S. Basu B. M. Cetegen 1 Fellw ASME Mechanical Engineering Department, University f Cnnecticut, Strrs, CT 06269-3139 Analysis f Hydrdynamics and Heat Transfer in a Thin Liquid Film Flwing Over a Rtating Disk by the Integral Methd An integral analysis f hydrdynamics and heat transfer in a thin liquid film flwing ver a rtating disk surface is presented fr bth cnstant temperature and cnstant heat flux bundary cnditins. The mdel is fund t capture the crrect trends f the liquid film thickness variatin ver the disk surface and cmpare reasnably well with experimental results ver the range f Reynlds and Rssby numbers cvering bth inertia and rtatin dminated regimes. Nusselt number variatin ver the disk surface shws tw types f behavir. At lw rtatin rates, the Nusselt number exhibits a radial decay with Nusselt number magnitudes increasing with higher inlet Reynlds number fr bth cnstant wall temperature and heat flux cases. At high rtatin rates, the Nusselt number prfiles exhibit a peak whse lcatin advances radially utward with increasing film Reynlds number r inertia. The results als cmpare favrably with the full numerical simulatin results frm an earlier study as well as with the reprted experimental results. DOI: 10.1115/1.2150836 Keywrds: liquid film cling, rtatin, heat transfer enhancement, integral analysis Intrductin Many investigatins have been perfrmed in the past n flw and heat transfer characteristics in thin liquid films due t the fact that high heat transfer rates can be btained in thin films as reviewed by Webb and Ma 1. Hydrdynamic characteristics f thin liquid films flwing ver statinary and rtating disk surfaces are imprtant in understanding the majr factrs affecting the heat transfer perfrmance. A better understanding f the fluid mechanics f the liquid film flw allws better design and ptimizatin f high perfrmance cmpact heat transfer systems such as thse emplyed in space applicatins, where cnsideratins n efficiency, size, and weight are f critical imprtance. The study f rtating thin-film fluid physics and heat transfer is als f fundamental interest in the develpment f cmpact vapr absrptin systems. Watsn 2 was first t analyze a free-falling jet impinging n a hrizntal statinary plate. The film flw was divided int fur regins. The first regin is the impingement zne, fllwed by a secnd regin f a grwing bundary layer in the liquid film which eventually reaches t the film surface. The third regin is cmprised f the transitin t a fully develped film flw fllwed by a fully develped flw regime identified as the furth regin. Fr statinary and rtating liquid films at lw flw rates and rtatin speeds, a hydraulic jump culd exist that has been studied bth cmputatinally and experimentally under nrmal and zer gravity cnditins by Faghri and c-wrkers 3,4 and Avedisian and Zha 5. The hydraulic jump phenmenn was fund t disappear at zer gravity cnditins. Labus and Dewitt 6 cnducted a cmbined numerical and experimental study t determine the free surface f a circular jet impinging n a flat plate in micrgravity. The gverning ptential flw equatins were slved numerically. The experimental study was perfrmed using a drptwer. The flw patterns f the free surface were examined 1 Crrespnding authr; e-mail: cetegen@engr.ucnn.edu Cntributed by the Heat Transfer Divisin f ASME fr publicatin in the JOUR- NAL OF HEAT TRANSFER. Manuscript received April 18, 2005; final manuscript received September 12, 2005. Review cnducted by Prf. Bengt Sunden. experimentally, cncluding that the surface tensin and inertia were the dminant frces acting n the liquid. The experimental results were fund t be in a gd agreement with the numerical results. Thmas et al. 3 perfrmed a useful ne-dimensinal analysis f the film thickness including the hydraulic jump phenmenn. They predicted significant thinning f the film with increased levels f rtatin. They neglected inertia in their analysis. Miyasaka 7 perfrmed a similar theretical and experimental study f the thickness f a thin viscus liquid film n a rtating disk. He btained the film thickness by slving the gverning equatins f mtin in the inviscid and viscus limits, with the latter utilizing the bundary layer apprximatin. Miyasaka als carried ut experiments by using a liquid jet falling nt the center f a rtating disk. He deduced the film height frm the electrical resistance f the liquid n the disk by cmparing it t the electrical resistance f a standard thickness f the fluid. The cmputed values were fund t agree with the experimental results. Rahman et al. 4 was the first t reprt a full numerical slutin