Theoretical study on impingement heat transfer with single-phase free-surface slot jets

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1 International Jornal of Heat and Mass Transfer 4 (25) Theoretical stdy on impingement heat transfer with single-phase free-srface slot jets Y.C. Chen, C.F. Ma *, M. Qin, Y.X. Li Enhanced Heat Transfer and Energy Conservation, The Key Laboratory of Ministry of Edcation, China Heat Transfer and Energy Conversion, The Key Laboratory of Beijing Mnicipality, College of Environmental and Energy Engineering, Beijing University of Technology, Beijing 122, China Received 1 Jly 24; received in revised form 6 Janary 25 Abstract A theoretical analysis was condcted to characterize heat transfers from horizontal srfaces to normally impinging slot jets nder arbitrary-heat-fl condition. The thermal and hydralic bondary layers of laminar flow were divided into for regions of flow along heat transfer srfaces inclding a stagnation zone and three wall jet zones, from which general epressions of heat transfer coefficients were obtained. Frthermore, these reslts were compared with eperimental and analytical data available in pblished literatre. Good agreements were observed from the comparisons. Ó 25 Elsevier Ltd. All rights reserved. Keywords: Slot jet impingement; Heat transfer; Laminar bondary layer; Theoretical analysis 1. Introdction * Corresponding athor. Tel./fa: address: machf@bjt.ed.cn (C.F. Ma). Impinging jets have been widely sed in indstrial cooling or heating processes as a means of providing very high heat and mass transfer rates. In most cases air is employed as a working medim in jet impingement and a large amont of research on impinging air jets has been reported [1,2]. Recently, more attention has been directed to the stdy of liqid jet impingement as the heat transfer rates can be increased several orders of magnitde in comparison with that of gas jets. Research on liqid jets was stimlated by their possible application to the cooling of heat engines [3,4] or electronic devices [5 7], as well as in the thermal treatment of metals [,9]. Two principal jet configrations are relevant to the application of jet impingements: circlar jets and slot jets, bt for which flow and heat transfer mechanics are distinctly different. A significant amont of research has been pblished on the heat and mass transfer with circlar impinging jets [1,2,1,11]. Recently, however stdies of heat and mass transfer with slot jet impingement have attracted more attention from scientists and engineers. Slot jet impingement becomes remarkable as it offers more beneficial featres, sch as higher cooling effectiveness, greater niformity and more controllability. All these factors are sitable to contering the trend of continosly increasing heat fl and decreasing dimensions in compact electronics packages [12 16]. Schwarz [14] presented measrements and an analysis on the flow characteristics of two-dimensional jet /$ - see front matter Ó 25 Elsevier Ltd. All rights reserved. doi:1.116/j.ijheatmasstransfer

2 332 Y.C. Chen et al. / International Jornal of Heat and Mass Transfer 4 (25) Nomenclatre B C p H h k L N Pr Q q Re T T w T T 1 U 1 V v slot nozzle width specific heat liqid film thickness heat transfer coefficient thermal condctivity sectional length of slot nozzle =hb/k, Nsselt nmber =lc p /k, Prandtl nmber jet flow rate wall heat fl =V B/m, Reynolds nmber local temperatre local wall temperatre jet temperatre at nozzle eit temperatre otside thermal bondary layer velocity otside flow bondary layer lateral velocity component jet velocity at nozzle eit vertical velocity component coordinate parallel to impingement plate t v y Z lateral location where thermal bondary layer reaches free srface lateral location where viscos bondary layer reaches free srface coordinate normal to impingement plate nozzle-to-plate spacing Greek symbols d hydrodynamic bondary layer thickness D thermal bondary layer thickness D 1 see Eq. (21) e ¼ R qd=q g =y/d h dimensionless temperatre l dynamic viscosity m kinematic viscosity n =D/d q flid density 2Re K ¼ d B a ¼ Hq kðt w T 1Þ impingement, bt only for the trblent wall-jet zone. Kendosh [15] stdied the heat and mass transfer mechanics of an impinging slot jet by means of bondary-layer theory for laminar flow. The reslts were restricted jst to the stagnation zone. Park [16] and Chiriac [17] reported the nmerical soltions of stream and heat transfers in stagnation and wall jet zones for laminar flow, respectively, whereas the soltions to the confined slot jets were available only for a limited range of Reynolds nmbers. The objective of the present stdy is to provide detailed theoretical soltions on laminar flow for free-srface slot jet impinging onto horizontal srfaces nder arbitrary-heat-fl conditions. The inflence of Prandtl nmbers on heat and mass transfers is also considered. 2. Theoretical analyses The stream of slot jet impingement is shown in Fig. 1, in which the flow field along a heated srface is divided into for regions inclding a stagnation zone and three wall-jet zones based on the characteristics of flow and heat transfer for Pr P 1, respectively. In this paper predicative formlas are provided for the entire range of lateral flow nder arbitrary-heat-fl conditions. According to bondary layer theory [1], the flow in a liqid film layer satisfies the following conservation eqations: B V T y v Fig. 1. Bondary layers of a free-srface slot jet impinging onto a horizontal plate. o >< o þ ov oy ¼ o o þ v o ð1 2Þ oy ¼ 1 dp q d þ m o2 >: oy 2 with the general conditions: < y ¼ ; ¼ v ¼ : y ¼ dðy!1þ; ¼ U 1 ðþ; t Z H 2L dy ¼ Q ¼ V ðblþ I II III IV o oy ¼ U I: BII:B< v III: v < t IV: t < T ð3 4Þ ð5þ

