Effect of Magnetic Field on Forced Convection between Two Nanofluid Laminar Flows in a Channel

Transcription

1 CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 9aNo. 6a DOI: /CJME available online at Eect o Magnetic Field on Forced Convection between Two Nanoluid Arasiab Raisi* and Ahmad Qanbary Engineering Faculty Shahrekord University Shahrekord PO Box 115 Iran Received August ; revised May ; accepted June Abstract: This paper provides a numerical study o orced convection between hot and cold nanoluid laminar lows that are separated by a thin membrane in a horizontal channel. Outer surace o channels walls are thermally insulated and divide into two parts; namely NMP and MP. NMP is the channel s wall rom the entrance section to the middle section o channel that is not inluenced by magnetic ield. MP is the channel s wall rom the middle section to the exit section o channel which is inluenced by a uniorm-strength transverse magnetic ield. The governing equations or both hot and cold lows are solved together using the SIMPLE algorithm. The eects o pertinent parameters such as Reynolds number (10 Re 500) Hartman number (0 Ha 60) and the solid volume raction o copper nano-particles (0 0.05) are studied. The results are reported in terms o streamlines isotherms velocity and temperature proiles and local and average Nusselt number. The results o the numerical simulation indicate that the increase in Reynolds number and the solid volume raction lead to increase in Nusselt number. Meanwhile the results also show that the rate o heat transer between the lows increases as the Hartmann number increases especially at higher values o the Reynolds number. Keywords: magnetic ield orced convection nanoluid membrane channel 1 Introduction In order to save the energy consumption in various industries including the automotive electronics aerospace ood processing and manuacturing industries the heat exchangers that are used must be smaller and lighter and also with higher thermal perormance. Due to restrictions on the size o the heat exchangers the heat exchange surace cannot be increased. The use o luids that have better heat transer characteristics is a good way to increase the eiciency o such heat exchangers. Nanoluid is a suspension o solid nanoparticles in conventional base luids such as water oil or ethylene glycol and has a higher thermal conductivity than the base luid depending on the shape size and type o the solid nanoparticles [1 5]. In recent years orced convection o nanoluids in dierent channels has been investigated by several researchers. IZADI et al [6] numerically investigated laminar orced convection o alumina-water nanoluid in an annulus. Their indings showed that adding nanoparticles to the base luid increased the convective heat transer coeicient but had a little eect on the velocity proile. BIANCO et al [7] numerically studied the developing laminar orced convection low o a water Al O 3 nanoluid in a circular tube submitted to a constant and uniorm heat lux at the wall. They showed that the single and two-phase * Corresponding author. raisi@eng.sku.ac.ir Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 016 models led to the same results and the temperature dependent models presented higher values o heat transer coeicient and Nusselt number. SANTRA et al [8] examined laminar orced convective heat transer o copper-water nanoluid in a horizontal parallel plate channel. For this purpose the nanoluid was considered as Newtonian as well as non-newtonian. They concluded that the thermal behavior o Newtonian and non- Newtonian nanoluid was more or less the same. Also the sharp increase in the wall shear stress with increasing the solid volume raction or non- Newtonian nanoluid was one o their outcomes. MANCA et al [9] reported the results o an investigation on orced convection with Al O 3 -water nanoluids in a two dimensional channel under uniorm heat lux condition. In their report an increase in the heat transer coeicient and required pumping power as solid volume raction increases was conirmed. SAHA et al [10] presented a numerical investigation o turbulent orced convection low o water based Al O 3 and TiO nanoluids lowing through a horizontal circular pipe under uniorm heat lux boundary condition applied to the wall. Based on the results o this study with increasing Reynolds number and solid volume raction and decreasing size o nanoparticles Nusselt number increased. In experimental research PEYGHAMBARZADEH et al [11] and HUSSEIN et al [1] experimentally examined the riction actor and orced convection heat transer enhancement o a water based nanoluid with various nanoparticles in the automotive cooling system. In general