f the mmentum equatins using a finite difference scheme. The methd utilized a bundary-fitted crdinate gridding scheme with a k- mdel fr turbulence clsure and an iterative technique t define the free surface. They predicted the liquid-film thickness in the vicinity f the hydraulic jump reasnably well and evaluated the effects at the uter edge f the disk. Rahman and Faghri 8 investigated the hydrdynamic behavir f a thin liquid film flwing ver a rtating disk. They used a three-dimensinal bundary-fitted crdinate system t perfrm the calculatins. The cmputed film thickness agreed well with the existing experimental measurements. It was als cncluded that the flw was dminated by inertia near the entrance and by centrifugal frce near the uter edge the disk. The hydrdynamic characteristics f a radially spreading liquid jet n a hrizntal plate were als predicted numerically by Buyevich and Ustinv 9. Hwever they reprted n cmparisn with ther studies. Ra and Arakeri 10 perfrmed an analytical study f free liquid jets n surfaces including circular plates, cnes, and spheres. They used a bundary layer apprximatin and a third rder plynmial fr the velcity prfile. The equatins were Jurnal f Heat Transfer Cpyright 2006 by ASME MARCH 2006, Vl. 128 / 217
slved by the integral methd. Their wrk hwever did nt include heat transfer in the film. Nevertheless, they were the first t use the integral apprach t predict relevant parameters like film thickness. They als did nt present any experimental validatin f their data. Liu and Lienhard 11 perfrmed an integral analysis f a liquid jet impingement heat transfer n a unifrm heat flux surface withut rtatin. They reprted Nusselt number crrelatins fr a range f Prandtl numbers. Azuma and Hshin 12 examined the laminar-turbulent transitin, liquid-film thickness, velcity prfile, stability, and wall pressure fluctuatins f thin liquid films n a statinary hrizntal disk. The laminar-turbulent transitin was determined as a functin f the nzzle inside diameter, the gap height, and the vlumetric flw rate. The liquid-film thickness measurements were perfrmed using a needle prbe. Mre recently, Ozar et al. 13 measured the liquid film thickness by a laser light reflectin technique with which the spatial distributins f film thickness were captured including the hydraulic jump. Heat transfer in liquid films was first analyzed by Chaudury 14 wh incrprated Watsn s results int the heat transfer analysis. Wang et al. 15 develped a tw-dmain slutin in which the heat transfer at the liquid film and the slid disk were treated separately and matched at the liquid-disk interface. Rahman and Faghri 8 used mixed numerical and analytical methds t predict the heat transfer. Fr develping flw and heat transfer, a three-dimensinal numerical mdel was utilized. Als, a twdimensinal analytical slutin was frmulated fr develping heat transfer and fully develped flw assuming slid bdy rtatin. Fr the case f fully develped heat transfer and fluid flw, a clsed frm slutin was develped. This slutin predicted that the Nusselt number, based n film thickness, apprached a cnstant value in the fully develped regime. Carper et al. 16 evaluated the cnvective heat transfer frm a jet f cling il t an apprximately isthermal rtating disk. Crrelatins were presented fr the average Nusselt number as influenced by rtatinal Reynlds number, jet Reynlds number and radius f impingement. Carper et al. 17 later extended their study t include the effect f Prandtl numbers. Vader et al. 18 studied the effects f jet velcity and temperature n the heat transfer between a planar water jet and a statinary heated plate. They cncluded that the heat transfer perfrmance was affected by the free stream turbulence intensity and the Prandtl number. A similar study was presented by Stevens and Webb 19 where the influence f jet Reynlds number, nzzle t plate spacing, and jet diameter were evaluated. Empirical crrelatins were develped fr stagnatin pint, lcal and average Nusselt numbers. Faghri et al. 20 presented heat transfer results fr a cntrlled liquid impinging jet n a statinary disk. They presented a numerical study shwing gd agreement between heat transfer predictins and experimental data. Aune and Ramshaw 21 perfrmed heat and mass transfer experiments n a liquid flwing ver a rtating disk. They predicted the heat transfer cefficients analytically by adapting the slutin that Nusselt 22 used fr the film cndensatin under the influence f gravity. Ozar et al. 23 published an experimental study f heat transfer and reprted the lcal and disk surface averaged Nusselt