The study of heat transfer and laminar flow of kerosene/ multi-walled carbon nanotubes (MWCNTs) nanofluid in the microchannel heat sink with slip boundary condition

In this investigation, the laminar heat transfer of kerosene nanofluid/multi-walled carbon nanotubes in the microchannel heat sink is studied. The considered microchannel is two layers in which the length of bottom layer is truncated and is equal to
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  The study of heat transfer and laminar flow of kerosene/ multi-walled carbon nanotubes (MWCNTs) nanofluidin the microchannel heat sink with slip boundary condition Abedin Arabpour 1 • Arash Karimipour 2 • Davood Toghraie 1 Received: 25 May 2017/Accepted: 12 August 2017/Published online: 4 September 2017   Akade´miai Kiado´, Budapest, Hungary 2017 Abstract  In this investigation, the laminar heat transfer of kerosene nanofluid/multi-walled carbon nanotubes in themicrochannel heat sink is studied. The consideredmicrochannel is two layers in which the length of bottomlayer is truncated and is equal to the half of the length of bottom layer. The length of microchannel bottom layer is  L   =  3 mm, and the length of top layer is  L  1  =  1.5 mm.The microchannel is made of silicon, and each layer of microchannel has the thickness of   t   =  12.5  l m. Along theexternal bottom wall, the sinusoidal oscillating heat flux isapplied. The top external and lateral walls are insulated,and they do not have heat transfer with the environment.The results of this research revealed that in differentReynolds numbers, applying oscillating heat flux signifi-cantly influences the profile figure of Nusselt number andthis impressionability is obvious in Reynolds numbers of 10 and 100. Also, by increasing the slip velocity coefficienton the solid surfaces, the amount of minimum temperaturereduces significantly which behavior remarkably entails theheat transfer enhancement. Keywords  Heat transfer    Kerosene/multi-walled carbonnanotubes  Microchannel heat sink   Oscillating heat flux  Slip velocity coefficient List of symbols  A  Area (m 2 )  B  =  b  /   H   Dimensionless slip velocity C  f   Skin friction factor C  p  Heat capacity (J kg - 1 K  - 1 )  H   Microchannel height (m) K   Thermal conductivity coefficient(W m - 1 K  - 1 )  L   Down-layer microchannel length(m)  L  1  Top-layer microchannel length(m)  Nu  Nusselt number P  Fluid pressure (Pa) Pe  =  ( u s d  s  /  a f  ) Peclet number Pr   =  t f   /  a f   Prandtl number q 00 (  X  ) Oscillating heat flux (W m - 2 ) q 0 00 Constant heat flux (W m - 2 )  R  Thermal resistance (K W - 1 )  Re  =  q f  u c d   /  l f   Reynolds number T   Temperature (K)( U  ,  V  )  =  ( u  /  U  0 ,  v  /  U  0 ) Dimensionless velocitycomponents in  x ,  y  directions(  X  ,  Y  )  =  (  x  /  d  ,  y  /  d  ) Cartesian dimensionlesscoordinates u ,  v  Velocity components in  x ,  y  directions (m s - 1 ) u c  (m/s) Inlet velocity in  x  directions(m s - 1 ) u s  (m/s) Brownian motion velocity(m s - 1 ) Greek symbols b  Slip velocity coefficient (m) u  Nanoparticles volume fraction &  Arash Karimipourarashkarimipour@gmail.com 1 Department of Mechanical Engineering, KhomeinishahrBranch, Islamic Azad University, Khomeinishahr, Iran 2 Department of Mechanical Engineering, Najafabad Branch,Islamic Azad University, Najafabad, Iran  1 3 J Therm Anal Calorim (2018) 131:1553–1566https://doi.org/10.1007/s10973-017-6649-x  k l  =  L  1  /   L   Dimensionless length ration l  Dynamic viscosity (Pa s - 1 ) h  =  ( T   -  T  C )/  D T   Dimensionless temperature q  Density (kg m - 3 ) s  Shear stress (N m - 2 ) t  