ScienceDirect. Thermal convection measurements inside aluminum foam and comparison to existing analytical solutions

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1 Avilble online t ScienceDirect Procedi Mterils Science 4 (14 ) th Interntionl Conference on Porous Metls nd Metllic Foms, Metfom 13 Therml convection mesurements inside luminum fom nd comprison to existing nlyticl solutions Nihd Dukhn*, Muntdher A. Al-Rmmhi, Ahmed S. Suleimn Deprtment of Mechnicl Engineering, University of Detroit Mercy, 41W. McNichols Rd., Detroit, MI 481, USA Abstrct Metl foms hve high therml conductivity nd lrge surfce re per unit volume. The internl structure of the foms promotes vigorous mixing of moving fluid inside the foms. As such, metl foms re very suited for convection het trnsfer designs. Cler models for forced convection het trnsfer inside the fom, s well s relible therml mesurements re indispensble for convection-bsed therml system designs. This pper present direct experiment for Drcy irflow nd fluid s temperture inside heted luminum cylinder filled with luminum fom. The experimentl fluid temperture is compred to the vilble nlyticl solutions for the two-eqution model for fully-developed forced convection. Peculir, physiclly-unexplinble behvior is displyed when plotting the existing nlyticl solution in the literture. An error ws discovered nd corrected. Good greement is obtined with the correct solution. 14 Published The Authors. by Elsevier Published Ltd. This by Elsevier is n open Ltd. ccess rticle under the CC BY-NC-ND license ( Peer-review under responsibility of Scientific Committee of North Crolin Stte University. Peer-review under responsibility of Scientific Committee of North Crolin Stte University Keywords: Convection; experiment; fluid temperture; nlyis ; het trnsfer 1. Introduction Due to the complex internl structure of the fom, exct solutions of the complete trnsport equtions re virtully impossible. Exmples of numericl, experimentl nd nlyticl work re given in Clmidi et l. (), Hwng et l. (), Lee nd Vfi (1999), nd Vfi nd Kim (1989). Lu et l. (6) nlyzed forced convection in tube filled with porous medium subjected to constnt het flux using the two-eqution model. Zho et l. (6) presented n nlyticl solution for tube-in-tube het exchnger, * Corresponding uthor. Tel.: ; fx: E-mil ddress: nihd.dukhn@udmercy.edu Published by Elsevier Ltd. This is n open ccess rticle under the CC BY-NC-ND license ( Peer-review under responsibility of Scientific Committee of North Crolin Stte University doi:1.116/j.mspro

2 34 Nihd Dukhn et l. / Procedi Mterils Science 4 ( 14 ) for which the inner tube nd the nnulus were filled with metl fom. Anlyticl solutions in porous medi continue to be sought (Xu et l. (11), Qu et l. (1)). Direct comprisons to experimentl vlues of key vribles, e.g., the fluid nd solid tempertures inside the fom, is lcking in the literture. In recent comprehensive review, Zho (1) indicted tht there is lck of relible experimentl het trnsfer dt for open-cell metl fom. Reserchers typiclly mesure substrte (wll) temperture, nd/or the tempertures t the inlet nd outlet of the fom. Averge surfce het trnsfer coefficient, volumetric het trnsfer coefficient nd/or Nusselt number re determined nd used for compring nlyticl nd numericl results to experimentl dt (Clmidi et l. (), Hwng et l. (), Bhttchry et l. (), Boomsm et l. (3), Kurbs nd Celik (9), Zho et l. (4), nd Kim et l. ()) Solution of Two-Eqution Model A cylindricl porous medium of rdius r is bounded by wll nd heted by constnt het flux q, Fig. 1. There is fully-developed one-dimensionl flow in the z-direction with volume-verged velocity component u. From the Brinkmn-Drcy eqution, the velocity is (Vfi nd Kim (1989), Dukhn (1)): U I( ωr) = 1 I ( ω) (1) Where R = r/r,d = K/r (Drcy number), K is the