number ver a range f flw rates and rtatin speeds fr water. Recently, Rice et al. 24 published full numerical cmputatin f heat transfer in a thin liquid film ver a rtating disk simulating the experimental cnditins f Ozar et al. 23. Review f the literature indicates that the analysis f the prblem has prgressed alng tw main themes. One theme has been numerical slutin f the full prblem taking int accunt its salient features. The ther is f analytical nature utilizing simplifying assumptins t btain simple analytical results. In between lies the studies that utilize the Karman-Phlhausen-type integral analyses. This type f analysis is capable f capturing sufficient details f the slutin while aviding versimplificatins needed in analytical studies. Fr example, many f the analytical analyses had nt cnsidered the liquid film inertia effects in the presence f Fig. 1 Schematics f the thin film ver a rtating disk rtatin. It was thus the bjective f this study t present a cmprehensive integral analysis f the thin liquid film flwing radially utward n a circular disk including effects f inertia and rtatin. Capability f the integral methd t predict the flw and heat transfer is demnstrated by cmparisns with experimental and numerical results fr the same prblem. Prblem Frmulatin The rtating disk is schematically shwn in Fig. 1 which resembles the experimental set-up that has been utilized by Thmas et al. 3 and Ozar et al. 13,23 in a series f experimental studies. In the experiments, the flw is intrduced frm a central cllar that directs the liquid radially utward ver a gap height f h. The liquid flws ver the rtating disk while being heated frm underneath by an electric resistance heater. In these experimental studies, the liquid film thickness and heat transfer cefficients were measured. Liquid film thickness measurements were made by either a capacitance prbe 3 r an ptical technique 13. The heat transfer cefficients were determined frm the difference between the measured disk surface temperature and the liquid inlet temperatures and the cnstant heat flux supplied t the disk. The prblem is cnsidered in the radial r and axial z crdinates assuming azimuthal symmetry. Fr this situatin, the gverning equatins in cylindrical crdinate system ver a rtating circular disk are: Cntinuity: 1 r r ru r + u z z =0 1 r-mmentum: 1 r r ru ru r + u ru z = r 2 + 2 u r z z 2 u r r 2 + 1 r rr u r 2 r Energy: 1 r r ru rt + z u zt = 1 r rr r 2 T + 2 T 3 z Defining nndimensinal parameters as, r r 0, T T T T T i, ũ r u r u, z z h, ũ z u z u, q k q T T i, where r 0 is the inlet radius f the disk, u 0 is the inlet velcity f the liquid jet, h 0 is the cllar height, T is the nndimensinal temperature fr the cnstant wall temperature, and q is its cunterpart fr cnstant wall heat flux. T 0 is the temperature f the disk surface fr cnstant wall temperature case, T i is the inlet temperature f the liquid jet, q 0 is the heat flux supplied t the disk fr the cnstant wall heat flux case. Nndimensinalizing the gverning equatins we btain, 218 / Vl. 128, MARCH 2006 Transactins f the ASME
1 ũ r 1 2 + r ũ r + r 1 ũ r + r ũ z z =0 z ũ rũ z = Re R + 1 1 2 ũ r z ũ z = Pe 1 1 ũ r + r + r 4 2 2 ũ r 2 2 z 2 where Reynlds, Rssby, and Peclet numbers are defined as Re u r, R u 2 2 2 r, Pe u r z 2 where is the kinematic viscsity and is the thermal diffusivity, bth f which are assumed t be cnstant in the cntext f this analysis. Since r h, then, r 2 /h 2 1, we can infer that r 2 2 ũ r z 2 1 1 ũ r and r 2 2 ũ r z 2 ũ r 2 and r Hence the mmentum and energy equatins take the frm 1 ũ r 1 2 + r z ũ rũ z = Re R + 1 r ũ r + r z ũ z = 1 Pe r Liquid Film Hydrdynamics 2 2 ũ r 2 2 z 2 z 2 2 2 z 2 Integrating the mmentum equatin with respect t z frm 0 t =/h, we get, 1 ũ 2 r dz + r ũ r ũ z = h 0 R Re 1 r 2 ũ r 9 z 0 Frm the cntinuity equatin, we can write ũ z = h ũ r dz r 0 A parablic radial velcity prfile is assumed as, 5 6 7 8 10 ũ r = a 0 + a 1 z + a 2 z 2 11 subject t the bundary cnditins f n slip at the wall ũ r z =0=0 and n shear at the free surface ũ r /z =0 at z =. In additin, the ttal vlume flw at any crss sectin perpendicular t r has t be equal t the inlet flw rate fr the case f n vaprizatin r mass lss, these cnditins allw determinatin f cefficients in the velcity prfile leading t, ũ r =3 2 z 2 3 3 z 2 12 Substituting this prfile int the integral mmentum equatin and integrating with respect t z, ne gets d d + + 5 6R 3 3 = 2Re 5 r 2 13 In this equatin, the first tw terms are due t advectin, the third term is a result f rtatin and the term n the right-hand side represents