Kinematics viscosity (m 2 s - 1 ) Super- and subscripts Ave Averagec ColdEff Effectivef Base fluid (pure water)H HotIn InletMax MaximumMin Minimumnf NanofluidOut OutletS Solid nanoparticles Introduction The cooling of miniature equipment in the micro electromechanical and nano-electromechanical industries hasincreased the need of understanding the fluid flow and heattransfer in the micro- and nanogeometrics. The behavior of fluid flow and heat transfer in the miniature scales and byusing nanofluid, due to the improvement in heat transfermechanisms in nano- and microdimensions, comparing tothe custom scales, is far different. Numerous numerical andempirical studies have been done for investigating the flowand heat transfer of custom fluids and nanofluid in themicrochannels whose main purpose is increasing the heattransfer [1–5]. The investigation of heat transfer enhance- ment in different industrial and experimental fluids byusing novel methods has been expanded as the study fieldsamong the adherents of this issue [6]. The microchannelheat sink as an applicable miniature equipment has highimportance in heat transfer of electronic industries. Thisequipment has been suggested by Tukerman and Pease [7]for cooling the electronic chips. In recent decades, thisequipment has been investigated and optimized byresearchers in different structures and arrangements forenhancing the cooling of electronic chips [8–10]. Kulkami et al. [11] numerically studied the multi-purposedoptimization of double-layer microchannel heat sink withthe cross-figured inlet section. Their results evidenced thatthe microchannel with narrower design has lower thermalresistance and higher pumping power and the pumpingpower by increasing the heat flux reduces significantly.Husain and Kim [12, 13] optimized the indented microchannel heat sink and indicated that the thermalresistance of microchannel heat sink by optimizationreduces considerably. Xie et al. [14] studied the efficiencyof double-layer microchannel heat sink with the wavy wallin the states of parallel and contrary flows. They investi-gated the effects of wavy wall limitation and the ratio of mass flow on the thermal resistance and pressure dropparameters. Seyf and Nikaaein [15] by using Al 2 O 3 , zincand Cu nanoparticles in the ethylene glycol/water fluidnumerically studied the effects of nanoparticles dimensionsand Brownian motion of nanoparticles on the thermalperformance of a rectangular microchannel heat sink. Theirresults showed that the amount of nanofluid conductivitywithout considering the Brownian motion reduces almostto 6.5%. Wu et al. [16] numerically studied the thermalresistance, pumping power and thermal distribution on thewall surface of double-layer microchannel heat sink (DL-MCHS). In their research, different parameters of microchannel dimensions and different flow conditionshave been studied. The results of his study showed that theimprovement in total efficiency of double-layermicrochannel heat sink depends on the pumping power.Chen and Chung [17] used the water/Cu nanofluid. In theirinvestigation, the absorbed energy by the nanofluid wasmore than the absorbed energy by water, and it has beenobserved that by enhancing volume fraction of nanoparti-cles, the high-temperature differences accomplish betweenthe inlet and outlet sections of microchannel heat sink in alow flow rate. Jang and Choi [18] by using nanofluidnumerically studied the cooling performance of amicrochannel heat sink. They reported that the nanofluidcauses the reduction in thermal resistance and dimension-less temperature difference in microchannel heated walland cooling