THERMAL RADIATION EFFECTS ON THE ONSET OF UNSTEADINESS OF FLUID FLOW IN VERTICAL MICROCHANNEL FILLED WITH HIGHLY ABSORBING MEDIUM

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1 THEMA ADIATION EFFECTS ON THE ONSET OF UNSTEADINESS OF FUID FOW IN VETICA MICOCHANNE FIED WITH HIGHY ABSOBING MEDIUM by M. Y. ABDOAHZADEH JAMAABADI a,b, Jae Hyun PAK *, an Chang Yeop ee a Assistant Professor, Maritime University of Chabahar, Chabahar , Iran b Senior esearher, Grauate Shool of Mehanial an Aerospae Engineering, Gyeongsang National University, Jinju, Gyeongnam , South Korea Assistant Professor, Department of Aerospae an System Engineering an esearh Center for Airraft Parts Tehnology, Gyeongsang National University, Jinju, Gyeongnam , South Korea Thermohemial Energy System &BD Group Korea Institute of Inustrial Tehnology Cheonan-si,Chungheongnam-o -, South Korea *orresponing author; parkj@gnu.a.kr This stuy presents the effet of thermal raiation on the steay flow in a vertial miro hannel fille with highly absorbing meium. The governing equations (mass, momentum an energy equation with osselan approximation an slip bounary onition) are solve analytially. The effets of thermal raiation parameter, the temperature parameter, eynols number, Grashof number, veloity slip length, an temperature jump on the veloity an temperature profiles, Nusselt number, an skin frition oeffiient are investigate. esults show that the skin frition an the Nusselt number are inrease with inrease in Grashof number, veloity slip, an pressure graient while temperature jump an eynols number have an averse effet on them. Furthermore, a riterion for the flow unsteainess base on the temperature parameter, thermal raiation parameter, an the temperature jump is presente. Keywors: Thermal raiation; rarefation effets, mixe onvetion, miro hannel, flow unsteainesss

2 Introution Heat transfer in free an mixe onvetion in vertial hannels is foun in many appliations [- ] suh ooling systems for eletroni evies [5], hemial proessing equipment [6], miroeletroni ooling [7], et. Solar thermal olletor, whih absorbs sunlight an onvert to heat, is a typial example utilizing suh onfiguration [-9]. Different from the other appliations [0-], in the solar olletor the raiative heat transfer is signifiant in aition to onution an onvetion. Also, the signifiane of raiative heat transfer is foun in the inustrial proesses an MHD flow [-]. The raiative heat transfer is ommon. eently, the use of mirohannel in solar olletor has been inreasing beause the miniaturize evies proue higher performane [9-]. Even though many researhers have performe the steay-state analysis of solar olletor onsiering ombine heat transfer of onution, onvetion, an raiation an flui flow, the onset of flow unsteainess has been harly pai attention to. Upon the unsteainess of the system, many aspets in numerial analysis oul be hange; therefore the ientifiation of limit of flow unsteainess is essential to hoose an effiient omputational metho []. When flui flow is onfine by small system suh as miro/nano-hannels, some isontinuity in veloity an temperature profiles might happen at the interfae of flui an soli surfae [-]. Ulmanella an Ho [] experimentally observe the veloity for various miro-size hannels, whih is a funtion of shear rate, type of liqui an surfae morphology. Boquet an Barrat [] esribe a possibility of temperature jump along with veloity slip. The veloity slip an temperature jump relations shoul be use as the bounary onitions for liqui-soli interfae of miro/nano-size hannel [5]. The purpose of solar thermal olletor is to absorb the sunlight as muh as possible an a number of stuies have been fouse to inrease the absorptivity of the flui meium in the solar olletor. eently, the researhers have foun that the absorptivity an be signifiantly enhane by using nanoflui that is a mixture of liqui an the nanopartiles suspene [9]. With seeing the nanopartile in base flui, the extintion oeffiient of the meium inreases over 00 m - [9]. It is apparently optially thik meium even for the mirohannel whose harateristi length is in mirosale an therefore the optially thik meium approximation is appliable to the analysis of raiative transfer. Consiering all the above, in this stuy we theoretially investigate the thermal raiation effets on the onset of unsteainess of flui flow in vertial mirohannel fille with highly absorbing meium. This kin of onfiguration is ommonly foun in thermal storage with solar olletor [7] an thermo-syphon solar water heaters []. The veloity slip an temperature jump are moele by introuing veloity slip length an temperature slip length, respetively. The raiative heat transfer through the meium is simplifie by the iffusion approximation (or, osselan approximation). Besies the thermal raiation an rarefation effets, the flow harateristis are ientifie by using the lassial thermo-flui parameters suh as eynols number, Grashof number, Nusselt number, an skin frition oeffiient are also fouse. Finally, the influene of thermal raiation an rarefation effets on the flow unsteainess is examine.