permebility, γ = μ/μ e, μ is the fluid viscosity, μ e is the effective viscosity, U = u/u, u is the velocity outside the boundry lyer, ω= γ / D nd I is the modified Bessel function of order zero. Fig. 1. Schemtic of metl fom heted cylinder. In the thermlly fully-developed region, the volume-verged two-eqution model is (Clmidi nd Mhjn (), Lee nd Vfi (1999), Krishnn et l. (4), nd (Alzmi nd Vfi )): 1 θs ( R ) Bi( θs θf) = R R R λ θ f ( R ) Bi( θs θf) = U R R R θ θ s f = = At R = R R θ θ s f = λ = 1 At R = 1 R R nd θ s = θ f = () (3) (4) (5) where θ f = (T f T w ) / q r / k s, θ s = (T s T w ) / q r / k s, Z = z / r, R = r / r, λ = k f / k s nd Bi = h σ r / k s is the Biot number. Also, k s nd k f re the effective therml conductivities, T s nd T f re tempertures of the solid nd the fluid, respectively, h is the interstitil het trnsfer coefficient, σ is the surfce re per unit volume nd ρ nd c re the density nd the specific het of the fluid is the wll temperture which is not known priori (Lee nd Vfi

3 Nihd Dukhn et l. / Procedi Mterils Science 4 ( 14 ) (1999)). The equlity of the tempertures t the heted wll s described in Eq. (5) hs been used in the literture (Lu et l. (6), Zho et l. (6), Xu et l. (11)). θ f / Z is constnt for thermlly fully-developed conditions, nd is obtined by integrting the sum of Eqs. () nd (3) over the cross section, which gives θ f / Z = q /(ρcu m r ). Substituting for U from Eq. (1) in the sum yields: 1 I ( ωr) R θ + λθ = 1 R R R I s f ( ω) which is n ordinry differentil eqution (ODE) with θ s +λθ f s the dependent vrible. The solution is 1 I( ωr) θs + λθf = ( R 1) 1 + ω I( ω) Solving Eq. (7) for θ s in terms of θ f, substituting for θ s, θ f / Z nd U in Eq. (3) nd rerrnging: d θ d I ( R) + θ = [ ( ) + + ( ) ω ] (8) dr R dr λ λ ω ω I ( ω) f 1 f Bi( λ + 1) 1 1 Bi ω Bi θ f Bi R The generl solution of Eq. (8) is obtined by first rerrnging the homogenous prt in the generl form of Bessel eqution (Myers (1998)). The complete solution of Eq. (8) is θf = I ( ψr) + I ( ωr) + R + (9) (6) (7) where, di ( ψ ) + d + d =, I( ψ ) ( Bi ω ) = I Bi ω ( ω)[ λω ( λ+ 1)], 3 1 = ( λ 1) +, Bi Bi( λ+ 1) Bi( λ+ 1) λ λ Bi(1/ / ω ) ( λ 1) 4 = +, ψ = + It is importnt to note tht, while obtining the homogenous solution, complex root ± i Bi( λ + 1)/ λ is encountered. For this cse, Bessel function Jo must be replced by the modified Bessel function I o in the solution (Xu et l. (11), Qu et l. (1), nd (Meyers 1998)) which ws not done in Lu et l. (6) nd Zho et l. (6) nd resulted in physiclly-unexplined behvior of the fluid temperture s will be shown lter.. Experiment The experimentl model ws essentilly n luminum tube filled nd brzed to n luminum fom core. The tube hd length of 15.4 cm, inside dimeter of 5.56 cm nd thickness of 6.4 mm. The 91%-porous -ppi (pores per liner inch) fom ws obtined commercilly (ERG). To mesure the fluid tempertures inside the fom, ech thermocouple ws shielded in specilly designed smll perforted luminum tube. A thermocouple ws inserted in ech perforted tube nd fixed using epoxy, blocking s few holes s possible. The bed of ech thermocouple did not extend out of the smll tube, nd remined shielded, s seen in Fig.. As such, the bed would not touch the solid ligments of the fom or the wll of the smll tube. A set of ten holes were drilled through the wll of the fom reching.3, 3.3, 4.57, 5.84, 7.11, , 1.9, 1.19 nd cm, mesured from the outer surfce of the tube. The set ws t distnce of 6.35 cm form the fom entrnce. The holes were rrnged round the cross section with n ngle of 36o between ech two