2 136 Arasiab Raisi et al: Eect o Magnetic Field on Forced Convection between Two Nanoluid the results showed that the use o nanoluids increased the riction actor and orced convective heat transer coeicient so that the percentage increase in heat transer coeicient was more than the percentage increase o the riction actor. In most o the existing equipment in various industries a magnetic ield is created due to the electric ield. Thus the low o electrically conducting luid through a channel or a duct is exposed to a transverse magnetic ield. The results o the research conducted have shown that the luid low subjected to a magnetic ield experience a Lorentz orce. Lorentz orce directly aects the low ield and consequently the heat transer is also aected. Thereore understanding o the low and thermal behaviors o the channels that are cooled by electrically conducting luids and are exposed to a magnetic ield has long been o interest to researchers. SIEGEL [13] was among the irst researchers who have studied the hydromagnetic lows. He studied convective heat transer or the ully developed and laminar low o an incompressible and electrically conducting luid in a parallel plate channel submitted to a constant and uniorm heat lux at the walls and exposed to a transverse magnetic ield. The results showed that the low ield and the heat transer in the channel were aected by the magnetic ield. ALPHER [14] continued and extended the earlier study done by SIEGEL [13]. In particular the earlier analysis o Siegel involving non-conducting plates were corrected and extended to plates o inite conductivity. BACK [15] analytically studied the uniorm laminar low o an electrically conducting luid in a constant cross-section parallel-plate channel with isothermal electrically nonconducting walls. In this study the eect o the Joule heating parameter was also considered. The results showed that the thermal entrance lengths were proportional to Reynolds and Prandtl numbers similar to the case o no applied magnetic ield. Additionally these lengths decreased with an increase in Joule heating parameter. LAHJOMRI et al [16] analytically investigated the thermal developing laminar Hartmann low through a parallel plate channel including viscous dissipation Joule heating and axial conduction with a step change in wall temperature. The results showed that in the absence o viscous dissipation and Joule heating the Hartman eect led to a reduction o the bulk temperature. This study also showed that viscous dissipation and Joule heating played a signiicantly dierent role in the heat transer depending on the electrical condition o the walls. In another study LAHJOMRI et al [17] reiterated his earlier work under a constant heat lux boundary condition. This time they showed that the heat transer increased with the increasing Hartmann number until inally a saturated state was achieved. With the arrival o nanoluids to the technology arena the interest to examine the hydro-magnetic low o nanoluids has greatly increased. AMINOSSADATI et al [18] numerically examined the laminar orced convection o a water Al O 3 nanoluid lowing through a partially heated microchannel. They indicated that or any value o the Reynolds number velocity and temperature along the centerline o the microchannel decreased as the Hartmann number increased. Their results also showed that the Nusselt number increased as the Reynolds and Hartman numbers increased. MAHIAN et al [19] analytically investigated the eects o using TiO /water nanoluid on the entropy generation in an isolux vertical annulus under MHD low eects. In this study the low was assumed as laminar steady and ully developed