3 Governing Equations an Numerial metho Figure. Shemati of system In orer to unerstan the funamental basis of an infinitely long vertial solar olletor in miro-sale, we onsiere a simple onfiguration as shown in Fig.. An optially thik inompressible flui is onfine by two parallel planar walls that are separate by the istane of. Sine we assume the infinitely long geometry, the flow veloity u is only the funtion of y-iretion aross the hannel an the origin of the system is plae at the mile point. The left an right walls (y=±) are maintaine at the uniform temperatures of T an T, respetively. In the steay state, this system an be esribe by momentum an energy equations employing Boussinesq approximation for buoyany fore: Momentum equation: Energy equation [6]: u y p r g T T µ x µ 0 = β T y σ T = kχ y ( ref ) () () where p/x is a pressure graient in the flow iretion, i.e. x-iretion, whih is perpeniular to the y-iretion, T is the meium temperature, k is the thermal onutivity of the flui, β is the thermal expansion oeffiient, μ is the ynami visosity, ρ is the flui ensity, an T ref is a onstant referene temperature efine as T ref = (T T )/. σ is the Stefan-Boltzmann onstant an χ is the mean absorption oeffiient of the meium. Sine the hannel size is in miro-sale, the veloity slip an the temperature jump nee to be onsiere as the bounary onitions at the soli walls:

4 T T u ( y = ) ( y = ) u u( y = ) = lv y = ( y ) y y= () u = = lv () y T σ T = T lt y k y (5) χ y= T σ T = T lt y k y (6) χ y= where l v is the veloity slip length an l T is the harateristi length of temperature jump, whih is referre to as temperature jump length hereafter. The moelling of veloity slip an temperature jump at flui-soli interfae is not easy ue to the omplexity of the phenomena themselves. However, in this stuy we have use the simplest moels of equations (-6), whih resemble those for gas-soli interfae, beause our purpose is to investigate the physial essene rather than the quantitative etails. Nevertheless, we an still obtain the orret physial essene from the simplest veloity slip an temperature jump moels even though they oul have lear limitation in preiting the aurate quantities. In orer to larify the physial essene, in this stuy the following non-imensionalize variables are introue: ( Y ) x X = (7) y Y = () u( y) U Y = (9) ( ) u m T T T = (0) T T p P = () m u m ( T T ) gβ Gr = ν () u m e = ν () ( T T ) σ = () 6kχ

5 T = T T T (5) where the u = u( y) m y l v l = (6) v l T l T = (7) is a onstant, Gr is the Grashof number, e is the eynols number, is the raiation parameter, an is the temperature parameter. Then, the momentum an energy equations of () an () an be rewritten as: U Gr P = Y e X () ( ) 0 = Y (9) U U ( Y = ) = λ v Y (0) Y = U U ( Y = ) = λ v Y Y = () ( Y = ) = λt [ ( ) ] Y = Y () ( Y = ) = λt [ ( ) ] () Y = Y where the equations from (0) to () are the bounary onitions. Also, the skin frition rag oeffiient an the Nusselt number an they are expresse as U Y Y = C = () f y= e U Y Y = C = (5) f y= e Nu y= ( ) Y = U ( Y = ) Y Y = (6) 5

