Magneto-thermal convection of low concentration nanofluids

Transcription

1 MATEC Web o Conerences 8, 6 () DOI:./ mateccon/ 86 C Owned by the authors, ublished by EDP Sciences, Magneto-thermal convection o low concentration nanoluids Aleksandra Roszko, Elzbieta Fornalik-Wajs, Janusz Donizak, Jan Wajs, Anna Kraszewska, Lukasz Pleskacz, and Sasa Kenjeres AGH University o Science and Technology, Deartment o Fundamental Research in Energy Engineering, Poland Gdansk University o Technology, Deartment o Energy and Industrial Aaratus, Poland Delt University o Technology, Deartment o Chemical Engineering, The Netherlands Abstract. The main aim o this aer was to analyze ossible utilization o the low concentration nanoluids and the magnetic ield to enhance heat transer. The studied luids were based on water with an addition o coer arties (-6 nm diameter). They belonged to the diamagnetic grou o materials. As a irst attemt to stated target the analysis o enosure laced in the maximal value o square magnetic induction gradient was carried out. The maximum was in the centre o investigated cavity and it caused the most comlex system o gravitational and magnetic buoyancy orces. In the lower art o cavity both orces acted in the same direction, while in the uer art they counteracted. Thereore an enhancement and attenuation o heat transer could be observed. Due to the artie concentration and magnetic ield action the character o low was changed. In the case o m nanoluid the low was steady end the strong magnetic ield didn t change much in its structure excet or the suression o some vortices. In the case o m nanoluid the low was not steady even without magnetic ield, but increasing magnetic induction caused change o its structure towards the inertial-convective regime o turbulent low. Introduction Due to a technical develoment, the energy consumtion increases. A lot o eorts are ut on the better energy utilization and higher eiciency o devices. At the same time the researchers are struggling with the challenges related to the high heat lux transer in the devices cooling rocesses. The most imortant limitation in obtaining high heat lux transer by the luids is their relatively low thermal conductivity []. Thereore, there are trials to obtain higher heat transer by various methods. In this aer a combination o two methods will be taken into account the alication o magnetic ield to intensiy heat transer and the addition o coer nanoarties to base luid (changing the roerties o susension), which is called nanoluid. In general, the nanoluids are the mixtures o base luid and disersed arties, which sizes do not exceed nm. As a base luid very oten the water and ethylene glycol are used []. The variety o arties, their kind, size and shae is very wide []. The arties used or susension roduction belong to a grou o cerami, carbon based, metals and alloys []. From the heat transer oint o view articularly interesting are two last grous o arties with the emhasis laced on the ure metallic arties. In the literature it was mentioned, that the susension thermal conductivity even with retty low arties concentration is much higher than in the case o only base luid []. Numerous exerimental studies were carried out in the ield o suerior thermal roerties o the nanoluids comared to tyical working luids used in the heat exchangers. Taking into account that the most imortant arameter resonsible or enhanced heat transer is thermal conductivity, many studies connected with this roblem were conducted. Researchers observed enhancement o thermal conductivity in comarison with conventional coolants []. A lot o exerimental investigations concerned the thermal roerties o nanoluids containing nano-arties o alumina (Al O ) and coer (CuO) oxides. The results demonstrated that adding the nano-arties to the base luid led to an enhancement o heat transer. The 6% imrovement o the thermal conductivity was obtained, in comarison with the corresonding base luids, or only vol% o the nanoarties. Also, comarison with various data indicated that the decrease in arties size resulted in increase o thermal conductivity o nanoluid. Eective thermal conductivity raised by % at vol.% o nanoarties []. In [6] the review o exerimental data concerning the thermal conductivity o nanoluids was resented. Many research results were analyzed and comared with each other. The major conusion rom above survey is that the results or similar nanoluids diered rom one another a Corresonding author: ela@agh.edu.l This is an Oen Access artie distributed under the terms o the Creative Commons Attribution License., which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Artie available at htt:// or htt://dx.doi.org/./mateccon/86