4 344 Nihd Dukhn et l. / Procedi Mterils Science 4 ( 14 ) djcent holes, nd were orgnized in order to minimize interference with the ir flow through the fom. The smll perforted tubes with their thermocouples were inserted in these holes, Fig. b. Other smll shllow holes were drilled in the wll of the cylinder for mesuring the wll temperture t vrious loctions in the flow direction. Thermfoil heters were ttched nd covered the outside surfce re. Ech heter could provide up to 645 Wtts. The ssembly ws insulted. Two 7.9-mm holes were drilled t the top of the test section t distnce of 5.8 cm for pressure drop mesurement using differentil pressure trnsmitter with rnge of to 746 P. Experiments were performed in n open-loop wind tunnel, Fig. 3. A suction unit which could produce ir flow to 17 m3/min inducted room ir through the fom smple. A reducing nozzle connected the exit of the test section to flow-mesurement section. A gs flow meter tht could mesure speeds up to 35 m/s ws used. The pressure trnsducer nd the flow meter were connected to dt cquisition system, which delivered the redings to computer. The flow rte ws vried to relize different velocities in the test section. For ech velocity, the stedy-stte sttic pressure drop ws mesured. The flow rte ws djusted gin, such tht the desired speed ws relized inside the fom. The surfce heters were powered, nd the power input ws djusted using vric, so tht the desired het flux ws chieved. The ir tempertures inside the fom, s well s the wll temperture, were monitored until stedy-stte conditions were chieved, which took bout 4 minutes. The stedy-stte ir nd wll tempertures were recorded, s ws the mbient ir temperture. The uncertinty in the non-dimensionl temperture of the fluid ws determined to be ± 14.3% (Figliol nd Besley ()). The effective therml conductivity of the fom ws obtined from Clmidi nd Mhjn (1999) nd is reported in Tble Results For velocities below.3 m/s, the flow regime ws Drcy, s ws indicted by the behvior of the pressure drop. So, for het trnsfer mesurements, velocity of. m/s ws chosen. The permebility ws determined from the Drcy-regime pressure-drop dt points only (Dukhn nd Ali (1)). The surfce re density of the fom ws clculted form correltion given by the mnufcturer. The interstitil het trnsfer coefficient inside the fom, h, ws computed bsed on correltion given in Kuwrhr et l. (1). Fig. 4 shows the fluid temperture s function of rdil distnce. The solution given by Eq. (9) is plotted using solid line, while the experimentl dt points re given using the blck solid circles. In generl, the two temperture curves decrese grdully s the distnce from the heted wll increses. Also plotted (dshed line) is the solution of Lu et l. 6. The fluctutions in this solution hve no physicl reson. They re due to using the wrong Bessel function (J) in the solution of Lu et l. 6, insted of using I, since the roots of Bessel eqution re imginry. Fig.. Experimentl het trnsfer model construction: ) Thermocouple in its smll perforted tube, b) Themocouple ssemblies inserted into holes.

5 Nihd Dukhn et l. / Procedi Mterils Science 4 ( 14 ) Fig. 3. Photogrph of the experimentl set-up. Tble 1: Prmeters for the metl-fom filled model. Prmeter Vlue r o u m.18 m. m/s K 1.16X1-7 m q W/m k s 6.6 W/m.K h 7.8 W/m.K Bi 68.9 σ m /m 3 ω 34.8 ψ 33.1 λ.44 θ f R Fig. 4. Experimentl nd nlyticl fluid tempertures. The experimentl nd the nlyticl temperture curves of the current study re reltively close to ech other, with the nlyticl solution predicting slightly lower temperture. In ddition to experimentl errors, other errors my hve existed in the het trnsfer coefficient nd the effective therml conductivity. Another possible error my be the existence of temperture slip t the wll, nd physicl non-uniformities ner the wll, which lters the vlue of the effective conductivities of the solid nd fluid (Shroui nd Kviny (1993)). The nture of the wll boundry condition when constnt het flux is pplied is still subject to debte, i.e., how the totl het flux is split between the solid nd the fluid phses (Vfi nd Yng (13)).