the viscus shear. Equatin 13, subject t the initial cnditin =1=1, was numerically integrated t determine the film thickness ver the disk surface. Equatin 13 becmes an algebraic equatin fr the case f negligible inertia in which case the first tw terms disappear. The film thickness can be written in exact frm as, = 3R r 2 Re 1/3 1 2/3 14 This result is identical t that btained based n the falling film analysis with gravity being replaced by the centrifugal frce. Having the liquid film hydrdynamics established, we nw prceed with the heat transfer analysis in the film. Heat Transfer in the Liquid Film The heat transfer in the liquid film is analyzed by cnsidering the tw cases f cnstant disk surface temperature and cnstant disk surface heat flux. In either case, there is a thermal entry regin where the thermal bundary layer lies belw the film surface as shwn in Fig. 1. As it is shwn in Appendix A fr cnstant disk surface temperature case, the thermal entry regin length scales as, * =1+ Pe h 2 6 r 1/2 15 Fr mderate Peclet numbers, *1 r r * r since h r, thus allwing us t neglect the entry length. Cnstant Disk Surface Temperature Case. Fr this case, the temperature prfile can be cnstructed as a secnd rder plynmial given by, = a 0 + a 1 z T + a 2 z 2 T 16 which has t satisfy the bundary cnditins: =0 at z=0, d/dz=0 at z= T. The prfile satisfying these cnditins is, = z 2 T 2 z T Þ =1 T where T = Þ = T r T T T i 17 Substituting this prfile alng with the velcity prfile int the integral energy equatin, ne btains, d d = 5 2Pe r 2 5 d 18 4 d Integrating and applying the cnditin that =*=1,weget = exp 2 5 r 2 1 d Per * + 5 4 * 19 Defining the Nusselt number as, Nu T hr 0 k = r d Nu T =2 r h 2 exp 5 r 0 h 0 2 1 Per * dz z =0 d + 5 4 * 20 The area-averaged Nusselt number can be defined as Jurnal f Heat Transfer MARCH 2006, Vl. 128 / 219
1 Nu avg = 2 Nu T 2d 11 21 Cnstant Heat Flux Case. The temperature prfile fr the cnstant heat flux case is subject t and d q = 1 dz z =0 d q =0, dz z = i.e., prescribed heat flux at the wall, q and adiabatic free surface. With these cnditins, the temperature prfile becmes, = a + ẑ2 ẑ 22 2 where a 0 is a parameter which is a functin f r. Substituting this temperature prfile int the integral frm f the energy equatin and integrating, ne gets, Pe da 0 d d + a 0 d = r 0 2 h 0 + 2 d 5 d + 2 d d 23 subject t the initial value f a 0 =1. While this value can be taken as a =0 based n q =1=0 in Eq. 22, a better estimate is prvided by integrating the energy flux at =1, 0 1 q h q kt i +1dz =1, yielding a =1=1/3. Fr the case f negligible inertia, an analytical slutin fr a can be btained as described in Appendix B, a = r 2 1 Re Pr 20 7 r 2/3 3R 1/3 2/3 1 + 1 Re 3 24 The Nusselt number fr the case f cnstant disk surface heat flux can be fund frm, Nu q hr k = q r ktz =0 T i = r 25 a h Fr the case f negligible inertia, 1 1/3 R 1/3 Nu q = r + 1 3 h Re Pr 31/3 7 r 20 1 r Re 2/3 1 26 The averaged Nusselt number is btained by integrating the lcal Nusselt number ver the disk surface based n Eq. 21. In the fllwing, the results btained frm this analysis are presented and discussed. Results and Discussin The calculatin f the liquid film thickness and Nusselt numbers fr cnstant wall temperature and cnstant heat flux cases were perfrmed fr a range f inlet Reynlds numbers Re u r / and Rssby numbers Ru 2 / 2 r 2. The range f parameters were chsen t cver thse reprted in the experiments f Ozar et al. 13,23 and the recent numerical simulatins f Rice et al. 24. The value f the gemetric parameter r /h was taken t be same as in the experiments with a value f 200. In the fllwing, the cmputed results are presented and cmpared with numerical and experimental results mentined abve. Figure 2 shws the variatin f the film thickness ver the disk surface with Reynlds number fr tw values f Rssby number. Fig. 2 Variatin f nrmalized film thickness as a functin f Reynlds number fr a R=1000 and b R=0.5 At lw rtatin speeds R=1000, the film thickness initially increases alng the disk radius and then begins t thin ut at uter radii due t effect f rtatin as seen in Fig. 2a. The film thickness decreases with increasing inlet velcity r Reynlds number and the peak film thickness lcatin shifts t larger radii with increasing Re. The analytical result given by Eq. 14 is als shwn in Fig. 2a fr Re=10 4. It is seen that the tw results agree at large radii where the inertial effects are diminished and the negligible inertia result agrees with the full slutin. At high rtatin rates R=0.5, film thickness exhibits a radial decay with the film