fluid. Sui et al. [19] numerically investigatedthe fluid flow in the wavy microchannels. Their numericalresults indicated that with the uniform cross section, thethermal performance of wavy microchannel is higher thanthe rectangular flat one. Ho et al. [20] studied the forcedconvection cooling performance of a Copper microchannelheat sink with water/Al 2 O 3  nanofluid as the cooling fluid.Their results showed that the heat sink cooled by nanofluid,comparing to the heat sink cooled by water, has moreaverage heat transfer coefficient. Till now, numerousresearches about the heat transfer in the microchannels andnanofluid have been presented, and sometimes, the slipvelocity conditions, the effects of magnetic field and theforced heat transfer under the influence of constant tem-perature or constant heat flux have been investigated dis-parately [21–35]. Nikkhah et al. [36] numerically studied the water nanofluid/functional multi-walled carbon nan-otubes in a two-dimensional microchannel with slip andno-slip boundary conditions. They concluded that theaugment of solid nanoparticles weight fraction and slip 1554 A. Arabpour et al.  1 3  velocity coefficient cause the increase in Nusselt number,and in higher Reynolds numbers, this enhancement is moreconsiderable. In their research, the computational fluiddynamics and laminar heat transfer of kerosenenanofluid/multi-walled carbon nanotubes in the double-layer microchannel heat sink are simulated in the two-di-mensional domain. By considering the effect of slipboundary condition on the outcome results of numericalsimulation, in this study, the slip velocity boundary con-dition on the solid walls is used. The results of this researchare presented for different volume fractions of nanoparti-cles, slip velocity coefficients and different ranges of Reynolds numbers. The main purpose of this study isinvestigating the behavior of temperature domain andhydrodynamic of laminar flow of nanofluid in the two-dimensional double-layer microchannel. Problem statement In the present study, the laminar flow of kerosenenanofluid/multi-walled carbon nano tubes in volume frac-tions of 0, 4 and 8% of nanoparticles is investigated. Fig-ure 1 indicates the studied geometrics of this paper. In thisresearch, the material of microchannel is silicon. In Fig. 1,the bottom layer of microchannel is  L   =  3 mm and theheight is  H   =  50  l m. The top layer of microchannel withthe length of   L  2  is equal to  L  2  =  1.5 mm, and by placing onthe bottom layer at the interface area, the heat transferswith it and in this region, the amount of heat generation isconstant and is equal to 100 kw/m 3 . In each layer of microchannel, the silicon material with the thickness of  t   =  12.5  l m has surrounded the layers. The external areasof top layer with the length of   L  2  are insulated, and thebottom area of microchannel, on the external wall with thelength of   L  , is under the influence of sinusoidal flux withthe equation of   q 00  X  ð Þ¼ 2 q 00 0 þ q 00 0  sin  p  X  4    in which theamount is calculated from the equation of ( q 0 00 ). With thedefinition of dimensionless slip velocity coefficient as(  B  =  b  /   H  ), the ratio of slip velocity coefficient to theheight of microchannel, in this research, the numericalsimulation is done for the dimensionless slip velocitycoefficients (  B  =  b  /   H  ) of 0.001, 0.01 and 0.1 and Reynoldsnumbers of 1, 10 and 100. The inlet fluid at the top andbottom layers enters with the temperature of 301 K asshown in Fig. 1. All of the internal walls which are incontact with fluid have the slip velocity boundary condi-tion. The