6 Nu y= [ ( ) ] Y = U ( ( Y = )) Y Y = (7) esults an Disussion In the simplest onition in whih rarefation effets an the thermal raiation are not inlue (λ v = λ t = = 0), the equations (-) have the following analytial solutions: = Y () ( ) Gr U = Y Y 6e (9) P / X = (0) In Figure (a), the veloity profiles are plotte by hanging Gr/e from 0 to 5 by the inrement of 9. Sine the Grashof number, Gr, is the ratio of buoyany to visous fore an the eynols number, e, is the ratio of momentum to visous fores, the Gr/e represents the ominane of buoyany over momentum. When Gr/e < 9, the veloity profile has a paraboli shape with a single maximum of / near the origin. Then, with further inrease in Gr/e, two loal extrema are observe at e e Y = Gr Gr (loal minimum) an e e Y = Gr Gr (loal maximum). Figure. The effet of the ratio of Grashof number to eynols number (Gr/e) on (a) veloity an (b) shear stress without rarefation effets an thermal raiation (λ v = λ t = = 0) 6

7 Figure (b) shows the imensionless shear stress profiles without rarefation effets an thermal raiation. Following Figure (a), Gr/e varies from 0 to 5 by the inrement of 9. The imensionless shear stress is efine as τ = τ mum. The maximum shear stress is observe as mum.5e Gr τ max = Gr 6e at e Y =. The skin frition oeffiient at the right an left walls are Gr Gr ifferent as C f = ue to the presene of the pressure graient. The relation of C y =± f e e iniates that the natural onvetion augments the flow in the hot wall sie an inreases the wall skin fration while it reues the flow in the ol wall sie an ereases the wall skin fration. Without the thermal raiation, Nusselt number at the wall an be expresse as e Nu =. When the Grashof number equals to forty five times of eynols number, y =± Gr /90 0.5e the average temperature is the same as right wall temperature an the Nusselt number at the right wall beomes infinity. Figure. Effet of raiation parameter,, an temperature parameter,, on (a) temperature an (b) veloity without rarefation effets (λ v =λ l =0, Gr/e=5). Figure show the effet of raiation parameter,, an temperature parameter, on the temperature an the veloity profiles without inluing rarefation effets (λ v =λ l =0). The ratio of Grashof number to eynols number is fixe at 5. Without raiation effet, the temperature profile is linear. However, with the inrease in the meium temperature inreases in overall while it has a suen rop near the left ol wall. Suh rop is also observe with the reue temperature parameter,. The effet of is saturate for > 0: The profile with = 0 is not muh ifferent 7

8 from the urve with = 000. Different from the temperature, the veloity profile is remarkably hange with. When = 0, the veloity in the right sie is observe larger than the value in the left sie (sinusoial form). However, with the inrease in the profile onverte to a paraboli shape along with a signifiant inrease of the maximum value. Suh behavior is also observe with reue. Next, we onsier the ase in whih the rarefation effets are inlue while the thermal raiation is still ignore ( =0). In this ase, the temperature an veloity profiles are obtaine as: = Y λ T () U Gr P λ Gr P λ Gr λvgr = v v () 6e X e X e e T T ( Y ) Y ( Y ) λ ( Y λ ) P X λvgr Gr = λ e () T e λ / v The effet of temperature jump appears only in the temperature as inrease the profile by λ T, i.e. It oes not affet the flui veloity. Figure ompares the veloity profiles by hanging λ v while Gr/e is fixe at 5. The veloity profile is signifiantly influene by the slip length : When λ v <, the inrease in λ v auses the inrease in the veloity, while when λv, the inrease in λ v auses the erease in the veloity. The maximum veloity is foun either at the wall or at the following position: e λv λv e λ e v λv Gr Gr Gr 7 Y = λv () where the tangent of veloity is zero. The veloities at the left an right walls given as P λt Gr λvgr U ( ± ) = λv ( ± λv ) (5) X e e

9 Figure. The effet of slip length, λ v, on the veloity profile without thermal raiation ( = 0, Gr/e=5) From the imensionless pressure graient relation of Eq. (), we obtain the following relation: λv ρgβ T T p = λρ ( ) 6 T gβt T x λ / ( ) v mu m (6) It iniates that for a given flow rate, the pressure graient inreases with larger veloity slip an temperature jump. The graient is also augmente as Gr/e inreases. Finally, the skin frition oeffiient an the Nusselt number an be erive as: C f y=± v λvgr λ Gr Gr λ Gr v = e e e e e / Nu y=± ( λv ) Gr λv = 5e 6 Gr λ 0. 5 / 5e v (7) () With inrease in Gr/e an veloity slip, both skin frition oeffiient an Nusselt number inrease. It is noteworthy that the temperature jump has no effet on the skin frition oeffiient an the Nusselt number. Without veloity slip an temperature jump (λ v = λ T = 0), from eqs. (-) we an erive an impliit formula for the temperature as: ( ) ( ) ( ) = Y 6 (9) 9