thickness still decreasing with increasing inertia as shwn in Fig. 2b. The variatin f the film thickness fr high levels f rtatin is well represented by Eq. 14 as seen by the agreement at Re=10 4. The effect f varying the rtatin speed n the film thickness at a cnstant value f inlet velcity r Reynlds number is depicted in Fig. 3. With increasing rtatin speed i.e., decreasing R, the film thickness decreases. The trend at high R, which exhibits a maximum in the film thickness, gradually changes t a radially decaying film thickness at high rtatin speeds i.e., lw R. This change appears t happen at arund R=10 fr this case. The cmputed results are in reasnably gd agreement with the experimental results reprted by Ozar et al. 13 as shwn in Fig. 4. The integral mdel captures the trend f liquid film thickness variatin at different Reynlds and Rssby numbers. Due t the uncertainties in the experimentally measured film thickness, a clser agreement can nt be expected. Figure 5 shws the variatin f the Nusselt number ver the disk surface fr the case f cnstant wall temperature. At lw rtatin rates R=1000, shwn in Fig. 5a, the Nusselt number 220 / Vl. 128, MARCH 2006 Transactins f the ASME
Fig. 3 Nrmalized film thickness variatin ver the disk surface fr different Rssby numbers fr medium flw rate Re =10 5 decreases cntinuusly with increasing radial distance. The magnitude f Nusselt number is highest at the entrance since the temperature gradient, and heat flux is the highest there at the liquiddisk interface. With increasing radial distance, the liquid film temperature increases and cnsequently the heat flux diminishes leading t a decrease in the heat transfer cefficient and Nusselt number. The value f the Nusselt number appraches a cnstant value at large radii. The Nusselt number increases with increasing inlet velcity r Reynlds number as expected. At high rtatin speeds, shwn in Fig. 5b, the trend f Nusselt number is quite different in that it increases frm the inlet t a maximum value and decreases frm that pint n. The magnitude f Nusselt number increases with increasing Reynlds number with the peak Nusselt number lcatin shifting radially utward. This prnunced increase in Nusselt number is due t the thinning f the liquid film due t inertia and rtatin. The decrease at larger radii is due t the effect f heating f the liquid film reducing the temperature gradient and heat flux. At high rtatin rates, the Nusselt number increases significantly abve the values fr lw rtatin rates suggesting the strng enhancement f heat transfer due t rtatin. T better visualize the effects f rtatin at a fixed value f inlet velcity r Reynlds number, Fig. 6 shws the effect f Rssby number n Nusselt number variatin. At lw rtatin speeds i.e., Fig. 5 Cnstant wall temperature Nusselt number variatin ver the disk surface fr different Reynlds numbers fr a lw rtatin R=1000 and b high rtatin R=0.5 cases high R, the Nusselt number is highest at the entrance and decreases with increasing radial distance. At abut R=1, the trend exhibits an increase f Nusselt number frm the inlet reaching a maximum value fllwed by a decay. At higher rtatin rates R=0.5, this trend f Nusselt number becmes mre prnunced with the maximum values f Nusselt number being an Fig. 4 Cmparisn f film thickness fr experimental and theretical data fr several Reynlds number and Rssby numbers. Case 1: Re=2.4Ã10 5, R=361; Case 2: Re=4.8Ã10 4,R =361; Case 3: Re=4.8Ã10 4,R=0.4 Fig. 6 Nusselt number variatin fr different Rssby numbers fr Re=10 5 Jurnal f Heat Transfer MARCH 2006, Vl. 128 / 221
Fig. 8 Nusselt number variatin fr different Rssby numbers fr Re=10 5 maximum deviatins f abut 20%. It is cnceivable that the differences are due t the mre detailed treatment affrded in the numerical mdel and ther effects nt captured by the integral mdel. Figure 10 displays the cmparisn f experimental results with the integral mdel fr tw cases. It is fund that the experimental results are in gd agreement with the mdel fr these tw Fig. 7 Cnstant wall heat flux Nusselt number variatin ver the disk surface fr different Reynlds numbers fr a lw rtatin R=1000 and b high rtatin R=0.5 cases rder f magnitude higher. The peak Nusselt number lcatin shifts t smaller radii indicating that the rtatinal effects are felt at smaller radii with increasing rtatin speed. In Fig. 7, the results are shwn fr the cnstant heat flux case at tw values f Rssby number. At lw rtatin rates i.e., high R, the Nusselt number variatin is similar t that f cnstant wall temperature case shwn in Fig. 5a except that the Nusselt number