inside the annulus. The results showed that using TiO /water nanoluid reduced the entropy generation in the annulus while an increase in the Hartmann number increased the entropy generation number. SYAMSUNDAR et al [0] experimentally examined the orced convection heat transer and riction actor in a tube with Fe 3 O 4 magnetic nanoluid. In this study the heat transer coeicient increased by 30.96% and riction actor by 10.01% at 0.6% volume concentration compared to low o water at similar operating conditions. GHOFRANI et al [1] established an experimental investigation on laminar orced convection heat transer o erroluids inside a circular copper tube under an alternating magnetic ield. They investigated the eects o magnetic ield volume concentration and Reynolds number on the convective heat transer coeicient. They ound that increasing the requency o the alternating magnetic ield and the solid volume raction led to better heat transer enhancement. MALVANDI et al [] presented a theoretical study on the laminar low and orced convective heat transer o water/alumina nanoluid inside a parallel plate channel in the presence o a uniorm magnetic ield. They applied a modiied two component model or water/alumina nanoluid and looked at the migration o nanoparticles. The results indicated that nanoparticles moved rom the heated walls toward the core region o the channel and constructed a non-uniorm nanoparticles distribution. It is worth mentioning that previous studies have solely investigated heat transer in a single channel. However in many engineering applications such as heat exchangers hot and cold luids low through a pair o adjacent channels and orced convective heat transer takes place between them. In act this system can be a simple model o a heat exchanger which is exposed to a uniorm magnetic ield to increase the rate o heat transer. The main aim o the present work is a numerical investigation o heat transer perormance o a Cu-water nanoluid in a pair o horizontal channels that is under the inluence o a transverse magnetic ield. It is assumed that the lows are separated by a thin plat (membrane) which its mass is negligible. Problem Description Fig. 1 shows the schematic diagram o a two-dimensional horizontal channel wherein hot and cold lows are separated by a membrane. Dimensionless height

3 CHINESE JOURNAL OF MECHANICAL ENGINEERING 137 and length are H = h/ h= 1 and L= l/ h= 30 respectively. U i æ ö X Y RePr X Y i i n i VCi i i + Vi = + èç ø (4) where VC1 = uc / uh and VC = 1. Fig. 1. A schematic diagram o the physical model Both upper and lower walls o the channel are thermally insulated. The irst part o the channel which is not inluenced by magnetic ield (Non-Magnetic part i.e. NMP here ater) has a dimensionless length o L 1 = l1 / h= 15 and the second part inluenced by magnetic ield (magnetic part i.e. MP rom now on) has the same dimensionless length o L = l / h= 15. Hot low with velocity u h and temperature T h = 303 K and cold low with velocity u c and temperature T c = 93 K enter lower and upper channels respectively. Both hot and cold nanoluid lows contain water and Cu nanoparticles and are assumed to be laminar Newtonian and incompressible. Constant thermophysical properties are considered or the nanoluid except or the density variation in the buoyancy orces that is determined by the Boussinesq approximation. The thermophysical properties o pure water (base luid) and the Cu-nanoparticles at a ixed temperature o 5 are presented in Table 1 [3]. The Prandtl number o water is assumed to be Pr = 6.. Table 1. Thermophysical properties o water and Cu at 5 Property (kg/m 3 ) C p (J/(kg K)) k(w/(m K)) Pure water Copper(Cu) Mathematical Formulation The non-dimensional governing equations o continuity momentum and energy or hot and cold lows can be written as ollow ( i = 1 is or hot low and i = is or cold low: Ui Vi + = 0 X Y Ui Ui Pi VC i n i Ui + Vi =- + X Y X Re n i æ Ui U ö æ i n i Ha ö + - U i èç X Y ø ç n i Re è ø V V P VC æ V V ö Ui + =- + ç + X Y Y Re ç X Y i i i i n i i i Vi n i çè (1) () ø (3) ui vi Ti- Tc Ui = Vi = i = uh uh Th - Tc p1 p P1 = P = u u n 1 h n c li d 1 h Li = = 15 D= = H = = 1 h h h x y X = Y. h = h (5) The Reynolds Hartman and Prandtl numbers are: uh Re = Ha = B h Pr =. (6) c 0 where = k /( Cp) is the thermal diusivity coeicient o the base luid (pure water). It is noticeable that in the hot low equations the VC1 = uc / uh and since the Reynolds number is deined into the inlet velocity o cold low this coeicient appears in the hot low equations. The eective density heat capacitance and the thermal diusivity coeicient o the nanoluid are deined based on the properties o water and Cu [3] : = (1- ) + (7) n s ( C ) = (1- )( C ) + ( C ) (8) p n p p s kn n =. ( C ) (9) The electrical conductivity o nanoluid was presented by MAXWELL [4] as below: n æ ö æ ö s èç ø = 1 +. æ ö æ s ö s çè çè ø çè ø ø p n (10) The eective dynamic viscosity o the nanoluid can be modeled by BRINKMAN [5] : n =. (11).5 (1 - ) The thermal conductivity o the nanoluid k e was determined by PATEL et al [6]. For the two-component

4 138 Arasiab Raisi et al: Eect o Magnetic Field on Forced Convection between Two Nanoluid entity o spherical-particle suspension the model gives: where é ka s s A ù s ke = k 1 cks Pe + + k A k A êë úû As A d = d s 1- and Pe is the Peclet number deined as: (1) (13) ud s s Pe = (14) where u s is the Brownian motion velocity o the nanoparticles and is bt us =. (15) d In the above-mentioned equations b is the Boltzmann -3 constant ( b = J/ K) and c is an experimental constant (c=36 000) which was suggested by PATEL et al [6]. The molecular diameter o the solid nanoparticles is d s = 100 nm and the molecular diameter o the base luid is d = A. The non-dimensional boundary conditions used to solve Eqs. (1) (4) are as ollows: U = 1 V = 0 or X = 0 and 0 Y H i i 1 = 1 = 0 or X = 0 and 0 Y H Ui i Vi = 0 = = 0 or X = L and 0 Y H X X i Ui = Vi = = 0 ory = 0 H and 0 X L. (16) Y Since the membrane is thin and its mass is negligible the thermal boundary conditions on it are as ollows: s 1 1 = kn 1 = kn. Y Y 4 Numerical Approach (17) The non-dimensional governing equations or hot and cold lows (Eqs. (1) (4)) with the boundary conditions given in Eqs. (16) and (17) are solved simultaneously by the control volume ormulation using Patankar s SIMPLE algorithm [7]. The convection-diusion terms are discretized by a power-law scheme and the system is numerically modeled in FORTRAN. A regular rectangular domain with a uniorm grid is used. The convergence criteria are to reduce the maximum mass residual o the grid control volume below Ater solving both the hot and the cold governing equations or U1 V1 1 and U V other useul quantities such as the Nusselt number can be determined. The local Nusselt number ( Nu x ) at the membrane can be expressed as: h Nux = (18) k where is the convection heat transer coeicient and is calculated as ollows: q = (19) T - T h where q is the heat lux which passes through the massless membrane rom the hot low to the cold low and is as ollows: æ T ö æ T ö q =- k ç =-k ç c 1 n 1 n. ç y y è ø ç y= d è øy= d (0) Thus the local Nusselt number by using the dimensionless parameters given in Eq. (5) is as ollows: Nu x k æ ö k æ ö =- ç =- ç k Y k Y n 1 1 n. ç è ø ç Y= D è øy= D (1) Finally the average Nusselt number ( Nu m ) is determined by integrating Nu along the membrane separator: x 1 L Num =- NuXd X. L ò () 0 5 Grid Independency and Code Validation Table indicates the solution grid independence or two dierent Hartmann numbers ( Ha = 0 and 40). Table. Grid points Grid independency study at Re=100 =0.03 D=1/ VC 1 = ( m1 ) out ( m1 ) out Nu m Ha= ( m ) out Nu m Ha= ( m ) out The average Nusselt number and the dimensionless bulk mean temperature o hot and cold lows at channels outlet