6 346 Nihd Dukhn et l. / Procedi Mterils Science 4 (14) The two points closest to the center of the fom re seen to produce higher physicl temperture. The rrngement of the thermocouples cused blockge leding to higher tempertures t these two loctions. 4. Conclusions Direct mesurements of the fluid temperture inside heted metl-fom cylinder were conducted using specilly-designed technique. Scrutiny of literture reveled tht such mesurements hve been lcking. The therml-non-equilibrium two-eqution model ws revisited, nd n error in previously published solution ws identified. Reltively good greement between the nlyticl nd experimentl fluid temperture ws displyed, with the nlyticl solution under-predicting the fluid temperture over most of the cross section. Nonetheless, the mesurements give n opportunity to look into the issue of how volume-verged vlues compre to physicllymesured locl vlues of the temperture inside porous medi. The experimentl technique cn be of utility for het trnsfer designs, nd for vlidting complex numericl modes of the het trnsfer in open-cell meso-porous medi. References Boomsm, K., Poulikkos, D., Zwick, F., 3. Metl foms s compct high performnce het exchngers. Mechnics of Mterils 35, Clmidi, V. V., Mhjn, R. L., The effective therml conductivity of high porosity fibrous metl foms, Journl of Het Trnsfer 11, Clmidi, V. V., Mhjn, R. L.,. Forced convection in high porosity metl foms, Journl of Het Trnsfer 1, Dukhn, N., 1. Anlysis of Brinkmn-extended Drcy flow in porous medi nd experimentl verifiction using metl fom, Journl Fluids Engineering 134, 711. Dukhn, N., Ali, M., 1. Strong wll nd trnsverse size effects on pressure drop of flow through open-cell metl fom, Interntionl Journl of Therml Science 57, Figliol, R., Besley, D.,. Theory nd Design for Mechnicl Mesurements, John Wiley nd Sons, New York, NY, Hwng, J. J., Hwng, G. J., Yeh, R. H., Cho, C. H., Mesurement of the interstitil convection het trnsfer nd frictionl drg for flow cross metl fom, Journl of Het Trnsfer 14, Kim, S.Y., Pek, J.W., Kng, B.H.,. Flow nd het trnsfer correltions for porous fin in plte-fin het exchnger, Journl of Het Trnsfer 1, Krishnn, S., Murthy, J. Y., Grimell, S. V., 4. A two temperture model for the nlysis of pssive therml control system. Journl of Het Trnsfer 16, Kurbs, I.,Celik, N., 9. Experimentl investigtion on forced nd mixed convection het trnsfer in fom-filled horizontl rectngulr chnnel. Interntionl Journl of Het nd Mss Trnsfer 5, Kuwhr, F., Shirot, M., Nkym, A., 1. A numericl study of interfcil convective het trnsfer coefficient in two-energy eqution model for convection in porous medi. Interntionl Journl of Het nd Mss Trnsfer 44, Lu, W., Zho, C. Y., Tssou, S. A., 6. Therml nlysis on metl-fom filled het exchngers: Prt I: metl-fom filled pipes, Interntionl Journl of Therml Science 49, Myers, G. E., Anlyticl Methods in Conduction Het Trnsfer. nd Ed., AMCHT Publictions, Mdison, 46. Qu, Z. G., Xu, H. J., To, W. Q., 1, Fully developed forced convective het trnsfer in n nnulus prtilly filled with metllic foms: An nlyticl solution, Interntionl Journl of Het nd Mss Trnsfer 55, Shroui, M., Kviny, M., Slip nd no-slip temperture boundry condition t the interfce of porous/plin medi, Interntionl Journl of Het nd Mss 36, Vfi, K., Kim, S. J., Forced convection in chnnel filled with porous medium: n exct solution. Journl of Het Trnsfer 111, Vfi, K., Yng, K., 13. A Note on the therml non-equilibrium in porous medi nd het flux bifurction phenomenon in porous medi. Trnsport in Porous Medi 96, Xu, H. J., Qu, Z. G., To, W. Q., 11. Anlyticl solution of forced convection het trnsfer in tubes prtilly filled with metllic fom using the two-eqution model. Interntionl Journl of Het nd Mss Trnsfer 54,