magnitude is higher than that fr the cnstant wall temperature. The increase in Nusselt number is expected since the heat flux at the disk surface is maintained in this case. Fr the high rtatin speed R=0.5, Nusselt number exhibits a decay with increasing radial distance at lw Reynlds number as the temperature f the liquid at the disk surface increases with increasing radial distance, similar t the cnstant wall temperature case. With increasing Reynlds number, the Nusselt number magnitude increases and it exhibits a maximum. Figure 8 illustrates the influence f rtatin n the Nusselt number variatin with at a fixed value f inlet velcity r Reynlds number. It is fund that rtatin enhances the heat transfer int the liquid film very significantly. As the rtatin rate reduces, the maximum in the Nusselt number variatin disappears and it decays cntinuusly with increasing radial distance. The lcatin f the peak shifts t smaller radii with increasing rtatin rate i.e., decreasing R since the effects f rtatin are felt clser t the center with increasing rtatin. Figure 9 shws the cmparisn f the integral methd results with the full numerical simulatin results f Rice et al. 24. The cmparisns are shwn fr Re=1.4210 4 and 2.8410 4 fr tw rtatin speeds f 50 and 100 rpm. The agreement between the integral mdel and numerical simulatins appears t be gd with Fig. 9 Cmparisn f calculated Nusselt Number with numerical data f Rice et al. 24 fr a Re=2.84Ã10 4 ; b Re=1.42 Ã10 5 and rtatin speeds f 50, 100 rpm 222 / Vl. 128, MARCH 2006 Transactins f the ASME
Fig. 12 Variatin f average Nusselt number based n area with Reynlds number fr different rtatin rates crrespnding t the cnstant wall heat flux case Fig. 10 Cmparisn f Nusselt Number frm the integral analysis and the experimental data f Ozar et al. 23 fr a Re=2.85Ã10 4 and b Re=1.42Ã10 5 cases with the radial variatin f experimental Nusselt number being slightly less. Cnsidering the uncertainty f the experimental data and the simplificatins emplyed in the mdel, the level f agreement is remarkable. The area averaged Nusselt numbers ver the disk surface are shwn in Figs. 11 and 12, respectively, fr the cases f cnstant wall temperature and cnstant heat flux. The trend f average Nusselt number is apprximately linear fr bth cases with slightly decreasing curvature fr the cnstant wall heat flux at lw rtatin rates i.e., high R. The Nusselt numbers fr cnstant heat flux are always higher than thse fr the cnstant wall temperature. The influence f Rssby number appears t be nnlinear and the average Nusselt number is fund t scale as NuR m, where m0.55 fr cnstant wall temperature and m0.18 fr cnstant heat flux. The strnger dependence fr cnstant wall temperature can be ratinalized based n the fact the thinning f the liquid film has a strnger effect n the temperature gradient at the wall fr the cnstant wall temperature case than that fr the cnstant heat flux. The Reynlds number dependence f the average Nusselt number was fund scale as NuRe 1.2. Finally, Fig. 13 shws the cmparisn f the analytical slutin given by Eq. 26 fr the negligible inertia case with the full slutin fr Re=10 4 and R=0.5. It is seen that the tw slutins differ substantially near the entrance regin with similar trends f Nusselt number. Cncluding Remarks A detailed integral analysis f flw and heat transfer in a thin liquid film flwing ver a rtating disk was frmulated t determine the liquid film thickness and Nusselt numbers fr bth cnstant wall temperature and heat flux cases. The results are pre- Fig. 11 Variatin f average Nusselt number based n area with Reynlds number fr different rtatin rates crrespnding t the cnstant wall temperature Fig. 13 The cmparisn f the Nusselt number fr cnstant heat flux btained frm full slutin and the apprximatin f negligible inertia Jurnal f Heat Transfer MARCH 2006, Vl. 128 / 223
sented fr a range f inlet liquid flw rates r Reynlds numbers and Rssby numbers. It is fund that the integral mdel captures the variatin f film thickness ver the disk radius fr a range f parameters representing bth inertia and rtatin dminated regimes. At lw rtatin rates crrespnding t high Rssby numbers, the film thickness grws radially until the centrifugal effect becmes sufficiently strng at large radii. Fr high inlet flw rates r Reynlds numbers, the film thickness decreases radially with decay being strnger with increasing inertia. Fr lw inertia, the film thickness increases radially due t significant retardatin f the film flw by viscsity. In cases where bth inertial and rtatinal effects cme int play, the film thickness