5 无法显示链接的图像 该文件可能已被移动 重命名或删除 请验证该链接是否指向正确的文件和位置 无法显示链接的图像 该文件可能已被移动 重命名或删除 请验证该链接是否指向正确的文件和位置 CHINESE JOURNAL OF MECHANICAL ENGINEERING 139 i.e. ( m1 ) out and ( m ) out are presented at dierent grid points when =0.03 VC 1 =1 and D=1/. The obtained results rom Table illustrates that the grid size o satisies grid independency. Fig. (a) compares the isotherms o this work with those o SANTRA et al [8] to validate our code. Moreover the variation o temperature along the centerline o the channel or Re=100 and =0.0 o Al O 3 -Water Nanoluid is compared with that o AMINOSSADATI et al [18] in Fig. (b). Furthermore in Fig. (c) the ully developed velocity proiles at three values o Hartman number contrasted with BACK s indings [15] presented by Eq. (3). Good agreements are seen in all Figs. (a) (b) and (c). é æhaö ææhaö öù cosh cosh cosh ( Y 1) - - æhaöê èç ø èç èç ø ø ú U = ë û. ç è øéæhaö æhaö æhaöù cosh cosh - sinh sinh ê ç ç ç ëè ø è ø è øú û (3) 6 Results The eects o the Reynolds number (10 Re 500) the Hartman number (0 Ha 60) and the solid volume raction (0 0.05) on the thermal perormance o the channel are studied and are presented in the ollowing sections or VC 1 = 1 and D = Eects o Reynolds and Hartman numbers In this section the eects o Re and Ha on heat transer isotherms and streamlines are studied when =0.03. Streamlines and isotherms are shown in Fig. 3 at dierent values o Re or Ha=0 and 40. In all values o Re streamlines match together at NMP whereas at Ha=40 both the hot and the cold lows aected by magnetic ield at MP where streamlines expand due to Lorentz orce. Fig.. Validation code against: (a) Isotherms in Re. [8] (b) Variation o temperature along the centerline o the channel in Re. [18] and (c) Analytical velocity proile modeled in Re. [15] Fig. 3. Streamlines (top) and isotherms (bottom) or Cu-water nanoluid (=0.03) or dierent Re at Ha=40 and Ha=0

6 140 Arasiab Raisi et al: Eect o Magnetic Field on Forced Convection between Two Nanoluid Isotherms are not aected by the magnetic ield at Re=10 and they are the same or Ha=0 and 40. This is due to the small amounts o hot and cold lows velocities at Re=10 so both lows reach a balanced temperature very soon. Other indings show that heat transer occurs throughout the channel length at Re=100 and 500. Thus the temperature patterns do penetrate into the exit section and at Ha=40 isotherms are inluenced by magnetic ield at MP. The comparison between the isotherms o the nanoluid near the membrane shows that the heat transer increases as the Re increases. Fig. 4 presents the variations o dimensionless velocities along the centerline o hot or cold lows at Re= and 500. It is observed that both lows have the same velocity variations along the low centerline in all values o because o their similar geometric and hydrodynamic properties. In this igure it can be seen that at Re=10 centerline velocity stabilizes much sooner in comparison with other values o Re. In addition the length o hydrodynamic entrance increases as the Re increases. However U max decreases due to the magnetic ield eect by entering hot and cold lows to the MP. The gradient o this decrease depends on Re. higher Nu x. Regardless o the values o Re and Ha the maximum amount o Nu x is obtained at the channel inlet due to the highest temperature gradient between hot and cold lows. Ater heat exchange between two nanoluid lows along the membrane length Nu x starts to decrease. For Re=10 two lows reach a thermal equilibrium immediately. Thus in this case magnetic ield has no signiicant eect on isotherms variations and Nu x. For Re=100 and 500 an increasing jump can be observed in Nu x diagram at the entrance o MP. This happens because in the existence o magnetic ield the velocity o two lows near the walls and membrane increases. Fig. 5. Variation o horizontal velocity along the centerline o the channel or various Hartman numbers at Re=100 =0.03 Fig. 4. Variation o horizontal velocity along the centerline o the channel or various Reynolds numbers at Ha=40 =0.03 In Fig. 5 U max variations are presented at our dierent values o Hartman number (Ha= and 60) and Re=100. In the absence o magnetic ield (Ha=0) the maximum o horizontal low velocity is constant at ully developed conditions due to Lorentz orce eect however U max decreases at Ha ¹ 0. The highest decrease o U max achieves or the largest amount o (see Eq. ()). Fig. 6 shows the velocity proiles at our dierent cross sections o channel; namely X= and 30 or hot and cold nanoluid lows at Ha=40 and Re=100. According to this igure the velocity proile at X=7.5 is parabolic and ully developed. The magnetic ield starts at X=15 where the shape o velocity proile begins to change. In two sections o X=.5 and 30 magnetic ield yields a