10 If the optial thikness of meium is remarkably large (χ >> ) an subsequently <<, from the perturbation metho in the first orer (=Y 0 ) the above equation an be simplifie as ( Y ) ( ) ( Y ) = 0. (0) The first-orer approximation of Eq. (0) appears as = Y () 0 ( ) with subjet to the bounary onitions of ( ) 0 () 0 = 0 () = 0 () Then, the first-orer solution an be obtaine as ( ) ( ) ( ) ( ) = Y ( ) Y Y () / Eq. (9) has a minimum as =( ), whih is unonitionally greater than zero. Therefore, the equations (9) an () always have the real solutions. Figure 5 ompares -( =0) an U- U( =0) from perturbation metho an from exat formulation, for various an uner the fixe Gr/e value of 5. From the figures, it is observe that the perturbe solution is similar to the exat solution. Figure 5. Comparison of -( =0) an U-U( =0) from perturbation metho an from exat formulation, for various an (Gr/e=5) 0

11 If both thermal raiation an rarefation effets are onsierable, the analytial solution of equation (-) an be expresse as ) ( } ){ ( ) ( = Y (5) where =λ T /. Eq. (5) has a real solution only when the following onition is satisfie: ( ) ( ) 7 > (6) Therefore, if both an are small, / ) 7( / > an the flow in the hannel oul beome unsteay. Eq. (5) has the minimum as / ) ( an it shoul be less than zero to have a real solution. Therefore, / / / ) ( < (7 )

12 The above inequality an ientify the stable zone in ( λ T /, λ T )-plane as illustrate in Figure 6. With erease in, the values of an satisfying unsteay onition ereases. Figure 6. Stability iagrams in ( λ T /, λ T )-plane for (a) =000, (b) =00, an () =0. Conlusions an eommenation In this stuy, the effet of thermal raiation on the steay flow in a vertial miro hannel fille with highly absorbing meium has been thoroughly investigate. The results an be summarize as follows: () When the thermal raiation is ignore, the behaviors of temperature an veloity are ifferent: The temperature profile remains linear. At small Gr/e, the veloity profile is paraboli while at large Gr/e, it is hange as a sinusoial shape. () Grashof number, veloity slip, an pressure graient inrease skin frition an the Nusselt number whereas temperature jump an eynols number reue their values. () As inreases an ereases, the temperature profile beomes more non-linear. () In the presene of thermal raiation, the maximum veloity inreases with inrease in. However, the veloity profile is not influene by the thermal raiation. (5) With inlusion of both thermal raiation an rarefation effets, the values of an satisfying unsteay onition ereases with erease in. Aknowlegement This work was supporte by the Power Generation & Eletriity Delivery Core Tehnology Program of the Korea Institute of Energy Tehnology Evaluation an Planning (KETEP), grante finanial resoure from the Ministry of Trae, Inustry & Energy, epubli of Korea.(No.00000).