used nanofluid properties of this simulation andthe material of microchannel wall are described, respec-tively, in Table 1.In this simulation, the fluid flow and heat transfer areconsidered as laminar and fully developed. The nanofluidproperties are considered as constant and independent fromthe temperature. The solid–liquid suspension in less den-sities is modeled as single-phased, and on the channelwalls, the oscillating heat flux is applied. The slip boundarycondition is used on the microchannel. The numericalsimulation domain is two dimensional. Governing equations The dimensionless governing equations on the simulationdomain are defined as follows [39, 40]: Continuity equation : o U  o  X  þ o V  o Y  ¼ 0  ð 1 Þ  Momentum equation : U   o U  o  X   þ V   o U  o Y   ¼ o P o  X  þ  l nf  q nf  m f  1  Re o 2 U  o  X  2  þ o 2 U  o Y  2    ð 2 Þ U   o V  o  X  þ V   o V  o Y  ¼ o P o Y  þ  l nf  q nf  m f  1  Re o 2 V  o  X  2 þ o 2 V  o Y  2    ð 3 Þ Energy equation : U   o h o  X  þ V   o h o Y   ¼ l nf  a f  1  Re Pr  o 2 h o  X  2 þ o 2 h o Y  2    ð 4 Þ For non-dimensioning Eqs. (1)–(4), following parame- ters are used [36]: Insoulation Solid Heat generation L 1 L 1 LC B D H t A U  in , T  in U  in , T  in q  ′′ (  X  ) = 2 q 0 ′′ + q 0 ′′ sin (   π  X 4 ) MWCNT//kerosene nanofluid Xy Fig. 1  The studied schematics of this research Table 1  The thermophysical properties of base fluid and nanoparti-cle of multi-walled carbon nanotubes and silicon [37, 38] u  /%  q  / kg m - 3 C  p  / J kg - 1 K  - 1 k   / W m - 1 K  - 1 l  /Pa s  Pr  0 783 2090 0.145 0.001457 214 815 1989 0.265 0.001613 12.18 845 1895 0.390 0.001795 8.72Silicon 2329 702 124 – –The study of heat transfer and laminar flow of kerosene/multi-walled carbon nanotubes (MWCNTs) …  1555  1 3   X   ¼  x H Y   ¼  y H V   ¼  vu c h ¼ T   T  c D T  U   ¼  t u c  B ¼  b  H  D T   ¼ q 00 0  H k  f  Pr   ¼  t f  a f  P ¼  P q nf  u 2c ð 5 Þ Another parameter for investigating the microchannelperformance is the friction coefficient which is calculatedfrom the following equation [41]: C  f   ¼ 2  s w q u 2in ð 6 Þ The average Nusselt number can be obtained as follows[42, 43]:  Nu x  ¼ h   H k  f  !  Nu ave  ¼ 1  L  Z   L  0  Nu x  X  ð Þ d  X   ð 7 Þ The amounts of thermal resistance [44, 45] of bottom wall of microchannel and pressure drop are calculated fromthe following equation:  R ¼ T  max  T  min q 00 0   A ¼ T  max  T  in q 00 0   A !  A ¼ W    L  !  R  W  ¼ T  max  T  min q 00 0   L   ð 8 Þ D P ¼ P in  P out  ð 9 Þ In Eq. (9),  T  max ,  T  min ,  A  and  q 0 00 are, respectively, themaximum temperature of bottom wall, the minimum tem-perature (the temperature of inlet fluid), cross section andthe applied heat flux to the AB wall. The governing boundary conditionson the problem-solving The hydrodynamic and thermal boundary conditions usedin this problem are as follows: U   ¼ 1 ;  V   ¼ 0 and  h ¼ 0 for  X   ¼ 0 and0 : 25  Y   1 : 25 and  X   ¼ 60 ;  1 : 75  Y   2 : 75 V   ¼ 0 and  o h o  X   ¼ o U  o  X   for  X   ¼ 60 and0 : 25  Y   1 : 25 and  X   ¼ 30 ;  1 : 75  Y   2 : 75 V   ¼ 0 ;  U   ¼ 0 and  o h o Y  ¼ 2 q 00 0 þ q 00 0  sin  p  X  4   for  Y   ¼ 0 and 0   X   60 V   ¼ 0 ;  U  s  ¼  B  o h o Y  and  k  nf  o h o Y  ¼ k  s o h o Y  for  Y   ¼ 0 : 25 and 0   X   60 V   ¼ 0 ;  U   ¼ 0 and  o h o Y  ¼ 0 for  Y   ¼ 1 : 5 and0   X   30 and  Y   ¼ 3 and 0   X   60 V   ¼ 0 ;  U  s  ¼  B o U  o Y  and  k  nf  o h o Y  ¼ k  s o h o Y  for Y   ¼ 1 : 25 and 0   X   60 V   ¼ 0 ;   U  s  ¼  B o U  o Y  and  k  nf  o h o Y  ¼ k  s o h o Y  for Y   ¼ 1 : 75 and 30   X   60 V   ¼ 0 ;   U  s  ¼  B o U  o Y  and  k  nf  o