2 MATEC Web o Conerences and the reason or that may be the use o various theoretical conductivity models that were alied in calculations. Several studies on the convective heat transer o nanoluids have been ublished in recent years. Khanaer et al. [7] studied the buoyancy-driven heat transer enhancement o Cu-based nanoluid with various concentrations o susended arties. Hwang et al. [8] investigated Rayleigh-Benard convection o nanoluids in a rectangular cavity. They alied emirical ormulas to evaluate the heat transer coeicient and reorted contrary nanoarties eect on heat transer. Santra et al. [9] studied heat transer enhancement in dierentially heated square cavity. They addressed Cu-water luid as non-newtonian and observed a decrease o heat transer by enlargement o nanoarties volume raction. Because research concerning the heat transer o nanoluids have been conducted relatively recent, since early nineties, there are a lot o inconsistencies and unear mechanisms, which require urther studies. The reort by Schwab et al. [] aears to be the irst one to have shown the enhancement o heat transer by magnetic ield in weakly magnetic (aramagnetic or diamagnetic), one-hase luid. They studied the Rayleigh-Benard convection o the aqueous solution o gadolinium nitrate and reorted the Nusselt number versus the temerature dierence. Their data distributed either above or below the reerence heat transer rate or a non-magnetic ield, i.e., in a terrestrial gravitational state. They laced the exerimental enosure at the location either to enhance or suress the gravitational acceleration. Furthermore, they roosed a magnetic Rayleigh number using actual gravitational and magnetic acceleration summation. Tagawa et al. [] emloyed the Boussinesq aroximation or magnetic orce and derived the simle model or the momentum equation and gave numerical analyses or convection o air in an enosure with a magnetic orce. They studied various cases subsequently []. The magnetic convection, rom the engineering oint o view, was resented in a book by Ozoe [], while Fornalik-Wajs [] derived theoretical and exerimental analysis o aramagnetic luid magnetic convection. Bednarz et al. [6] and Filar et al. [7] conducted very interesting exerimental analysis and obtained results showing intensiication o the heat transer rocesses in the cubical and thermosyhon-like geometry, resectively. Wrobel et al. [8] ublished results o exerimental and numerical analysis considering a thermo-magnetic convective low o aramagnetic luid in an annular enosure with a round rod core and a cylindrical outer wall. They resented the eect o the magnetic ield on the convection o the aramagnetic luid in the annular vessel in various ositions. Kenjeres et al. [9] erormed exerimental and numerical studies o combined eects o thermal buoyancy and magnetization orce alied on a cubical enosure o aramagnetic luid heated rom below and cooled rom to. The magnetization orce roduced signiicant changes in low, enhancement in wall-heat transer and numerical simulations rovided a detailed insights into changes o the local wall-heat transer and low structure. Stability diagram, containing three characteristic states: steady, oscillatory and turbulent regimes, was resented. It was already shown that magnetic ield is able to control low o weakly magnetic substances as the aramagneti and diamagneti. In recent years only the numerical studies about alying the external magnetic ield to the nanoluids aeared. Examined luids contained various concentrations o the nanoarties (usually above % vol.) with dierent kinds o arties [] in diverse shaes o exerimental enosures (e.g.): cubical [], u-shaed [] and l-shaed []. No aers concerning the exerimental studies o nanoluids in the external magnetic ield were ound. In the resent aer the irst ste o analysis veriying the otential heat transer enhancement in the nanoluids by the magnetic ield is resented. Exerimental stand. Exerimental aaratus The exerimental setu is resented in Figure. It consisted o an exerimental enosure laced in the bore o a suerconducting magnet, a heater control system, a constant temerature bath and a data acquisition system connected to a ersonal comuter. The exerimental enosure is shown schematically in Figure. The Plexiglas cubical enosure o size equal to. [m] was heated (with constant heat lux) rom one horizontal wall and isothermally cooled rom the oosite one, while the our remaining walls were insulated. A rubber-coated nichrome wire was used as a heater and connected to a DC ower suly. The heating ower was monitored with multimeters. Water was umed rom a thermostatic water bath through a small cooling chamber built in the cooling wall. The temerature o heated and cooled walls was measured with six T-tye thermocoules (three in each wall). The temerature was also measured inside the cube six K-tye thermocoules were inserted and arranged as it is shown in Figure. The thermocoules signals were stored in a comuter through a data acquisition system. The enosure was illed with working luid. The enosure osition in the magnet test section was correlated with the location o maximal value o grad B, where B indicates the magnetic induction. This maximum was in the central art o exerimental cavity, in the distance o. [m] rom the magnet to. Such enosure osition in the magnetic ield caused comlex system o gravitational and magnetic buoyancy orces. In the lower (heated) art o cavity both orces were acting in the same direction, enhancing each other. In the uer (cooled) art orces counteracted, attenuating each other. The detail analysis and understanding will come rom the numerical analysis as a next ste o resented work. 6-.