first increases reaching a maximum fllwed by radial decay. The lcatin f maximum film thickness changes based n the values f Reynlds and Rssby numbers. Fr cases dminated either by high rtatin r lw inertia, the film thickness is well represented by the analytical expressin btained fr negligible inertia. The analysis f the heat transfer in the liquid film indicate that the Nusselt number exhibits a radial decay at lw rtatin rates i.e., high R fr bth cases f cnstant wall temperature and cnstant heat flux. The dependence f Nusselt number is slightly strnger than linear n inlet Reynlds number fr bth cases. Hwever, the Nusselt number values fr cnstant wall heat flux are unifrmly greater than thse fr the cnstant wall temperature. At high rtatin speeds and inlet Reynlds numbers, the Nusselt number first increases, reaches a peak and then displays a radial decay. The radial lcatin f the peak Nusselt number shifts t larger radii with increasing Reynlds number and this is due t the cmpeting effects f inertia and rtatin n the film thickness and heat transfer characteristics. The effect f Rssby number appears t be significantly nnlinear and high rtatin rates can prduce significant enhancement in heat transfer fr bth cases f cnstant wall temperature and heat flux. In additin t lcal Nusselt number variatin, the disk surface area averaged Nusselt numbers were cmputed fr bth cases. They als shw an apprximately linear variatin with respect t inlet flw rate r Reynlds number and inverse pwer law dependence n Rssby number. Rssby number dependence is strnger fr the cnstant wall temperature case. The results btained frm this integral analysis were cmpared with thse frm a recent numerical study f Rice et al. 24 as well as the experiments f Ozar et al. 13,23. Bth cmparisns indicate gd agreement with the integral mdel. Finally, the integral analysis prvided analytical and semianalytical expressins fr the lcal Nusselt number in the limit f negligible inertia fr the cases f cnstant wall heat flux and cnstant wall temperature, respectively. Acknwledgment The wrk presented in this article was funded by NASA Micrgravity Fluid Physics Prgram under Grant N. NCC3-789 with Dr. S. Sankaran as the grant mnitr. Nmenclature a cnstant in Eq. 22 C heat capacity J/kg K h cllar height m h cnvective heat transfer cefficient W/m 2 K k thermal cnductivity W/m K Nu Nusselt number hr /k Pe Peclet number u r / q Cnstant wall heat flux W/m 2 r radial crdinate m nrmalized radial crdinate r/r r cllar radius m Re Reynlds number u r / R Rssby number u 2 / 2 r 2 T temperature K Greek Appendix A T temperature f the disk surface K T i inlet temperature f the liquid K u liquid inlet velcity at the cllar exit m/s u r velcity cmpnent in the radial directin m/s u z velcity cmpnent in the directin nrmal t disk surface m/s ũ nndimensinal velcity u/u z crdinate nrmal t disk surface z nrmalized crdinate z/h ẑ nrmalized crdinate with respect t thermal bundary thickness thermal diffusivity, k/c m 2 /s rati f thermal t hydrdynamic bundary layer thickness / T hydrdynamic bundary layer r film thickness m T thermal bundary layer thickness m nrmalized film thickness /h T nrmalized film thickness T /h T nndimensinal temperature fr cnstant wall temperature, T T /T T i q nndimensinal temperature fr cnstant wall heat flux, kt T i /q h rtatinal speed rev/s Fr determining the thermal entry length fr the cnstant disk surface temperature case, the temperature prfile in this regin is taken as, = z 2 T 2 z T A1 Intrducing, ẑz / T and T /, the previusly determined velcity prfile and the nndimensinal temperature prfiles becme, ẑ 2 ũ r =3 ẑ 2 2 3 2 3 3 = ẑ 2 2ẑ A2 T T substituting int the integral energy equatin and integrating t ẑ=1, ne gets d d 2 4 20 3 = Pe 2 r 2 A3 which can be integrated subject t the cnditin =1=0 as, 18 31 80 = Pe 12 r * d A4 1 It seen that the secnd term in the bracket n the left is small fr liquids with high Prandtl numbers. Fr example, water with a Prandtl number 5 9 leads t a value f arund 0.5. Upn neglecting this secnd term, ne btains = Pe 12 r 2 * 1/3 d A5 1 2 If we assume that 1 near the entrance, then the integral can be evaluated and the result becmes, 224 / Vl. 128, MARCH 2006 Transactins f the ASME