h o Y  ¼ k  s o h o Y  for Y   ¼ 2 : 75 and 30   X   60 ð 10 Þ The mesh study and numerical solving procedure In order to ensure the results independency of thisresearch, the rectangular organized grids have changedfrom the number of 30,000 to 100,000. The studiedparameters in the validation of present investigation areincluding Nusselt number along the AB wall and theamount of pressure drop. The changes in these twoparameters are investigated in Reynolds numbers of 10and 100 and volume fraction of 8% of nanoparticles inthe slip velocity coefficient of 0.01. According to Table 2,by choosing grid number of 100,000, comparing to othergrid numbers, more accurate results can be obtained.However, the grid number of 63,000, compared to thegrid number of 100,000, has acceptable error and lessdemanded time for solving the numerical domain; there-fore, in this numerical simulation, the grid number of 63,000 has been used. In this study, in order to enhancethe solving accuracy, to couple velocity and pressure,SIMPLEC algorithm [46, 47] has been used, and the maximum loss for results convergence of this simulationhas been chosen 10 - 6 [48–50]. Table 2  The changes in studied grid numbers in the present study  Re  Parameters Grid point30,000 50,000 63,000 100,000  Re  =  100  Nu ave  9.786 10.2103 10.4661 10.623Error 7.9% 3.89% 1.48% Base grid D P  /Pa 101,231 97,635 94,908.5 94,851Error% 6.72% 2.94% 0.06% Base grid  Re  =  10  Nu ave  3.68 4.011 4.0248 4.101Error 10.27% 2.2% 1.86% Base grid D P  /Pa 9845 9271 9161.3 9100.5Error 8.2% 1.88% 0.67% Base grid1556 A. Arabpour et al.  1 3  Results and discussion Validation The results of the present study have been validated withthe numerical study of Nikkhah et al. [36] in Reynoldsnumber of 100 for the dimensionless temperature param-eter at central section of flow. Nikkhah et al. [36] numer-ically investigated the laminar flow and heat transfer of water nanofluid/functional carbon nanotubes in a rectan-gular microchannel with the ratio of length to the height of channel equal to 32. Their investigation has been done inReynolds numbers of 1–100 for volume fractions of 0–0.25% of nanoparticles. According to Fig. 2 and propercoincidence of the results of the present research with thestudy of Nikkhah et al. [36], it can be said that the solvingprocedure and the applied boundary conditions areaccurate.  X  051015202530         θ    H   /   2 My study Nikkhah et al. [36] ϕ  = 0.12  Re = 100 Fig. 2  The validation with numerical study of Nikkhah et al. [36] 0.275091 0.825272 1.37545 1.92563 2.47582 3.026 4.126363.57618 B   = 0.1 B   = 0.01 B   = 0.001 0 0.001 0.002 0.003 x   /m 0 0.001 0.002 0.003 x   /m 0 0.001 0.002 0.003 x   /m Fig. 3  The changes in dimensionless temperature in Reynoldsnumber of 1 and different dimensionless slip coefficients in volumefraction of 0% 0.275091 0.825272 1.37545 1.92563 2.47582 3.026 4.126363.57618 B   = 0.1 B   = 0.01 B   = 0.001 0 0.001 0.002 0.003 x   /m 0 0.001 0.002 0.003 x   /m 0 0.001 0.002 0.003 x   /m Fig. 5  The changes in dimensionless temperature in Reynoldsnumber of 1 and different dimensionless slip coefficients in volumefraction of 8% 0.275091 0.825272 1.37545 1.92563 2.47582 3.026 4.126363.57618 B   = 0.1 B   = 0.01 B   = 0.001 0 0.001 0.002 0.003 x   /m 0 0.001 0.002 0.003 x   /m 0 0.001 0.002 0.003 x   /m Fig. 4  The changes in dimensionless temperature in Reynoldsnumber of 1 and different dimensionless slip coefficients in volumefraction of 4%The study of heat transfer and laminar flow of kerosene/multi-walled carbon nanotubes (MWCNTs) …  1557  1 3
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