3 HEAT luid in Table the resented roerty values are taken as or water. Table. Thermo-hysical roerties o working luids. * indicates values taken as or water, ** - values calculated rom Eqs., *** values measured Proerty Thermal conductivity Unit k n [W (mk) - ] Cu (m)* Cu (m)**.67.7 Density ρ n [kg m - ] Figure. Exerimental setu.. Working luid Two low arties concentration nanoluids were analyzed. Water was the base luid, while the arties were made o coer. The arties size was in the range o -6 [nm]. First luid s artie concentration was [m], while the second one s [m]. Prearation o nanoluids caused a lot o roblems, mentioned also in the literature []. Among the others there are mainly the agglomeration and sedimentation o arties. The [m] concentration luid was already reared and it didn t showed those roblem, whereas the [m] luid was reared or the needs o exeriment. Mechanical stirrer was used to mix the base luid and arties. The rocess lasted 8 hours and rom the transarent luid the oaque one was obtained. In the next arts o aer the ollowing abbreviations will be alied to indicate the luids: Cu luid o [m] arties concentration, Cu luid o [m] arties concentration. Seciic heat Thermal exans. coe. Dynamic viscosity Electrical conductivity Mass magn. suscetibility c n [J (kg K) - ] β n [K - ] μ n [kg (m s) - ] σ n [S m - ] χ m [m kg - ] *** The measurements o thermohysical roerties (excet the magnetic suscetibility) o Cu were very diicult due to the sedimentation o arties. The values were changing with time as the artie were settling. Thereore these measurements were taken as erroneous. For the heat transer analysis the ormulas listed in Table were taken to calculate roerties o Cu. The ollowing subscrits are alied to indicate the comonents: b base luid, arties and n nanoluid. Symbol reresents the volume ercentage o arties. Table. Formulas alied or the calculation o Cu thermohysical roerties [] Proerty Formula, source Figure. Exerimental enosure. Following roerties necessary or the analysis were measured: magnetic suscetibility, density, viscosity, thermal exansion coeicient. However in the case o Cu measured roerties didn t show any dierences (within the accuracy range o alied devices) in comarison with the water roerties. Thereore or this Thermal k k ( k kb ) b kn kb conductivity k k ( k k ) Density and Seciic heat roduct Thermal exans. coe. Dynamic viscosity Electrical conductivity n c ( c ) ( )( c ) n ( ) n. ( ) n (. ) <. ( ) n [ ] ( ) ( ) b b 6-.