3 Y.C. Chen et al. / International Jornal of Heat and Mass Transfer 4 (25) In region (I), stagnation zone, the speed otside the flow bondary layer is U 1 ¼ V ð=bþ ð6þ So, the classic for-power polynomial can be sed to approimate the velocity profile satisfying Eqs. (3 4): ¼ f ðgþ ¼ð2g 2g 3 þ g 4 Þþ K gð1 gþ3 ð7þ U 1 6 where g ¼ y, K ¼ d2 U d m 1 ¼ d2 Re. With the conditions that B 2 d is finite at =,d and can be obtained by integrating Eq. (2): d ¼ 2.749BRe.5 ; K ¼ 7.55 ðþ Conseqently, we obtain f ðgþ ¼ð2g 2g 3 þ g 4 Þþ1.26gð1 gþ 3 ð9þ In region(ii), the speed U 1 remains almost constant, eqal to V, becase the liqid here is naffected by the viscos stresses. The velocity profile may be epressed by ¼ 3 V 2 g 1 2 g3 ð1þ Sbstitting Eq. (1) into Eq. (2) yields dd 2 d ¼ 2 m ð11þ 13 V Using the continos condition of d between regions (I) and (II) the vale of d in region II can be determined: d ffi B Re.5 ð12þ Then, the hydrodynamic bondary layer of liqid film flow will be developed ntil right p to the free srface of stream where lateral distance v can be derived from Eqs. (1) (12): v ¼.1627BRe.695 ð13þ In regions (III) and (IV), as > v, the viscos stresses become appreciable right p to the free srface, and the entire flow is of bondary layer type. We are able to se the velocity profile (1), bt the free stream speed varies with. The integral momentm eqation shold be revised, becase in their eqation the pressre gradient is zero: dd 2 d þ d 2 2 R H U 1 dy d 2! 1 U 1 du 1 d ¼ s w qu 2 1 From Eq. (5), we have 5 HU 4 1 ¼ BV Then, we get H ¼ þ. B Re Re.35 U 1 ¼ V þ 1 Re Re.35 ð14þ ð15þ ð16þ From region (I) to region (III), the for order polynomial are sed to approach the temperatre profile which satisfies the bondary conditions: >< T ¼ T w ; >: T ¼ T 1 ; o 2 T ¼ ; as y ¼ oy2 ot oy ¼ o2 T oy ¼ 2 as y ¼ D ð17 1Þ The temperatre profile is h ¼ T T 1 ¼ 1 2 y T w T 1 D þ 2 y 3 y 4 ð19þ D D The integral energy eqation is R qd ¼ D 1 ðt w T 1 Þ ð2þ qu 1 C p Z 1 T T 1 where D 1 ¼ dy ð21þ U 1 T w T 1 By the Forier law, Eq. (19) yields the heat fl at wall q ¼ 2k D ðt w T 1 Þ ð22þ By solving Eqs. 9 and (19) (22) simltaneosly, a fororder eqation for D in stagnation zone is obtained n n 4 ¼ e ð23þ Pr where n = D/d,d = 2.749BRe.5 and e ¼ R qd=q. The soltions of Eq. (23) with the parameter Pre 1 yields, within 5% errors, approimately >< 1.245e.47 Pr.47 ; 1. 6 Pr e 1 < 3. n ¼ 1.147e.3 Pr.3 ; 3. 6 Pr e 1 < 1. ð24þ >: 1.2e 1=3 Pr 1=3 ; 1. 6 Pr e 1 So, N =.727n 1 Re.5 ><.54e.47 Pr.47 Re.5 ; 1. 6 Pr e 1 < 3. ¼.634e.3 Pr.3 Re.5 ; 3. 6 Pr e 1 < 1. ð25þ >:.672e 1=3 Pr 1=3 Re.5 ; 1. 6 Pr e 1 In region (II), D can be solved from Eqs. (1), (13), (19), (21) and (22) and obtained: D ¼ 3.94B ere.5 Pr 1=3 ð26þ Conseqently,.24 N ¼.5136e 1 B Re.5 Pr 1=3 ð27þ Similar to above calclations, D and N can be obtained for region (III) as follows: D ¼ 2.924e 1=3 B 1=3 Pr 1=3 Re 1= = þ. B ð2þ Re Re.35