similar lat velocity proile or both sections. Reynolds and Hartman numbers eects on Nu x are shown in Figs. 7(a) and 7(b). Higher Re corresponds to Fig. 6. Horizontal velocity proiles across various sections o channel at =0.03 Re=100 Ha=40 The eects o Ha on Nusselt number can be better understood by studying values presented in Table 3. As mentioned beore the increase in leads to the increase in Nu m at higher values o Re. Ater running the computer code or dierent values o the Reynolds and Hartmann numbers the ollowing correlation was ound between the average Nusselt Reynolds and Hartmann numbers when the solid volume raction =0.03. where m = ( ) + ( ) + ( ) (4) Nu Re Ha g Re Ha h Re -5-5 ( Re) = ( 10 Re Re+.6) -8-6 gre ( ) = 5 10 Re + 10 Re hre ( ) = 0.446Re. (5)

7 CHINESE JOURNAL OF MECHANICAL ENGINEERING 141 conditions and variations o lows temperature along the channel are symmetrical. Fig. 7. Variation o local Nusselt number along the channel length at =0.03 and at (a) various Reynolds numbers at Ha=40 (b) various Hartman number at Re=100 Table 3. Eects o Ha on the Nu m and its relative increase percentage (%Incr.) at various Re and =0.03 Re Item Ha Nu m %Incr Nu m %Incr Nu m %Incr énum Nu ù m Ha 0 Note: %Incr. means relative increase percentage - = ú 100. Nu êë mha = 0 úû 6. Eect o solid volume raction Figs. 8(a) and 8(b) show that the dimensionless bulk mean temperatures o the hot and the cold lows start to decrease and increase respectively as soon as they enter the channel. The rate o temperature change is higher at low values o Re so that or Re=10 the dimensionless bulk mean temperatures o the two lows become equal beore reaching the exit section or both the nanoluid and pure water while or Re=100 and 500 these temperatures are substantially dierent rom each other. For all values o Re the changes in dimensionless bulk mean temperature are more rapid or nanoluid low than that o pure water. Figs. 8(a) and 8(b) also show that in all values o Re the hot and the cold lows have the same hydrodynamic Fig. 8. Variation o dimensionless balk mean temperature along the channel length or nanoluid (=0.03) and pure water at various Reynolds numbers and Ha=40 or (a) hot low and (b) cold low Nu m at dierent values o copper nanoparticles volume raction (0 0.05) or Re=0 and 40 is presented in Table 4 at dierent values o. In all cases higher values o Nu m can be achieved at higher amounts o. As a result it can be said that the eects o and on increasing Nu m are more signiicant or higher values o Re. For example at Re=500 and Ha=40 an increase o 3% in the volume raction o copper nanoparticles results in approximately 18.84% increase in the Nu m whereas this values is 1.0% or Re=10 and Ha=40. Ater running the computer code or dierent values o the Reynolds numbers and solid volume ractions the ollowing correlation was ound between the average Nusselt and Reynolds numbers and solid volume ractions when the Hatrmann number is Ha=0 and 50 Re 500. where Nu = ( Re) + g( Re) (6) m -7 3 ( Re) = 3 10 Re Re Re = Re (7) gre ( )

8 14 Arasiab Raisi et al: Eect o Magnetic Field on Forced Convection between Two Nanoluid Table 4. Eects o on the Nu m and its relative increase percentage or dierent values o Re and Ha Ha Re Item Nu m %Incr Nu m %Incr Nu m %Incr Nu m %Incr Nu m %Incr Nu m %Incr énumn Nu ù m Note: %Incr. means relative increase percentage - ú 100. Nu êë m úû 7 Conclusions This research studied the orced convection between two laminar Cu-water nanoluid lows with dierent temperatures. The hot and the cold lows were separated by a very thin membrane in a horizontal channel. The heat transer occurred only between two lows along the membrane because the outer surace o the channel was thermally insulated. Also channel's wall rom the middle section to the exit section was inluenced with a uniorm-strength transverse magnetic ield (MP). The low and temperature ields and the heat transer rate o the channel were studied at various values o Reynolds and Hartman numbers and solid volume ractions. The Brownian motion o nanoparticles was taken into account in the model used to determine thermal conductivity o nanoluid. The results o the numerical