4 MATEC Web o Conerences The base luid water was diamagnetic, the same as the coer arties. Obtained nanoluids were then also diamagnetic. Due to the low electrical conductivity they were considered as electrically non-conductive. It meant that the Lorentz orce in this system could be neglected [6]. The only orces governing the luid low were: the gravitational buoyancy and magnetic buoyancy orces. Exerimental rocedure For both analyzed nanoluids the rocedure was the same. First ste was connected with the analysis o conduction state, thereore the enosure was laced in such coniguration that the heated late was at the to, while the cooled one at the bottom. The enosure was located in the magnet working section to kee the same temerature o environment (about 8 [ o C]). The analysis o conduction state was necessary or determination o the Nusselt number. The way o Nusselt number calculation will be described in section.. The next ste was connected with analysis o thermomagnetic convection. At irst the natural convection o nanoluids was analyzed. The ower suly was set to obtain chosen temerature dierence between heated and cooled walls about ΔT ~. [ C] (the value came rom the reliminary measurements and the calibration o setu). The enosure was then let to obtain the stable state. It lasted about [h]. Ater this time the controlled arameters (temerature, electrical current, voltage) were recorded and the magnetic ield was alied to the system. The value o magnetic induction was changed by stages o [T] u to [T]. At each stage the system had to reach the steady state beore recording the analyzed signals outut. The time needed to obtain the steady state slightly varied or each case (deending on the low mode) but it was always in the range o - [h]. Signal analysis Recorded temerature time series enabled investigations o two asects o thermo-magnetic convection: the heat transer and low structure. The heat transer was determined through calculation o the Nusselt number, while the low structure trough the Fast Fourier Transorm (FFT) and sectral analysis.. Heat transer The Nusselt number deinition based on the convection and conduction heat luxes can be written as: Q net _ conv Nu. () Q net _ cond Estimation o the net convection ( Q net _ conv ) and net conduction ( Q net _ cond ) heat luxes was done according to a method invented by Ozoe and Churchill [7] and based on the ollowing equations: Q net conv Q conv Q _, () It was assumed that the heat deends strictly on the temerature o the heated wall and does not deend on the heat transer mode inside the enosure. As a irst ste in the Nusselt number estimation the conduction exeriment was done. The Q was estimated rom the ollowing equation: Q Q cond Q theor _ cond, () where Q theor _ cond lkn T, () (l indicates the size o enosure equal to. [m], k n is the thermal conductivity o nanoluid and ΔT is the temerature dierence between heated and cooled late) Q theor _ cond was calculated rom Fourier s law or the conduction area o l. The estimated heat was linearly aroximated and the equations or both luids are resented below Q.69 T, () _ Cu Q _ Cu.86 T. (6) There are two dierent equations, because the heat changed due to the higher thermal conductivity o Cu. Thereore in the heat transer analysis each o the luids was considered searately with an utilization o the characteristic or it equation. Alying equations (), () and () to equation (), the deinition or the Nusselt number could be rewritten Q conv Q Nu. (7) l kn T The convection heat lux ( Q conv ) was calculated as a roduct o the electrical current and voltage o heater suly. Having the convection heat lux ( Q conv ), the heat ( Q ) estimated rom equations () and theoretical conduction heat lux ( Q theor _ cond ) it was ossible to estimate the Nusselt number. The results are resented in Figure.. Flow structure The sectral analysis o scalar ield (or examle temerature or concentration) are useul in the analysis o turbulent transort mechanisms. Assuming that the turbulence is homogeneous, the sectral unctions can be calculated with utilization o FFT. In the resented analysis the temerature signals not the temerature luctuations were considered. In deendence on the low the requency sectra can resent various character. For the steady convective low in osed cavity they can reresent number o vertical structures rotating with articular requencies. The requency sectra may show then the characteristic shar eaks, which can be related with eriodic coherent structures [8]. The smaller requency is indicated the larger vortical structure aears. In the case o turbulent low the vortices are continuously changing rom the large to the smallest ones. This is only the qualitative not quantitative inormation, however it can hel to describe the low structure. 6-.