* =1+ Pe h 2 1/2 0 6 r 0 A6 Since h r, then, *1 rr * r. The smallness f the entry regin length is utilized in the heat transfer analysis presented in the main bdy f the paper. Appendix B Cnsidering the integral mmentum equatin given by Eq. 9, neglecting the inertial terms n the left-hand side allws the direct integratin between z =0 and z = t yield the velcity prfile, ũ r = R Re h r 2 z z 2 B1 2 The crrespnding film thickness can be btained upn integratin fr the ttal vlumetric flw rate f liquid at any radius leading t the expressin fr given by Eq. 14. Substitutin f this velcity prfile int the energy integral equatin leads t the fllwing differential equatin fr a, da 0 d = 1 Re Pr r 2 + 31/3 7 R 30 Re 1/3 r 2/3 5/3 2/3 2 h 3 r 1/3 Re R 1/3 a 8/3 B2 Since r h, the last term n the right-hand side is small cmpared t the thers. If neglected, Eq. B2, becmes directly integrable. Applying the cnditin a =1=1/3, we get, a = r 2 1 Re Pr 31/3 7 r 20 2/3 Re R 1/3 2/3 1 + 1 3 B3 References 1 Webb, B. W., and Ma, C. F., 1995, Single Phase Liquid Impingement Heat Transfer, Adv. Heat Transfer, 26, pp. 105 217. 2 Watsn, E. J., 1964, The Radial Spread f a Liquid Jet Over a Hrizntal Plane, J. Fluid Mech., 20, pp. 481 499. 3 Thmas, S., Hankey, W., Faghri, A., and Swansn, T., 1990, One- Dimensinal Analysis f the Hydrdynamic and Thermal Characteristics f Thin Film Flws Including Hydraulic Jump and Rtatin, ASME J. Heat Transfer, 112, pp. 728 735. 4 Rahman, M. M., Faghri, A., and Hankey, W., 1991, Cmputatin f Turbulent Flw in a Thin Liquid Layer f Fluid Invlving a Hydraulic Jump, J. Fluids Eng., 113, pp. 411 418. 5 Avedisian, C. T., and Zha, Z., 2000, The Circular Hydraulic Jump in Lw Gravity, Prc. R. Sc. Lndn, Ser. A, 456, pp. 2127 2151. 6 Labus, T. L., and DeWitt, K. J., 1978, Liquid Jet Impingement Nrmal t a Disk in Zer Gravity, J. Fluids Eng., 100, pp. 204 209. 7 Miyasaka, Y., 1974, On the Flw f a Viscus Free Bundary Jet n a Rtating Disk, Bull. JSME, 17, pp. 1469 1475. 8 Rahman, M. M., and Faghri, A., 1992, Numerical Simulatin f Fluid Flw and Heat Transfer in a Thin Liquid Film Over a Rtating Disk, Int. J. Heat Mass Transfer, 35, pp. 1441 1453. 9 Buyevich, Y. A., and Ustinv, V. A., 1994, Hydrdynamic Cnditins f Transfer Prcesses Thrugh a Radial Jet Spreading Over a Flat Surface, Int. J. Heat Mass Transfer, 37, pp. 165 173. 10 Ra, A., and Arakeri, J. H., 1998, Integral Analysis Applied t Radial Film Flws, Int. J. Heat Mass Transfer, 41, pp. 2757 2767. 11 Liu, X., and Lienhard, J. H., 1989, Liquid Jet Impingement Heat Transfer n a Unifrm Flux Surface, Heat Transfer Phenmena in Radiatin, Cmbustin and Fires, ASME HTD, 106, pp. 523 530. 12 Azuma, T., and Hshin, T., 1984, The Radial Flw f a Thin Liquid Film, 1st 4th Reprts, Bull. JSME, 27, pp. 2739 2770. 13 Ozar, B., Cetegen, B. M., and Faghri, A., 2003, Experiments n the Flw f a Thin Liquid Film Over a Hrizntal Statinary and Rtating Disk Surface, Exp. Fluids, 34, pp. 556 565. 14 Chadhury, Z. H., 1964, Heat Transfer in a Radial Liquid Jet, J. Fluid Mech., 20, pp. 501 511. 15 Wang, X. S., Dagan, Z., and Jiji, L. M., 1989, Heat Transfer Between a Circular Free Impinging Jet and a Slid Surface with Nn-Unifrm Wall Temperature f Wall Heat Flux: 1: Slutin fr the Stagnatin Regin, Int. J. Heat Mass Transfer, 32, pp. 1351 1360. 16 Carper, H. J., and Defenbaugh, D. M., 1978, Heat Transfer frm a Rtating Disk with Liquid Jet Impingement, Prceedings f the 6th Internatinal Heat Transfer Cnference, Trnt, pp. 113 118. 17 Carper, Jr., H. J., Saavedra, J. J., and Suwanprateep, T., 1986, Liquid Jet Impingement Cling f a Rtating Disk, ASME J. Heat Transfer, 108, pp. 540 546. 18 Vader, D. T., Incrpera, F. P., and Viskanta, R., 1991, Lcal Cnvective Heat Transfer Frm a Heated Surface t an Impinging, Planar Jet f Water, Int. J. Heat Mass Transfer, 34, pp. 611 623. 19 Stevens, J., and Webb, B. W., 1991, Lcal Heat Transfer Cefficients Under and Axisymmetric, Single-Phase Liquid Jet, ASME J. Heat Transfer, 113, pp. 71 78. 20 Faghri, A., Thmas, S., and Rahman, M. M., 1993, Cnjugate Heat Transfer frm a Heated Disk t a Thin Liquid Film Frmed by a Cntrlled Impinging Jet, ASME J. Heat Transfer, 115, pp. 116 123. 21 Aune, A., and Ramshaw, C., 1999, Prcess Intensificatin: Heat and Mass Transfer Characteristics f Liquid Films n Rtating Discs, Int. J. Heat Mass Transfer, 42, pp. 2543 2556. 22 Nusselt, W. Z., 1916, Die Oberflachenkndensatin des Wasserdampfes, Z. Ver Deut. Ing., 60, pp. 541 546. 23 Ozar, B., Cetegen, B. M., and Faghri, A., 2004, Experiments n Heat Transfer in a Thin Liquid Film Flwing Over a Rtating Disk, ASME J. Heat Transfer, 126, pp. 184 192. 24 Rice, J., Faghri, A., and Cetegen, B. M., 2005, Analysis f a Free Surface Film Frm a Cntrlled Liquid Impingement Jet Over a Rtating Disk Including Cnjugate Effects With and Withut Evapratin, Int. J. Heat Mass Transfer, 48, pp. 5192 5204. Jurnal f Heat Transfer MARCH 2006, Vl. 128 / 225