4 334 Y.C. Chen et al. / International Jornal of Heat and Mass Transfer 4 (25) N ¼.64e 1=3 Pr 1=3 Re 1= = þ. ð29þ ReB Re.35 From Eqs. (15) and (2), the lateral position t can be determined as t ¼ 6.4e.27Re þ.163re.695 B ð3þ Pr In region (IV), when lateral distance is larger than t, both flow and thermal bondary layers are flly developed throgh the whole liqid film. In this region, the free srface temperatre of liqid film varies with. Another temperatre profile is given by satisfying the bondary condition of heat fl at wall: h ¼ 1 a y H þ 1 3 a y 3 ð31þ H where a = Hq/[k(T w T 1 )]. By similar calclations, we can get N ¼ 2e Pr Re þ 272H 1 ð32þ 525B where H can be obtained from Eq. (15). If the heat fl at the wall is niform, q = constant or e = 1, the heat transfer coefficients can be simplified from region I to region IV. 3. Comparison of analytical and eperimental reslts Theoretical epressions of heat transfer coefficients for free-srface slot jets were obtained by ndertaking the mathematical analyses shown above. Comparisons of heat transfer coefficients in both the stagnation zone and the wall-jet zone were made between the present stdy and the analytical and eperimental reslts in pblished literatre. Heat transfer coefficients at the stagnation line for slot jet impingement are shown in Fig. 2, in which both eqations from this stdy and epressions from other researches are plotted. McMrray et al. [19], Miyasaka et al. [2], Zmbrnnen et al. [21] and Vader et al. [22] presented either analytical or empirical correlations of heat transfer for free-srface slot jet impingement. In their eperiments water was mostly sed as the working flid together with nozzles of different slot widths. Good agreements between present theoretical reslts and the correlations from other stdies were observed even for the high Reynolds nmbers (Re > 1). A deviation below 25% between the present eqations and other empirical correlations eists ecept for those by Zmbrnnen with B = 1.2 mm de to different trblence level of jet flow. Another plotted line based on the theoretical eqation by Kendosh [15], also shown in the figre, shows somewhat a higher Nsselt nmbers than N 1 1 Pr=5 Present theory Vader etal[22] B=1.2mm Vader etal[22] analysis Zmbrnnen etal[21] B=2.3mm Zmbrnnen etal[21] B=1.2mm Kendosh[15] Z/B=4 Miyasaka & Inada[2] B=1mm McMrray etal[19] B=6.4mm Re Fig. 2. Comparisons of stagnation heat transfer for free-srface slot jet impingement. that of present theoretical reslts within a deviation of 27%. The relative deviations can be eplained by the inflence of the trblent intensity of the free jet in those eperiments. The level of trblent flow at the eit of the nozzle is distinctly different for varied Prandtl nmbers, Reynolds nmbers and geometry of nozzles. Figs. 3 and 4 ehibit comparisons of local heat transfer coefficients along heated srfaces between present reslts and the eperimental data by Vader et al. [22], Zmbrnnen et al. [21]. The eperimental reslts of Vader et al. [22] and Zmbrnnen et al. [21] with water as the working flid are shown in Figs. 3 and 4, in which N Vader et al. Water Z/B=. Ep. Theory Re=52174 Re= /B Fig. 3. Comparison of theoretical reslts with eperimental reslts by Vader et al. [22] with water.