5 HEAT There is an analogy between the sectral analysis o scalar arameter and the Kolmogorov energy sectrum [8,9]. The FFT analysis was conducted with an algorithm []. The sectral unctions o temerature generally deends on the energy dissiation, temerature, kinematic viscosity and thermal diusivity. In some range, the sectral unction does not deend on the diusion rocesses, thereore do not deend on the kinematic viscosity or thermal diusivity. This range is called inertial-convective and the sectral unction is similar to the energy sectrum as the ower unction o wave number with exonent -/ [8]. When the thermal diusivity becomes more signiicant, then the sectral unction is roortional to the inverse value o wave number (k - ) [8]. This turbulent range is called viscousdiusive [9]. Both o the unctions are marked in Figures and as the straight lines with sloes o -/ and -. (a) Nusselt number, Nu (b) 9 8,8,7 Cu Cu -,x 9-9,x 8-6,x 8 -,x 8, Rayleigh number, Ra TM Results and discussion The results o heat transer analysis are resented in Figure. The Nusselt number as a unction o the thermo-magnetic Rayleigh number is shown in Figure (a). The thermo-magnetic Rayleigh number is deined as: B z Ra TM Ra T[ Bz ( )], (8) z where Ra T is the thermal Rayleigh number described by the ollowing ormula g n n c n Ra T l T, (9) k n while γ is the magnetization number written as: mbmax () m gd and g gravitational acceleration; β n thermal exansion coeicient; ρ n density; c n seciic heat; μ n dynamic viscosity; k n thermal conductivity; l characteristic dimension; ΔT temerature dierence; χ m mass magnetic suscetibility; B max magnetic induction in the central art o coil; B z magnetic induction in the osition o enosure centre; μ m vacuum magnetic ermeability. The value o B z and gradb z were calculated numerically based on the suerconducting magnet coil technical data [] and led to the relations: B z,66 B, () B z,87 Bmax. () z The negative sign o thermo-magnetic Rayleigh number was coming rom the relation between the natural and magnetic convection in accordance to Eq. (8) and suggested revalence o magnetic art o this deinition. At T o magnetic induction the Nusselt number got higher values or Cu, so or the luid o lower arties concentration. It could be exlained with the Nusselt number deinition (Eqs. () and (7)). Increasing thermal conductivity (as or Cu) caused increase o the conduction heat lux and decrease o the max Heat lux [W] (c) Heat lux [W],6,,,8,7,6,, Convection heat lux Conduction heat lux 6 8 Magnetic induction, B [T] Convection heat lux Conduction heat lux 6 8 Magnetic induction, B [T] Figure.(a) the Nusselt number versus the magnetic Rayleigh number, (b) the convection and conduction heat luxes versus the magnetic induction or Cu; (c) the convection and conduction heat luxes versus the magnetic induction or Cu Nusselt number, when the other conditions were ket the same. Analysis o system under the magnetic ield should be done searately or each luid. Alication o the magnetic ield to the system with Cu changed slightly the situation, at magnetic induction o [T] (Ra TM = -7 ) increase o the Nusselt number could be observed. Further increase o magnetic induction u to 9 [T] caused continuous decrease o the Nusselt number. In the case o Cu at [T] o magnetic induction (Ra TM = -9.9 ), there is no measurable change in the Nusselt number, but at [T] o magnetic induction (Ra TM = ) small increase o its value could be ound. 6-.