13 eferenes [] Barletta, A., Analysis of Combine Fore an Free Flow in a Vertial Channel with Visous Dissipation an Isothermal-isoflux Bounary Conitions, ASME J. Heat Transfer, (999), pp.9-56 [] Boulama, K., Galanis, N., Analytial Solution for Fully Develope Mixe Convetion Between Parallel Vertial Plates with Heat an Mass Transfer, ASME J. Heat Transfer, 6, (00), pp.- [] Hanafi, A. S., et al.,thermal an Hyroynami Charateristis of Fore an Mixe Convetion Flow Through Vertial etangular Channels, Thermal Siene, (00),, pp. 0-7 [] Babaelahi, M., et al., Analytial Treatment of Mixe Convetion Flow Past Vertial Flat Plate, Thermal Siene, (00),, pp [5] Zhao, C. Y., u, T. J., Analysis of Mirohannel Heat Sinks for Eletronis Cooling, Int. J. Heat Mass Transfer, 5, (00), pp [6] Barletta, A., Fully Develope Mixe Convetion an Flow eversal in a Vertial etangular Dut with Uniform Wall Heat Flux, Int. J. Heat Mass Transfer, 5, (00), pp.6-65 [7] Kanlikar, S. G., High flux Heat emoval with Mirohannels a oamap of Challenges an Opportunities, Heat Transfer Eng, 6 (005), pp.5 [] Hossain, M. S., et al., eview on Solar Water Heater Colletor an Thermal Energy Performane of Cirulating Pipe, enew. Sust. Energy ev., 5 (0),, pp. 0- [9] anguri,e. M., et al.,an Energy an Exergy Stuy of a Solar Thermal Air Colletor, Thermal Siene, (009),, pp [0] iu, D., Garimella, S. V., Investigation of iqui Flow in Mirohannels, AIAA J. Thermophys. Heat Transfer, (00), pp [] Juy, J., et al., Charaterization of Fritional Pressure Drop for iqui Flows through Mirohannels, Int. J. Heat Mass Transfer, 5 (00), pp [] Steinke, M. E., Kanlikar, S. G., Single-Phase iqui Frition Fators in Mirohannels, Proeeing, th ASME Conferene on Mirohannels an Minihannels, ohester, NY, 00 [] Kanlikar, S. G., et al., Charaterization of Surfae oughness Effets on Pressure Drop in Mirohannels, Phys. Fluis, 7 (005), [] Abollahzaeh Jamalabai, M. Y., et al., Two-Dimensional Simulation of Thermal oaing with Horizontal Heat Soures, Pro. Inst. Meh. Eng. C J. Meh. Eng. Si., 6 (0), pp.0-0 [5] Abollahzaeh Jamalabai, M. Y., et al., Numerial Investigation of Thermal aiation Effets on Open Cavity with Disrete Heat Soures, Int. J. Numer Metho Heat Flui Flow, (0),, pp [6] Abollahzaeh Jamalabai, M. Y., Experimental Investigation of Thermal oaing of a Horizontal Thin Plate Using Infrare Camera, J. King Sau Univ. Eng. Si., 6 (0),, pp.59-67

14 [7] Abollahzaeh Jamalabai, M. Y., Park, J. H., Thermal aiation, Joule Heating, an Visous Dissipation Effets on MHD Fore Convetion Flow with Uniform Surfae Temperature, Open J. Flui Dyn., (0),, pp. 5- [] Uin, Z., et al., Influene of Thermal aiation an Heat Generation/Absorption on MHD Heat Transfer Flow of a Miropolar Flui Past a Wege with Hall an Ion Slip Currents, Thermal Siene, (0),, pp [9] Taylor,. A., et al., Nanoflui Optial Property Charaterization: Towars Effiient Diret Absorption Solar Colletors, Nanosale es. ett., 6 (0), 5 [0] Horvath, J. A., Webb,. N., Experimental Stuy of aiation Absorption by Mirohannels of Varying Aspet atios, Solar Energy, 5 (0), pp [] Hossain, M. S., et al., eview on Solar Water Heater Colletor an Thermal Energy Performane of Cirulating Pipe, enew. Sust. Energy ev., 5 (0), pp. 0- [] Crouh, J.D., et al., Preiting the Onset of Flow Unsteainess Base on Global Instability, J. Comput. Phys., (007), pp [] Ulmanella, U., Ho, C. M., Moleular Effets on Bounary Conition in Miro/Nanoliqui Flows, Phys. Fluis, 0 (00), 05,. [] Boquet,., Barrat, J.-., Flow Bounary Conitions From Nano- to Miro-Sales, Soft Matter, (007), pp [5] Karniaakis, G., Beskok, A., Aluru., N., Miroflows an Nanoflows: Funamentals an Simulation, Springer, New York, USA, 005 [6] Moest, M. F., aiative Heat Transfer (n eition), Aaemi Press, New York, USA, 00

Chapter 2: One-dimensional Steady State Conduction

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