analyses are summarized below. The lows were becoming ully developed later at higher values o the Reynolds number. Streamliners were expanded at MP due to applying the magnetic ield. The Hartman number had no eect on heat transer rate at Re=10 because two lows were reached an equal temperature immediately. This is while at higher values o Reynolds number the average Nusselt number increased as the Hartman number increased. Moreover a more noticeable increase in the Nusselt number with solid volume raction was evident at higher values o Reynolds number. Nomenclature B 0 Magnetic ield strength c Experimental constant Eq. (1) C p Speciic heat J kg 1 K 1 d Distance between channel s bottom wall and membrane m D Dimensionless distance between channel s bottom wall and membrane D=dh h Channel height m H Dimensionless channel height H=h/h=1 Ha Hartman number Ha = B0 h k Thermal conductivity W m 1 K 1 l Length o channel m L Dimensionless length o channel L= l/ h= 30 L 1 Dimensionless length o irst region L1= l1/ h= 15 without o magnetic ield L Dimensionless length o second region ( L = l/ h= 15) with o magnetic ield MP Part o channel s wall that is inluenced by magnetic ield (magnetic part) NMP Part o channel s wall that is not inluenced by magnetic ield (non-magnetic part) Nu x Local Nusselt number on the membrane Nu m Average Nusselt number along the membrane p 1 Hot low pressure Pa p Cold low pressure Pa Modiied pressure p= p+ gy P 1 Dimensionless pressure or hot low P1= p1 / n uh P Dimensionless pressure or cold low P = p / n uc Pe Peclet number Pe= ud s s / Pr Prandtl number Pr= / Re Reynolds number Re= uh c / T 1 T Temperature o hot and cold lows respectively K T c Inlet temperature o cold low K T h Inlet temperature o hot low K u s Brownian motion velocity m/s u v Velocity components in x y directions m/s U 1 V 1 Dimensionless velocity o hot low U1= u/ uh V1= v/ uh U V Dimensionless velocity o cold low U = u/ uc V = v/ uc VC 1 Velocity coeicient VC1 = uc/ uh x y Cartesian coordinates m X Y Dimensionless coordinates X=x/h Y=y/h Thermal diusivity (k/cp) m /s Solid volume raction b Boltzman constant J/K 1 Dimensionless temperature o hot low 1= ( T1 - Tc) / ( Th-Tc) Dimensionless temperature o cold low = ( T - Tc) / ( Th-Tc) m1 Dimensionless balk mean temperature o hot 1 D low m 1= U11d Y D ò 0 m3 Dimensionless balk mean temperature o cold 1 H low m = Ud Y H - Dò D Heat transer coeicient W m K 1 Dynamic viscosity N s m - Kinematic viscosity (/) m /s Density kg m 3

9 CHINESE JOURNAL OF MECHANICAL ENGINEERING 143 Electrical conductivity S/cm. Subscripts C Cold h 1 Hot e Eective Pure luid n Nanoluid out Channel s outlet s Nanoparticle. Reerences [1] CHOI SUS. Enhancing thermal conductivity o luids with nanoparticles[j]. Journal o Developments and Applications o Non-Newtonian Flows ASME : [] EASTMAN J A CHOI SUS LI S et al. Anomalously increased eective thermal conductivities o ethylene glycol-based nanoluids containing copper nanoparticles[j]. Applied Physics Letters (6): [3] ZHOU D W. Heat transer enhancement o copper nanoluid with acoustic cavitation[j]. International Journal o Heat and Mass Transer : [4] MOGHADASSI A R MASOUDHOSSEINI S HENNEKE D E. Eect o CuO nanoparticles in enhancing the thermal conductivities o mono ethylene glycol and parain luids[j]. Industrial and Engineering Chemistry Research : [5] SAHOOLI M SABBAGHI S Investigation o thermal properties o CuO nanoparticles on the ethylene glycol water mixture[j]. Journal o Materials Letters : [6] IZADI M BEHZADMEHR A JALALI-VAHIDA D. Numerical study o developing laminar orced convection o a nanoluid in an annulus[j]. International Journal o Thermal Sciences (4): [7] BIANCO V CHIACCHIO F MANCA O et al. Numerical investigation o nanoluids orced convection in circular tubes[j]. Applied Thermal Engineering 009 9: [8] SANTRA A K SeN S CHAKRABORTY N. Study o heat transer due to laminarlow o copper water nanoluid through two isothermally heated parallel plates[j]. International Journal o Thermal Sciences : [9] MANCA O NARDINI S RICCI D. A numerical study o nanoluid orced convection in ribbed channels[j]. Applied Thermal Engineering 01 37: [10] SAHA G PAUL CM. Numerical analysis