6 MATEC Web o Conerences (a) (b),89 c,, E- E- c -/ sloe - sloe E- (c),,,, (d) E-6 E-,,,89 c,, E- E- c -/ sloe - sloe E- (e),,,, () E-6 E-,,,89 c,, E- E- E- c -/ sloe - sloe,,,, E-6 E-,, Figure. The ower sectrum versus requency or Cu (a) at [T] o magnetic induction, Ra TM =. 6, (b) log-log scale at [T] o magnetic induction, Ra TM =. 6, (c) at 6 [T] o magnetic induction, Ra TM =-.7 8, (b) log-log scale at 6 [T] o magnetic induction, Ra TM =-.7 8, (c) at 8 [T] o magnetic induction, Ra TM =-.6 8, (b) log-log scale at 8 [T] o magnetic induction, Ra TM = Further increase o magnetic induction resulted in increase o the Nusselt number. It should be emhasized that the boundary conditions were the same only the concentration o luids was dierent. Increase o the Cu thermal conductivity inluenced the temerature ield and indirectly the orces acting on the luid and then the heat transer. As it was mentioned, the orce system in the analysed enosure was comlex, that is why the tendency o Nusselt number changes was dierent or each luid. In Figures (b) and (c) comarison between the convective and conduction heat luxes or Cu and Cu (resectively) are resented. In both cases the convection heat lux was higher than the conduction one. For Cu the convection heat lux was higher than or Cu, what was also reresented by higher value o the Nusselt number (Figure (a)). However, the conduction heat lux was higher or Cu o about %. The results o sectral analysis are resented in Figures and. The analysis was carried out or all recorded signals rom all thermocoules laced inside the enosure. Figure shows the thermal ower sectrum versus the requency or Cu in deendence on the magnetic induction. In Figure (a) the magnetic induction was equal to T (natural convection without magnetic ield), in Figure (c) 6 T and in Figure (e) 8 T. 6-.6

7 HEAT (a) (b) c,, E- E- c -/ sloe - sloe E- (c),,,, (d) E-6 E-,, c,, E- E- c -/ sloe - sloe E- (e),,,, () E-6 E-,, c,, E- E- c -/ sloe - sloe E-,,,, E-6 E-,, Figure. The ower sectrum versus requency or Cu (a) at [T] o magnetic induction, Ra TM =.8 6, (b) log-log scale at [T] o magnetic induction, Ra TM =.8 6, (c) at 6 [T] o magnetic induction, Ra TM = -.8 8, (b) log-log scale at 6 [T] o magnetic induction, Ra TM = -.8 8, (c) at 8 [T] o magnetic induction, Ra TM = , (b) log-log scale at 8 [T] o magnetic induction, Ra TM = In the let-hand side column (Figures (a), (c) and (e)) characteristic eak could be seen. As it was mentioned in Section., or the steady convective low such eaks could reresent vortical structure rotating with articular requency. In all igures the requency about.8 [Hz] reeated onesel. It suggested one large vortex in whole volume o analyzed enosure. The lower the requency the larger should be the vertical structure. What is more imortant, this requency was ound or almost all thermocoules. Increasing magnetic induction rom [T] to 8 [T] caused decrease in the ower sectrum, what corresonded with decreased values o the Nusselt number (Figure (a)). Both results indicated the suression o convection. The right-hand side column in Figure ((b), (d) and ()) resents the ower sectrum versus the requency but in the log-log scale. The straight lines o -/ and - sloes were also marked. These sloes reresented the inertial-convective and viscous-diusive turbulent regimes, resectively. They did not it the lat ower sectrum distribution. The lat ower sectrum distribution is characteristic or steady low and conirmed the conusion o attenuating inluence o magnetic buoyancy orce on the low o Cu. 6-.7

