  A.R. Rahmati et al. Thermal Science and Engineering Progress 5 (2018) 263–277  265  3. Governing equations In this research, the single-phase model has been considered forinvestigating heat transfer and non-Newtonian nano fl uid  fl ow. The di-mensional governing equations included of mass, momentum and en-ergy conservation as follows [41]: ∂∂ + ∂∂ = u x1r r(rv) 0 (1) ⎜ ⎟ ⎛⎝∂∂ + ∂∂ ⎞⎠ = −∂∂ + ⎛⎝∂∂ + ∂∂ ⎞⎠ ρ u u x v  urp x τ  x1r r(rτ )  xxTrxT (2) ⎜ ⎟ ⎛⎝∂∂ + ∂∂ ⎞⎠ = −∂∂ + ⎛⎝∂∂ + ∂∂ − ⎞⎠ ρ u v  x v  v rpr τ  x1r r(rτ ) τ r  xrTrrT θθT (3) ∂∂ + ∂∂ = ∂∂ ⎛⎝∂∂ ⎞⎠ + ∂∂ ⎛⎝∂∂ ⎞⎠ + ⎛⎝∂∂ + ∂∂ ⎞⎠ ρcu T xρcv  Tr1r rkr Tr xk T x τ  ur v  x rx (4)The dimensionless parameters in provide the results, and derivationnon-dimensional forms of governing equations are as follow: = = = = −= ″= == ∗ X xD,R  rD,U uuθ T T∆T,∆ T q Dk,P pρ uβ β/D ,LLD h h inin hf  nf 2 h 1 (5)The non-dimensional equations as shown below [42], ∂∂+ ∂∂= UX V R 0 (6) ⎜ ⎟ ∂∂ + ∂∂ = −∂∂ + ⎛⎝∂∂ ⎛⎝∂∂ ⎞⎠ + ∂∂ ⎛⎝∂∂ ⎞⎠⎞⎠ U UX V  UR PX1Re XUX R UR  nf n n (7) ⎜ ⎟ ∂∂ + ∂∂ = −∂∂ + ⎛⎝∂∂ ⎛⎝∂∂ ⎞⎠ + × ∂∂ ⎛⎝∂∂ ⎞⎠⎞⎠ − U V X V  V R PR 1Re X V X1R R R  V R  V R  nf n n2 (8) ⎜ ⎟ ∂∂ + ∂∂ = ⎛⎝∂∂ ⎛⎝∂∂ ⎞⎠ + × ∂∂ ⎛⎝∂∂ ⎞⎠⎞⎠ θ U θX V  θR 1Pr Re X X1R R R  θR  nf nf n n (9)Due to the using of non-Newtonian power-law model, the shearstress ( τ ) is described as follow [43], = ⇒ = ⎡⎣⎢⎛⎝∂∂ ⎞⎠ + ⎛⎝ ⎞⎠ + ⎛⎝∂∂ ⎞⎠⎤⎦⎥ + ⎡⎣∂∂ + ∂∂ ⎤⎦ γ γ   τ K| ̇| | ̇| 2 v r v ru x v  xur n2 2 2 2 (10)In this equation,  τ , K,  γ  and n are respectively shear stress, con-sistency index, shear rate and power-law index. In order to describeReynolds number for the non-Newtonian  fl uid, coe ffi cient and power-law index are very important in determining Reynolds number andinitial velocity of   fl uid in the inlet of tube. Reynolds number is de-scribed for the non-Newtonian  fl uid as follow [44], =  − Re ρU DK 2 n n (11)Peclet number and Prandtl number for the non-Newtonian  fl uid aresuch as follow [16], ⎜ ⎟ = ⎛⎝⎞⎠ − υ Pr Cp .rk D nf nf nf hn 1 (12) = = Pe Re.Prρ . C υ Dk nf  p nf hnf  nf  (13)In this numerical study, power-law model has been used for in-vestigating the rheological behavior of   fl uid. Therefore, in spite of de-termining the viscosity of nano fl uid, K and n are respectively power-lawindex and consistency index. For determining K and n in 1 and 1.5% of volume fraction of CuO nanoparticles in the non-Newtonian solution of  Fig. 4.  The validation of present study with the study of Raisi [65].  A.R. Rahmati et al. Thermal Science and Engineering Progress 5 (2018) 263–277  266  0.5% wt fraction of CMC in Water, the empirical  fi gure from Hojjatet al. [45] has been used (Fig. 2). This  fi gure is for determining K and ncoe ffi cients related to the non-Newtonian Water-CMC/CuO nano fl uid,for volume fractions of 0 and 4%. For each considered temperature(which in this paper 15K has been considered as the temperature dif-ferences between surface and  fl uid), each volume fraction of solid na-noparticle (horizontal axis), power-law coe ffi cient (vertical axis) can bedeciphered from these  fi gures. The determination of accomplishedcoe ffi cients in the equations of which K and n are demanded for theircalculation is necessary.In order to access the nano fl uid properties, the empirical equationssuggested by researchers are used. For calculating the density of na-no fl uid and the speci fi c heat capacity, the following equations are used[46,47], = − + φ φ ρ (1 )ρ ρ nf f s  (14) = − + φ φ (ρC ) (1 )(ρC ) (ρC ) p nf p f p s  (15)In the above equations,  φ  is the volume fraction of nanoparticle andf, s and nf indexes are respectively the indicators of   fl uid, solid andnano fl uid. In order to determine the thermal conductivity of nano fl uid,the equations of Chon [48] are used, ⎜ ⎟  ⎜ ⎟ = + ⎛⎝⎞⎠ × ⎛⎝⎞⎠ φ kk1 64.7 ddkkPr Re nf f 0.7460 f p0.3690pf 0.74760.9955 1.2321 (16)In Eq. (16), Pr and Re parameters are de fi ned as, Fig. 5.  Chart bars of average Nusselt number in di ff  erent volume fractions and Reynolds numbers and slip velocity coe ffi cients.  A.R. Rahmati et al. Thermal Science and Engineering Progress 5 (2018) 263–277  267
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