o the heat transer behaviour o water based Al O 3 and TiO nanoluids in a circular pipe under the turbulent low condition[j]. International Journal o communications in Heat and Mass Transer : [11] PEYGHAMBARZADEH S M HASHEMABADI S H SEIFIJAMNANI M et al. Improving the cooling perormance o automobile radiator with Al O 3 /water nanoluid[j]. Applied Thermal Engineering : [1] HUSSEIN A M BAKAR R A KADIRGAMA K. Study o orced convection nanoluid heat transer in the automotive cooling system[j]. Case Studies in Thermal Engineering 014 : [13] SIEGEL R. Eect o magnetic ield on orced convection heat transer in a parallel plate channel[j]. Journal o Applied Mechanics : [14] ALPHER R A. Heat transer in magneto hydrodynamic low between parallel plates[j]. International Journal o Heat and Mass Transer (): [15] BACK L H. Laminar heat transer in electrically conducting luids lowing in parallel plate channels[j]. International Journal o Heat and Mass Transer : [16] LAHJOMRI J OUBARRA A ALEMANY A. Heat transer by laminar Hartmann low in thermal entrance region with a step change in wall temperatures: the Graetz problem extended[j]. International Journal o Heat and Mass Transer 00 45(5): [17] LAHJOMRI J ZNIBER K OUBARRA A et al. Heat transer by laminar Hartmann s low in thermal entrance region with uniorm wall heat lux: the Graetz problem extended[j]. Energy Conversion and Management (1): [18] AMINOSSADATI S M RAISI A GHASEMI B. Eects o magnetic ield on nanoluid orced convection in a partially heated microchannel. International Journal o Non-Linear Mechanics : [19] MAHIAN O POP I SAHIN Z A et al. Irreversibility analysis o a vertical annulus using TiO /water nanoluid with MHD low eects[j]. International Journal o Heat and Mass Transer : [0] SYAMSUNDAR L NAIK M T SHARMA K V et al. Experimental investigation o orced convection heat transer and riction actor in a tube with Fe 3 O 4 magnetic nanoluid[j]. Experimental Thermal and Fluid Science : [1] GHOFRANI A DIBAEI M H HAKIM-SIMA A et al. Experimental investigation on laminar orced convection heat transer o erro-luids under an alternating magnetic ield[j]. Experimental Thermal and Fluid Science : [] MALVANDI A GANJI D D. Magnetic ield eect on nanoparticles migration and heat transer o water/alumina nanoluid in a channel [J]. Journal o Magnetism and Magnetic Materials : [3] RAISI A GHASEMI B AMINOSSADATI S M. A numerical study on the orced convection o laminar nanoluid in a microchannel with both slip and no-slip conditions[j]. Numerical Heat Transer Part A: Applications (): [4] SHEIKHOLESLAMI M SOLEIMANI S GORJI-BANDPY M et al. Natural Convection o nanoluids in an enclosure between a circular and a sinusoidal cylinder in the presence o magnetic ield[j]. International Communications in Heat and Mass Transer 01 39: [5] BRINKMAN H C. The viscosity o concentrated suspensions and solution[j]. The Journal o Chemical Physics 195 0: [6] PATEL H E SUNDARARAJAN T PRADEEP T et al. A micro-convection model or thermal conductivity o nanoluids[j]. Journal o Physics (5): [7] PATANKAR S V. Numerical heat transer and luid low[m]. New York: Hemisphere Publishing Corporation-Taylor and Francis Group Biographical notes Arasiab Raisi was born in Iran and currently is an associate proessor at Department o Mechanical Engineering Shahrekord University Shahrekord Iran. He received his B. Eng. Degree in Mechanical Engineering rom Ferdowsi University o Mashhad Iran in 199 and the M.Sc. and Ph.D. degrees in Mechanical Engineering rom Isahan University o Thechnology (IUT) Iran in 1995 and 001 respectively. In 001 he joined the Department o Mechanical Engineering Shahrekord University Iran as an assistant Proessor became an associate proessor in 015. His current research interests include orced and natural convection heat transer convective heat transer in nanoluids and non-newtonian luids melting and solidiication and conduction microscale heat transer. Tel: ; araisi@gmail.com; raisi@eng.sku.ac.ir Ahmad Qanbary was born in Iran in 1980 and currently is an engineer at National Iranian Gas Company. He received his M.Sc. degree in Mechanical Engineering (CFD branch) rom Shahrekord University Iran in ahmadqanbary@yahoo.com