8 MATEC Web o Conerences In Figure the results o signal analysis or Cu are shown. The let-hand side column (Figures (a), (c) and (e)) consists o the ower sectrum distribution versus requency, while the right hand side column (Figures (b), (d) and ()) o log-log scale reresentation o ower sectrum. In Figures (a) and (b) the results were obtained at [T] o magnetic induction, in Figures (c) and (d) at 6 [T] o magnetic induction and in Figures (e) and () at 8 [T] o magnetic induction. Increasing magnetic ield did not change much the distribution o ower sectrum but in Figure (c) the most visible were two eaks, while in Figure (e) one. It looked like the magnetic ield ordered slightly the low. The ower sectrum reresentation in log-log scale also conirmed dierent characteristic o Cu low. The straight line o -/ sloe was in good agreement with the ower sectrum, what suggested the changes in low were going towards the inertial-convective turbulent regime. The straight line o - sloe did not it the ower sectrum distribution in any range. 6 Summary In this aer the exerimental analysis o thermomagnetic convection o low concentration nanoluids was resented. Two luids o [m] and [m] coer arties concentration in the water were investigated. The inluence o various magnetic induction values on the transort rocesses was checked. Estimation o the heat transer and low structure was able due to the signal analysis. The results discussion led to the ollowing conusions: () resented results did not show one ear tendency, what deended on the alied boundary conditions, () the heat transer enhancement strongly deended on the osition o exerimental enosure in the magnetic ield, () analyzed osition o exerimental enosure was characterized by the most comlex orce system rom ossible ones. It caused (a) increase o the Nusselt number at low magnetic induction values, u to [T] or Cu, (b) decrease o the Nusselt number at higher magnetic induction values, < [T] or Cu, while increase or Cu, (c) comlex low structure with local convection suression, () reaction o the analyzed system on the magnetic ield and additional magnetic buoyancy orce is romising and shows its otential in the heat transer enhancement area even with low concentration nanoluids. Reerences. A.M.L. Bethancourt, M. Hashiguchi, K. Kuwahara, J.M. Hyun, Int. J. Heat Mass Tran. (999). X.Q. Wang, A.S. Mujumdar, Braz. J. Chem. Eng. (8). E. Goharshadi, H. Ahmadzadeh, S. Samiee, M. Hadadian, Phys. Chem. Research (). X.Q. Wang, A.S. Mujumdar, Braz. J. Chem. Eng. (8). R. Saidur, K. Leong, H. Mohammad, Renew. Sust. Energ. Rev. () 6. A. Sobti, R.K. Wanchoo, Mater. Sci. Forum, 77 () 7. K. Khanaer, K. Vaai, M. Lightstone, Int. J. Heat Mass Tran. 6 () 8. K.S. Hwang, J.H. Lee, S.P. Jang, Int. J. Heat Mass Tran. (7) 9. A.K. Santra, S. Sen, N. Chakraborty, Int.J. Therm. Sci. 7 (8). L. Schwab et al., J. Magn. Magn. Mater. 9 (98). T. Tagawa, R. Shigemitsu, H. Ozoe, Int. J. Heat Mass Tran. (). J. Huang, B. Edwards, Phys. Rev. E 7 (998). H. Ozoe, Magnetic Convection (Imerial College Press, ). E. Fornalik, Magnetic convection o aramagnetic luid in an enosure (AGH Uczelniane Wydawnictwa Naukowo-Dydaktyczne, monograh, 9). K. Ezaki, M. Kaneda, T. Tagawa, H. Ozoe, ISIJ International () 6. T. Bednarz, E. Fornalik, T. Tagawa, H. Ozoe J.S. Szmyd, Int. J. Therm. Sci. () 7. P. Filar, E. Fornalik, T. Tagawa, H. Ozoe, J.S. Szmyd, J. Heat Trans. 8 (6) 8. W. Wrobel, E. Fornalik-Wajs, J.S. Szmyd, Int. J. o Heat Fluid Fl. () 9. S. Kenjeres, L. Pyrda, E. Fornalik-Wajs, J.S. Szmyd, Flow Turbulence Combust. 9 (). D.D. Ganji, A. Malvandi, Powder Technol. 6 (). B. Ghasemi, S.M. Aminossadati, A. Raisi, Int. J. Therm. Sci. (). B. Ghasemi, Numer. Heat. Tr. A-al, 6 (). M. Sheikholeslami, M. Gorji-Bandy, D.D. Ganji, S. Soleimani, IJST, Transactions o Mechanical Engineering 8 (). D. Wu, H. Zhu, L. Wang, L. Liu, Curr. Nanosci. (9). Y. Xuan, W. Roetzel, Int. J. Heat Mass Tran. () 6. J. Qi, N.I. Wakayama, A. Yabe, Int. J. Heat Mass Tran., () 7. H. Ozoe, S. W. Churchill AIChE Symosium Series, Heat Transer J.W. Elsner, Turbulence o Flows (in Polish) (PWN, Warszawa, 989) 9. W.D. McComb, The hysi o luid turbulence (Oxord University Press, Oxord, 99). htt:// Suerconducting Magnet Manual Acknowledgement The resent work was suorted by Polish National Science Centre Project No. /7/B/ST8/