5 Y.C. Chen et al. / International Jornal of Heat and Mass Transfer 4 (25) N generally good agreements between eperimental reslts and the present correlations are observed before the transition from laminar to trblent flow. As illstrated in the figres, the hmps in heat transfer distribtion crves are clearly observable from the eperimental reslts, and are de to the transition at flow bondary layer from laminar flow to trblence. The local heat transfer may be well reglated by the present theory, focsed on the laminar flow of bondary layer, even for initially trblent impinging jet with high Reynolds nmbers (Re > 1). 4. Conclsion Zmbrnnen et al. Water Z/B=5.5 Ep. Theory Re=493 Re= /B Fig. 4. Comparison of theoretical reslts with eperimental reslts by Zmbrnnen et al. [21] with water. A theoretical analysis has been condcted to stdy heat transfer characteristics of free-srface liqid slot jet impingement. Eqations for local heat transfer coefficients are obtained along heated srfaces with slot jets impingement based on the theory of laminar bondary layer. Also, the formla is compared with eperimental reslts for both different working flids and different nozzle sizes. Present theory shows good agreements with eperimental reslts of for slot jet impingement. Frther research shold be developed to correlate heat and mass transfers of slot jet impingement with high trblence. Acknowledgements The present research is spported by National Key Fndamental Research Plan (973) Project of China (no ) and National Natral Science Fondation of China. 12 References [1] H. Martin, Heat and mass transfer between impinging gas jets and solid srfaces, in: j.p. hartnett, t.f. irvine (Eds.), Adv. Heat Transfer, Vol. 13, Academic Press, 1977, pp [2] S.J. Dovns, E.H. James, Jet Impingement Heat Transfer A literatre Srvey, ASME paper 7-HT-35, 197. [3] M. Kiry, Development of oil-cooled 75 cc motorcycle engine, Atomobile Technol. (in Japanese) 4 (9) (196) [4] C.F. Ma, A.E. Bergles, Convective heat transfer on a small vertical heated srface in an impingement circlar liqid jet, in: B.X. Wang (Ed.), Heat Transfer Science and Technology 19, Hemisphere, 199, pp [5] C.F. Ma, Fndamental research on etremely small size liqid jet impingement heat transfer, in: Proceeding of 3rd International Thermal Energy Congress, Kitakysh Japan, Jly 2 Agst 1, 1997, pp [6] L.M. Jiji, Z. Dagan, Eperiment investigation of single phase mlti-jet impingement cooling of an array of microelectronic heat sorces, in: Win Ang (Ed.), Cooling Technology for Electronic Eqipment, Hemisphere pblishing Corp., 19, pp [7] C.F. Ma, A.E. Bergles, Boiling jet impingement cooling of simlated microelectronic chips, ASME Pbl. HTD 2 (193) [] F.C. Kohing, Waterwall: Water cooling system, Iron-Steel Eng. 62 (Jne) (195) [9] C.F. Ma, J. Y, D.H. Lei, Y.P. Gan, F.K. Tso, H. Aracher, Transient jet impingement boiling heat transfer on hot srfaces, in: X.J. chen et al. (Eds.), Mltiphase Flow and Heat Transfer-second International Symposim, vol. 1, Hemisphere, 199, pp [1] X. Li, V.J.H. Lienhard, J.S. Lombara, Convective heat transfer by impingement of circlar liqid jets, J. Heat Transfer 113 (1991) [11] C.F. Ma, Y.H. Zhao, T. Masoka, T. Gomi, Analytical stdy on impingement heat transfer with single-phase free-srface circlar liqid jets, J. Thermal Sci. 5 (4) (1996) [12] Y. Zhang, C.F. Ma, M. Qin, Eperimental stdy on local heat transfer with liqid impingement flow in two-dimensional micro-channels, Int. J. Heat Mass Transfer 4 (17) (1997) [13] Z.H. Lin, Y.J. Cho, Y.H. Hng, Heat transfer behaviors of a confined slot jet impingement, Int. J. Heat Mass Transfer 4 (5) (1997) [14] W.H. Schwarz, W.P. Cosart, The two-dimensional trblent wall-jet, J. Flid Mech. 1 (1961) [15] A.A. Kendosh, Theory of stagnation region heat and mass transfer to flid jets impinging normally on solid srfaces, Chem. Eng. Process. 37 (199) [16] V.A. Chiriac, A. Ortega, A nmerical stdy of the nsteady flow and heat transfer in a transitional confined slot jet impinging on an isothermal srface, Int. J. Heat Mass Transfer 45 (22) [17] T.H. Park, H.G. Choi, J.Y. Yoo, S.J. Kim, Streamline pwind nmerical simlation of two-dimensional confined impinging slot jets, Int. J. Heat Mass Transfer 46 (23)

6 336 Y.C. Chen et al. / International Jornal of Heat and Mass Transfer 4 (25) [1] H.Z. Li, H.J. Sh, Theory of Bondary Layer, PeopleÕs Jiaotong Press of China, [19] D.C. McMrray, P.S. Myers, O.A. Uychara, Inflence of impinging jet variables on local heat transfer coefficients along a flat srface with constant heat fl, in: Proc. 3rd Int. Heat Transfer Conf., New York, AIChE J 2, 1966, pp [2] Y. Miyasaka, S. Inada, The effect of pre forced convection on the boiling heat transfer between a two-dimensional sbcooled water jet and a heated srface, J. Chem. Eng. Japan 15 (19) [21] D.A. Zmbrnnen, F.P. Incropera, R. Viskanta, Convective heat transfer distribtion on a plate cooled by planar water jets, J. Heat Transfer 111 (199) [22] D.T. Vader, F.P. Incropera, R. Viskanta, Local convective heat transfer from a heated srface to an impinging planar jet of water, Int. J. Heat Mass Transfer 34 (1991)

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